Chapter 1 Number Toolbox
Transcript of Chapter 1 Number Toolbox
Chapter 1 Number Toolbox
Chapter 2 Introduction to Algebra Chapter 2 Introduction to Algebra
Chapter 3 Decimals
Ch t 4 N b Th d F tiChapter 4 Number Theory and Fractions
Chapter 5 Fraction Operations
Chapter 6 Collect and Display Data
Chapter 7 Plane Geometry
Chapter 8 Ratio, Proportion, and Percent
Chapter 9 Integers
Chapter 10 Perimeter, Area, and Volume
Chapter 11 Probabilityp y
Chapter 12 Functions and Coordinate Geometry
1-1 Comparing and Ordering Whole Numbers
1-2 Estimating with Whole Numbers1-2 Estimating with Whole Numbers
1-3 Exponents
1 4 O d f O ti1-4 Order of Operations
1-5 Mental Math
1-6 Choose the Method of Computation
1-7 Find a Pattern
2-1 Variables and Expressions
2-2 Translate Between Words and Math2-2 Translate Between Words and Math
2-3 Equations and Their Solutions
2 4 S l i Additi E ti2-4 Solving Addition Equations
2-5 Solving Subtraction Equations
2-6 Solving Multiplication Equations
2-7 Solving Division Equations
3-1 Representing, Comparing, and Ordering Decimals
3-2 Estimating Decimals3-2 Estimating Decimals
3-3 Adding and Subtracting Decimals
3 4 D i l d M t i M t3-4 Decimals and Metric Measurement
3-5 Scientific Notation
3-6 Multiplying Decimals
3-7 Dividing Decimals by Whole Numbers
3-8 Dividing by Decimals
3-9 Interpret the Quotient
3-10 Solving Decimal Equations
4-1 Divisibility
4-2 Factors and Prime Factorization4-2 Factors and Prime Factorization
4-3 Greatest Common Factor
4 4 D i l d F ti4-4 Decimals and Fractions
4-5 Equivalent Fractions
4-6 Comparing and Ordering Fractions
4-7 Mixed Number and Improper Fractions
4-8 Adding and Subtracting with Like Denominators
4-9 Multiplying Fractions by Whole Numbers
5-1 Multiplying Fractions
5-2 Multiplying Mixed Numbers5-2 Multiplying Mixed Numbers
5-3 Dividing Fractions and Mixed Numbers
5 4 S l i F ti E ti M lti li ti d Di i i5-4 Solving Fraction Equations: Multiplication and Division
5-5 Least Common Multiple
5-6 Estimating Fraction Sums and Differences
5-7 Adding and Subtracting with Unlike Denominators
5-8 Adding and Subtracting Mixed Numbers
5-9 Renaming to Subtract Mixed Numbers
5-10 Solving Fraction Equations: Addition and Subtraction
6-1 Make a Table
6-2 Range Mean Median and Mode6-2 Range, Mean, Median, and Mode
6-3 Additional Data and Outliers
6 4 B G h6-4 Bar Graphs
6-5 Frequency Tables and Histograms
6-6 Ordered Pairs
6-7 Line Graphs
6-8 Misleading Graphs
6-9 Stem-and-Leaf Plots
7-1 Points, Lines, and Planes
7-2 Angles7-2 Angles
7-3 Angle Relationships
7 4 Cl if i Li7-4 Classifying Lines
7-5 Triangles
7-6 Quadrilaterals
7-7 Polygons
7-8 Geometric Patterns
7-9 Congruence
7-10 Transformations
7-11 Symmetryy y
7-12 Tessellations
8-1 Ratios and Rates
8-2 Proportions8-2 Proportions
8-3 Proportions and Customary Measurement
8 4 Si il Fi8-4 Similar Figures
8-5 Indirect Measurement
8-6 Scale Drawings and Maps
8-7 Percents
8-8 Percents, Decimals, and Fractions
8-9 Percent Problems
8-10 Using Percents
9-1 Understanding Integers
9-2 Comparing and Ordering Integers9-2 Comparing and Ordering Integers
9-3 The Coordinate Plane
9 4 Addi I t 9-4 Adding Integers
9-5 Subtracting Integers
9-6 Multiplying Integers
9-7 Dividing Integers
9-8 Solving Integer Equations
10-1 Finding Perimeter
10-2 Estimating and Finding Area10-2 Estimating and Finding Area
10-3 Break into Simpler Parts
10 4 C i P i t d A10-4 Comparing Perimeter and Area
10-5 Circles
10-6 Solid Figures
10-7 Surface Area
10-8 Finding Volume
10-9 Volume of Cylinders
11-1 Introduction to Probability
11-2 Experimental Probability11-2 Experimental Probability
11-3 Theoretical Probability
11 4 M k O i d Li t11-4 Make an Organized List
11-5 Compound Events
11-6 Making Predictions
12-1 Tables and Functions
12-2 Graphing Functions12-2 Graphing Functions
12-3 Graphing Translations
12 4 G hi R fl ti12-4 Graphing Reflections
12-5 Graphing Rotations
12-6 Stretching and Shrinking
1-1 Comparing and Ordering Whole Numbers
Warm UpWarm Up
Lesson PresentationLesson Presentation
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
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1-1 Comparing and Ordering Whole Numbers
Warm UpWarm UpCompare. Use <, >, or =.
1 8 9 2 27 14< >1. 8 9 2. 27 143. 56 23 4. 10 155. 11 12 6. 37 16
< >> << >5. 11 12 6. 37 16< >
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1-1 Comparing and Ordering Whole Numbers
Problem of the DayProblem of the Day
Subtract your age from your age multiplied by 100 Divide the result by multiplied by 100. Divide the result by 11, and then divide the quotient by 9. What number do you get?What number do you get?
The answer will be the student’s age.
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1-1 Comparing and Ordering Whole Numbers
Learn to compare and order whole pnumbers using place value or a number line.
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1-1 Comparing and Ordering Whole Numbers
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1-1 Comparing and Ordering Whole NumbersAdditi l E l 1 U i Pl V l t Additional Example 1: Using Place Value to
Compare Whole Numbers
Belize’s 2000 population was 249 183 people Belize s 2000 population was 249,183 people. Iceland’s 2000 population was 276,365 people. Which country had more people?
249,183 BelizeStart at the left and compare digits in the same place value position Look for the first
276,365 Icelandposition. Look for the first place where the values are different.
40 thousand is less than 70 thousand.
249,183 is less than 276,365.
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Iceland had more people.
1-1 Comparing and Ordering Whole Numbers
T Thi E l 1Try This: Example 1
In 2000, the population of San Diego, California was 1 223 400 people In 2000 the population was 1,223,400 people. In 2000, the population of Dallas, Texas was 1,188,580 people. Which city had more people?
Start at the left and compare 1,223,400 San Diego
1 188 580 Dallas
Start at the left and compare digits in the same place value position. Look for the first
1,188,580 Dallas
200 th d i t th 100 th d
place where the values are different.
200 thousand is greater than 100 thousand.
1,223,400 is greater than 1,188,580.
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San Diego had more people.
1-1 Comparing and Ordering Whole Numbers
To order numbers, you can compare them , y pusing place value and then write them in order from least to greatest. You can also
h h b b lgraph the numbers on a number line. As you read the numbers from left to right, they will be ordered from least to greatestthey will be ordered from least to greatest.
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1-1 Comparing and Ordering Whole Numbers
Remember!< means
“is less than.”3 < 5 120 < 504
> means “is greater than.”
17 > 9 212 > 83
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1-1 Comparing and Ordering Whole NumbersAdditional E ample 2 Using a N mber Line Additional Example 2: Using a Number Line
to Order Whole NumbersOrder the numbers from least to greatest:
Graph the numbers on a number line:The number 675 is between 600 and 700
675; 1,044; 497
The number 675 is between 600 and 700.The number 1,044 is between 1,000 and 1,100.The number 497 is between 400 and 500The number 497 is between 400 and 500.
400 600 800 1,000
497 675 1,044
The numbers are ordered when you read the number line from left to right.The numbers in order from least to greatest are
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The numbers in order from least to greatest are 497, 675, and 1,044.
1-1 Comparing and Ordering Whole Numbers
Try This: Example 2
Order the numbers from least to greatest: 732 923 502Graph the numbers on a number line:The number 732 is between 700 and 800
732; 923; 502
The number 732 is between 700 and 800.The number 923 is between 900 and 1,000.The number 502 is between 500 and 600The number 502 is between 500 and 600.
400 600 800 1,000
502 732 923
The numbers are ordered when you read the number line from left to right.The numbers in order from least to greatest are
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The numbers in order from least to greatest are 502, 732, and 923.
Insert Lesson Title Here1-1 Comparing and Ordering Whole Numbers
Lesson QuizCompare. Write < or >.
1. 47,328 47,238
2. 933,826 933,520 >
>
, ,
3. Write the numbers in order from least to greatest: 726 847 221 221 726 847greatest: 726, 847, 221.
4. The are of Panama is 78,200 square kil d h f Li h i i 65 200
221, 726, 847
kilometers, and the area of Lithuania is 65,200 square kilometers. Which country is smaller?Lith iLithuania
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1-2 Estimating with Whole Numbers
Warm UpWarm Up
Lesson PresentationLesson Presentation
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
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1-2 Estimating with Whole Numbers
Warm UpWarm UpFind each sum.
1 3 214 + 5 4901. 3,214 + 5,4902. 9,225 + 8,652
8,70417,877
3. 3,210 + 1,2004. 8,774 + 2,156
4,41010,9300,930
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1-2 Estimating with Whole Numbers
Problem of the DayProblem of the Day
Continue the number pattern below. Explain the pattern you foundExplain the pattern you found.3, 6, 10, 15, ___, ___
21, 28; One possible pattern is to increase the difference between consecutive terms b th th diff b t by one more than the difference between preceding consecutive terms.
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1-2 Estimating with Whole Numbers
Learn to estimate with whole numbers.
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Insert Lesson Title Here1-2 Estimating with Whole Numbers
Vocabularycompatible numberunderestimateoverestimate
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1-2 Estimating with Whole Numbers
Sometimes in math you do not need an t I t d exact answer. Instead, you can use an
estimate. Estimates are close to the exact answer but are usually easier and faster to answer but are usually easier and faster to find.
When estimating, you can round the numbers in the problem to compatible numbers Compatible numbers are close numbers. Compatible numbers are close to the numbers in the problem, and they can help you do math mentally.
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can help you do math mentally.
1-2 Estimating with Whole Numbers
Remember!When rounding, look at the digit to the right of the place to which you are
Remember!
right of the place to which you are rounding.
If that digit is 5 or greater round up• If that digit is 5 or greater, round up.
• If that digit is less than 5, round down.
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Insert Lesson Title Here1-2 Estimating with Whole Numbers
Additional Example 1A: Estimating a Sum or Difference by Rounding
E i h b di h l Estimate the sum by rounding to the place value indicated.
A 12 345 + 62 167;A. 12,345 + 62,167;ten thousands
10 000 Round 12 345 down10,000 Round 12,345 down.Round 62,167 down.+ 60,000__________
70 000
The sum is about 70,000.
70,000
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,
Insert Lesson Title Here1-2 Estimating with Whole Numbers
Additional Example 1B: Estimating a Sum or Difference by Rounding
E i h diff b di h Estimate the difference by rounding to the place value indicated.
B 4 983 2 447;B. 4,983 – 2,447;thousands
5 000 Round 4 983 up5,000 Round 4,983 up.Round 2,447 down.– 2,000__________
3 000
The difference is about 3,000.
3,000
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,
Insert Lesson Title Here1-2 Estimating with Whole Numbers
Try This: Example 1A
Estimate the sum by rounding to the place Estimate the sum by rounding to the place value indicated.
A. 13,235 + 41,139;, , ;
ten thousands
10 000 Round 13 235 down10,000 Round 13,235 down.Round 41,139 down.+ 40,000__________
50 000
The sum is about 50,000.
50,000
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,
Insert Lesson Title Here1-2 Estimating with Whole Numbers
Try This: Example 1B
Estimate the difference by rounding to the y gplace value indicated.
B. 5,723 – 1,393;
thousands
6 000 Round 5 723 up6,000 Round 5,723 up.Round 1,393 down.– 1,000__________
5 000
The difference is about 5,000.
5,000
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,
1-2 Estimating with Whole Numbers
An estimate that is less than the exact answer is an underestimate.
An estimate that is greater than the exact answer is an overestimate.
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1-2 Estimating with Whole Numbers
Remember!The area of a rectangle is found by multiplying the length by the width.
Remember!
g y
A = l w
l
w
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Insert Lesson Title Here1-2 Estimating with Whole NumbersAdditi l E l 2 E ti ti P d t b Additional Example 2: Estimating a Product by
Rounding
The sixth grade class wants to paint a wall The sixth-grade class wants to paint a wall with white paint. The wall is a rectangle 9 feet tall and 18 feet wide. One quart of paint will cover 100 square feet. How many quarts of paint should the students buy?
9 18 9 20 Overestimate the area of the wall9 18 9 20 Overestimate the area of the wall.The actual area is less than 180 square feet.
9 20 = 180square feet.
If one quart of paint will cover 100 square feet, then 2 quarts will cover 200 square feet. The
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q qstudents should buy 2 quarts of paint.
Insert Lesson Title Here1-2 Estimating with Whole Numbers
Try This: Example 2
The seventh-grade class wants to paint a wall g pwith blue paint. The wall is a rectangle 9 feet tall and 29 feet wide. One quart of paint will cover 100 square feet How many quarts of cover 100 square feet. How many quarts of paint should the students buy?
9 29 9 30 Overestimate the area of the wall9 29 9 30 Overestimate the area of the wall.The actual area is less than 270 square feet.
9 30 = 270square feet.
If one quart of paint will cover 100 square feet, then 3 quarts will cover 300 square feet. The
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q qstudents should buy 3 quarts of paint.
Insert Lesson Title Here1-2 Estimating with Whole NumbersAdditi l E l 3 E ti ti Q ti t Additional Example 3: Estimating a Quotient
Using Compatible Numbers
Mr Dehmel will drive 243 miles to the fair at Mr. Dehmel will drive 243 miles to the fair at 65 mi/h. About how long will his trip take?
240 and 60 are compatible 243 ÷ 65 240 ÷ 60
240 and 60 are compatible numbers. Underestimatethe speed.
Because he underestimated the speed, the actual time will be less than 4 hours
240 ÷ 60 = 4will be less than 4 hours..
The trip will take about 4 hours.
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The trip will take about 4 hours.
Insert Lesson Title Here1-2 Estimating with Whole Numbers
Try This: Example 3
Mrs. Blair will drive 103 miles to the airport at p55 mi/h. About how long will her trip take?
100 and 50 are compatible 103 ÷ 55 100 ÷ 50
100 and 50 are compatible numbers. Underestimatethe speed.
Because she underestimated the speed, the actual time will be less than 2 hours
100 ÷ 50 = 2will be less than 2 hours..
The trip will take about 2 hours.
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The trip will take about 2 hours.
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Insert Lesson Title Here1-2 Estimating with Whole NumbersLesson Quiz
Estimate each sum or difference by rounding to the place value indicated.to the place value indicated.
1. 7,420 + 3,527; thousands 11,000
2. 47,821 + 19,925; ten thousands
3. 8,254 – 5,703; thousands
70,000
2,000, , ;
4. 66,845 – 24,782; ten thousands
,000
50,000
5. One quart of paint covers an area of 100 square feet. How many quarts are needed to paint a wall 8 f t t ll d 19 f t id ?feet tall and 19 feet wide?
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2
1-3 Exponents
Warm UpWarm Up
Lesson PresentationLesson Presentation
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
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1-3 Exponents
Warm UpWarm UpMultiply.
1 3 3 31. 3 3 32. 4 4 4
2764
3. 2 2 2 24. 5 5 5 5
16625625
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1-3 Exponents
Problem of the DayProblem of the Day
Replace the letters a, b, and c with the numbers 3 4 and 5 to make a true numbers 3, 4, and 5 to make a true statement.2a + 2a = bc 2a + 2a = bc
25+ 25 = 43
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1-3 Exponents
Learn to represent numbers by using p y gexponents.
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Insert Lesson Title Here1-3 Exponents
Vocabularyexponentbaseexponential form
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1-3 Exponents
An exponent tells how many times a number called the base is used as a factor.
A number is in exponential formwhen it is written with a base and when it is written with a base and an exponent.
7733BaseExponent
= 7 7 7= 343
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77 = 7 7 7= 343
1-3 Exponents
Exponential Form 101
Read “10 to the 1st power”Multiply 10Value 10
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1-3 Exponents
Exponential Form 102
Read“10 squared” or “10 to the 2nd power”
M l i l 10 10Multiply 10 10Value 100
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1-3 Exponents
Exponential Form 103
Read“10 cubed” or “10 to the 3rd power”
M l i l 10 10 10Multiply 10 10 10Value 1,000
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1-3 Exponents
Exponential Form 104
Read “10 to the 4th power”Multiply 10 10 10 10Value 10,000
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Insert Lesson Title Here1-3 Exponents
Additional Example 1: Writing Numbers in Exponential Form
Write each expression in exponential form.
A. 5 5 5 5
5 is a factor 4 times.54
B. 3 3 3 3 3
35 3 is a factor 5 times.35
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Insert Lesson Title Here1-3 Exponents
Try This: Example 1
Write each expression in exponential form.Write each expression in exponential form.
A. 7 7 7
7 is a factor 3 times73 7 is a factor 3 times.73
B. 6 6 6 6 6 6
66 6 is a factor 6 times.66
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Insert Lesson Title Here1-3 Exponents
Additional Example 2: Finding the Value of Numbers in Exponential Form
Find each value.
A. 26
26 = 2 2 2 2 2 2= 64
B. 45
45 4 4 4 4 4 45 = 4 4 4 4 4 = 1,024
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Insert Lesson Title Here1-3 Exponents
Try This: Example 2
Find each value.
A. 34
34 = 3 3 3 3 = 81
B. 25
25 2 2 2 2 2 25 = 2 2 2 2 2 = 32
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1-3 Exponents
A phone tree is used to contact families at P l’ h l Th t ll 4 f ili
Additional Example 3: Problem Solving Application
Paul’s school. The secretary calls 4 families. Then each family calls 4 other families, and so on. How many families will notified during the y gfourth round of calls?
11 Understand the Problem11 Understand the ProblemThe answer will be the number of families called in the 4th round.
List the important information:
• The secretary calls 4 families.
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The secretary calls 4 families.
• Each family calls 4 families.
1-3 Exponents
You can draw a diagram to see how many
22 Make a Plan
g ycalls are in each round.
SecretarySecretary
1st round – 4 calls1st round 4 calls
2nd round–16 calls
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1-3 Exponents
Solve33Notice that in each round, the number of calls is a power of 4is a power of 4.
1st round: 4 calls = 4 = 41
2nd round: 16 calls = 4 x 4 = 42
So during the 4th round, there will be 44 calls. 44 4 4 4 4 25644 = 4 4 4 4 = 256
During the 4th round of calls, 256 families will have been notifiedhave been notified.
Look Back44Drawing a diagram helps you see how to use
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Drawing a diagram helps you see how to use exponents to solve the problem.
1-3 ExponentsTr This E ample 3Try This: Example 3
A phone tree is used to contact families at Paul’s school The secretary calls 3 families Paul s school. The secretary calls 3 families. Then each family calls 3 other families, and so on. How many families will notified during
f fthe fourth round of calls?
11 Understand the Problem
The answer will be the number of families called in the 3rd round.
List the important information:
• The secretary calls 3 families.
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The secretary calls 3 families.
• Each family calls 3 families.
1-3 Exponents
You can draw a diagram to see how many
22 Make a Plan
g ycalls are in each round.
SecretarySecretary
1st round – 3 calls1st round 3 calls
2nd round–9 calls
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1-3 Exponents
Solve33Notice that in each round, the number of calls is a power of 3is a power of 3.
1st round: 3 calls = 3 = 31
2nd round: 9 calls = 3 x 3 = 32
So during the 4th round, there will be 34 calls. 34 3 3 3 3 8134 = 3 3 3 3 = 81
During the 4th round of calls, 81 families will have been notifiedhave been notified.
Look Back44Drawing a diagram helps you see how to use
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Drawing a diagram helps you see how to use exponents to solve the problem.
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Insert Lesson Title Here1-3 ExponentsLesson Quiz
Write each expression in exponential form.
1. 12 12 12
2. 9 9 9 9 9 9 9 97
123
2. 9 9 9 9 9 9 9
Find each value.
9
3. 202 4. 64
5. In a phone tree, each of three people will call
400 1,296
p , p pthree people, and then each of those will call three more. If there are five levels of the tree, how many people will be called?
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243
1-4 Order of Operations
Warm UpWarm Up
Lesson PresentationLesson Presentation
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
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1-4 Order of Operations
Warm UpWarm UpPerform the operations in order from left to right.
1. 8 + 4 – 22 9 3 + 1
10282. 9 3 + 1
3. 7 – 3 + 54 20 ÷ 4 + 6
289114. 20 ÷ 4 + 6 11
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1-4 Order of Operations
Problem of the DayProblem of the Day
0 1 2 3 4 5 6 7 8 9 = 1P t th i t l i i Put the appropriate plus or minus signs between the numbers so that the total equals 1equals 1.
0 + 1 – 23 + 45 + 67 – 89 = 1
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1-4 Order of Operations
Learn to use the order of operations.p
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Insert Lesson Title Here1-4 Order of Operations
Vocabularynumerical expressionsevaluateorder of operations
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1-4 Order of Operations
A numerical expression is a mathematical phase that includes only numbers and operation phase that includes only numbers and operation symbols.
Numerical Expressions 4 + 8 ÷ 2 6 371 – 203 + 2 5,006 19
When you evaluate a numerical expression, you find its value.
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1-4 Order of Operations
When an expression has more than one operation, you must know which operation to do first. To make sure that everyone gets the same answer, we use the order of operations.
ORDER OF OPERATIONS1. Perform operations in parentheses.1. Perform operations in parentheses.2. Find the values of numbers with exponents.3. Multiply or divide from left to right as ordered in the problem.4. Add or subtract from left to right as ordered in the problem
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in the problem.
1-4 Order of Operations
The first letters of these words Helpful Hint
can help you remember the order of operations.
Pl P thPlease Parentheses
Excuse Exponents
My Multiply
Dear Divide
Aunt Add
Sally Subtract
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y
1-4 Order of Operations
Additional Example 1A: Using the Order of Operations
Evaluate each expression.
A. 15 – 10 ÷ 2 There are no parentheses or exponents.
15 – 5 Divide.
10 Subtract.10 Subtract.
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1-4 Order of OperationsAdditi l E l 1B U i th O d f Additional Example 1B: Using the Order of
Operations
B. 8 + 14 ÷ 2 + 6 33There are no parentheses. Find the value of the number with exponents.8 + 14 ÷ 2 + 6 27Divide.8 + 7 + 6 27
Multiply.8 + 7 + 162
Add15 + 162
177
Add.15 + 162
Add.
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1-4 Order of Operations
Additional Example 1C: Using the Order of Operations
C. 3 (4 + 7) + 22Perform operations within parentheses 3 11 22 parentheses.
3 11 + 4
3 11 + 22
Find the value of the number with exponents
Multiply.33 + 4
number with exponents.
37 Add.
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1-4 Order of Operations
Try This: Example 1A
Evaluate each expressionEvaluate each expression.
There are no parentheses or A. 12 – 6 ÷ 2 There are no parentheses or exponents.
12 – 3 Divide.
9 Subtract.9 Subtract.
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1-4 Order of Operations
Try This: Example 1B
B. 4 + 14 ÷ 2 + 4 23 There are no parentheses. Find the value of the 4 10 2 4 8 Find the value of the number with exponents.
4 + 10 ÷ 2 + 4 8
Divide.4 + 5 + 4 8
Multiply.4 + 5 + 32
Add9 + 32
41
Add.9 + 32
Add.
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1-4 Order of Operations
Try This: Example 1C
2C. 2 (3 + 6) + 42
Perform operations within parentheses 2 9 42 parentheses.
2 9 + 16
2 9 + 42
Find the value of the number with exponents
Multiply.18 + 16
number with exponents.
34 Add.
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1-4 Order of Operations
Additional Example 2: Consumer Application
Mr. Kellett bought 6 used CDs for $4 each and 5 used CDs for $3 each. Evaluate the following expression to find the amount Mr. Kellett spent on CDs.Kellett spent on CDs.
6 4 + 5 3
24 + 15
39
Mr. Kellett spent $39 on CDs.
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1-4 Order of Operations
Try This: Example 2
Ms. Nivia bought 4 new CDs for $8 each and 6 used CDs for $4 each. Evaluate the following expression to find the amount Ms. Nivia spent on CDs.on CDs.
4 8 + 6 432 + 2432 + 24
56
Ms. Nivia spent $56 on CDs.
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Insert Lesson Title Here1-4 Order of OperationsLesson Quiz
Evaluate each expression.1 15 + 4 2 231. 15 + 4 2
2. (12 – 5)2 – 10 39
23
( )
3. 3 + 9 2 – 5 16
4. 43 – 30 ÷ 2 49
5. Chaz bought 4 football cards for $2 each and 8 baseball cards for $3 each. Evaluate the expression to find the amount Chaz spent on p pcards: 4 2 + 8 3.
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$32
1-5 Mental Math
Warm UpWarm Up
Lesson PresentationLesson Presentation
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
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1-5 Mental Math
Warm UpWarm UpFind each sum or product.
1 17 + 151. 17 + 152. 29 + 39
3268
3. 8(24)4. 7(12)
19284
5. 3(91)6. 6(15)
827390( )
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90
1-5 Mental Math
Problem of the Day
Determine the secret number from the following clues:• The number is a multiple of 5.• It is divisible by 3.• It is less than 200.• Its tens digit equals the sum of its g qother two digits.165
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1-5 Mental Math
Learn to use number properties to p pcompute mentally.
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Insert Lesson Title Here1-5 Mental Math
VocabularyCommutative PropertyAssociative Propertyp yDistributive Property
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1-5 Mental Math
l h “d hMental math means “doing math in your head.”
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1-5 Mental Math
COMMUTATIVE PROPERTY (Ordering)Words NumbersWords Numbers
You can add or multiply numbers in
18 + 9 = 9 + 18multiply numbers in any order. 15 2 = 2 15
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1-5 Mental Math
ASSOCIATIVE PROPERTY (Grouping)ASSOCIATIVE PROPERTY (Grouping)Words Numbers
Wh l When you are only adding or only multiplying, you (17 + 2) + 9 = 17 + (2 + 9) multiplying, you can group any of the numbers
h
( ) ( )(12 2) 4 = 12 (2 4)
together.
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1-5 Mental MathAdditi l E l 1A U i P ti t Additional Example 1A: Using Properties to
Add and Multiply Whole Numbers
A Evaluate 17 + 5 + 3 + 15A. Evaluate 17 + 5 + 3 + 15.
17 + 5 + 3 + 15 Look for sums that are multiples of 10multiples of 10
Use the Commutative Property. 17 + 3 + 5 + 15
(17 + 3) + (5 + 15)Use the Associative Property to make groups of compatible numbers
40 Use mental math to add.
20 + 20 of compatible numbers.
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1-5 Mental MathAdditional Example 1B: Using Properties to Additional Example 1B: Using Properties to
Add and Multiply Whole Numbers
B E l t 4 13 5B. Evaluate 4 13 5.
4 13 5 Look for products that are multiples of 10multiples of 10
Use the Commutative Property.13 4 5
13 (4 5)
3 20
Use the Associative Property to group compatible numbers.
260
Use mental math to multiply.13 20
Course 1
1-5 Mental Math
Try This: Example 1A
A. Evaluate 12 + 5 + 8 + 5.A. Evaluate 12 + 5 + 8 + 5.
12 + 5 + 8 + 5 Look for sums that are multiples of 10multiples of 10
Use the Commutative Property.12 + 8 + 5 + 5
(12 + 8) + (5 + 5)Use the Associative Property to make groups of compatible numbers
30 Use mental math to add.
20 + 10 numbers.
Course 1
1-5 Mental Math
Try This: Example 1B
B. Evaluate 8 3 5.
8 3 5 Look for products that are multiples of 10multiples of 10
Use the Commutative Property.3 8 5
3 (8 5) Use the Associative Property to group compatible numbers.
120
Use mental math to multiply.3 40
Course 1
1-5 Mental Math
DISTRIBUTIVE PROPERTYWords NumbersWords Numbers
When you multiply a number times a sum 6 (10 + 4) = 6 14number times a sum, you can• find the sum first
6 (10 + 4) = 6 14= 84
6 (10 + 4) = (6 10) + (6 4)and then multiply, or• multiply by each number in the sum
( ) ( ) ( )= 60 + 24= 84
number in the sum and then add.
Course 1
1-5 Mental Math
When you multiply two numbers, you can y p y , y“break apart” one of the numbers into a sum and then use the Distributive Property.
Helpful HintBreak the greater factor into a sum that contains a multiple of 10 and a one-digit number. You can add
Helpful Hint
and multiply these numbers mentally.
Course 1
1-5 Mental MathAdditional Example 2A: Using the Distributive Additional Example 2A: Using the Distributive
Property to Multiply
Use the Distributive Property to find the productUse the Distributive Property to find the product.
A. Evaluate 6 35.
6 35 = 6 (30 + 5) “Break apart” 35 into 30 + 5.
Use the Distributive Property.= (6 30) + (6 5) Use the Distributive Property. (6 30) + (6 5)
= 180 + 30 Use mental math to multiply.= 210 Use mental math to add.
Course 1
1-5 Mental MathAdditi l E l 2B U i th Di t ib ti Additional Example 2B: Using the Distributive
Property to Multiply
Use the Distributive Property to find the Use the Distributive Property to find the product.
B Evaluate 9 x 87B. Evaluate 9 x 87.
9 87 = 9 (80 + 7) “Break apart” 87 into 80 + 7.
h i ib i (9 80) (9 7) Use the Distributive Property.= (9 80) + (9 7)
= 720 + 63 Use mental math to multiply.
= 783 Use mental math to add.
Course 1
1-5 Mental Math
Try This: Example 2A
Use the Distributive Property to find the p yproduct.
A. Evaluate 4 x 27.
4 27 = 4 (20 + 7) “Break apart” 27 into 20 + 7.
h i ib i (4 20) (4 7) Use the Distributive Property.= (4 20) + (4 7)
= 80 + 28 Use mental math to multiply.
= 108 Use mental math to add.
Course 1
1-5 Mental Math
Try This: Example 2B
Use the Distributive Property to find the p yproduct.
B. Evaluate 6 43.
6 43 = 6 (40 + 3) “Break apart” 43 into 40 + 3.
h i ib i (6 40) (6 3) Use the Distributive Property.= (6 40) + (6 3)
= 240 + 18 Use mental math to multiply.
= 258 Use mental math to add.
Course 1
Insert Lesson Title Here1-5 Mental Math
Lesson QuizEvaluate.
1. 18 + 24 + 2 + 6 2. 10 5 3
3. 13 + 42 + 7 + 8
15050
703. 13 + 42 + 7 + 8
Use the Distributive Property to find each product
70
product.
4. 8 12 5. 6 1596 90
6. Angie wants to buy 3 new video games. How much will she need to save if each game costs g$27?
Course 1
$81
1-6 Choose the Method of Computation
Warm UpWarm Up
Lesson PresentationLesson Presentation
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
Course 1
1-6 Choose the Method of Computation
Warm UpWarm UpUse mental math to find each fraction of 80.
1. 2.40 8110__1
2
3. 4.2010
1614
15
Course 1
1-6 Choose the Method of Computation
Problem of the DayProblem of the Day
About 23% of 600 firefighters are not on duty on any particular day Steven on duty on any particular day. Steven estimates that about 200 firefighters have the day off. Is his estimate too have the day off. Is his estimate too high or too low? Explain.
too high; 25% of 600 is 150too high; 25% of 600 is 150
Course 1
1-6 Choose the Method of Computation
Learn to choose an appropriate method of pp pcomputation and justify your choice.
Course 1
1-6 Choose the Method of Computation
Additional Example 1
Find the sum: 4 + 3 + 2 + 10 + 8 + 2 + 5 + 1.
There are probably too many numbers to add mentally, but the numbers are small. You can y,use paper and pencil.35
Course 1
1-6 Choose the Method of Computation
Try This: Example 1
Find the sum: 3 + 7 + 2 + 18 + 7 + 1 + 3 + 1.
It might be hard to keep track of all of these numbers if you tried to add mentally. But the y ynumbers themselves are small. You can use paper and pencil.4242
Course 1
1-6 Choose the Method of Computation
Additional Example 2
Find the difference: 4,562 – 397.,
397 is close to a multiple of 100.
You can use mental mathYou can use mental math.
(4,562 + 3) – (397 + 3)Think: Add 3 to 397 to make 400. Add 3 to 4 562 to
4,565 – 400
4,165
to 4,562 to compensate.
,
Course 1
1-6 Choose the Method of Computation
Try This: Example 2
Find the difference: 3,442 – 298.,
298 is close to a multiple of 100.
You can use mental math Think: Add 2 to 298 You can use mental math.
(3,442 + 2) – (298 + 2)
Think: Add 2 to 298 to make 300. Add 2 to 3,442 to
3,444 – 300
3,144
compensate.
,
Course 1
1-6 Choose the Method of Computation
Additional Example 3
Find the quotient: 9,288 ÷ 24.q ,
The numbers are not compatible, so mental math is not a good idea. You could use paper g p pand pencil, but long division requires several steps. Using a calculator will probably be fasterfaster.
9,288 ÷ 24 = 387
Course 1
1-6 Choose the Method of Computation
Try This: Example 3
Find the quotient: 9,248 ÷ 32.q ,
The numbers are not compatible, so mental math is not a good idea. You could use paper and pencil, but long division requires several steps. Using a calculator will probably be faster.faster.
9,248 ÷ 32 = 289
Course 1
L Q i
Insert Lesson Title Here1-6 Choose the Method of Computation
Lesson Quiz
Evaluate the expression and state the method f t ti dof computation you used.
1. 17 + 6 + 24 + 35 + 3 + 5 90; mental math
2. 63 197
3 It takes Jupiter approximately 4 344 days to
12,411; paper and pencil
3. It takes Jupiter approximately 4,344 days to complete one revolution around the Sun. It takes Earth 365 days to revolve around the Sun How Earth 365 days to revolve around the Sun. How many more days does it take Jupiter to revolve around the Sun than Earth? 3,979 days3,979 days
Course 1
1-7 Find a Pattern
Warm UpWarm Up
Lesson PresentationLesson Presentation
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
Course 1
1-7 Find a Pattern
Warm UpWarm UpDetermine what could come next.
1 3 4 5 6 71. 3, 4, 5, 6, ___2. 10, 9, 8, 7, 6, ___3. 1, 3, 5, 7,
7593. 1, 3, 5, 7, ___
4. 2, 4, 6, 8, ___5. 5, 10, 15, 20, ___
91025
Course 1
1-7 Find a Pattern
Problem of the Day
How can you place the numbers 1 through 6 in the circles so that the
l h d lsums along each side are equal?
612
6
534
Course 1
1-7 Find a Pattern
Learn to find patterns and to recognize, p gdescribe, and extend patterns in sequences.
Course 1
Insert Lesson Title Here1-7 Find a Pattern
Vocabularyperfect squaresequenceqterm
Course 1
1-7 Find a Pattern
Whole numbers raised to the second power are called perfect squares. This is because they can be represented by objects arranged in the shape be represented by objects arranged in the shape of a square.
1 4 9 16
Course 1
1-7 Find a Pattern
The perfect squares can be written as a sequence. A sequence is an ordered set of numbers. Each number in the sequence is called a term. In a sequence, there is often a pattern between one term and the next.term and the next.
Course 1
1-7 Find a Pattern
1 4 9 161 t 2 d 3 d 4th
Sequence1st term 2nd term 3rd term 4th term Term
+3 +5 +73 5 7
You can use this pattern to find the fifth and sixth terms in the sequence. To get the fifth term, add 9. te s t e seque ce o get t e t te , add 9To get the sixth term, add 11.
1, 4, 9, 16, , , . . . 1, 4, 9, 16, , , . . .
16 + 9 = 25 25 + 11 = 36
Course 1
So the next two perfect squares are 25 and 36.
1-7 Find a Pattern
Look for a relationship between the 1st term and Helpful HintLook for a relationship between the 1st term and the 2nd term. Check if this relationship works between the 2nd term and the 3rd term, and so on.
Course 1
1-7 Find a PatternAdditional Example 1A: Extending Sequences
with Addition and Subtraction
Id tif tt i h d Identify a pattern in each sequence and name the next three terms.
A 48 42 36 30 A. 48, 42, 36, 30, , , , . . .
–6 –6 –6 –6 –6 –6
A pattern is to subtract 6 from each term to get the next term.30 – 6 = 24 24 – 6 = 18 18 – 6 = 12
So 24, 18, and 12 will be the next three terms.
Course 1
, ,
1-7 Find a PatternAdditional Example 1B: Extending Sequences
with Addition and Subtraction
Id tif tt i h d Identify a pattern in each sequence and name the next three terms.
B 24 34 31 41 38 48 B. 24, 34, 31, 41, 38, 48, , , , . . .
+10 –3 +10 –3 +10 –3 +10 –3
A pattern is to add 10 to one term and subtract 3 from the next.48 – 3 = 45 45 + 10 = 55 55 – 3 = 52
So 45, 55, and 52 will be the next three terms.
Course 1
, ,
1-7 Find a PatternTry This: Example 1A
Identify a pattern in each sequence and name Identify a pattern in each sequence and name the next three terms.
A 39 34 29 24 A. 39, 34, 29, 24, , , , . . .
–5 –5 –5 –5 –5 –5
A pattern is to subtract 5 from each term to get the next term.24 – 5 = 19 19 – 5 = 14 14 – 5 = 9
So 19, 14, and 9 will be the next three terms.
Course 1
, ,
1-7 Find a Pattern
Try This: Example 1B
Identify a pattern in each sequence and name Identify a pattern in each sequence and name the next three terms.
B 12 23 17 28 22 33 B. 12, 23, 17, 28, 22, 33, , , , . . .
+11 –6 +11 –6 +11 –6 +11 –6
A pattern is to add 11 to one term and subtract 6 from the next.33 – 6 = 27 27 + 11 = 38 38 – 6 = 32
So 27, 38, and 32 will be the next three terms.
Course 1
, ,
1-7 Find a PatternAdditional Example 2A: Completing
Sequences with Multiplication and Division
Id tif tt i h d Identify a pattern in each sequence and name the missing terms.
A 2 6 18 162 A. 2, 6, 18, , 162, ,…
3 3 3 3 3
A pattern is to multiply each term by 3.
18 3 54 162 3 48618 3 = 54 162 3 = 486
So 54 and 486 are the missing terms.
Course 1
g
1-7 Find a PatternAdditi l E l 2B C l ti S Additional Example 2B: Completing Sequences
with Multiplication and Division
Identify a pattern in each sequence and name Identify a pattern in each sequence and name the missing terms.
A 12 6 24 12 48 96 48 96 A. 12, 6, 24, 12, 48, , 96, 48, , 96, . . .
÷2 4 ÷2 4 ÷2 4 ÷2 4 ÷2 A pattern is to divide one term by 2 and multiply the next by 4.
48 ÷ 2 = 24 48 4 = 192
So 24 and 192 are the missing terms.
Course 1
g
1-7 Find a PatternTry This: Example 2A
Identify a pattern in each sequence and name th i i tthe missing terms.
A 6 12 24 96 A. 6, 12, 24, , 96, , . . .
2 2 2 2 2
A pattern is to multiply each term by 2.
24 2 = 48 96 2 = 192
So 48 and 192 are the missing terms.
Course 1
g
1-7 Find a PatternTry This: Example 2B
Identify a pattern in each sequence and name th i i tthe missing terms.
A 8 2 16 4 32 64 16 A. 8, 2, 16, 4, 32, , 64, 16, ,…
÷4 8 ÷4 8 ÷4 8 ÷4 8
A pattern is to divide one term by 4 and multiply the next by 8.
32 ÷ 4 = 8 16 8 = 128
So 8 and 128 are the missing terms.
Course 1
g
Insert Lesson Title Here1-7 Find a Pattern
Lesson Quiz
Identify a pattern in each sequence, and name the next three terms.
1. 12, 24, 36, 48, , , , … add 12; 60, 72, 84
2. 75, 71, 67, 63, , , ,…
Id tif tt i h d
subtract 4; 59, 55, 51
; , ,
Identify a pattern in each sequence, and name the missing terms.
3. 1000, 500, , 125,…
4. 100, 50, 200, , 400, ,…
divide by 2; 250
divide by 2 then , , , , , ,multiply by 4; 100, 200
Course 1
Course 1
Warm Up
Lesson Presentation
Problem of the Day
2-1 Variables and Expressions
Course 1
Warm Up Simplify. 1. 4 + 7 × 3 − 1 2. 87 − 15 ÷ 5 3. 6(9 + 2) + 7 4. 35 ÷ 7 × 5
24 84 73 25
2-1 Variables and Expressions
Course 1
Problem of the Day
How can the digits 1 through 5 be arranged in the boxes to make the greatest product?
× 4 3 1
5 2
2-1 Variables and Expressions
Course 1
Learn to identify and evaluate expressions.
2-1 Variables and Expressions
Course 1
Vocabulary variable constant algebraic expression
2-1 Variables and Expressions
Course 1
A variable is a letter or symbol that represents a quantity that can change.
A constant is a quantity that does not change.
2-1 Variables and Expressions
Course 1
An algebraic expression contains one or more variables and may contain operation symbols. p × 7 is an algebraic expression.
Algebraic Expressions NOT Algebraic Expressions
150 + y 85 ÷ 5
35 × w + z 10 + 3 × 5
To evaluate an algebraic expression, substitute a number for the variable and then find the value.
2-1 Variables and Expressions
Course 1
Evaluate the expression to find the missing values in the table.
Additional Example 1A: Evaluating Algebraic Expressions
Substitute for y in 5 × y. y 5 × y
16
27
35
80 y = 16; 5 × 16 = 80
135
175
27 135
35 175
y = 27; 5 × =
y = 35; 5 × =
2-1 Variables and Expressions
Course 1
The missing values are 135 and 175.
Evaluate the expression to find the missing values in the table.
Additional Example 1B: Evaluating Algebraic Expressions
Substitute for z in z ÷ 5 + 4. z z ÷ 5 + 4
20
45
60
8 z = 20; 20 ÷ 5 + 4 = 8
z = 45; __ ÷ 5 + 4 = __
z = 60; __ ÷ 5 + 4 = __
13
16
45 13
60 16
2-1 Variables and Expressions
Course 1
The missing values are 13 and 16.
Try This: Example 1A
Evaluate the expression to find the missing values in the table.
Substitute for x in x ÷ 9. x x ÷ 9
18
36
54
2 x = 18; 18 ÷ 9 = 2
4
6
4
54 6 x = 36; ÷ 9 = x = 54; ÷ 9 =
36
2-1 Variables and Expressions
Course 1
The missing values are 4 and 6.
Evaluate the expression to find the missing values in the table.
Try This: Example 1B
Substitute for z in 8 × z + 2. z 8 × z + 2
7
9
11
58 z = 7; 8 × 7 + 2 = 58
z = 9; 8 × __+ 2 = __
z = 11; 8 × __ + 2 = __ 74
90
9 74
11 90
2-1 Variables and Expressions
Course 1
The missing values are 74 and 90.
Multiplication and division can be written without using the symbols × and ÷.
Instead of . . . You can write . . .
x × 3 x • 3
35 ÷ y
x(3)
3x
35 y
When you are multiplying a number times a variable, the number is written first. Write “3x” not “x3.” Read 3x as “three x.”
Writing Math
2-1 Variables and Expressions
Course 1
Find an expression for the table.
Additional Example 2A: Finding an Expression
39 ÷ 13 = 3
n
39
52
65
3
4
5 65 ÷ 13 = 5
52 ÷ 13 = 4
n ÷ 13
An expression is n ÷ 13, or . n 13
2-1 Variables and Expressions
Course 1
Find an expression for the table.
Additional Example 2B: Finding an Expression
7 • 4 = 28
p
4
6
8
28
42
56 7 • 8 = 56
7 • 6 = 42
7 • p
An expression is 7 • p, or 7p.
2-1 Variables and Expressions
Course 1
Find an expression for the table.
21 ÷ 3 = 7
u
21
42
63
7
14
21 63 ÷ 3 = 21
42 ÷ 3 = 14
u ÷ 3
An expression is u ÷ 3, or . u 3
Try This: Example 2A
2-1 Variables and Expressions
Course 1
Find an expression for the table.
Try This: Example 2B
4 • 5 = 20
n
5
8
11
20
32
44 4 • 11 = 44
4 • 8 = 32
4 • n
An expression is 4 • n, or 4n.
2-1 Variables and Expressions
Course 1
1. Evaluate each expression to find the missing values in the tables.
Lesson Quiz: Part 1
x 10 7 5
x 1 3 5
7 21 35
x – 5
2. Find an expression for the table.
5 2
0
7x
2-1 Variables and Expressions
Course 1
Lesson Quiz: Part 2
3. Evaluate 8x for x = 5.
4. Evaluate 4x – 1 for x = 12.
40
47
2-1 Variables and Expressions
Course 1
Course 1
2-2 Translating Between Words and Math
Course 1
Warm Up
Lesson Presentation
Problem of the Day
16 36
19
4
Course 1
2-2 Translating Between Words and Math
Warm Up Evaluate each expression for x = 9.
1. 7 + x 2. 4x 3. 2x + 1 4. 36
x
Problem of the Day
Draw a square around the numbers of 4 adjacent days on the calendar for this month. Add all the numbers in the square and subtract 4 times the first number. What number do you get? 16
Solving Subtraction Equations
Course 1
2-2 Translating Between Words and Math
Learn to translate between words and math.
Course 1
2-2 Translating Between Words and Math
In word problems, you may need to translate words to math.
Course 1
2-2 Translating Between Words and Math
Put together or combine
Find how much more or less
Put together groups of equal parts
Separate into equal groups
Add
Subtract
Multiply
Divide
Action Operation
Lake Superior is the largest lake in North America. Let a stand for the area in square miles of Lake Superior. Lake Erie has an area of 9,910 square miles. Write an expression to show how much larger Lake Superior is than Lake Erie.
Additional Example 1A: Social Studies Applications
Course 1
2-2 Translating Between Words and Math
To find how much larger, subtract the area of Lake Erie from the area of Lake Superior.
a – 9,910 Lake Superior is a – 9,910 square miles larger than Lake Erie.
Additional Example 1B
Course 1
2-2 Translating Between Words and Math
Let p represent the number of colored pencils in a box. If there are 26 boxes on the shelf, write an algebraic expression to represent the total number of pencils on the shelf.
To put together 26 equal groups of p, multiply 26 times p.
26p
There are 26p pencils on the shelf.
The Nile River is the world’s longest river. Let n stand for the length in miles of the Nile. The Paraná River is 3,030 miles long. Write an expression to show how much longer the Nile is than the Paraná.
Try This: Example 1A
Course 1
2-2 Translating Between Words and Math
To find how much longer, subtract the length of the Paraná from the length of the Nile.
n – 3,030
The Nile is n – 3,030 miles longer than the Paraná.
Try This: Example 1B
Course 1
2-2 Translating Between Words and Math
Let p represent the number of paper clips in a box. If there are 125 boxes in a case, write an algebraic expression to represent the total number of paper clips in a case.
To put together 125 equal groups of p, multiply 125 times p.
125p
There are 125p paper clips in a case.
Course 1
2-2 Translating Between Words and Math
37 + 28
• 28 added to 37 • 37 plus 28 • the sum of 37 and 28 • 28 more than 37
• 28 added to x • x plus 28 • the sum of x and 28 •28 more than x
x + 28
+ Operation
Numerical Expression
Words
Algebraic Expression
Words
Course 1
2-2 Translating Between Words and Math
• 12 subtracted from k • 12 minus k • the difference of k and 12 • take away 12 from k
k — 12
Operation
Numerical Expression
Words
Algebraic Expression
Words
− 90 — 12
• 12 subtracted from 90 • 90 minus 12 • the difference of 90 and 12 • 12 less than 90 • take away 12 from 90
Course 1
2-2 Translating Between Words and Math
• 8 times w • w multiplied by 8 • the product of 8 and w • 8 groups of of w
8 • w or (8)(w) or 8w
Operation
Numerical Expression
Words
Algebraic Expression
Words
8 × 48 or 8 • 48
• 8 times 48 • 48 multiplied by 8 • the product of 8 and 48
× or (8)(48) or
8(48) or (8)48
Course 1
2-2 Translating Between Words and Math
• n divided by 3 • the quotient of n and 3
n ÷ 3 or
Operation
Numerical Expression
Words
Algebraic Expression
Words
37 ÷ 3 or
• 327 divided by 3 • the quotient of 327 and 3
÷ 327 3
n 3
Write each phrase as a numerical or algebraic expression.
Additional Example 2: Translating Words into Math
Course 1
2-2 Translating Between Words and Math
A. 987 minus 12
987 − 12
B. x times 45
45 • x or 45x
Write each phrase as a numerical or algebraic expression.
Try This: Example 2
Course 1
2-2 Translating Between Words and Math
A. 42 less than 79
79 − 42
B. y divided by 22
y ÷ 22 or y 22
Write two phrases for each expression.
Additional Example 3: Translating Math into Words
Course 1
2-2 Translating Between Words and Math
A. 16 + b
• 16 added to b
B. (75)(32)
• 75 times 32
• b more than 16
• the product of 75 and 32
Write two phrases for each expression.
Try This: Example 3
Course 1
2-2 Translating Between Words and Math
A. 17 – 14
• 14 subtracted from 17
• 16 divided by b
• 17 minus 14
• the quotient of 16 and b
B. 16 b
Write an expression 1. Let x represent the number of minutes Kristen works out in one week. Write an expression for the number of minutes she works out in 4 weeks.
Lesson Quiz
Write each phrase as a numerical or algebraic expression 2. 7 less than x 3. The product of 12 and w
4x
x − 7 12w
x more than 17 or x added to 17 n divided by 12 or the quotient of n and 12
Course 1
2-2 Translating Between Words and Math
Write a phrase for each expression 4. 17 + x 5. n ÷ 12
Course 1
2-3 Equations and Their Solutions
Course 1
Warm Up
Lesson Presentation
Problem of the Day
29 16 9
16
Warm Up Evaluate each expression for x = 8.
1. 3x + 5 2. x + 8 3. 2x – 7 4. 8x ÷ 4 5. 7x – 1 6. x – 3
Course 1
2-3 Equations and Their Solutions
55 5
Problem of the Day
Complete the magic square so that every row, column, and diagonal add up to the same total.
Solving Subtraction Equations
10 7
4
9 2
8 6
12 5
Course 1
2-3 Equations and Their Solutions
Learn to determine whether a number is a solution of an equation.
Course 1
2-3 Equations and Their Solutions
equation solution
Course 1
2-3 Equations and Their Solutions
Vocabulary
An equation is a mathematical statement that two quantities are equal. You can think of a correct equation as a balanced scale.
Course 1
2-3 Equations and Their Solutions
3 + 2 5
Equations may contain variables. If a value for a variable makes an equation true, that value is a solution of the equation.
Course 1
2-3 Equations and Their Solutions
12 + 15 27
s + 15 = 27
s = 12 s = 10 10 + 15
27
s = 12 is a solution because 12 + 15 = 27.
s = 10 is not a solution because 10 + 15 ≠ 27.
Determine whether the given value of the variable is a solution.
Additional Example 1A: Determining Solutions of Equations
b — 447 = 1,203 for b = 1,650
Because 1,203 = 1,203, 1,650 is a solution to b — 447 = 1,203.
Course 1
2-3 Equations and Their Solutions
b — 447 = 1,203 Substitute 1,650 for b.
1,203 = 1,203 ?
1,650 — 447 = 1,203 ?
1,203 1,203
Subtract.
Determine whether the given value of the variable is a solution.
Additional Example 1B: Determining Solutions of Equations
27x = 1,485 for x = 54
Because 1,458 ≠ 1,485, 54 is not a solution to 27x = 1,485.
Course 1
2-3 Equations and Their Solutions
27x = 1,485 Substitute 54 for x.
1,458 = 1,485 ?
27 • 54 = 1,485 ?
1,458 1,485
Multiply.
Determine whether the given value of the variable is a solution. u + 56 = 139 for u = 73
Because 129 ≠ 139, 73 is not a solution to u + 56 = 139.
Course 1
2-3 Equations and Their Solutions
u + 56 = 139 Substitute 73 for u.
129 = 139 ?
73 + 56 = 139 ?
Try This: Example 1A
129 139
Add.
Determine whether the given value of the variable is a solution.
45 ÷ g = 3 for g = 15
Because 3 = 3, 15 is a solution to 45 ÷ g = 3.
Course 1
2-3 Equations and Their Solutions
45 ÷ g = 3 Substitute 15 for g.
3 = 3 ?
45 ÷ 15 = 3 ?
Try This: Example 1B
3 3
Divide.
Paulo says that his yard is 19 yards long. Jamie says that Paulo’s yard is 664 inches long. Determine if these two measurements are equal.
Additional Example 2
Because 684 ≠ 664, 19 yards are not equal to 664 inches.
Course 1
2-3 Equations and Their Solutions
Substitute 19 for y
684 = 664 ?
Multiply.
36 • yd = in.
36 • 19 = 664 ? 36 • y = 664
Anna says that the table is 7 feet long. John says that the table is 84 inches long. Determine if these two measurements are equal.
Because 84 = 84, 7 feet is equal to 84 inches.
Course 1
2-3 Equations and Their Solutions
12f = 84 Substitute 7 for f.
84 = 84 ?
12 • 7 = 84 ?
Try This: Example 2
12 • ft. = in.
Multiply.
Determine whether the given value of the variable is a solution. 1. 85 = 13x for x = 5 2. w + 38 = 210 for w = 172 3. 8y = 88 for y = 11 4. 16 = w ÷ 6 for w = 98
Lesson Quiz
no yes
yes no
no
5. The local pizza shop charged Kylee $172 for 21 medium pizzas. The price of a medium pizza is $8. Determine if Kylee paid the correct amount of money. (Hint: $8 • pizzas = total cost.)
Course 1
2-3 Equations and Their Solutions
Course 1
2-4 Solving Addition Equations
Course 1
Warm Up
Lesson Presentation
Problem of the Day
yes no yes yes
Warm Up Determine whether each value is a solution. 1. 86 + x = 102 for x = 16 2. 18 + x = 26 for x = 4 3. x + 46 = 214 for x = 168 4. 9 + x = 35 for x = 26
Course 1
2-4 Solving Addition Equations
Problem of the Day
After Renee used 40 m of string for her kite and gave 5 m to her sister for her wagon, she had 8 m of string left. How much string did she have to start with?
Solving Subtraction Equations
53 m
Course 1
2-4 Solving Addition Equations
Learn to solve whole-number addition equations.
Course 1
2-4 Solving Addition Equations
The equation h + 14 = 82 can be represented as a balanced scale.
h + 14 82
Course 1
2-4 Solving Addition Equations
To find the value of h, you need h by itself on one side of the scale.
h ?
To get h by itself, first take away 14 from the left side of the scale. Now the scale is unbalanced.
h 68 To rebalance the scale, take away 14 from the other side.
h + 14 82
Taking away 14 from both sides of the scale is the same as subtracting 14 from both sides of the equation.
h + 14 = 82 –14
Course 1
2-4 Solving Addition Equations
–14 h = 68
Subtraction is the inverse, or opposite, of addition. If an equation contains addition, solve it by subtracting from both sides to “undo” the addition.
Solve the equation. Check your answer.
Additional Example 1A: Solving Addition Equations
x + 87 = 152
Check x + 87 = 152
x + 87 = 152 87 is added to x.
x = 65
Course 1
2-4 Solving Addition Equations
– 87 – 87 Subtract 87 from both sides to undo the addition.
65 + 87 = 152 ?
152 = 152 ?
Substitute 65 for x in the equation.
65 is the solution.
Solve the equation. Check your answer.
Additional Example 1B: Solving Addition Equations
72 = 18 + y
Check 72 = 18 + y
72 = 18 + y 18 is added to y.
54 = y
Course 1
2-4 Solving Addition Equations
–18 –18 Subtract 18 from both sides to undo the addition.
72 = 18 + 54 ?
72 = 72 ?
Substitute 54 for y in the equation.
54 is the solution.
Solve the equation. Check your answer.
Try This: Example 1A
u + 43 = 78
Check u + 43 = 78
u + 43 = 78 43 is added to u.
u = 35
Course 1
2-4 Solving Addition Equations
– 43 – 43 Subtract 43 from both sides to undo the addition.
35 + 43 = 78 ?
78 = 78 ?
Substitute 35 for u in the equation.
35 is the solution.
Solve the equation. Check your answer.
Try This: Example 1B
68 = 24 + g
Check 68 = 24 + g
68 = 24 + g 24 is added to g.
44 = g
Course 1
2-4 Solving Addition Equations
–24 –24 Subtract 24 from both sides to undo the addition.
68 = 24 + 44 ?
68 = 68 ?
Substitute 44 for g in the equation.
44 is the solution.
Johnstown, Cooperstown, and Springfield are located in that order in a straight line along a highway. It is 12 miles from Johnstown to Cooperstown and 95 miles from Johnstown to Springfield. How far is it from Cooperstown to Springfield?
Additional Example 2: Social Studies Application
It is 83 miles from Cooperstown to Springfield.
95 = 12 + d
Course 1
2-4 Solving Addition Equations
distance between Johnstown and Springfield
distance between Johnstown and Cooperstown
distance between Cooperstown and Springfield
= +
95 = 12 + d –12 –12 83 = d
12 is added to d.
Subtract 12 from both sides to undo the addition.
Patterson, Jacobsville, and East Valley are located in that order in a straight line along a highway. It is 17 miles from Patterson to Jacobsville and 35 miles from Patterson to East Valley. How far is it from Jacobsville to East Valley?
Try This: Example 2
The distance between Jacobsville and East Valley is 18 miles.
35 = 17 + d
Course 1
2-4 Solving Addition Equations
distance between Patterson and East Valley
distance between Patterson and Jacobsville
distance between Jacobsville and East Valley
= +
35 = 17 + d –17 –17 18 = d
17 is added to d.
Subtract 17 from both sides to undo the addition.
Solve each equation. 1. x + 15 = 72 2. 81 = x + 24 3. x + 22 = 67 4. 93 = x + 14
Lesson Quiz
x = 57 x = 57 x = 45
x = 79
56 inches 5. Kaitlin is 2 inches taller than Reba. Reba is 54 inches tall. How tall is Kaitlin?
Course 1
2-4 Solving Addition Equations
Course 1
2-5 Solving Subtraction Equations
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Simplify.
1. x + 7 = 22 2. 18 + x = 105 3. 16 = x + 9 4. 23 = x + 4
x = 15 x = 87
x = 7 x = 19
2-5 Solving Subtraction Equations
Course 1
Problem of the Day
Bruce has 25 CD’s remaining after giving 14 to John, 17 to Mary, and 25 to Sue. How many CD’s did he begin with? 81
Solving Subtraction Equations 2-5 Solving Subtraction Equations
Course 1
Learn to solve whole-number subtraction equations.
2-5 Solving Subtraction Equations
Course 1
When John F. Kennedy became president of the United States, he was 43 years old, which was 8 years younger than Abraham Lincoln was when Lincoln became President. How old was Lincoln when he became president?
2-5 Solving Subtraction Equations
Course 1
Abraham Lincoln’s age
John F. Kennedy’s age – = 8
Let a represent Abraham Lincoln’s age.
a – = 8 43
a – 8 = 43 + 8 + 8
a = 51 Abraham Lincoln was 51 years old when he became president.
2-5 Solving Subtraction Equations
Course 1
y – 23 = 39 23 is subtracted from y. + 23
y = 62 Add 23 to both sides to undo the subtraction.
Check y – 23 = 39 Substitute 62 for y in the equation.
Solve y – 23 = 39. Check your answer.
62 is the solution. 39 = 39 ?
Additional Example 1A: Solving Subtraction Equations
62 – 23 = 39 ?
+ 23
2-5 Solving Subtraction Equations
Course 1
78 = s – 15 15 is subtracted from s. + 15
93 = s Add 15 to both sides to undo the subtraction.
Check 78 = s – 15 Substitute 93 for s in the equation.
Solve 78 = s – 15. Check your answer.
93 is the solution. 78 = 78 ?
Additional Example 1B: Solving Subtraction Equations
78 = 93 – 15 ?
+ 15
2-5 Solving Subtraction Equations
Course 1
z – 3 = 12 3 is subtracted from z. + 3
z = 15 Add 3 to both sides to undo the subtraction.
Check z – 3 = 12 Substitute 15 for z in the equation.
Solve z – 3 = 12. Check your answer.
15 is the solution. 12 = 12 ?
Additional Example 1C: Solving Subtraction Equations
15 – 3 = 12 ?
+ 3
2-5 Solving Subtraction Equations
Course 1
a – 4 = 7 4 is subtracted from a. + 4
a = 11 Add 4 to both sides to undo the subtraction.
Check a – 4 = 7 Substitute 11 for a in the equation.
Solve a – 4 = 7. Check your answer.
11 is the solution. 7 = 7 ?
11 – 4 = 7 ?
+ 4
Try This: Example 1A
2-5 Solving Subtraction Equations
Course 1
57 = c – 13 13 is subtracted from c. + 13
70 = c Add 13 to both sides to undo the subtraction.
Check 57 = c – 13 Substitute 70 for c in the equation.
Solve 57 = c – 13. Check your answer.
70 is the solution. 57 = 57 ?
57 = 70 – 13 ?
+ 13
Try This: Example 1B
2-5 Solving Subtraction Equations
Course 1
g – 62 = 14 62 is subtracted from g. + 62
g = 76 Add 62 to both sides to undo the subtraction.
Check g – 62 = 14 Substitute 76 for g in the equation.
Solve g – 62 = 14. Check your answer.
76 is the solution. 14 = 14 ?
76 – 62 = 14 ?
+ 62
Try This: Example 1C
2-5 Solving Subtraction Equations
Course 1
1. x – 9 = 21
2. 14 = x – 3
3. x – 7 = 11
4. 16 = x – 14
5. x – 9 = 11
Lesson Quiz
6. Susan is taller than James. The difference in their height is 4 inches. James is 62 inches tall. How tall is Susan?
x = 30
x = 17
x = 18
x = 30
x = 20
66 inches
2-5 Solving Subtraction Equations
Course 1
Course 1
2-6 Solving Multiplication Equations
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Divide. 1. 2.
3.
4.
5.
6.
6
13
10
4
72 12 65 5 60 6 40 10 130 5 91 7
26
13
Course 1
2-6 Solving Multiplication Equations
Problem of the Day
Katie’s little brother is building a tower with blocks. He adds 6 blocks and then removes 4 blocks. If he adds 8 more blocks, the tower will have twice as many blocks as when he started. How many blocks did he start with? 10
Course 1
2-6 Solving Multiplication Equations
Learn to solve whole-number multiplication equations.
Course 1
2-6 Solving Multiplication Equations
Division is the inverse of multiplication. To solve an equation that contains multiplication, use division to “undo” the multiplication.
Course 1
2-6 Solving Multiplication Equations
4m = 32 4m = 32 4 4 m = 8
Remember! 4m means “4 × m.”
5p = 75 p is multiplied by 5.
5 p = 15
Divide both sides by 5 to undo the multiplication.
Check 5p = 75 Substitute 15 for p in the equation.
Solve 5p = 75. Check your answer.
15 is the solution. 75 = 75 ?
Additional Example 1A: Solving Multiplication Equations
5(15)= 75 ?
5
Course 1
2-6 Solving Multiplication Equations
5p = 75
16 = 8r r is multiplied by 8.
8 2 = r
Divide both sides by 8 to undo the multiplication.
Check 16 = 8r Substitute 2 for r in the equation.
Solve 16 = 8r. Check your answer.
2 is the solution. 16 = 16 ?
Additional Example 1B: Solving Multiplication Equations
16 = 8(2) ?
8
Course 1
2-6 Solving Multiplication Equations
16 = 8r
8a = 72 a is multiplied by 8.
8 x = 9
Divide both sides by 8 to undo the multiplication.
Check 8a = 72 Substitute 9 for a in the equation.
Solve 8a = 72. Check your answer.
9 is the solution. 72 = 72 ?
Try This: Example 1A
8(9)= 72 ?
8
Course 1
2-6 Solving Multiplication Equations
8a = 72
18 = 3w w is multiplied by 3.
3 6 = w
Divide both sides by 3 to undo the multiplication.
Check 18 = 3w Substitute 6 for w in the equation.
Solve 18 = 3w. Check your answer.
6 is the solution. 18 = 18 ?
Try This: Example 1B
18 = 3(6) ?
3
Course 1
2-6 Solving Multiplication Equations
18 = 3w
Course 1
2-6 Solving Multiplication Equations
The area of a rectangle is 56 square inches. Its length is 7 inches. What is its width?
Additional Example 2: Problem Solving
Course 1
2-6 Solving Multiplication Equations
Additional Example 2 Continued
1 Understand the Problem
The answer will be the width of the rectangle in inches.
List the important information:
• The area of the rectangle is 56 square inches.
• The length of the rectangle is 7 inches.
Draw a diagram to represent this information.
56
7
w
Course 1
2-6 Solving Multiplication Equations
Additional Example 2 Continued
2 Make a Plan
You can write and solve an equation using the formula for area. To find the area of a rectangle, multiply its length by its width.
w
l
A = lw 56 = 7w
Course 1
2-6 Solving Multiplication Equations
Additional Example 2 Continued
56 = 7w 56 = 7w
Solve 3
7 7 8 = w
So the width of the rectangle is 8 inches.
w is multiplied by 7. Divide both sides by 7 to undo the multiplication.
Course 1
2-6 Solving Multiplication Equations
Additional Example 2 Continued
Look Back
Arrange 56 identical squares in a rectangle. The length is 7, so line up the squares in rows of 7. You can make 8 rows of 7, so the width of the rectangle is 8.
4
Try This: Example 2
Course 1
2-6 Solving Multiplication Equations
The area of a rectangle is 48 square inches. Its width is 8 inches. What is its length?
Course 1
2-6 Solving Multiplication Equations
Try This: Example 2
1 Understand the Problem
The answer will be the length of the rectangle in inches.
List the important information:
• The area of the rectangle is 48 square inches.
• The width of the rectangle is 8 inches.
Draw a diagram to represent this information.
48
l
8
Course 1
2-6 Solving Multiplication Equations
Try This: Example 2 Continued
2 Make a Plan
You can write and solve an equation using the formula for area. To find the area of a rectangle, multiply its length by its width.
w
l
A = lw 48 = l8
Course 1
2-6 Solving Multiplication Equations
Try This: Example 2 Continued
48 = l 8 48 = l 8
Solve 3
8 8 6 = l
So the length of the rectangle is 6 inches.
l is multiplied by 8. Divide both sides by 8 to undo the multiplication.
Course 1
2-6 Solving Multiplication Equations
Try This: Example 2 Continued
Look Back
Arrange 48 identical squares in a rectangle. The width is 8, so line up the squares in columns of 8. You can make 6 columns of 8, so the length of the rectangle is 6.
4
1. 10y = 300
2. 2y = 82
3. 63 = 9y
4. 78 = 13x
Lesson Quiz
5. The area of a board game is 468 square inches. Its width is 18 inches. What is the length?
y = 30
y = 41
y = 7
x = 6
26 inches
Course 1
2-6 Solving Multiplication Equations
Course 1
2-7 Solving Division Equations
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Solve.
1. 5x = 20
2. 7n = 84
3. 18 = y – 4
4. 21 = n + 3
x = 4
n = 12
y = 22
n = 18
Course 1
2-7 Solving Division Equations
Course 1
Problem of the Day
If 11 is 4 less than 3 times a number, what is the number? 5
2-7 Solving Division Equations
Course 1
Learn to solve whole-number division equations.
2-7 Solving Division Equations
Course 1
Multiplication is the inverse of division. When an equation contains division, use multiplication to “undo” the division.
2-7 Solving Division Equations
Course 1
x = 35
Multiply both sides by 7 to undo the division.
Substitute 35 for x in the equation.
Solve the equation. Check your answer.
35 is the solution. 5 = 5 ?
Additional Example 1A: Solving Division Equations
2-7 Solving Division Equations
Course 1
x 7 = 5
x is divided by 7. x 7 = 5
7 • x 7 = 7 • 5
Check x 7 = 5
35 7 = 5 ?
78 = p
Multiply both sides by 6 to undo the division.
Substitute 78 for p in the equation.
Solve the equation. Check your answer.
78 is the solution. 13 = 13 ?
Additional Example 1B: Solving Division Equations
2-7 Solving Division Equations
Course 1
p 6 13 =
6 • p 6
6 • 13 =
p is divided by 6. p 6 13 =
Check p 6 13 =
? 78 6 13 =
x = 18
Multiply both sides by 2 to undo the division.
Substitute 18 for x in the equation.
Solve the equation. Check your answer.
18 is the solution. 9 = 9 ?
Try This: Example 1A
2-7 Solving Division Equations
Course 1
x 2 = 9
x is divided by 2. x 2 = 9
2 • x 2 = 2 • 9
Check x 2 = 9
18 2 = 9 ?
288 = p
Multiply both sides by 4 to undo the division.
Substitute 288 for p in the equation.
Solve the equation. Check your answer.
288 is the solution. 72 = 72 ?
Try This: Example 1B
2-7 Solving Division Equations
Course 1
p 4 72 =
4 • p 4
4 • 72 =
p is divided by 4. p 4 72 =
Check p 4 72 =
? 288 4 72 =
An aspen tree is one-third the height of a pine tree at Elk Meadow Park.
Additional Example 2
2-7 Solving Division Equations
Course 1
height of aspen = height of pine 3
The aspen tree is 14 feet tall. How tall is the pine tree? Let h represent the height of the pine tree.
42 = h
Multiply both sides by 3 to undo the division.
3 • h 3
3 • 14 =
Substitute 14 for height of aspen. h is divided by 3.
h 3 14 =
The pine tree is 42 feet tall.
Try This: Example 2
2-7 Solving Division Equations
Course 1
Let w represent her father’s weight.
180 = w
Multiply both sides by 2 to undo the division.
2 • w 2
2 • 95 =
Substitute 95 for Jamie’s weight. w is divided by 2.
w 2 95 =
Jamie’s father weighs 180 pounds.
Jamie weighs one-half as much as her father.
Jamie’s weight = father’s weight 2
Jamie weighs 95 pounds. How many pounds does her father weigh?
1. = 7
2. 8 =
3. = 11
4. = 7
Lesson Quiz
5. The area of Sherry’s flower garden is one-fourth the area of her vegetable garden. The area of the flower garden is 17 square feet. Let x represent the area of her vegetable garden. Find the area of her vegetable garden?
x = 70
x = 32
x = 99
x = 105
68 square feet
2-7 Solving Division Equations
Course 1
x 4
x 9
Solve each equation. Check your answers. x 10
x 15
3-1 Representing, Comparing, and Ordering Decimals
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Order the numbers from least to greatest.
1. 242, 156, 224, 165 2. 941, 148, 914, 814, 721 3. 345, 376, 354, 397
156, 165, 224, 242
148, 721, 814, 914, 941
345, 354, 376, 397
Course 1
3-1 Representing, Comparing, and Ordering Decimals
Problem of the Day
Lupe is taller than Reba and shorter than Miguel. Tory is shorter than Lupe but taller than Reba. List the four brothers and sisters in order from tallest to shortest.
Miguel, Lupe, Tory, Reba
Course 1
3-1 Representing, Comparing, and Ordering Decimals
Learn to write, compare, and order decimals using place value and number lines.
Course 1
3-1 Representing, Comparing, and Ordering Decimals
Additional Example 1A & B: Reading and Writing Decimals
Write each decimal in standard form, expanded form, and words.
A. 1.07
B. 0.03 + 0.006 + 0.0009
1 + 0.07 Expanded form: one and seven hundredths Word form:
0.0369 Standard form: three hundred sixty-nine ten-thousandths Word form:
Course 1
3-1 Representing, Comparing, and Ordering Decimals
Additional Example C: Reading and Writing Decimals
Write the decimal in standard form and expanded form.
C. fourteen and eight hundredths 14.08 Standard form:
10 + 4 + 0.08 Expanded form:
Course 1
3-1 Representing, Comparing, and Ordering Decimals
Try This: Example 1A & 1B
Write each decimal in standard form, expanded form, and words.
A. 1.12
B. 0.6 + 0.008 + 0.0007
1 + 0.12 Expanded form: one and twelve hundredths Word form:
0.6087 Standard form: six thousand eighty-seven ten-thousandths
Word form:
Course 1
3-1 Representing, Comparing, and Ordering Decimals
Try This: Example 1C
Write each decimal in standard form and expanded form.
C. eleven and two hundredths
11.02 Standard form:
10 + 1 + 0.02 Expanded form:
Course 1
3-1 Representing, Comparing, and Ordering Decimals
Additional Example 2: Earth Science Application
The star Wolf 359 has an apparent magnitude of 13.5. Suppose another star has an apparent magnitude of 13.05. Which star has the smaller magnitude?
13.50 Line up the decimal points.
Start from the left and compare the digits. 13.05 Look for the first place where the digits are
different. 0 is less than 5. 13.05 < 13.50 The star that has an apparent magnitude of 13.05 has the smaller magnitude.
Course 1
3-1 Representing, Comparing, and Ordering Decimals
Try This: Example 2
Tina reported on a star for her science project that has a magnitude of 11.3. Maven reported on another star that has a magnitude of 11.03. Which star has the smaller magnitude?
11.30 Line up the decimal points.
Start from the left and compare the digits. 11.03 Look for the first place where the digits are
different. 0 is less than 3. 11.03 < 11.30 The star that has a magnitude of 11.03 has the smaller magnitude.
Course 1
3-1 Representing, Comparing, and Ordering Decimals
Additional Example 3: Comparing and Ordering Decimals
Order the decimals from least to greatest.
16.67, 16.6, 16.07
16.67
16.60
Compare two of the numbers at a time.
Write 16.6 as “16.60.” 16.60 < 16.67 16.67
16.07 Start at the left and compare the digits. 16.07 < 16.67
Course 1
3-1 Representing, Comparing, and Ordering Decimals
16.60
16.07 Look for the first place where the digits are different. 16.07 < 16.60
Additional Example 3 Continued
Graph the numbers on a number line.
16.67 16.07 16.6
Course 1
3-1 Representing, Comparing, and Ordering Decimals
16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 17
The numbers are ordered when you read the number line from left to right. The numbers in order from least to greatest are 16.07, 16.6, and 16.67.
Try This: Example 3
Order the decimals from least to greatest.
12.42, 12.4, 12.02
12.42
12.40
Compare two of the numbers at a time.
Write 12.4 as “12.40.” 12.40 < 12.42
12.42
12.02 Start at the left and compare the digits. 12.02 < 12.42
Course 1
3-1 Representing, Comparing, and Ordering Decimals
12.40
12.02 Look for the first place where the digits are different. 12.02 < 12.40
Try This: Example 3 Continued
Graph the numbers on a number line.
Course 1
3-1 Representing, Comparing, and Ordering Decimals
12.42 12.02 12.4
12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13
The numbers are ordered when you read the number line from left to right. The numbers in order from least to greatest are 12.02, 12.4, and 12.42.
Lesson Quiz: Part 1 Write each in standard form, expanded form and words.
1. 8.0342
2. 18 + 0.3 + 0.006
3. eight and twelve hundredths
18.306; eighteen and three hundred six thousandths
8 + 0.03 + 0.004 + 0.0002; eight and three hundred forty-two ten thousandths
Insert Lesson Title Here
8.12; 8 + 0.1 + 0.02
Course 1
3-1 Representing, Comparing, and Ordering Decimals
Lesson Quiz: Part 2
4. It takes Pluto 246.7 years to orbit the Sun, and it takes Neptune 164.8 years. Which planet takes longer to orbit the Sun?
5. Order the decimals from least to greatest: 16.35, 16.3, 16.5.
Insert Lesson Title Here
Pluto
16.3, 16.35, 16.5
Course 1
3-1 Representing, Comparing, and Ordering Decimals
3-2 Estimating Decimals
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Order the decimals from least to greatest.
1. 18.74, 18.7, 18.47 2. 9.06, 9.66, 9.6, 9.076 Write each in words. 3. 3.072 4. 6.1258
18.47, 18.7, 18.74
9.06, 9.076, 9.6, 9.66
three and seventy-two thousandths
Course 1
3-2 Estimating Decimals
six and one thousand two hundred fifty-eight ten-thousandths
Problem of the Day
Calculate your age in months.
Possible answer: 11 yr 8 mo = 140 mo
Course 1
3-2 Estimating Decimals
Learn to estimate decimal sums, differences, products, and quotients.
Course 1
3-2 Estimating Decimals
Vocabulary clustering front-end estimation
Insert Lesson Title Here
Course 1
3-2 Estimating Decimals
Course 1
3-2 Estimating Decimals
When numbers are about the same value, you can use clustering to estimate. Clustering means rounding the numbers to the same value.
Course 1
3-2 Estimating Decimals Additional Example 1: Health Application
Nancy wants to cycle, ice skate, and water ski for 30 minutes each. About how many calories will she burn in all? (Cycling = 165.5 calories, ice skating = 177.5 calories, and water skiing = 171.5 calories) 165.5 170 The addends cluster around 170. 177.5 170 To estimate the total number of
calories, round each addend to 170.
171.5 + 170
Add. 510
Nancy burns about 510 calories
Course 1
3-2 Estimating Decimals Try This: Example 1
Abner wants to run, roller blade, and snow ski for 60 minutes each. About how many calories will he burn in all? (Running = 185.5 calories, roller blading = 189.5 calories, and snow skiing = 191.5 calories) 185.5 190 The addends cluster around 190. 189.5 190 To estimate the total number of
calories, round each addend to 190. 191.5 + 190
Add. 570
Abner burns about 570 calories
Course 1
3-2 Estimating Decimals
When rounding, look at the digit to the right of the place to which you are rounding.
• If it is 5 or greater, round up.
• If it is less than 5, round down.
Remember!
Course 1
3-2 Estimating Decimals Additional Example 2: Rounding Decimals to
Estimate Sums and Differences
Estimate by rounding to the indicated place value.
A. 7.13 + 4.68; ones
B. 9.705 – 0.2683; tenths
7.13 + 4.68 Round to the nearest whole number. 7 + 5 = 12 The sum is about 12.
9.705 9.7 Round to the tenths. Align.
9.4 Subtract. – 0.2683 –0.3
Course 1
3-2 Estimating Decimals
Try This: Example 2
Estimate by rounding to the indicated place value.
A. 6.09 + 3.72; ones
B. 8.898 – 0.4619; tenths
6.09 + 3.72 Round to the nearest whole number. 6 + 4 = 10 The sum is about 10.
8.898 8.9 Round to the tenths. Align.
8.4 Subtract. –0.4619 –0.5
Course 1
3-2 Estimating Decimals
Compatible numbers are close to the numbers that are in the problem and are helpful when you are solving the problem mentally.
Remember!
Course 1
3-2 Estimating Decimals
Additional Example 3: Using Compatible Numbers to Estimate Products and Quotients
Estimate each product or quotient.
A. 33.83 × 1.98
B. 72.77 ÷ 26.14
35 × 2 = 70 35 and 2 are compatible.
75 ÷ 25 = 3 75 and 25 are compatible.
So, 72.77 ÷ 26.14 is about 3.
So 33.83 × 1.98 is about 70.
Course 1
3-2 Estimating Decimals
Try This: Example 3
Estimate each product or quotient.
A. 22.12 × 4.98
B. 62.31 ÷ 18.52
20 × 5 = 100 20 and 5 are compatible.
60 ÷ 20 = 3 60 and 20 are compatible.
So, 62.31 ÷ 18.52 is about 3.
So 22.12 × 4.98 is about 100.
Course 1
3-2 Estimating Decimals
You can also use front-end estimation to estimate with decimals. Front-end estimation means to use only the whole-number part of the decimal.
Course 1
3-2 Estimating Decimals
Additional Example 4: Using Front-End Estimation
Estimate a range for the sum.
7.86 + 36.97 + 5.40
Use front-end estimation.
7.86 7 Add the whole numbers only. 36.97 36 The whole-number values of the
decimals are less than the actual numbers, so the answer is an underestimate.
5.40 + 5 at least 48
The exact answer of 7.86 + 36.97 + 5.40 is 48 or greater.
Course 1
3-2 Estimating Decimals
Additional Example 4 Continued
You can estimate a range for the sum by adjusting the decimal part of the numbers. Round the decimals to 0, 0.5, or 1.
0.86 1.00 Add the decimal part of the numbers.
0.97 1.00 Add the whole-number estimate and the adjusted estimate. 0.40 + 0.50
2.50
48.00 + 2.50 = 50.50
The adjusted decimals are greater than the actual decimal, so 50.50 is an overestimate.
The estimated range for the sum is from 48.00 to 50.50.
Course 1
3-2 Estimating Decimals
Try This: Example 4
Estimate a range for the sum.
8.92 + 47.88 + 3.41
Use front-end estimation.
8.92 8 Add the whole numbers only. 47.88 47 The whole-number values of the
decimals are less than the actual numbers, so the answer is an underestimate.
3.41 + 3 at least 58
The exact answer of 8.92 + 47.88 + 3.41 is 58 or greater.
Course 1
3-2 Estimating Decimals
Try This: Example 4
You can estimate a range for the sum by adjusting the decimal part of the numbers. Round the decimals to 0, 0.5, or 1.
0.92 1.00 Add the decimal part of the numbers.
0.88 1.00 Add the whole-number estimate and the adjusted estimate. 0.41 +0.50
2.50
58.00 + 2.50 = 60.50
The adjusted decimals are greater than the actual decimal, so 60.50 is an overestimate.
The estimated range for the sum is from 58.00 to 60.50.
Lesson Quiz: Part 1
Estimate by rounding to the indicated place value.
3
4.5
Insert Lesson Title Here
Course 1
3-2 Estimating Decimals
1. 3.07442 + 1.352; tenths
2. 7.305 – 4.12689; nearest whole number
Lesson Quiz: Part 2 Estimate each product or quotient.
3
14
Insert Lesson Title Here
80
Course 1
3-2 Estimating Decimals
3. 6.75 × 1.82
4. 10.5 ÷ 3.42
5. The snowfall in December, January, and February was 18.26 cm, 29.36 cm, and 32.87 cm, respectively. About how many total centimeters of snow fell during the three months?
3-3 Adding and Subtracting Decimals
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Estimate by rounding to the indicated place value.
1. 70.27 + 15.36; ones 2. 84.37 – 21.82; tenths Estimate each product or quotient. 3. 27.25 × 8.7 4. 44.52 ÷ 3.27
85 62.6
270
Course 1
3-3 Adding and Subtracting Decimals
15
Problem of the Day
Find a three-digit number that rounds to 440 and includes a digit that is the quotient of 24 and 3. Is there more than one possible answer? Explain your thinking.
438; no; the numbers that round to 440 are 435-444, 24 divided by 3 is 8, and 438 is the only number with 8 as a digit.
Course 1
3-3 Adding and Subtracting Decimals
Learn to add and subtract decimals.
Course 1
3-3 Adding and Subtracting Decimals
Course 1
3-3 Adding and Subtracting Decimals
Estimating before you add or subtract will help you check whether your answer is reasonable.
Helpful Hint
Course 1
3-3 Adding and Subtracting Decimals
Elise Ray’s Scores Event Points
Floor exercise 9.8
Balance beam 9.7
Vault 9.425
Uneven bars 9.85
American gymnast Elise Ray won the 2000 U.S. Championships in the all-around, uneven bars, and floor-exercise events.
To find the total number of points, you add all of the scores.
Course 1
3-3 Adding and Subtracting Decimals
Additional Example 1A: Sports Application
A. What was Elise Ray’s total for the events other than the floor exercise?
Estimate by rounding to the nearest whole number.
9.7 + 9.425 + 9.85
The total is about 29 points. 10 + 9 + 10 = 29
Find the sum of 9.7, 9.425, and 9.85.
Course 1
3-3 Adding and Subtracting Decimals
Additional Example 1A Continued
Add.
9.700
9.425 +9.850 28.975
Align the decimal points.
Use zeros as placeholders.
Add. Then place the decimal point.
Since 28.975 is close to the estimate of 29, the answer is reasonable. Elise Ray’s total for the events other than the floor exercise was 28.975.
Course 1
3-3 Adding and Subtracting Decimals
Additional Example 1B: Sports Application
B. How many more points did Elise need on the vault to have a perfect score of 10?
10.000 -9.425 0.575
Align the decimal points. Use zeros as placeholders. Subtract. Then place the decimal point.
Elise needed another 0.575 points to have a perfect score.
Find the difference between 10 and 9.425.
Course 1
3-3 Adding and Subtracting Decimals
Try This: Example 1A
A. What was Elise Ray’s total for the events other than the vault exercise?
Estimate by rounding to the nearest whole number. 9.8 + 9.7 + 9.85
The total is about 30 points. 10 + 10 + 10 = 30
Find the sum of 9.8, 9.7, and 9.85.
Course 1
3-3 Adding and Subtracting Decimals
Try This: Example 1A Continued
Add 9.800
9.700 +9.850 29.350
Align the decimal points.
Use zeros as placeholders.
Add. Then place the decimal point.
Since 29.350 is close to the estimate of 30, the answer is reasonable. Elise Ray’s total for the events other than the vault exercise was 29.350.
Course 1
3-3 Adding and Subtracting Decimals
Try This: Example 1B
B. How many more points did Elise need on the uneven bars to have a perfect score of 1
10.00 -9.85 0.15
Align the decimal points. Use zeros as placeholders. Subtract. Then place the decimal point.
Elise needed another 0.15 points to have a perfect score.
Find the difference between 10 and 9.85.
Course 1
3-3 Adding and Subtracting Decimals
Additional Example 2: Using Mental Math to Add and Subtract Decimals
Find each sum or difference.
A. 1.8 + 0.2
B. 4 – 0.7
Think: 0.8 + 0.2 = 1. 1.8 + 0.2 = 2
Think: What number added to 0.7 is 1?
4 – 0.7 = 3.3 0.7 + 0.3 = 1 So 1 – 0.7 = 0.3
Course 1
3-3 Adding and Subtracting Decimals
Try This: Example 2
Find each sum or difference.
A. 1.6 + 0.4
B. 6 – 0.3
Think: 0.6 + 0.4 = 1. 1.6 + 0.4 = 2.0
Think: What number added to 0.3 is 1?
6 – 0.3 = 5.7 0.3 + 0.7 = 1 So 1 – 0.3 = 0.7
Course 1
3-3 Adding and Subtracting Decimals
Additional Example 3A: Evaluating Decimal Expressions
Evaluate 6.73 – x for each value of x.
A. x = 3.8
Substitute 3.8 for x. 6.73 – x 6.73 – 3.8 6.73 Align the decimal points.
– 3.80 Use a zero as a placeholder.
2.93 Subtract. Place the decimal point.
Course 1
3-3 Adding and Subtracting Decimals
Additional Example 3B: Evaluating Decimal Expressions
Evaluate 6.73 – x for each value of x.
B. x = 2.9765
Substitute 2.9765 for x. 6.73 – x 6.73 – 2.9765 6.7300 Align the decimal points.
–2.9765 Use a zero as a placeholder.
3.7535 Subtract. Place the decimal point.
Course 1
3-3 Adding and Subtracting Decimals
Try This: Example 3A
Evaluate 7.58 – x for each value of x.
A. x = 3.8
Substitute 3.8 for x. 7.58 – x 7.58 – 3.8 7.58 Align the decimal points. –3.80 Use a zero as a placeholder. 3.78 Subtract. Place the decimal point.
Course 1
3-3 Adding and Subtracting Decimals
Try This: Example 3B
Evaluate 8.17 – x for each value of x.
B. x = 2.9765
Substitute 2.9765 for x. 8.17 – x 8.17 – 2.9765 8.1700 Align the decimal points. – 2.9765 Use a zero as a placeholder.
5.1935 Subtract. Place the decimal point.
Lesson Quiz
Find each sum or difference.
1. 8.3 + 2.7
2. 9.7 – 4
3. 22.6 + 8.4
4. Evaluate 12.76 – x for x = 8.41.
5. During an ice-skating competition, Dawn received the following scores: 4.8, 5.2, 5.4. What was Dawn’s total score?
5.7
11
Insert Lesson Title Here
31
4.35
Course 1
3-3 Adding and Subtracting Decimals
15.4
3-4 Decimals and Metric Measurement
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Multiply.
84
87.2 4.2
Course 1
3-4 Decimals and Metric Measurement
73,200 920
1. 8.4 × 10
2. 8.72 × 10 3. 0.42 × 10 4. 732 × 100 5. 9.2 × 100
Problem of the Day
A nurse must administer a 4 mL dose of a drug daily to a patient. If there is one liter of this drug on hand, will it last the patient 150 days? If so, how much will remain? If not, how much more will be needed?
0.4L will remain.
Course 1
3-4 Decimals and Metric Measurement
Learn to multiply and divide decimals by powers of ten and to convert metric measurements.
Course 1
3-4 Decimals and Metric Measurement
Course 1
3-4 Decimals and Metric Measurement
Additional Example 1A: Multiplying and Dividing by Powers of Ten
Multiply.
A. 7,126 × 1,000 7,126.000
= 7,126,000
There are 3 zeros in 1,000. To multiply, move the decimal point 3 places right. Write 3 placeholder zeros.
Course 1
3-4 Decimals and Metric Measurement Additional Example 1B & 1C: Multiplying and
Dividing by Powers of Ten
Divide.
B. 7,126 ÷ 1,000
C. 46.34 ÷ 104
7,126.
= 7.126
There are 3 zeros in 1,000. To divide, move the decimal point 3 places left.
0046.34
= 0.004634
The power of 10 is 4. Move the decimal point 4 places left. Write placeholder zeros.
Course 1
3-4 Decimals and Metric Measurement
Try This: Example 1A
Multiply.
A. 4,639 × 1,000
4,639.000
= 4,639,000
There are 3 zeros in 1,000. To multiply, move the decimal point 3 places right. Write 3 placeholder zeros.
Course 1
3-4 Decimals and Metric Measurement
Try This: Example 1B & 1C
Divide.
B. 4,639 ÷ 1,000
C. 32.08 ÷ 104
4,639.
= 4.639
There are 3 zeros in 1,000. To divide, move the decimal point 3 places left.
0032.08
= 0.003208
The power of 10 is 4. Move the decimal point 4 places left. Write placeholder zeros.
Course 1
3-4 Decimals and Metric Measurement
Unit Abbreviation Approximate Comparison
Length
Kilometer km Length of 10 football fields
Meter m Width of a door
Centimeter cm Width of your little finger
Millimeter mm Thickness of a dime
Course 1
3-4 Decimals and Metric Measurement
Unit Abbreviation Approximate Comparison
Mass Kilogram kg Mass of a
textbook
Gram g Mass of a small paperclip
Course 1
3-4 Decimals and Metric Measurement
Unit Abbreviation Approximate Comparison
Capacity Liter L Filled bottle of
sparkling water
Milliliter mL Half-filled eyedropper
Course 1
3-4 Decimals and Metric Measurement Additional Example 2A & 2B: Choosing
Appropriate Units
Use the abbreviation for the most appropriate metric unit.
A. A soda can is about 12 ____ tall.
B. The mass of a pen is about 5 ____.
A soda can is about 12 cm tall.
Think: A soda can is about the height of 12 little-finger widths.
?
The mass of a pen is about 5 g.
Think: A pen has a mass of about 5 small paper clips.
?
Course 1
3-4 Decimals and Metric Measurement
Additional Example 2C: Choosing Appropriate Units
Use the abbreviation for the most appropriate metric unit.
C. A sip of water is about 3 ____.
?
A sip of water is about 3 mL.
Think: A sip of water is about 3 half-filled eyedroppers.
Course 1
3-4 Decimals and Metric Measurement
Try This: Example 2A & 2B
Use the abbreviation for the most appropriate metric unit.
A. A one-car garage door is about 4 ____ wide.
B. The mass of an eraser is about 7 ____.
A garage door is about 4 m wide.
Think: A garage door is about the width of 4 door widths.
?
The mass of an eraser is about 7 g.
Think: An eraser has a mass of about 7 small paper clips.
?
Course 1
3-4 Decimals and Metric Measurement
Try This: Example 2C
Use the abbreviation for the most appropriate metric unit.
C. A telephone directory weighs about 3 ____.
?
A telephone directory weighs about 3 kg .
Think: A telephone directory is about 3 textbooks.
Course 1
3-4 Decimals and Metric Measurement
1,000 100 10 1 0.1 0.01 0.001
Thousands Hundreds Tens Ones Tenths Hundredths Thousandths
Kilo- Hecto- Deca- Base unit Deci- Centi- Milli-
In the metric system, each unit of measure is ten times greater than the unit to its right in a place-value chart.
To convert units within the metric system, multiply or divide by powers of ten.
Course 1
3-4 Decimals and Metric Measurement Additional Example 3A: Converting Within the
Metric System
Convert each measure.
A. The height of a door is about 2 m. 2 m = ___ km
2 m = (2 ÷ 1000)km 1 km = 1,000 m, so divide by 1,000.
?
2 m = 0.002 km Move the decimal point 3 places left.
To convert to smaller units, multiply.
To convert to larger units, divide.
Helpful Hint
Course 1
3-4 Decimals and Metric Measurement
Additional Example 3B: Converting Within the Metric System
Convert the measure.
B. The width of a door is about 0.85 m. 0.85 m = ___ cm
?
0.85 m = (0.85 × 100)cm 1 m = 100 cm, so multiply by 100.
0.85 m = 85 cm Move the decimal point 2 places right.
Course 1
3-4 Decimals and Metric Measurement
Try This: Example 3A
Convert each measure.
A. The height of a building is about 30 m. 30 m = ___ km
30 m = (30 ÷ 1000)km 1 km = 1,000 m, so divide by 1,000.
?
30 m = 0.03 km Move the decimal point 3 places left.
Course 1
3-4 Decimals and Metric Measurement
Try This: Example 3B
Convert the measure.
B. The width of a chair is about 0.98 m. 0.98 m = ___ cm ?
0.98 m = (0.98 × 100)cm 1 m = 100 cm, so multiply by 100.
0.98 m = 98 cm Move the decimal point 2 places right.
Lesson Quiz: Part 1
7.15731 1,604
Insert Lesson Title Here
m
cm
Course 1
3-4 Decimals and Metric Measurement
Multiply or divide.
1. 16.04 × 102 2. 715.731 ÷ 100
Use the abbreviation for the most appropriate metric unit.
3. The length of a soccer field is about 100 ___.
4. The width of a computer screen is about 23 ___.
?
?
Lesson Quiz: Part 2
Convert each measure.
5. The width of film for slides is 35 mm. 35 mm = ____ m.
6. The weight of an adult is about 70 kg. 70 kg = ____ g.
0.035
Insert Lesson Title Here
70,000
Course 1
3-4 Decimals and Metric Measurement
?
?
3-5 Scientific Notation
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Multiply.
1. 724 × 102
2. 837 × 10 3. 632.9 × 100 4. 18,256 × 10 5. 10 × 10 × 10
72,400 8,370 63,290
Course 1
3-5 Scientific Notation
182,560 1,000
Problem of the Day
A rope ladder is hanging from the back of a yacht. At 10:00 A.M., the water reaches the third step on the ladder. If every 2 hours the water rises the height of two steps, at what step would the water level be at 5:00 P.M.? Explain.
the third, because the boat will rise with the water
Course 1
3-5 Scientific Notation
Learn to write large numbers in scientific notation.
Course 1
3-5 Scientific Notation
Vocabulary scientific notation
Insert Lesson Title Here
Course 1
3-5 Scientific Notation
Course 1
3-5 Scientific Notation
Scientific notation is a shorthand method for writing large numbers like 3,456,000.
Course 1
3-5 Scientific Notation
To write the number 3,456,000 in scientific notation, do the following:
3,456,000 Move the decimal point left to form a number that is greater than 1 and less than 10.
3,456,000 Multiply that number by a power of ten.
3.456 × 106 The power of 10 is 6, because the decimal point is moved 6 places left.
Course 1
3-5 Scientific Notation
The number of zeros in the power of ten, or the exponents in the power of ten, tells you how many places to move the decimal point
Remember!
Course 1
3-5 Scientific Notation
A number written in scientific notation has two parts that are multiplied.
3.456 × 106
The first part is a number that is greater than 1 and less than 10.
The second part is a power of 10.
Course 1
3-5 Scientific Notation Additional Example 1A & 1B: Writing Numbers
in Scientific Notation
Write each number in scientific notation.
A. 6,000,000
B. 411,000
6,000,000 Move the decimal point 6 places left. The power of 10 is 6.
6,000,000 = 6 × 106
411,000 Move the decimal point 5 places left. The power of 10 is 5.
411,000 = 4.11 × 105
Course 1
3-5 Scientific Notation
Additional Example 1C: Writing Numbers in Scientific Notation
Write the number in scientific notation.
C. 79,000,000
79,000,000 Move the decimal point 7 places left. The power of 10 is 7.
79,000,000 = 7.9 × 107
Course 1
3-5 Scientific Notation
Try This: Example 1A & 1B
Write each number in scientific notation.
A. 8,000,000
B. 587,000
8,000,000 Move the decimal point 6 places left. The power of 10 is 6.
8,000,000 = 8 × 106
587,000 Move the decimal point 5 places left. The power of 10 is 5.
587,000 = 5.87 × 105
Course 1
3-5 Scientific Notation
Try This: Example 1C
Write the number in scientific notation.
C. 13,000,000
13,000,000 Move the decimal point 7 places left. The power of 10 is 7.
13,000,000 = 1.3 × 107
Course 1
3-5 Scientific Notation
You can write a large number written in scientific notation in standard form. Look at the power of 10 and move the decimal point that number of places to the right.
Course 1
3-5 Scientific Notation Additional Example 2A & 2B: Writing Numbers
in Standard From
Write each number in standard form.
A. 6.2174 × 103
B. 9.5 × 108
The power of 10 is 3.
6.2174 × 103 = 6,217.4
6.2174
The power of 10 is 8.
9.50000000
Move the decimal point 3 places right.
Move the decimal point 8 places right. Use zeros as placeholders.
9.5 × 108 = 950,000,000
Course 1
3-5 Scientific Notation Additional Example 2C: Writing Numbers in
Standard From
Write the number in standard form.
C. 4.83 × 105
The power of 10 is 5.
4.83 × 105 = 483,000
4.83000 Move the decimal point 5 places right. Use zeros as placeholders.
Course 1
3-5 Scientific Notation Try This: Example 2A & 2B
Write each number in standard form.
A. 8.1974 × 103
B. 2.3 × 106
The power of 10 is 3.
8.1974 × 103 = 8,197.4
8.1974
The power of 10 is 6. 2.300000
Move the decimal point 3 places right.
Move the decimal point 6 places right. Use zeros as placeholders.
2.3 × 106 = 2,300,000
Course 1
3-5 Scientific Notation
Try This: Example 2C
Write the number in standard form.
C. 9.77 × 105
The power of 10 is 5.
9.77 × 105 = 977,000
9.77000 Move the decimal point 5 places right. Use zeros as placeholders.
Lesson Quiz
Write each number in scientific notation.
1. 6,300
2. 70,400,000 Write each number in standard form. 3. 7.241 × 104
4. 8.2137 × 107
5. A Wall Street report indicated that a fast-moving stock had sold 3,295,000 shares. Write this number in scientific notation.
Insert Lesson Title Here
72,410
82,137,000
Course 1
3-5 Scientific Notation
6.3 × 103
7.04 × 107
3.295 × 106
3-6 Multiplying Decimals
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Multiply.
27,840 754,400 38,060
Course 1
3-6 Multiplying Decimals
120,700 14,112 62,760
1. 87 × 320 2. 943 × 800 3. 3,806 × 10 4. 1,207 × 100 5. 72 × 196 6. 120 × 523
Problem of the Day
Carmen and Rita sold homemade oatmeal cookies. After 8 days, they had sales totaling $70. Each day, their sales were $0.50 higher than the previous day. What were their sales on the first day?
$7.00
Course 1
3-6 Multiplying Decimals
Learn to multiply decimals by whole numbers and by decimals.
Course 1
3-6 Multiplying Decimals
Course 1
3-6 Multiplying Decimals Additional Example 1: Science Application
Something that weighs 1 lb on Earth weighs 0.17 lb on the Moon. How much would a 4 lb dumbbell weigh on the Moon?
4 × 0.17
0.17 You can think of multiplication by a whole number as a repeated addition. 0.17
0.17 0.17 + _____ 0.68
Course 1
3-6 Multiplying Decimals Additional Example 1 Continued
Something that weighs 1 lb on Earth weighs 0.17 lb on the Moon. How much would a 4 lb dumbbell weigh on the Moon?
You can also multiply as you would with whole numbers. Place the decimal point by adding the number of decimal places in the numbers multiplied.
0.17 4 × _____ 0.68
2 decimal places + 0 decimal places
2 decimal places
A 4 lb dumbbell on Earth weighs 0.68 lb on the Moon.
Course 1
3-6 Multiplying Decimals Try This: Example 1
Something that weighs 1 lb on Earth weighs 0.17 lb on the Moon. How much would a 7 lb dumbbell weigh on the Moon?
7 × 0.17 0.17 You can think of multiplication by a
whole number as a repeated addition.
0.17 0.17 0.17
+ _____ 1.19
0.17 0.17
0.17
Course 1
3-6 Multiplying Decimals Try This: Example 1 Continued
Something that weighs 1 lb on Earth weighs 0.17 lb on the Moon. How much would a 7 lb dumbbell weigh on the Moon?
You can also multiply as you would with whole numbers. Place the decimal point by adding the number of decimal places in the numbers multiplied.
0.17 7 × _____ 1.19
2 decimal places + 0 decimal places
2 decimal places
A 7 lb dumbbell on Earth weighs 1.19 lb on the Moon.
Course 1
3-6 Multiplying Decimals Additional Example 2A: Multiplying a Decimal by
a Decimal
Find the product.
A. 0.3 × 0.4 Multiply. Then place the decimal point.
0.3 0.4 ×
1 decimal place + 1 decimal place
2 decimal places 0.12
Course 1
3-6 Multiplying Decimals Additional Example 2B: Multiplying a Decimal by
a Decimal
B. 0.07 × 0.8
Multiply. Then place the decimal point.
0.07 0.8 ×
2 decimal places + 1 decimal place
3 decimal places; use a placeholder zero
0.056
Estimate the product. 0.8 is close to 1.
0.07 × 1 = 0.07
0.056 is close to the estimate of 0.07. The answer is reasonable.
Course 1
3-6 Multiplying Decimals Additional Example 2C: Multiplying a Decimal by
a Decimal
C. 1.34 × 2.5
Multiply. Then place the decimal point.
1.34 2.5 ×
2 decimal places + 1 decimal place
3 decimal places
670
Estimate the product. Round each factor to the nearest whole number.
1 × 3 = 3
3.350 is close to the estimate of 3. The answer is reasonable.
2680 3.350
Course 1
3-6 Multiplying Decimals
Try This: Example 2A
Find each product.
A. 0.5 × 0.2
Multiply. Then place the decimal point.
0.5 0.2 ×
1 decimal place + 1 decimal place
2 decimal places 0.10
Course 1
3-6 Multiplying Decimals
Try This: Example 2B
B. 0.03 × 0.9
Multiply. Then place the decimal point.
0.03 0.9 ×
2 decimal places + 1 decimal place
3 decimal places; use a placeholder zero
0.027
Estimate the product. 0.9 is close to 1.
0.03 × 1 = 0.03
0.027 is close to the estimate of 0.03. The answer is reasonable.
Course 1
3-6 Multiplying Decimals
Try This: Example 2C
C. 3.80 × 3.3
Multiply. Then place the decimal point.
3.80 3.3 ×
2 decimal places + 1 decimal places
3 decimal places
1140
Estimate the product. Round each factor to the nearest whole number.
4 × 3 = 12
12.54 is close to the estimate of 12. The answer is reasonable.
11400 12.540
Course 1
3-6 Multiplying Decimals
Additional Example 3A: Evaluating Decimal Expressions
Evaluate 5x for each value of x.
A. x = 3.062
3.062 5 ×
3 decimal places
+ 0 decimal places 3 decimal places 15.310
Substitute 3.062 for x. 5x = 5(3.062)
These notations all mean multiply 3 times x.
3 • x 3x 3(x)
Remember!
Course 1
3-6 Multiplying Decimals
Additional Example 3B: Evaluating Decimal Expressions
B. x = 4.79
4.79 5 ×
2 decimal places
+ 0 decimal places 2 decimal places 23.95
Substitute 4.79 for x. 5x = 5(4.79)
Course 1
3-6 Multiplying Decimals
Try This: Example 3A
Evaluate 5x for each value of x.
A. x = 2.012
2.012 5 ×
3 decimal places
+ 0 decimal places 3 decimal places 10.060
Substitute 2.012 for x. 5x = 5(2.012)
Course 1
3-6 Multiplying Decimals
Try This: Example 3B
B. x = 6.22
6.22 5 ×
2 decimal places
+ 0 decimal places 2 decimal places 31.10
Substitute 6.22 for x. 5x = 5(6.22)
Lesson Quiz Find each product.
1. 0.8
2. 0.006 × 0.07
Evaluate 8x for each value of x.
0.00042 0.056
Insert Lesson Title Here
21.64 6.432
Course 1
3-6 Multiplying Decimals
× 0.07
$87.40
3. x = 2.705 4. x = 0.804
5. “Pick your own” peaches sell for $0.95 per pound. You picked 92 pounds of peaches. How much were you charged?
3-7 Dividing Decimals by Whole Numbers
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Divide.
1. 56,000 ÷ 8 2. 5,219 ÷ 17 3. 9,180 ÷ 12
7,000 307 765
Course 1
3-7 Dividing Decimals by Whole Numbers
Problem of the Day
In his pocket, Bill has $0.77 made up of 10 coins. What are the coins?
1 quarter, 3 dimes, 4 nickels, and 2 pennies
Course 1
3-3 Dividing Decimals by Whole Numbers
Learn to divide decimals by whole numbers.
Course 1
3-7 Dividing Decimals by Whole Numbers
Course 1
3-7 Dividing Decimals by Whole Numbers
Additional Example 1A: Dividing a Decimal by a Whole Number
Find the quotient.
A. 0.84 ÷ 3
3 0.84 – 6
24 – 24
0
Place a decimal point in the quotient directly above the decimal point in the dividend. Divide as you would with whole numbers.
0. 2 8
Course 1
3-7 Dividing Decimals by Whole Numbers
Additional Example 1B: Dividing a Decimal by a Whole Number
Find the quotient.
B. 3.56 ÷ 4
4 3.56 – 32
36 – 36
0
Place a decimal point in the quotient directly above the decimal point in the dividend. Divide as you would with whole numbers.
0. 8 9
Course 1
3-7 Dividing Decimals by Whole Numbers
Try This: Example 1A
Find each quotient.
A. 0.72 ÷ 3
3 0.72 – 6
12 – 12
0
Place a decimal point in the quotient directly above the decimal point in the dividend. Divide as you would with whole numbers.
0. 2 4
Course 1
3-7 Dividing Decimals by Whole Numbers
Try This: Example 1B
Find the quotient.
B. 2.96 ÷ 4
4 2.96 – 28
16 – 16
0
Place a decimal point in the quotient directly above the decimal point in the dividend. Divide as you would with whole numbers.
0. 7 4
Course 1
3-7 Dividing Decimals by Whole Numbers
Additional Example 2A: Evaluating Decimal Expressions
Evaluate 0.936 ÷ x for each given value of x.
A. x = 9
9 0.936 – 9
3 – 0
36
Substitute 9 for x. 0. 1 0
0.936 ÷ x 0.936 ÷ 9
– 36 0
4 Sometimes you need to use a zero as a placeholder. 9 > 3, so place a zero in the quotient and divide 9 into 36.
Course 1
3-7 Dividing Decimals by Whole Numbers
Additional Example 2B: Evaluating Decimal Expressions
Evaluate 0.936 ÷ x for each given value of x.
B. x = 18
18 0.936 – 0
93 – 90
36
Substitute 18 for x. 0. 0 5
0.936 ÷ x 0.936 ÷ 18
– 36 0
2 Sometimes you need to use a zero as a placeholder. 18 > 9, so place a zero in the quotient and divide 18 into 93.
Course 1
3-7 Dividing Decimals by Whole Numbers
Try This: Example 2A
Evaluate 0.636 ÷ x for each given value of x.
A. x = 6
6 0.636 – 6
3 – 0
36
Substitute 6 for x. 0. 1 0
0.636 ÷ x 0.636 ÷ 6
– 36 0
6 Sometimes you need to use a zero as a placeholder. 6 > 3, so place a zero in the quotient and divide 6 into 36.
Course 1
3-7 Dividing Decimals by Whole Numbers
Try This: Example 2B
Evaluate 0.636 ÷ x for each given value of x.
B. x = 12
12 0.636 – 0
63 – 60
36
Substitute 12 for x. 0. 0 5
0.636 ÷ x 0.636 ÷ 12
– 36 0
3 Sometimes you need to use a zero as a placeholder. 12 > 6, so place a zero in the quotient and divide 12 into 63.
Course 1
3-7 Dividing Decimals by Whole Numbers
Multiplication can “undo” division. To check your answer to a division problem, multiply the divisor by the quotient.
Remember!
divisor dividend quotient
Course 1
3-7 Dividing Decimals by Whole Numbers
Additional Example 3: Consumer Application Jodi and three of her friends are making a tile design. The materials cost $10.12. If they share the cost equally, how much should each person pay?
4 $10.12 – 8
21 – 20
12
$2. 5 3
$10.12 should be divided into four equal groups. Divide $10.12 by 4.
– 12 0
Place a decimal point in the quotient directly above the decimal point in the dividend. Divide as you would with whole numbers. Check 2.53 × 4 = 10.12 Each person should pay $2.53.
Course 1
3-7 Dividing Decimals by Whole Numbers Try This: Example 3
Ty and three of his friends are building a fence. The lumber cost $21.44. If they share the cost equally, how much should each person pay?
4 $21.44 – 20
14 – 12
24
$5. 3 6
$21.44 should be divided into four equal groups. Divide $21.44 by 4.
– 24 0
Place a decimal point in the quotient directly above the decimal point in the dividend. Divide as you would with whole numbers. Check 5.36 × 4 = 21.44 Each person should pay $5.36.
Lesson Quiz
Find each quotient.
1. 3.12 ÷ 8 2. 5.68 ÷ 8
Evaluate the expression 1.25 ÷ x for the given value of x.
3. x = 5 4. x = 25
5. The tennis team is having 3 of their tennis rackets restrung. The total cost is $54.75. What is the average cost per racket?
0.71 0.39
Insert Lesson Title Here
0.25 0.05
Course 1
3-7 Dividing Decimals by Whole Numbers
$18.25
3-8 Dividing by Decimals
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Divide.
1. 4.8 ÷ 2 2. 16.1 ÷ 7 3. 0.36 ÷ 3 4. 25.28 ÷ 4 5. 6.25 ÷ 5
2.4 2.3 0.12
Course 1
3-8 Dividing by Decimals
6.32 1.25
Problem of the Day
In the following magic square, 3.375 is the product of the numbers in every row, column, and diagonal. Fill in the missing numbers.
Course 1
3-8 Dividing by Decimals
4.5 0.25
0.5
3
1 1.5 2.25
0.75 9
Learn to divide whole numbers and decimals by decimals.
Course 1
3-8 Dividing by Decimals
Course 1
3-8 Dividing by Decimals
Multiplying the divisor and the dividend by the same number does not change the quotient.
42 ÷ 6 = 7
×10 × 10
420 ÷ 60 = 7
× 10 × 10
4,200 ÷ 600 = 7
Helpful Hint
Course 1
3-8 Dividing by Decimals
Additional Example 1A: Dividing a Decimal by a Decimal
Find each quotient.
A. 5.2 ÷ 1.3
1.3 5.2 Multiply the divisor and dividend by the same power of ten.
There is one decimal place in the divisor. Multiply by 101, or 10.
Think: 1.3 x 10 = 13 5.2 x 10 = 52
Divide.
13 52 4
–52 0
Course 1
3-8 Dividing by Decimals
Additional Example 1B: Dividing a Decimal by a Decimal
B. 61.3 ÷ 0.36
0.36 61.3
Think: 0.36 x 100 = 36 61.3 x 100 = 6,130
Make the divisor a whole number by multiplying the divisor and dividend by 102, or 100.
36 6,130.000
Course 1
3-8 Dividing by Decimals
Additional Example 1B Continued
Divide. 36 6,130.000
7
-36
1
253 252
10 -0
0
100
.2
-72
-252
7
280 -252
280
7
28
170.277… = 170.27 __
When a repeating pattern occurs, show three dots or draw a bar over the repeating part of the quotient.
Course 1
3-8 Dividing by Decimals
Try This: Example 1A
Find each quotient.
A. 4.8 ÷ 1.2
1.2 4.8 Multiply the divisor and dividend by the same power of ten.
There is one decimal place in the divisor. Multiply by 10 , or 10.
Think: 1.2 x 10 = 12 4.8 x 10 = 48
1
Divide.
12 48 4
-48 0
Course 1
3-8 Dividing by Decimals
Try This: Example 1B
B. 51.2 ÷ 0.24
0.24 51.2
Think: 0.24 x 100 = 24 51.2 x 100 = 5,120
Make the divisor a whole number by multiplying the divisor and dividend by 102, or 100.
24 5120.000
Course 1
3-8 Dividing by Decimals
Try This: Example 1B Continued
Divide. 24 5,120.00
1
-48
2
32 24
80 -72
3
80
.3
-72
-72 80
3
213.33… = 213.33 __
When a repeating pattern occurs, show three dots or draw a bar over the repeating part of the quotient.
8
Course 1
3-8 Dividing by Decimals
Additional Example 2: Problem Solving Application
After driving 216.3 miles, the Yorks used 10.5 gallons of gas. On average, how many miles did they drive per gallon of gas?
1 Understand the Problem
The answer will be the average number of miles per gallon.
List the important information:
They drove 216.3 miles. They used 10.5 gallons of gas.
Course 1
3-8 Dividing by Decimals
2 Make a Plan Solve a simpler problem by replacing the decimals in the problem with whole numbers.
If they drove 10 miles using 2 gallons of gas, they averaged 5 miles per gallon. You need to divide miles by gallons to solve the problem.
Solve 3
10.5 216.3
Multiply the divisor and dividend by 10. Think: 10.5 x 10 = 105 216.3 x 10 = 2,163 Place the decimal point in the quotient. Divide.
The York family averaged 20.6 miles per gallon.
Course 1
3-8 Dividing by Decimals
Look Back 4
Use compatible numbers to estimate the quotient.
216.3 ÷ 10.5 220 ÷ 11 =20
The answer is reasonable since 20.6 is close to the estimate of 20.
Course 1
3-8 Dividing by Decimals
Try This: Example 2
After driving 191.1 miles, the Changs used 10.5 gallons of gas. On average, how many miles did they drive per gallon of gas?
1 Understand the Problem
The answer will be the average number of miles per gallon.
List the important information:
They drove 191.1 miles. They used 10.5 gallons of gas.
Course 1
3-8 Dividing by Decimals
2 Make a Plan Solve a simpler problem by replacing the decimals in the problem with whole numbers.
If they drove 10 miles using 2 gallons of gas, they averaged 5 miles per gallon. You need to divide miles by gallons to solve the problem.
Solve 3
10.5 191.1
Multiply the divisor and dividend by 10. Think: 10.5 x 10 = 105 191.1 x 10 = 1,911 Place the decimal point in the quotient. Divide.
The York family averaged 18.2 miles per gallon.
Course 1
3-8 Dividing by Decimals
Look Back 4
Use compatible numbers to estimate the quotient.
191.1 ÷ 10.5 190 ÷ 10 = 19
The answer is reasonable since 18.2 is close to the estimate of 19.
Lesson Quiz
Find each quotient.
1. 49 ÷ 0.7
2. 21.63 ÷ 2.1
3. 38.43 ÷ 6.1
4. 16.9 ÷ 5.2
5. John spent $13.44 renting 4 videos for the weekend. What was the cost per video?
10.3
70
Insert Lesson Title Here
6.3
3.25
Course 1
3-8 Dividing by Decimals
$3.36
3-9 Interpret the Quotient
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Divide.
1. 15.264 ÷ 3 2. 3.78 ÷ 3 3. 342 ÷ 7.6 4. 28.32 ÷ 4.8
5.088 1.26
45
Course 1
3-9 Interpret the Quotient
5.9
Problem of the Day
Divide your age in months by 12. What does the quotient tell you?
my age in years
Course 1
3-9 Interpret the Quotient
Learn to solve problems by interpreting the quotient.
Course 1
3-9 Interpret the Quotient
Course 1
3-9 Interpret the Quotient
To divide decimals, first write the divisor as a whole number. Multiply the divisor and dividend by the same power of ten.
Remember!
Course 1
3-9 Interpret the Quotient Additional Example 1: Measurement Application Suppose Mark wants to make bags of slime. If each bag of slime requires 0.15 kg of corn starch and he has 1.23 kg, how many bags of slime can he make? The question asks how many whole bags of slime can be made when the corn starch is divided into groups of 0.15 kg.
1.23 ÷ 0.15 = ? 1.23 ÷ 0.15 = 8.2
Think: The quotient shows that there is not enough to make 9 bags of slime that are 0.15 kg each. There is only enough for 8 bags. The decimal part of the quotient will not be used in the answer. Mark can make 8 bags of slime.
Course 1
3-9 Interpret the Quotient Try This: Example 1
Suppose Antonio wants to make bags of slime. If each bag of slime requires 0.15 kg of corn starch and he has 1.44 kg, how many bags of slime can he make? The question asks how many whole bags of slime can be made when the corn starch is divided into groups of 0.15 kg.
1.44 ÷ 0.15 = ? 1.44 ÷ 0.15 = 9.6
Think: The quotient shows that there is not enough to make 10 bags of slime that are 0.15 kg each. There is only enough for 9 bags. The decimal part of the quotient will not be used in the answer. Antonio can make 9 bags of slime.
Course 1
3-9 Interpret the Quotient Additional Example 2: Photography Application
There are 237 students in the seventh grade. If Mr. Jones buys rolls of film with 36 exposures each, how many rolls will he need to take every student’s picture? The question asks how many rolls are needed to take a picture of every one of the students.
Think: 6 rolls of film will not be enough to take every student’s picture. Mr. Jones will need to buy another roll of film The quotient must be rounded up to the next highest whole number. Mr. Jones will need 7 rolls of film.
237 ÷ 36 = 6.583 __
Course 1
3-9 Interpret the Quotient Try This: Example 2
There are 342 students in the seventh grade. If Ms. Tia buys rolls of film with 24 exposures each, how many rolls will she need to take every student’s picture? The question asks how many rolls are needed to take a picture of every one of the students.
Think: 14 rolls of film will not be enough to take every student’s picture. Ms. Tia will need to buy another roll of film. The quotient must be rounded up to the next highest whole number. Ms. Tia will need 15 rolls of film.
342 ÷ 24 = 14.25
Course 1
3-9 Interpret the Quotient
Additional Example 3: Social Studies Application
Gary has 42.25 meters of rope. If he cuts it into 13 equal pieces, how long is each piece?
The question asks how long each piece will be when the rope is cut into 13 pieces.
Think: The question asks for an exact answer, so do not estimate. Use the entire quotient.
Each piece will be 3.25 meters long.
42.25 ÷ 13 = 3.25
Course 1
3-9 Interpret the Quotient
Try This: Example 3
Ethan has 64.20 meters of rope. If he cuts it into 15 equal pieces, how long is each piece?
The question asks how long each piece will be when the rope is cut into 15 pieces.
Think: The question asks for an exact answer, so do not estimate. Use the entire quotient.
Each piece will be 4.28 meters long.
64.20 ÷ 15 = 4.28
Course 1
3-9 Interpret the Quotient
When the question asks You should
How many whole groups can be made when you divide?
Drop the decimal part of the quotient.
How many whole groups are needed to put all items from the dividend into a group?
Round the quotient up to the next highest whole number.
What is the exact number when you divide?
Use the entire quotient as the answer.
Lesson Quiz Solve. 1. The cross-country team’s goal is to run 26.25 mi next week. If they run only 5 days next week, how many miles would they have to run each day? 2. Shannon is having a surprise party for her parents. She wants to invite 22 friends. Invitations come in packages of 8. How many packages does Shannon need to buy?
3
5.25 mi
Insert Lesson Title Here
Course 1
3-9 Interpret the Quotient
3-10 Solving Decimal Equations
Course 1
Warm up
Lesson Presentation
Problem of the Day
Warm Up Solve.
1. x – 3 = 11 2. 18 = x + 4 3. = 42 4. 2x = 52 5. x – 82 = 172
x = 14
Course 1
3-10 Solving Decimal Equations
x 7
x = 14
x = 294
x = 26
x = 254
Problem of the Day
Find the missing entries in the magic square. 11.25 is the sum of every row, column, and diagonal.
3
Course 1
3-10 Solving Decimal Equations
3.75 5.25
6
6.75 1.5
2.25
0.75 4.5
Learn to solve equations involving decimals.
Course 1
3-10 Solving Decimal Equations
Course 1
3-10 Solving Decimal Equations
You can solve equations with decimals using inverse operations just as you solved equations with whole numbers.
$45.20 + m = $69.95 –$45.20 –$45.20
m = $24.75
Course 1
3-10 Solving Decimal Equations
Use inverse operations to get the variable alone on one side of the equation.
Remember!
Course 1
3-10 Solving Decimal Equations
Additional Example 1A: Solving One-Step Equations with Decimals
Solve the equation. Check your answer.
A. k – 6.2 = 9.5
k – 6.2 = 9.5 6.2 is subtracted from k. Add 6.2 to both sides to undo the subtraction.
+ 6.2 + 6.2 k = 15.7
Check k – 6.2 = 9.5
Substitute 15.7 for k in the equation. 15.7 – 6.2 = 9.5
?
9.5 = 9.5 ?
15.7 is the solution.
Course 1
3-10 Solving Decimal Equations
Additional Example 1B: Solving One-Step Equations with Decimals
Solve the equation. Check your answer. B. 6k = 7.2
6k = 7.2 k is multiplied by 6. Divide both sides by 6 to undo the multiplication.
k = 1.2
Check 6k = 7.2
Substitute 1.2 for k in the equation. 6(1.2) = 7.2
?
7.2 = 7.2 ?
1.2 is the solution.
6 6
6k = 7.2
Course 1
3-10 Solving Decimal Equations Additional Example 1C: Solving One-Step
Equations with Decimals
Solve the equation. Check your answer.
C. = 0.6
= 0.6
m is divided by 7.
Multiply both sides by 7 to undo the division.
m = 4.2
Check
Substitute 4.2 for m in the equation.
0.6 = 0.6 ?
4.2 is the solution.
· 7
m 7
m 7
· 7
= 0.6 m 7
= 0.6 4.2 7
?
Course 1
3-10 Solving Decimal Equations
Try This: Example 1A
Solve the equation. Check your answer.
A. n – 3.7 = 8.6
n – 3.7 = 8.6 3.7 is subtracted from n. Add 3.7 to both sides to undo the subtraction.
+ 3.7 + 3.7 n = 12.3
Check n – 3.7 = 8.6
Substitute 12.3 for n in the equation. 12.3 – 3.7 = 8.6
?
8.6 = 8.6 ?
12.3 is the solution.
Course 1
3-10 Solving Decimal Equations
Try This: Example 1B
Solve the equation. Check your answer.
B. 7h = 8.4
7h = 8.4 h is multiplied by 7. Divide both sides by 7 to undo the multiplication.
h = 1.2
Check 7h = 8.4
Substitute 1.2 for h in the equation. 7(1.2) = 8.4
?
8.4 = 8.4 ?
1.2 is the solution.
7 7
7h = 8.4
Course 1
3-10 Solving Decimal Equations
Try This: Example 1C
Solve the equation. Check your answer.
C. = 0.3
= 0.3
w is divided by 9.
Multiply both sides by 9 to undo the division.
w = 2.7
Check
Substitute 2.7 for w in the equation.
0.3 = 0.3 ?
2.7 is the solution.
· 9
w 9
w 9
· 9
= 0.3 w 9
= 0.3 2.7 9
?
Course 1
3-10 Solving Decimal Equations
The area of a rectangle is its length times its width.
A = lw
Remember!
w
l
Course 1
3-10 Solving Decimal Equations
Additional Example 2A: Measurement Application
Solve the equation. Check your answer. A. The area of Emily’s floor is 33.75 m2. If its length is 4.5 meters, what is its width?
33.75 = 4.5w
Write the equation for the problem. Let w be the width of the room.
Divide both sides by 4.5 to undo the multiplication.
7.5 = w 4.5 4.5
33.75 = 4.5 · w
area = length · width
33.75 = 4.5w
The width of Emily’s floor is 7.5 meters.
Course 1
3-10 Solving Decimal Equations
Additional Example 2B: Measurement Application
Solve the equation. Check your answer. B. If carpet costs $23 per square meter, what is the total cost to carpet the floor?
Let C be the total cost. Write the equation for the problem.
Multiply. C = 776.25
C = 33.75 · 23
total cost = area · cost of carpet per square meter
The cost of carpeting the floor is $776.25.
Course 1
3-10 Solving Decimal Equations
Try This: Example 2A
Solve the equation. Check your answer. A. The area of Yvonne’s bedroom is 181.25 ft2. If its length is 12.5 feet, what is its width?
181.25 = 12.5w
Write the equation for the problem. Let w be the width of the room.
Divide both sides by 12.5 to undo the multiplication.
14.5 = w 12.5 12.5
181.25 = 12.5 · w
area = length · width
181.25 = 12.5w
The width of Yvonne’s bedroom is 14.5 feet.
Course 1
3-10 Solving Decimal Equations
Try This: Example 2B
Solve the equation. Check your answer. B. If carpet costs $4 per square foot, what is the total cost to carpet the bedroom?
Let C be the total cost. Write the equation for the problem.
Multiply. C = 725
C = 181.25 · 4
total cost = area · cost of carpet per square foot
The cost of carpeting the bedroom is $725.
Lesson Quiz
Solve each equation. Check your answer.
1. x – 3.9 = 14.2
2. = 8.3
3. x – 4.9 = 16.2
4. 7x = 47.6
5. The area of the floor in Devon’s room is 35.7 m2. If the width is 4.2 m, what is the length of the bedroom?
Insert Lesson Title Here
Course 1
3-10 Solving Decimal Equations
x 4
x = 18.1
x = 33.2
x = 21.1
x = 6.8
8.5 m
4-1 Divisibility
Course 1
Warm Up
Problem of the Day
Lesson Presentation
Warm Up
1. 20 2. 48
1 x 20, 2 x 10, 4 x 5
1 x 48, 2 x 24, 3 x 16, 4 x 12, 6 x 8
Course 1
4-1 Divisibility
Write each number as a product of two whole numbers in as many ways as possible.
Problem of the Day
In this magic square, every row, column, and diagonal has the same sum, 34. Complete the square using the whole numbers from 1 to 16.
Course 1
4-1 Divisibility
6 7 10 11
16 3 2
14 1 15 4 9 12
8 13
5
Possible answer:
Learn to use divisibility rules.
Course 1
4-1 Divisibility
Vocabulary
divisibility composite number prime number
Insert Lesson Title Here
Course 1
4-1 Divisibility
Course 1
4-1 Divisibility
A number is divisible by another number if the quotient is a whole number with no remainder.
42 ÷ 6 = 7 Quotient
Course 1
4-1 Divisibility
Divisibility Rules A number is divisible by. . . Divisible Not Divisible
2 if the last digit is even (0, 2, 4, 6, or 8). 3,978 4,975
3 if the sum of the digits is divisible by 3. 315 139
4 if the last two digits form a number divisible by 4.
8,512 7,518
5 if the last digit is 0 or 5. 14,975 10,978
6 if the number is divisible by both 2 and 3 48 20
9 if the sum of the digits is divisible by 9. 711 93
10 if the last digit is 0. 15,990 10,536
Course 1
4-1 Divisibility
Additional Example 1A: Checking Divisibility
Tell whether 462 is divisible by 2, 3, 4, and 5.
2
3
4
5 Not divisible
So 462 is divisible by 2 and 3.
The last digit, 2, is even.
The sum of the digits is 4 + 6 + 2 = 12. 12 is divisible by 3.
The last two digits form the number 62. 62 is not divisible by 4.
Divisible
Divisible
Not divisible
The last digit is 2.
Course 1
4-1 Divisibility
Additional Example 1B: Checking Divisibility
Tell whether 540 is divisible by 6, 9, and 10.
6
9
10
So 540 is divisible by 6, 9, and 10.
The number is divisible by both 2 and 3.
The sum of the digits is 5 + 4 + 0 = 9. 9 is divisible by 9.
The last digit is 0.
Divisible
Divisible
Divisible
Course 1
4-1 Divisibility
Try This: Example 1A
Tell whether 114 is divisible by 2, 3, 4, and 5.
2
3
4
5 Not Divisible
So 114 is divisible by 2 and 3.
The last digit, 4, is even.
The sum of the digits is 1 + 1 + 4 = 6. 6 is divisible by 3.
The last two digits form the number 14. 14 is not divisible by 4.
Divisible
Divisible
Not Divisible
The last digit is 4.
Course 1
4-1 Divisibility
Try This: Example 1B
Tell whether 810 is divisible by 6, 9, and 10.
6
9
10
So 810 is divisible by 6, 9, and 10.
The number is divisible by both 2 and 3.
The sum of the digits is 8 + 1 + 0 = 9. 9 is divisible by 9.
The last digit is 0.
Divisible
Divisible
Divisible
Course 1
4-1 Divisibility
Any number greater than 1 is divisible by at least two numbers—1 and the number itself. Numbers that are divisible by more than two numbers are called composite numbers.
A prime number is divisible by only the numbers 1 and itself. For example, 11 is a prime number because it is divisible by only 1 and 11. The numbers 0 and 1 are neither prime nor composite.
Course 1
4-1 Divisibility
Click to see which numbers from 1 through 50 are prime.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
Tell whether each number is prime or composite.
Additional Example 2A & 2B: Identifying Prime and Composite Numbers
Course 1
4-1 Divisibility
A. 23 divisible by 1, 23 prime
B. 48 divisible by 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. composite
Additional Example 2C & 2D: Identifying Prime and Composite Numbers
Course 1
4-1 Divisibility
C. 31 divisible by 1, 31 prime
D. 18 divisible by 1, 2, 3, 6, 9, 18 composite
Tell whether each number is prime or composite.
Tell whether each number is prime or composite.
Try This: Example 2A & 2B
Course 1
4-1 Divisibility
A. 27 divisible by 1, 3, 9, 27 composite
B. 24 divisible by 1, 2, 3, 4, 6, 8, 12, 24 composite
Try This: Example 2C & 2D
Course 1
4-1 Divisibility
C. 11 divisible by 1, 11 prime
D. 8 divisible by 1, 2, 4, 8 composite
Tell whether each number is prime or composite.
Lesson Quiz Tell whether each number is divisible by 2, 3, 4, 5, 6, 9, and 10.
1. 256
2. 720
3. 615
Tell whether each number is prime or composite.
4. 47 5. 38
divisible by 2, 3, 4, 5, 6, 9, 10
divisible by 2, 4
Insert Lesson Title Here
divisible by 3, 5
Course 1
4-1 Divisibility
prime composite
4-2 Factors and Prime Factorization
Course 1
Warm Up
Problem of the Day
Lesson Presentation
Warm Up Identify each number as prime or composite.
1. 19 2. 82 3. 57 4. 85 5. 101 6. 121
prime composite
composite
Course 1
4-2 Factors and Prime Factorization
composite
composite prime
Problem of the Day
At the first train stop, 7 people disembarked. At the second stop, 8 people disembarked. At the fourth stop the last 6 people disembarked. If there were 28 people on the train before the first stop, how many people left at the third stop?
7 people left at the third stop
Course 1
4-2 Factors and Prime Factorization
Learn to write prime factorizations of composite numbers.
Course 1
4-2 Factors and Prime Factorization
Vocabulary
factor prime factorization
Insert Lesson Title Here
Course 1
4-2 Factors and Prime Factorization
Course 1
4-2 Factors and Prime Factorization
Whole numbers that are multiplied to find a product are called factors of that product. A number is divisible by its factors.
2 3 6 =
Factors Product
2 6 3 ÷ = 6 ÷ 2 = 3
6 is divisible by 3 and 2.
Course 1
4-2 Factors and Prime Factorization
When the pairs of factors begin to repeat, then you have found all of the factors of the number you are factoring.
Helpful Hint
Course 1
4-2 Factors and Prime Factorization
Additional Example 1A: Finding Factors List all factors of each number.
A. 16 Begin listing factors in pairs.
16 = 1 • 16 16 = 2 • 8
1 is a factor. 2 is a factor.
16 = 4 • 4 3 is not a factor. 4 is a factor. 5 is not a factor. 6 is not a factor. 7 is not a factor.
16 = 8 • 2 8 and 2 have already been listed so stop here.
The factors of 16 are 1, 2, 4, 8, and 16.
1 2 4 4 16 8 You can draw a diagram to illustrate the factor pairs.
Course 1
4-2 Factors and Prime Factorization
Additional Example 1B: Finding Factors List all factors of each number.
B. 19 Begin listing all factors in pairs.
19 = 1 • 19 19 is not divisible by any other whole number.
The factors of 19 are 1 and 19.
Course 1
4-2 Factors and Prime Factorization
Try This: Example 1A
List all factors of each number.
A. 12 Begin listing factors in pairs.
12 = 1 • 12 12 = 2 • 6
1 is a factor. 2 is a factor. 3 is a factor. 4 and 2 have already been listed so stop here.
12 = 4 • 3
1 2 4 3 12 6
The factors of 12 are 1, 2, 3, 4, 6, and 12
12 = 3 • 4
You can draw a diagram to illustrate the factor pairs.
Course 1
4-2 Factors and Prime Factorization
Try This: Example 1B
List all factors of each number.
B. 11 Begin listing all factors in pairs.
11 = 1 • 11 11 is not divisible by any other whole number.
The factors of 11 are 1 and 11.
Course 1
4-2 Factors and Prime Factorization
You can use factors to write a number in different ways.
Factorization of 12
2 • 6 1 • 12 3 • 4 3 • 2 • 2
The prime factorization of a number is the number written as the product of its prime factors.
Notice that these factors are all prime.
Course 1
4-2 Factors and Prime Factorization
You can use exponents to write prime factorizations. Remember that an exponent tells you how many times the base is a factor.
Helpful Hint
Course 1
4-2 Factors and Prime Factorization
Additional Example 2A: Writing Prime Factorizations
Write the prime factorization of each number.
A. 24 Method 1: Use a factor tree.
Choose any two factors of 24 to begin. Keep finding factors until each branch ends at a prime factor.
24
2 12 • 6
2
2 • 3 •
24
6 4 •
3 2 2 2
24 = 2 • 2 • 2 • 3
24 = 3 • 2 • 2 • 2
The prime factorization of 24 is 2 • 2 • 2 • 3, or 2 • 3 .
• •
3
Course 1
4-2 Factors and Prime Factorization
Additional Example 2B: Writing Prime Factorizations
Write the prime factorization of each number. B. 42
Method 1: Use a ladder diagram.
Choose a prime factor of 42 to begin. Keep dividing by prime factors until the quotient is 1.
3 42
2
1
14
7 7
42 = 3 • 2 • 7
The prime factorization of 42 is 2 • 3 • 7.
2 42
3
1
21
7 7
42 = 2 • 3 • 7
Course 1
4-2 Factors and Prime Factorization
In Example 2, notice that the prime factors may be written in a different order, but they are still the same factors. Except for changes in the order, there is only one way to write the prime factorization of a number.
Course 1
4-2 Factors and Prime Factorization
Try This: Example 2A
Write the prime factorization of each number. A. 28
Method 1: Use a factor tree.
Choose any two factors of 28 to begin. Keep finding factors until each branch ends at a prime factor.
28
2 14 • 7 2 •
28
7 4 • 2 2
28 = 2 • 2 • 7 28 = 7 • 2 • 2
The prime factorization of 28 is 2 • 2 • 7, or 22 • 7 .
•
Course 1
4-2 Factors and Prime Factorization
Try This: Example 2B
Write the prime factorization of each number.
B. 36 Method 1: Use a ladder diagram.
Choose a prime factor of 36 to begin. Keep dividing by prime factors until the quotient is 1.
3 36
2
1
12
6 2
36 = 3 • 2 • 2 • 3
The prime factorization of 36 is 3 • 2 • 2 • 3, or 32 • 23.
3 36
3
1
12
2 4
36 = 3 • 3 • 2 • 2
3 3 2 2
Lesson Quiz
List all the factors of each number.
1. 22
2. 40
3. 51
1, 2, 4, 5, 8, 10, 20, 40
1, 2, 11, 22
Insert Lesson Title Here
1, 3, 17, 51
Course 1
4-2 Factors and Prime Factorization
Write the prime factorization of each number.
4. 32
5. 120
25
23 × 3 × 5
4-3 Greatest Common Factor
Course 1
Warm Up
Problem of the Day
Lesson Presentation
Warm Up Write the prime factorization of each number.
1. 14 3. 63 2. 18 4. 54
2 × 7 32 × 7
2 × 32
Course 1
4-3 Greatest Common Factor
2 × 33
Problem of the Day
In a parade, there are 15 riders on bicycles and tricycles. In all, there are 34 cycle wheels. How many bicycles and how many tricycles are in the parade?
11 bicycles and 4 tricycles
Course 1
4-3 Greatest Common Factor
Learn to find the greatest common factor (GCF) of a set of numbers.
Course 1
4-3 Greatest Common Factor
Vocabulary
greatest common factor (GCF)
Insert Lesson Title Here
Course 1
4-2 Factors and Prime Factorization
Course 1
4-3 Greatest Common Factor
Factors shared by two or more whole numbers are called common factors. The largest of the common factors is called the greatest common factor, or GCF.
Factors of 24:
Factors of 36:
Common factors:
1, 2, 3, 4, 6, 8,
1, 2, 3, 4, 6,
The greatest common factor (GCF) of 24 and 36 is 12.
Example 1 shows three different methods for finding the GCF.
1, 2, 3, 4, 6, 9, 12, 12, 18,
24 36
12
Course 1
4-3 Greatest Common Factor
Additional Example 1A: Finding the GCF
Find the GCF of each set of numbers.
A. 28 and 42
Method 1: List the factors.
factors of 28:
factors of 42:
1, 2, 14, 7, 28
7, 1,
4,
3, 2, 42 6, 21, 14,
List all the factors.
Circle the GCF.
The GCF of 28 and 42 is 14.
Course 1
4-3 Greatest Common Factor
Additional Example 1B: Finding the GCF
Find the GCF of each set of numbers.
B. 18, 30, and 24
Method 2: Use the prime factorization.
18 =
30 =
24 =
2
5 •
3
2
2
3
2
3
2 3
Write the prime factorization of each number.
Find the common prime factors.
The GCF of 18, 30, and 24 is 6.
•
•
•
•
•
•
Find the product of the common prime factors.
2 • 3 = 6
Course 1
4-3 Greatest Common Factor
Additional Example 1C: Finding the GCF
Find the GCF of each set of numbers.
C. 45, 18, and 27
Method 3: Use a ladder diagram.
3 3
5 2 3
45 18 27 Begin with a factor that divides into each number. Keep dividing until the three have no common factors.
Find the product of the numbers you divided by.
3 • 3 =
The GCF of 45, 18, and 27 is 9.
9
15 6 9
Course 1
4-3 Greatest Common Factor
Try This: Example 1A
Find the GCF of each set of numbers.
A. 18 and 36
Method 1: List the factors.
factors of 18:
factors of 36:
1, 2, 9, 6, 18
6, 1,
3,
3, 2, 36 4, 12, 9,
List all the factors.
Circle the GCF.
The GCF of 18 and 36 is 18.
18,
Course 1
4-3 Greatest Common Factor
Try This: Example 1B
Find the GCF of each set of numbers.
B. 10, 20, and 30
Method 2: Use the prime factorization.
10 =
20 =
30 =
2
2 •
3
2
5
2
5
5
Write the prime factorization of each number.
Find the common prime factors.
The GCF of 10, 20, and 30 is 10.
•
•
•
•
Find the product of the common prime factors.
2 • 5 = 10
Course 1
4-3 Greatest Common Factor
Try This: Example 1C
Find the GCF of each set of numbers.
C. 40, 16, and 24
Method 3: Use a ladder diagram.
2 2 40 16 24 Begin with a factor that divides into
each number. Keep dividing until the three have no common factors.
Find the product of the numbers you divided by.
2 • 2 • 2 =
The GCF of 40, 16, and 24 is 8.
8
20 8 12
5 2 3 10 4 6 2
Course 1
4-3 Greatest Common Factor Additional Example 2:
Problem Solving Application Jenna has 16 red flowers and 24 yellow flowers. She wants to make bouquets with the same number of each color flower in each bouquet. What is the greatest number of bouquets she can make?
The answer will be the greatest number of bouquets 16 red flowers and 24 yellow flowers can form so that each bouquet has the same number of red flowers, and each bouquet has the same number of yellow flowers.
1 Understand the Problem
2 Make a Plan You can make an organized list of the possible bouquets.
Course 1
4-3 Greatest Common Factor
Solve 3
The greatest number of bouquets Jenna can make is 8.
Red Yellow Bouquets 2 3 RR
YYY
16 red, 24 yellow:
Every flower is in a bouquet
RR
YYY
RR
YYY
RR
YYY
RR
YYY
RR
YYY
RR
YYY
RR
YYY
Look Back 4 To form the largest number of bouquets, find the GCF of 16 and 24. factors of 16:
factors of 24:
1, 4, 2, 16 8,
1, 3, 24 8, 2, 4, 6, 12,
The GCF of 16 and 24 is 8.
Course 1
4-3 Greatest Common Factor
Try This: Example 2 Peter has 18 oranges and 27 pears. He wants to make fruit baskets with the same number of each fruit in each basket. What is the greatest number of fruit baskets he can make?
The answer will be the greatest number of fruit baskets 18 oranges and 27 pears can form so that each basket has the same number of oranges, and each basket has the same number of pears.
1 Understand the Problem
2 Make a Plan
You can make an organized list of the possible fruit baskets.
Course 1
4-3 Greatest Common Factor
Solve 3
The greatest number of baskets Peter can make is 9.
Oranges Pears Bouquets
2 3 OO
PPP
18 oranges, 27 pears:
Every fruit is in a basket
OO
PPP
OO
PPP
OO
PPP
OO
PPP
OO
PPP
OO
PPP
OO
PPP
Look Back 4 To form the largest number of bouquets, find the GCF of 18 and 27. factors of 18:
factors of 27:
1, 3, 2, 18 6,
1, 9, 3, 27
The GCF of 18 and 27 is 9.
OO
PPP
9,
Lesson Quiz: Part 1
1. 18 and 30
2. 20 and 35
3. 8, 28, 52
4. 44, 66, 88
5
6
Insert Lesson Title Here
4
Course 1
4-3 Greatest Common Factor
22
Find the greatest common factor of each set of numbers.
Lesson Quiz: Part 2
5. Mrs. Lovejoy makes flower arrangements. She has 36 red carnations, 60 white carnations, and 72 pink carnations. Each arrangement must have the same number of each color. What is the greatest number of arrangements she can make if every carnation is used?
Insert Lesson Title Here
Course 1
4-3 Greatest Common Factor
Find the greatest common factor of the set of numbers.
12 arrangements
4-4 Decimals and Fractions
Course 1
Warm Up
Problem of the Day
Lesson Presentation
Warm Up Find the GCF of each set of numbers.
1. 15, 24
2. 30, 60
3. 54, 102
4. 12, 16, 24
3
30
6
Course 1
4-4 Decimals and Fractions
4
Problem of the Day
Write the day of the month as the product of two factors.
Possible answer: 12th of the month, 6 × 2
Course 1
4-4 Decimals and Fractions
Learn to convert between decimals and fractions.
Course 1
4-4 Decimals and Fractions
Vocabulary mixed number terminating decimal repeating decimal
Insert Lesson Title Here
Course 1
4-4 Decimals and Fractions
Course 1
4-4 Decimals and Fractions
A number that contains both a whole number greater than 0 and a fraction, such as 1 , is called a mixed number. 3
4 __
__ 4 1 1 1 __
2 1
1 __ 4 3
2 __ 4 1
2 __ 2 1
Mixed numbers 12
14
34
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5
Course 1
4-4 Decimals and Fractions
Additional Example 1A: Writing Decimals as Fractions or Mixed Numbers
Write each decimal as a fraction or mixed number.
A. 0.67
Identify the place value of the digit farthest to the right.
The 7 is in the hundredths place, so use 100 as the denominator.
0.67
100 ____ 67 Remember!
One
s
Tent
hs
Hun
dred
ths
Thou
sand
ths
Place Value
Course 1
4-4 Decimals and Fractions
Additional Example 1B: Writing Decimals as Fractions or Mixed Numbers
Write each decimal as a fraction or mixed number.
B. 5.9
Identify the place value of the digit farthest to the right. Write the whole number, 1. The 9 is in the tenths place, so use 10 as the denominator.
5.9
10 ___ 9 5
Course 1
4-4 Decimals and Fractions
Try This: Example 1A
Write each decimal as a fraction or mixed number.
A. 0.73
Identify the place value of the digit farthest to the right.
The 7 is in the hundredths place, so use 100 as the denominator.
0.73
100 ____ 73
Course 1
4-4 Decimals and Fractions
Try This: Example 1B
Write each decimal as a fraction or mixed number.
B. 4.8
Identify the place value of the digit farthest to the right. Write the whole number, 1. The 8 is in the tenths place, so use 10 as the denominator.
4.8
10 ___ 8 4
Course 1
4-4 Decimals and Fractions
Additional Example 2A: Writing Fractions as Decimals
Write each fraction or mixed number as a decimal.
A.
Divide 3 by 20. Add zeros after the decimal point.
0 20 ____ 3
The remainder is 0.
.00 - 20
.1 3 20
100 - 100
5
0
20 ____ 3 = 0.15
Course 1
4-4 Decimals and Fractions
To write a repeating decimal, you can show three dots or draw a bar over the repeating part: 0.666… = 0.6
Writing Math
Course 1
4-4 Decimals and Fractions
Additional Example 2B: Writing Fractions as Decimals
A.
Divide 1 by 3. Add zeros after the decimal point.
0 3 __ 1
The three repeats in the quotient.
.000 - 9
.3 1 3
10 - 9
3
10
=
6
- 9
3
1 3 __ 1
6 6.333… = 6.3 _
Write each fraction or mixed number as a decimal.
Course 1
4-4 Decimals and Fractions
Try This: Example 2A
Write each fraction or mixed number as a decimal.
A. Divide 5 by 20. Add zeros after the decimal point.
0 20 ____ 5
The remainder is 0.
.00 - 40
.2 5 20
100 - 100
5
0
20 ____ 5 = 0.25
Course 1
4-4 Decimals and Fractions
Try This: Example 2B
Write each fraction or mixed number as a decimal.
B. Divide 2 by 3. Add zeros after the decimal point.
0 3 __ 2
The six repeats in the quotient.
.000 - 18
.6 2 3
20 - 18
6
20
=
7
- 18
6
2 3 __ 2
7 7.666… = 7.6 _
Course 1
4-4 Decimals and Fractions
A terminating decimal, such as 0.75, has a finite number of decimal places. A repeating decimal, such as 0.666…, has a block of one or more digits that repeat continuously
Common Fractions and Equivalent Decimals
0.2 0.25 0.3 0.4 0.5 0.6 0.6 0.75 0.8
1 3 __ 1
5 __ 1
4 __ 2
5 __ 1
2 __ 3
5 __ 2
3 __ 3
4 __ 4
5 __
Course 1
4-4 Decimals and Fractions
Additional Example 3: Comparing and Ordering Fractions and Decimals
Order the fractions and decimals from least to greatest.
, 0.8, 7 __ 10
3 __ 4
First rewrite the fractions as a decimal. = 0.75 = 0.7
Order the three decimals.
3 __ 4
0
7 __ 10
__ 10 7
3 __ 4
0.75 0.8 0.7
The numbers in order from least to greatest are , , and 0.8.
7 10
3 4
Course 1
4-4 Decimals and Fractions
Try This: Example 3
Order the fractions and decimals from least to greatest.
, 0.35, 1 __ 4
1 __ 2
First rewrite the fractions as decimals. = 0.5 = 0.25
Order the three decimals.
1 __ 2
0
1 __ 4
1 __ . 2
The numbers in order from least to greatest are , 0.35, and 1 __ 4
1 __ 2
0.35 0.5 0.25
1 __ 4
Lesson Quiz: Part 1 Write each decimal as a fraction or mixed number.
1. 0.24
2. 6.75
Write each fraction or mixed number as a decimal.
3. 2 4.
Insert Lesson Title Here
2.6 0.875
Course 1
4-4 Decimals and Fractions
3 5
__ 8 __ 7
6 25
____
6 3 4
__
Lesson Quiz: Part 2
5. Order the fractions and decimals from least to greatest.
0.38, ,
6. Jamal ran the following distances on three different days: 0.87 mile, mile, and 0.13 mile. Order the distances from least to greatest.
0.13, , 0.87
Insert Lesson Title Here
Course 1
4-4 Decimals and Fraction
3 4 __ 1
5 __
2 3 __
1 5 __ , 0.38, 3
4 __
2 3 __
4-5 Equivalent Fractions
Course 1
Warm Up
Problem of the Day
Lesson Presentation
Warm Up List the factors of each number.
1. 8
2. 10
3. 16
4. 20
5. 30
1, 2, 4, 8
1, 2, 5, 10
1, 2, 4, 8, 16
Course 1
4-5 Equivalent Fractions
1, 2, 4, 5, 10, 20
1, 2, 3, 5, 6, 10, 15, 30
Problem of the Day
John has 3 coins, 2 of which are the same. Ellen has 1 fewer coin than John, and Anna has 2 more coins than John. Each girl has only 1 kind of coin. Who has coins that could equal the value of a half-dollar? Ellen and Anna
Course 1
4-5 Equivalent Fractions
Learn to write equivalent fractions.
Course 1
4-5 Equivalent Fractions
Vocabulary equivalent fractions simplest form
Insert Lesson Title Here
Course 1
4-5 Equivalent Fractions
Course 1
4-5 Equivalent Fractions
Fractions that represent the same value are equivalent fractions. So , , and are equivalent fractions.
= =
1 2 __ 2
4 __ 4
8 __
12
24
48
Course 1
4-5 Equivalent Fractions
Additional Example 1: Finding Equivalent Fractions
Find two equivalent fractions for . 10 12 ___
10 12 ___ 5
6 __ 15
18 ___
10 12 ___ 15
18 ___ 5
6 __
= =
So , , and are all equivalent fractions.
Course 1
4-5 Equivalent Fractions
Try This: Example 1
Find two equivalent fractions for . 4 6 __
4 6
__ 2 3 __ 8
12 ___
4 6
__ 8 12 ___ 2
3 __
= =
So , , and are all equivalent fractions.
Course 1
4-5 Equivalent Fractions
Additional Example 2A: Multiplying and Dividing to Find Equivalent Fractions
Find the missing number that makes the fractions equivalent.
A. 3 5 __
20 ___ =
3 5
______
In the denominator, 5 is multiplied by 4 to get 20.
• 4 • 4
Multiply the numerator, 3, by the same number, 4.
= 12 20
____
So is equivalent to . 3 5
__ 12 20 ___
3 5
__ 12 20 ___ =
Course 1
4-5 Equivalent Fractions
Additional Example 2B: Multiplying and Dividing to Find Equivalent Fractions
Find the missing number that makes the fractions equivalent.
B. 4 5 __ 80 ___ =
4 5 ______
In the numerator, 4 is multiplied by 20 to get 80.
• 20 • 20
Multiply the denominator by the same number, 20.
= 80 100 ____
So is equivalent to . 4 5
__ 80 100 ___
4 5
__ 80 100 ___ =
Course 1
4-5 Equivalent Fractions
Try This: Example 2A
Find the missing number that makes the fraction equivalent.
A. 3 9 __
27 ___ =
3 9
______
In the denominator, 9 is multiplied by 3 to get 27.
• 3 • 3
Multiply the numerator, 3, by the same number, 3.
= 9 27
____
So is equivalent to . 3 9
__ 9 27 ___
3 9
__ 9 27 ___ =
Course 1
4-5 Equivalent Fractions
Try This: Example 2B
Find the missing number that makes the fraction equivalent.
B. 2 4 __ 40 ___ =
2 4 ______
In the numerator, 2 is multiplied by 20 to get 40.
• 20 • 20
Multiply the denominator by the same number, 20.
= 40 80
____
So is equivalent to . 2 4
__ 40 80 ___
2 4
__ 40 80 ___ =
Course 1
4-5 Equivalent Fractions
Every fraction has one equivalent fraction that is called the simplest form of the fraction. A fraction is in simplest form when the GCF of the numerator and the denominator is 1.
Example 3 shows two methods for writing a fraction in simplest form.
Course 1
4-5 Equivalent Fractions
Additional Example 3A: Writing Fractions in Simplest Form
Write the fraction in simplest form.
A. 20 48 ___
The GCF of 20 and 48 is 4, so is not in simplest form.
20 48 ___
Method 1: Use the GCF. 20 48 _______ ÷ 4
÷ 4 Divide 20 and 48 by their GCF, 4. = 5
12 __
Course 1
4-5 Equivalent Fractions
Additional Example 3A: Writing Fractions in Simplest Form
Write the fraction in simplest form.
Method 2: Use a ladder diagram.
Use a ladder. Divide 20 and 48 by any common factor (except 1) until you cannot divide anymore
20 48 ___
2 20/48 2 10/24
5/12 5 12 ___
So written in simplest form is .
Method 2 is useful when you know that the numerator and denominator have common factors, but you are not sure what the GCF is.
Helpful Hint
Course 1
4-5 Equivalent Fractions
Additional Example 3B: Writing Fractions in Simplest Form
Write the fraction in simplest form.
B. 7 10 ___
The GCF of 7 and 10 is 1 so is already in simplest form.
7 10 ___
Course 1
4-5 Equivalent Fractions
Try This: Example 3A
Write the fraction in simplest form.
A. 12 16 ___
The GCF of 12 and 16 is 4, so is not in simplest form.
12 16 ___
Method 1: Use the GCF. 12 16 _______ ÷ 4
÷ 4 Divide 12 and 16 by their GCF, 4. = 3
4 __
Course 1
4-5 Equivalent Fractions
Try This: Example 3A
Write the fraction in simplest form.
Method 2: Use a ladder diagram.
Use a ladder. Divide 20 and 48 by any common factor (except 1) until you cannot divide anymore
2 12/16 2 6/8
3/4
12 16 ___ 3
4 ___
So written in simplest form is .
Course 1
4-5 Equivalent Fractions
Try This: Example 3B
Write the fraction in simplest form.
B. 3 10 ___
The GCF of 3 and 10 is 1, so is already in simplest form.
3 10 ___
Lesson Quiz Write two equivalent fractions for each given fraction.
1. 2.
Find the missing number that makes the fractions equivalent.
3. 4.
Write each fraction in simplest form.
5. 6.
Insert Lesson Title Here
6
Course 1
4-5 Equivalent Fractions
4 10 ___ 7
14 ___
2 7
__ ___ 21
= 4 15 __ ___ 20 =
4 8
__ 7 49 ___ 1
7 ___ 1
2 __
75
1 2
___ 14 28 ___ , 8
20 ___ 2
5 ___ ,
Possible answers
4-6 Comparing and Ordering Fractions
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Find the missing number that makes the fraction equivalent.
1. 2. 3. 4.
9 8
15
Course 1
4-6 Comparing and Ordering Fractions
3 5
__ 15 =
3 4
__ 20
=
2 3
__ ___ 12 =
5 8
__ 24 = 15
Course 1
4-6 Comparing and Ordering Fractions
Problem of the Day
From 4:00 to 5:30, Carlos, Lisa, and Toni took turns playing the same computer game. Carlos played for hour, and Lisa played for hour. For how many minutes did Toni play the game?
3 4
___
1 2
___
15 minutes
Course 1
4-6 Comparing and Ordering Fractions
Learn to use pictures and number lines to compare and order fractions.
Course 1
4-6 Comparing and Ordering Fractions
Vocabulary like fractions unlike fractions common denominator
Course 1
4-6 Comparing and Ordering Fractions
When you are comparing fractions, first
check their denominators. When
fractions have the same denominator,
they are called like fractions. For
example, and are like fractions. 6 7
___ 4 7
___
Course 1
4-6 Comparing and Ordering Fractions
Additional Example 1A: Comparing Like Fractions
Compare. Write <, >, or =.
6 7 __ 4 __ A.
7
6 7
__ __ 7 4 >
From the model, > . 6 7
__ __ 7 4
Course 1
4-6 Comparing and Ordering Fractions
Additional Example 1B: Comparing Like Fractions
Compare. Write <, >, or =.
1 9 __ 5 __ B.
9
1 9
__ __ 9 5 <
From the model, < . 1 9
__ __ 9 5
Course 1
4-6 Comparing and Ordering Fractions
Try This: Example 1A
Compare. Write <, >, or =.
4 6 __ 5 __ A.
6
4 6
__ __ 6 5 <
From the model, < . 4 6
__ __ 6 5
Course 1
4-6 Comparing and Ordering Fractions
Try This: Example 1B
Compare. Write <, >, or =.
3 9 __ 2 __ B.
9
3 9
__ __ 9 2 >
From the model, > . 3 9
__ __ 9 2
Course 1
4-6 Comparing and Ordering Fractions
When two fractions have different denominators, they are called unlike fractions. To compare unlike fractions, first rename the fractions so they have the same denominator. This is called finding a common denominator.
Course 1
4-6 Comparing and Ordering Fractions
Additional Example 2: Cooking Application
2 3
__
3 4
__ Compare and . 2 3
__
3 4
__
Find a common denominator by multiplying the denominators.
3 • 4 = 12
Ray has cup of nuts. He needs cup to
make cookies. Does he have enough nuts for the recipe?
Course 1
4-6 Comparing and Ordering Fractions
Additional Example 2 Continued
Find equivalent fractions with 12 as the denominator.
2 3
__ 12 __ =
2 3 ______
• 4 = • 4 __ 12 8
3 4
__ 12 __ =
3 4 ______
• 3 • 3 __
12 9
2 3
__ 3 4
__ __ 12 8 =
=
= __ 12 9
Course 1
4-6 Comparing and Ordering Fractions
Additional Example 2 Continued
Compare the like fractions. __ 12 8 __
12 9 2
3 __ 3
4 __ < , so < .
Since cup is less than cup, he does not
have enough.
2 3
__ 3 4
__
Course 1
4-6 Comparing and Ordering Fractions
Try This: Example 2
1 3
__
1 4
__ Compare and . 1 3
__
1 4
__
Find a common denominator by multiplying the denominators.
3 • 4 = 12
Trevor has cup of soil. He needs cup to
fill a small planter. Does he have enough soil to fill the planter?
Course 1
4-6 Comparing and Ordering Fractions
Find equivalent fractions with 12 as the denominator.
1 3
__ 12 __ =
1 3 ______
• 4 = • 4 __ 12 4
1 4
__ 12 __ =
1 4 ______
• 3 • 3 __
12 3
1 3
__ 1 4
__ __ 12 4 =
=
= __ 12 3
Try This: Example 2 Continued
Course 1
4-6 Comparing and Ordering Fractions
Compare the like fractions.
__ 12 4 __
12 3 1
3 __ 1
4 __
> , so > .
Since cup is more than cup, he does have
enough.
1 3
__ 1 4
__
Try This: Example 2 Continued
Course 1
4-6 Comparing and Ordering Fractions
Additional Example 3: Ordering Fractions
Order , , and from least to greatest. 4 5 __
4 5
_______
2 3 __
2 3
_______ 1 3
_______ • 3 • 3
• 5 • 5
• 5 • 5 10
15 __ 12
15 __ = = = 5
15 __ Rename with like
denominators.
1 3 __
1 3
__ 2 3
__ The fractions in order from least to greatest are , , . 4
5 __
5 15
1 3
10 15
2 3
0 1
12 15
4 5
Course 1
4-6 Comparing and Ordering Fractions
Numbers increase in value as you move from left to right on a number line.
Remember!
Course 1
4-6 Comparing and Ordering Fractions
Try This: Example 3
Order , , and from least to greatest. 4 7 __
4 7
_______
3 4 __
3 4
_______ 1 4
_______ • 4 • 4
• 7 • 7
• 7 • 7 21
28 __ 16
28 __ = = = 7
28 __ Rename with like
denominators.
1 4 __
1 4
__ 4 7
__ The fractions in order from least to greatest
are , , . 3 4
__
7 28
4 7
16 28
3 4
21 28
1 4
0 1
Lesson Quiz
Compare. Write <, >, or =.
1. 2.
3. You drilled three holes in a piece of wood. The diameters of the holes are , , and inches. Which hole is the largest?
Order the fractions from least to greatest.
4. 5.
=
Insert Lesson Title Here
Course 1
4-6 Comparing and Ordering Fractions
7 8
__,
__ 8 4 > 5
8 __
16 9 __
1 8
__ 3 8
__ 3 16 __
3 6
__
5 8
__ , 2 3
__ 8 __, 5 2
3 __, 7
8 __ 3
4 __ , 5
8 __ , 5
6 __
8 5 5
6 __ __, 3
4 __ ,
3 8
__
4-7 Mixed Numbers and Improper Fractions
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Order the fractions from least to greatest.
1. 2. 3.
Course 1
4-7 Mixed Numbers and Improper Fractions
2 9
__ 1 6
__ 2 3
__
7 12 __ 5
6 __ 2
3 __
5 8
__ 1 2
__ 4 11 __
, ,
, ,
, , 5 8
__ 1 2
__ 4 11 __ , ,
2 3
__ 5 6
__ 7 12 __ , ,
2 9
__ 2 3
__ 1 6
__ , ,
Problem of the Day
Two numbers have a product of 48. When the larger number is divided by the smaller, the quotient is 3. What are the numbers? 4 and 12
Course 1
4-7 Mixed Numbers and Improper Fractions
Learn to convert between mixed numbers and improper fractions.
Course 1
4-7 Mixed Numbers and Improper Fractions
Course 1
4-7 Mixed Numbers and Improper Fractions
Vocabulary improper fraction proper fraction
Course 1
4-7 Mixed Numbers and Improper Fractions
An improper fraction is a fraction in which the numerator is greater than or equal to the denominator such as .
11 4 __
is read as “eleven-fourths.”
Reading Math 11 4
__
Course 1
4-7 Mixed Numbers and Improper Fractions
Whole numbers can be written as improper fractions. The whole number is the numerator, and the denominator is 1. For example, 7 = .
When the numerator is less than the denominator, the fraction is called a proper fraction.
7 1 __
Course 1
4-7 Mixed Numbers and Improper Fractions
Improper and Proper Fractions Improper Fractions • Numerator equals denominator fraction is equal to 1 •Numerator greater than denominator fraction is greater than 1
Proper Fractions •Numerator less than denominator fraction is less than 1
3 3 __
2 5 __
9 5 __
=
>
1 102 102 ____
13 1 __
102 351 ____
= 1
1 > 1
< 1 < 1
You can write an improper fraction as a mixed number.
Course 1
4-7 Mixed Numbers and Improper Fractions
Additional Example 1: Application
Ella hiked for hours yesterday. Write as a mixed number.
9 4 __ 9
4 __
Method 1: Use a model.
Draw squares divided into fourth sections. Shade 9 of the fourth sections.
1 4 __
1 4
1 4
1 4
1 4
There are 2 whole squares and 1 fourth square, or 2 squares, shaded.
1 4
1 4
1 4
1 4
1 4
1 4
1 4
1 4
1 2 1 4 __
Course 1
4-7 Mixed Numbers and Improper Fractions
Additional Example 1 Continued
Method 2: Use division. 1 4 __
Divide the numerator by the denominator.
To form the fraction part of the quotient, use the remainder as the numerator and the divisor as the denominator.
4 9 - 8
2
1
Ella hiked for 2 hours. 1 4 __
Course 1
4-7 Mixed Numbers and Improper Fractions
Try This: Example 1
Arnold biked for hours yesterday. Write
as a mixed number.
7 4 __ 7
4 __
Method 1: Use a model. Draw squares divided into fourth sections. Shade 7 of the fourth sections.
3 4 __
1 4
1 4
1 4
1 4
There is 1 whole square and 3 fourth squares, or 1 squares, shaded.
1 4
1 4
1 4
1 4
1 3 4 __
Course 1
4-7 Mixed Numbers and Improper Fractions
Try This 1 Continued
Method 2: Use division.
3 4 __
Divide the numerator by the denominator.
To form the fraction part of the quotient, use the remainder as the numerator and the divisor as the denominator.
4 7 - 4
1
3
Arnold biked for 1 hours. 3 4 __
Course 1
4-7 Mixed Numbers and Improper Fractions
Additional Example 2: Writing Mixed Numbers as Improper Fractions
Write 3 as an improper fraction. 2 3 __
Method 1: Use a model. You can draw a diagram to illustrate the whole and fractional parts.
There are 11 thirds or . 11 3 __ Count the thirds in the diagram.
Course 1
4-7 Mixed Numbers and Improper Fractions
Additional Example 2 Continued Method 2: Use multiplication and addition. When you are changing a mixed number to an improper fraction, spiral clockwise as shown in the picture. The order of operations will help you remember to multiply before you add.
2 3 __ 3 =
(3 • 3) + 2 __________ 3
Multiply the whole number by the denominator and add the numerator.
Keep the same denominator.
= 9 + 2 _____ 3
= 11 3 __
Multiply.
Then add.
2 3 3
Course 1
4-7 Mixed Numbers and Improper Fractions
Try This: Example 2
Write 4 as an improper fraction. 1 3 __
Method 1: Use a model. You can draw a diagram to illustrate the whole and fractional parts.
There are 13 thirds, or . 13 3 __ Count the thirds in the diagram.
Course 1
4-7 Mixed Numbers and Improper Fractions
Try This: Example 2 Continued Method 2: Use multiplication and addition. When you are changing a mixed number to an improper fraction, spiral clockwise as shown in the picture. The order of operations will help you remember to multiply before you add.
1 3 __ 4 =
(3 • 4) + 1 __________ 3
Multiply the whole number by the denominator and add the numerator.
Keep the same denominator.
= 12 + 1 ______ 3
= 13 3 __
1 3 __ 4
Multiply.
Then add.
Lesson Quiz Write each mixed number as an improper fraction.
1. 3 2. 6
3. 10 4. 4
5. Janet watches hours of television per day. Write as a mixed number.
Insert Lesson Title Here
Course 1
4-7 Mixed Numbers and Improper Fractions
3 5 __ 1
9 __
1 2 __ 1
8 __
7 4 __
7 4 __
55 9 __ 18
5 __
21 2 __ 33
8 __
3 4 __ 1
4-8 Adding and Subtracting with Like Denominators
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Course 1
4-8 Adding and Subtracting with Like Denominators
Warm Up Write each mixed number as an improper fraction.
1. 6 2. 2 3. 3 4. 8 5. 9 6. 4
1 3
__ 1 5
__
1 2
__
8 11 __
2 7
__
4 5
__
19 3
__ 11 5
__
23 7
__ 17 2
__
49 5
__ 52 11 __
Problem of the Day
Marge has six coins that total $1.15. She cannot make change for $1.00, a half-dollar, a quarter, a dime, or a nickel. What coins does she have? half dollar, quarter, 4 dimes
Course 1
4-8 Adding and Subtracting with Like Denominators
Learn to add and subtract fractions with like denominators.
Course 1
4-8 Adding and Subtracting with Like Denominators
Additional Example 1: Application
1 4 __
1 4 __
1 2 __
1 2 __
1 4 __
2 4 __
+
1 4 __ 1
4 __ + =
=
After 2 hours inch of snow fell
Add the numerators. Keep the same denominator
Write your answer in simplest terms.
Snow was falling at a rate of inch per hour. How
much snow fell after two hours? Write your answer in simplest form.
+ =
Course 1
4-8 Adding and Subtracting with Like Denominators
Try This: Example 1
1 8 __
1 8 __
1 4 __
1 8 __
1 8 __
2 8 __
+
1 8 __ 1
8 __ + =
=
After 2 hours inch of rain fell.
Add the numerators. Keep the same denominator.
Write your answer in simplest terms.
Rain was falling at a rate of inch per hour. How
much rain fell after two hours? Write your answer in simplest form.
+ =
Course 1
4-8 Adding and Subtracting with Like Denominators
Additional Example 2A: Subtracting Like Fractions and Mixed Numbers
Subtract. Write the answer in simplest form. A. 1 –
5 5 __ 3
5 __
3 5 __
2 5 __ = –
To get a common denominator, rewrite 1 as a fraction with a denominator of 5.
Subtract the numerators. Keep the same denominator.
= ―
Course 1
4-8 Adding and Subtracting with Like Denominators
Additional Example 2B: Subtracting Like Fractions and Mixed Numbers
Subtract. Write the answer in simplest form. B. 5 – 2
1 12 __
4 12 __
– Subtract the fractions.
Then subtract the whole numbers.
= -
5 12 __
5 12 __ 5 2
1 12 __
3
3 1 3 __ Write your answer in simplest form.
Course 1
4-8 Adding and Subtracting with Like Denominators
Try This: Example 2A
Subtract. Write the answer in simplest form. A. 1 –
6 6 __ 2
6 __
2 6 __
4 6 __ = –
To get a common denominator, rewrite 1 as a fraction with a denominator of 6. Subtract the numerators. Keep the same denominator.
= –
Course 1
4-8 Adding and Subtracting with Like Denominators
Try This: Example 2B
4 12 __
3 12 __
– Subtract the fractions.
Then subtract the whole numbers.
= -
7 12 __
7 12 __ 4 2
4 12 __
2
2 1 4 __ Write your answer in simplest form.
Subtract. Write the answer in simplest form. B. 4 – 2
Course 1
4-8 Adding and Subtracting with Like Denominators
Evaluate the expression for x = . Write each answer in simplest form.
5 9 __ A.
2 9 __
5 9 __
5 9 __
–
–
– x
x
2 9 __ 3
9 __ =
1 3 __ =
Write the expression.
Substitute for x and subtract the
numerators. Keep the same denominator.
Write your answer in simplest form.
2 9 __
Course 1
4-8 Adding and Subtracting with Like Denominators
Additional Example 3A: Evaluating Expressions with Fractions
Evaluate the expression for x = . Write each answer in simplest form.
4 9 __ B.
2 9 __
4 9 __
4 9 __
+
+
+ x
x
2 9 __ 6
9 __ =
2 3 __
=
Write the expression.
Substitute for x. Add the
fractions. Then add the whole numbers.
Write your answer in simplest form.
2 9 __
2
2
2 2
2 , or 8 3 __
Course 1
4-8 Adding and Subtracting with Like Denominators
Additional Example 3B: Evaluating Expressions with Fractions
Evaluate the expression for x = . Write each answer in simplest form.
8 9 __ C.
2 9 __
8 9 __
8 9 __
+
+
+ x
x
2 9 __ 10
9 __ =
Write the expression.
Substitute for x and subtract
the numerators. Keep the same denominator.
2 9 __
1 , or 1 9 __
Course 1
4-8 Adding and Subtracting with Like Denominators
Additional Example 3C: Evaluating Expressions with Fractions
Try This: Example 3A
Evaluate the expression for x = . Write each answer in simplest form.
4 6 __ A.
2 6 __
4 6 __
4 6 __
–
–
– x
x
2 6 __ 2
6 __ =
1 3 __ =
Write the expression.
Substitute for x and subtract the
numerators. Keep the same denominator. Write your answer in simplest form.
2 6 __
Course 1
4-8 Adding and Subtracting with Like Denominators
Try This: Example 3B
Evaluate the expression for x = . Write each answer in simplest form.
1 6 __ B.
2 6 __
1 6 __
1 6 __
+
+
+ x
x
2 6 __ 3
6 __ =
1 2 __
=
Write the expression.
Substitute for x and subtract
the numerators. Keep the same denominator. Write your answer in simplest form.
2 6 __
3
3
3 3
3
Course 1
4-8 Adding and Subtracting with Like Denominators
Try This: Example 3C
Evaluate the expression for x = . Write each answer in simplest form.
5 6 __ C.
2 6 __
5 6 __
5 6 __
+
+
+ x
x
2 6 __ 7
6 __ =
Write the expression.
Substitute for x and subtract
the numerators. Keep the same denominator.
2 6 __
1 , or 1 6 __
Course 1
4-8 Adding and Subtracting with Like Denominators
Lesson Quiz Add or subtract. Write each answer in simplest form.
1. 1 – 2. 1 –
4. Evaluate – x for x = .
9 11 __
3 10 __
8 9 __
1 10 __
6 21 __ 16
21 ___
3 4 __
2 11 __ 1
9 __
2 5 __
10 21 __
6
3 miles
3. 2 + 4
5. Weston walks of a mile every day. How far
will he have walked in 4 days?
Course 1
4-8 Adding and Subtracting with Like Denominators
4-9 Multiplying Fractions by Whole Numbers
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Multiply.
30 96 144
Course 1
4-9 Multiplying Fractions by Whole Numbers
66 120
1. 15 × 2
2. 12 × 8 3. 9 × 16 4. 6 × 11
5. 8 × 15
Problem of the Day
The length of three dowels are 13 inches, 27 inches, and 19 inches. How can you use the three dowels to mark off a length of 5 inches? 13 + 19 – 27
Course 1
4-9 Multiplying Fractions by Whole Numbers
Learn to multiply fractions by whole numbers.
Course 1
4-9 Multiplying Fractions by Whole Numbers
Course 1
4-9 Multiplying Fractions by Whole Numbers
Recall that multiplication by a whole number can be represented as repeated addition. For example, 4 • 5 = 5 + 5 + 5 + 5. You can multiply a whole number by a fraction using the same method.
+ + =
3 • = 3 4 __ 1
4 __
Course 1
4-9 Multiplying Fractions by Whole Numbers
3 1
__ 1 4
__ 3 • 1 1 • 4
______ 3 4
__ •
There is another way to multiply with fractions. Remember that a whole number can be written as an improper fraction with 1 in the denominator. So 3 = . 3
1 __
= = Multiply numerators Multiply denominators
Course 1
4-9 Multiplying Fractions by Whole Numbers
Additional Example 1A: Multiplying Fractions and Whole Numbers
Multiply. Write the answer in simplest form.
1 9
__
1 9
__ 7 1
__ 7 • 1 1 • 9
______
7 9
__
• =
=
Write 7 as a fraction. Multiply numerators and denominators.
A. 7 •
Course 1
4-9 Multiplying Fractions by Whole Numbers
Additional Example 1B: Multiplying Fractions and Whole Numbers
Multiply. Write the answer in simplest form.
1 8
__
1 8
__ 6 1
__ 6 • 1 1 • 8
______
6 8
__
• =
=
Write 6 as a fraction. Multiply numerators and denominators.
3 4
__ = Write your answer in simplest form.
B. 6 •
Course 1
4-9 Multiplying Fractions by Whole Numbers
Additional Example 1C: Multiplying Fractions and Whole Numbers
Multiply. Write the answer in simplest form.
2 3
__
2 3
__ 8 1
__ 8 • 2 1 • 3
______
16 3
___
• =
=
Write 8 as a fraction. Multiply numerators and denominators.
1 3
__ or 5
C. 8 •
Course 1
4-9 Multiplying Fractions by Whole Numbers
Try This: Example 1A
Multiply. Write the answer in simplest form.
1 5
__
1 5
__ 4 1
__ 4 • 1 1 • 5
______
4 5
__
• =
=
Write 4 as a fraction. Multiply numerators and denominators.
A. 4 •
Course 1
4-9 Multiplying Fractions by Whole Numbers
Try This: Example 1B
Multiply. Write the answer in simplest form.
1 6
__
1 6
__ 3 1
__ 3 • 1 1 • 6
______
3 6
__
• =
=
Write 3 as a fraction. Multiply numerators and denominators.
1 2
__ = Write your answer in simplest form.
B. 3 •
Course 1
4-9 Multiplying Fractions by Whole Numbers
Try This: Example 1C
Multiply. Write the answer in simplest form.
3 4
__
3 4
__ 9 1
__ 9 • 3 1 • 4
______
27 4
___
• =
=
Write 9 as a fraction. Multiply numerators and denominators.
3 4
__ or 6
C. 9 •
Course 1
4-9 Multiplying Fractions by Whole Numbers
Additional Example 2A: Evaluating Fraction Expression
Evaluate 4x for the value of x. Write the answer in simplest form.
1 10
___
4 1
__
1 10
___
4 10
___
•
=
Write the expression. 4x
4
• 1 10
___
= 2 5
__
Multiply.
Substitute for x. 1 10
___
Write your answer in simplest form.
A. x =
Course 1
4-9 Multiplying Fractions by Whole Numbers
Additional Example 2B: Evaluating Fraction Expression
Evaluate 4x for the value of x. Write the answer in simplest form.
3 8
__
4 1
__
3 8
__
12 8
___
•
=
Write the expression. 4x
4
• 3 8
__
= 3 2
__
Multiply.
Substitute for x. 3 8
__
Write your answer in simplest form. 1 2
__ ,or 1
B. x =
Course 1
4-9 Multiplying Fractions by Whole Numbers
Try This: Example 2A Evaluate 3x for the value of x. Write the answer in simplest form.
1 9
__
3 1
__
1 9
__
3 9
__
•
=
Write the expression. 3x
3
• 1 9
__
= 1 3
__
Multiply.
Substitute for x. 1 9
__
Write your answer in simplest form.
A. x =
Course 1
4-9 Multiplying Fractions by Whole Numbers
Try This: Example 2B Evaluate 3x for the value of x. Write the answer in simplest form.
4 7
__
3 1
__
4 7
__
12 7
___
•
=
Write the expression. 3x
3
• 4 7
__
=
Multiply.
Substitute for x. 4 7
__
Write your answer in simplest form. 5 7
__ 1
B. x =
Course 1
4-9 Multiplying Fractions by Whole Numbers
Sometimes the denominator of an improper fraction will divide evenly into the numerator. When this happens, the improper fraction is equivalent to a whole number, not a mixed number.
This makes sense if you remember that the fraction bar means “divided by”.
12 3
___ = 4
Course 1
4-9 Multiplying Fractions by Whole Numbers
Additional Example 3: Application There are 25 students in the music club. Of those
students, are also in the band. How many music
club students are in the band?
3 5
__
5 3 __ To find of 25, multiply.
25 1
___ 3 5
__ • 25 = • 3 5
__
75 5
___ =
= 15
There are 15 music club students in the band.
Course 1
4-9 Multiplying Fractions by Whole Numbers
Try This: Example 3 There are 18 saltwater fish in the aquarium. Of those
fish, are blue. How many fish in the aquarium
are blue?
2 3
__
2 3
__
To find of 18, multiply.
18 1
___ 2 3
__ • 18 = • 2 3
__
36 3
___ =
= 12
There are 12 blue fish in the aquarium.
Course 1
4-9 Multiplying Fractions by Whole Numbers
Lesson Quiz
Multiply. Write each answer in simplest form.
1. 10 • 2. 6 •
Evaluate 6x for each value of x. Write your answer in simplest form.
3. x = 4. x =
5. Alicia spent 15 minutes making a pizza. Of those minutes were spent rolling out the crust. How many
minutes did Alicia spend rolling out the crust?
2 5
__ 1 10
__
1 4
__ 5 6
__
1 3
__
5 minutes
5
4 3 5
__
, or 1 1 2
__ 3 2
__
5-1 Multiplying Fractions
Course 1
Warm Up
Problem of the Day Lesson Presentation
Warm Up
1. What is of 12? 2. What is of 100? 3. What is of 120? 4. What is of 100? 5. What is of 480?
6
50
60
Course 1
5-1 Multiplying Fractions
1 2 __
1 2 __
1 2 __
1 4 __
1 4 __
25
120
Problem of the Day
Your favorite uncle left one-fifth of his estate to each of his three children and the rest to his favorite charity. If his estate was worth $105,000, how much was given to charity? $42,000
Course 1
5-1 Multiplying Fractions
Learn to multiply fractions.
Course 1
5-1 Multiplying Fractions
Course 1
5-1 Multiplying Fractions
On average, people spend of their lives asleep. About
of the time they sleep, they dream. What fraction of
a lifetime does a person typically spend dreaming?
1 3 __
1 3 __
1 4 __
1 4 __
1 4 __ 1
3 __
1 4 __ 1
3 __ • =
1 12 __
One way to find of is to make a model. Find of .
Course 1
5-1 Multiplying Fractions
You can also multiply fractions without making a model.
1 4 __ 1
3 __ •
1 • 1 4 • 3
_____
1 12 __
=
=
Multiply the numerators. Multiply the denominators.
A person typically spends of his or her lifetime dreaming.
1 12 __
Course 1
5-1 Multiplying Fractions
You can look for a common factor in a numerator and a denominator to determine whether you can simplify before multiplying.
Helpful Hint
Course 1
5-1 Multiplying Fractions
Additional Example 1A: Multiplying Fractions Multiply. Write the answer in simplest form.
A.
1 4 __ 2
5 __ • _____ =
=
Multiply numerators. Multiply denominators.
2 20 __
1 • 2 4 • 5
1 4 __ •
2 5 __
The GCF of 2 and 20 is 2.
The answer is in simplest form. = 1 10 __
Course 1
5-1 Multiplying Fractions
Additional Example 1B: Multiplying Fractions Multiply. Write the answer in simplest form.
B.
5 7 __ 4
15 __ •
_____
=
=
Use the GCF to simplify the fractions before multiplying. The greatest common factor of 5 and 15 is 5.
4 21 __
1 • 4 7 • 3
5 7 __ •
4 15 __
Multiply numerators. Multiply denominators.
The answer is in simplest form.
1 7 __ •
4 3 __ 1
3
=
Course 1
5-1 Multiplying Fractions
Additional Example 1C: Multiplying Fractions Multiply. Write the answer in simplest form.
C.
4 9 __ 6
10 __ • ______ =
=
Multiply numerators. Multiply denominators.
24 90 __
4 • 6 9 • 10
4 9 __ •
6 10 __
The GCF of 24 and 90 is 6.
The answer is in simplest form. = 4 15 __
Course 1
5-1 Multiplying Fractions
Try This: Example 1A Multiply. Write the answer in simplest form.
A.
1 2 __ 3
6 __ • _____ =
=
Multiply numerators. Multiply denominators.
3 12 __
1 • 3 2 • 6
1 2 __ •
3 6 __
The GCF of 3 and 12 is 3.
The answer is in simplest form. = 1 4 __
Course 1
5-1 Multiplying Fractions
Try This: Example 1B Multiply. Write the answer in simplest form.
B.
3 7 __ 5
9 __ •
_____
=
=
Use the GCF to simplify the fractions before multiplying. The greatest common factor of 3 and 9 is 3.
5 21 __
1 • 5 7 • 3
3 7 __ •
5 9 __
Multiply numerators. Multiply denominators.
The answer is in simplest form.
1 7 __ •
5 3 __ 1
3
=
Course 1
5-1 Multiplying Fractions
Try This: Example 1C Multiply. Write the answer in simplest form.
C.
3 7 __ 5
15 __ • ______ =
=
Multiply numerators. Multiply denominators.
15 105 ____
3 • 5 7 • 15
3 7 __ •
5 15 __
The GCF of 15 and 105 is 15.
The answer is in simplest form. = 1 7 __
Course 1
5-1 Multiplying Fractions Additional Example 2A: Evaluating Fraction
Expressions 2 5 __
1 3 __ b • 2
5 __
2 5 __ 1
3 __ •
_____ 1 • 2 3 • 5
2 15 __
Substitute for b. 1 3 __
Multiply.
The answer is in simplest form.
A. b =
Evaluate the expression b • for each value of b. Write the answer in simplest form.
Course 1
5-1 Multiplying Fractions
Additional Example 2B: Evaluating Fraction Expressions
B. b = 3 8 __ b •
2 5 __
2 5 __ 3
8 __ •
_____ 3 • 1 4 • 5
3 20 __
Substitute for b. 3 8 __
Use the GCF to simplify.
The answer is in simplest form.
1
4
2 5 __ 3
8 __ •
Multiply.
Course 1
5-1 Multiplying Fractions
Additional Example 2C: Evaluating Fraction Expressions
C. b = 5 7 __ b • 2
5 __
2 5 __ 5
7 __ •
_____ 5 • 2 7 • 5 10 35 __
Substitute for b. 5 7 __
Multiply.
The GCF of 10 and 35 is 5.
The answer is in simplest form. 2 7 __
Course 1
5-1 Multiplying Fractions
Try This: Example 2A Evaluate the expression c • for each value of c.
Write the answer in simplest form.
A. c =
1 4 __
1 7 __ c • 1
4 __
1 4 __ 1
7 __ •
_____ 1 • 1 7 • 4
1 28 __
Substitute for c. 1 7 __
Multiply.
The answer is in simplest form.
Course 1
5-1 Multiplying Fractions
Try This: Example 2B
B. c = 2 7 __ c •
1 4 __
1 4 __ 2
7 __ •
_____ 1 • 1 7 • 2
1 14 __
Substitute for c. 2 7 __
Use the GCF to simplify.
The answer is in simplest form.
1
2
1 4 __ 2
7 __ •
Multiply.
Course 1
5-1 Multiplying Fractions
Try This: Example 2C
C. c = 4 7 __ c • 1
4 __
1 4 __ 4
7 __ •
_____ 4 • 1 7 • 4
4 28 __
Substitute for c. 4 7 __
Multiply.
The GCF of 4 and 28 is 4.
The answer is in simplest form. 1 7 __
Lesson Quiz
Multiply. Write each answer in simplest form.
1. 2.
Evaluate the expression x • for each value of x. Write the answer in simplest form.
3. x = 4. x =
5. At a particular college of the students take a math class. Of these students take basic algebra. What fraction of all students take basic algebra?
Insert Lesson Title Here
Course 1
5-1 Multiplying Fractions
3 5 __ 1
4 __ •
5 7 __ 3
10 __ •
1 8 __
4 5 __ 1
8 __
2 5 __
1 4 __
3 20 __ 3
14 __
1 10 __ 1
64 __
1 10 __
5-2 Multiplying Mixed Numbers
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Multiply.
1. 2. 3. 4.
Course 1
5-2 Multiplying Mixed Numbers
3 8 __ 5
8 __ • 2
• 4 5
__ 3 8
__
3 11 __
1 2 __ 1
3 __ •
•
1 6
__ 3 10 __
6 11 __ 15
64 __
Problem of the Day
Tracy and Zachary are sharing a large pizza. Tracy cuts the pizza in half. Zachary says he cannot eat that much. Zachary cuts his half into fourths and eats one piece. What fraction of the whole pizza does Zachary not eat?
Course 1
5-2 Multiplying Mixed Numbers
7 8 __
Learn to multiply mixed numbers.
Course 1
5-2 Multiplying Mixed Numbers
Course 1
5-2 Multiplying Mixed Numbers
To write a mixed number as an improper fraction, start with the whole number, multiply by the denominator, and add the numerator. Use the same denominator.
Remember!
1 5
__ 1 1 • 5 + 1 5
_________ 6 5 __ = =
Course 1
5-2 Multiplying Mixed Numbers
Additional Example 1A: Multiplying Fractions and Mixed Numbers
Multiply. Write the answer in simplest form.
A. • 1 1 4
__ 1 3
__
Write 1 as an improper fraction. 1 = 4 3 __ 1
3 __ 1
3 __ 1
4 __ 4
3 __ •
1 • 4 4 • 3
_____ Multiply numerators. Multiply denominators.
4 12 __
1 3 __ Write the answer in simplest form.
Course 1
5-2 Multiplying Mixed Numbers
Additional Example 1B: Multiplying Fractions and Mixed Numbers
Multiply. Write the answer in simplest form.
B. 3 • 1 2
__ 4 5
__
Write 3 as an improper fraction. 3 = 7 2 __ 1
2 __ 1
2 __ 7
2 __ 4
5 __ •
7 • 4 2 • 5
_____ Multiply numerators. Multiply denominators.
28 10 __
14 5 __ Write the answer in simplest form. You
can write the answer as a mixed number. 4 5 __ = 2
Course 1
5-2 Multiplying Mixed Numbers
Additional Example 1C: Multiplying Fractions and Mixed Numbers
Multiply. Write the answer in simplest form.
C. • 2 12 13 __ 3
8 __
Write 2 as an improper fraction. 2 = 19 8 __ 3
8 __ 3
8 __ 12
13 __ 19
8 __ •
3 • 19 13 • 2 ______
Use the GCF to simplify before multiplying.
57 26 __ You can write the answer as a mixed number. 5
26 __ = 2
3
2
12 13 __ 19
8 __ •
Course 1
5-2 Multiplying Mixed Numbers
Try This: Example 1A
Multiply. Write each answer in simplest form.
A. • 1 1 9
__ 1 2
__
Write 1 as an improper fraction. 1 = 3 2 __ 1
2 __ 1
2 __ 1
9 __ 3
2 __ •
1 • 3 9 • 2
_____ Multiply numerators. Multiply denominators.
3 18 __
1 6 __ Write the answer in simplest form.
Course 1
5-2 Multiplying Mixed Numbers
Try This: Example 1B
Multiply. Write each answer in simplest form.
B. 2 • 2 3
__ 5 6
__
Write 2 as an improper fraction. 2 = 8 3 __ 2
3 __ 2
3 __ 8
3 __ 5
6 __ •
8 • 5 3 • 6
_____ Multiply numerators. Multiply denominators.
40 18 __
20 9 __ Write the answer in simplest form. You
can write the answer as a mixed number. 2 9 __ = 2
Course 1
5-2 Multiplying Mixed Numbers
Try This: Example 1C
Multiply. Write each answer in simplest form.
C. • 3 14 15 __ 2
7 __
Write 3 as an improper fraction. 3 = 23 7 __ 2
7 __ 2
7 __ 14
15 __ 23
7 __ •
2 • 23 15 • 1 ______
Use the GCF to simplify before multiplying.
46 15 __ You can write the answer as a mixed
number. 1 15 __ = 3
2
1
14 15 __ 23
7 __ •
Course 1
5-2 Multiplying Mixed Numbers
Additional Example 2A: Multiplying Mixed Numbers
Find each product. Write the answer in simplest form. A. 1 • 2 2
3 __ 1
7 __
5 3 __ 15
7 __ •
5 • 15 3 • 7
______ Multiply numerators. Multiply denominators.
75 21 __
12 21 __ You can write the improper fraction as a
mixed number.
Write the mixed numbers as improper
fractions. 1 = 2 = 5 3 __ 15
7 __ 2
3 __ 1
7 __
3
Simplify. 4 7 __ 3
Course 1
5-2 Multiplying Mixed Numbers
Additional Example 2B: Multiplying Mixed Numbers
Find each product. Write the answer in simplest form. B. 1 • 2 3
8 __ 2
5 __
11 8 __ 12
5 __ •
11 • 3 2 • 5
______
Use the GCF to simply before multiplying.
33 10 __
Multiply numerators. Multiply denominators.
Write the mixed numbers as improper
fractions. 1 = 2 = 11 8 __ 12
5 __ 3
8 __ 2
5 __
Simplify. 3 10 __ 3
11 8 __ 12
5 __ •
3
2
Course 1
5-2 Multiplying Mixed Numbers
Additional Example 2C: Multiplying Mixed Numbers
Find each product. Write the answer in simplest form. C. 2 • 4 2
5 __
Use the Distributive Property.
Add.
4 5 __ 8 +
2 • 4 2 5 __
2 • (4 + ) 2 5 __
(2 • 4) + ( 2 • )
2 1 __
Multiply.
(2 • 4) + ( • ) 2 5 __
2 5 __
4 5 __ 8
Course 1
5-2 Multiplying Mixed Numbers
Try This: Example 2A Find each product. Write the answer in simplest form.
A. 1 • 2 3 4
__ 1 6
__
7 4 __ 13
6 __ •
7 • 13 4 • 6
______ Multiply numerators. Multiply denominators.
91 24 __
19 24 __ You can write the improper fraction as a
mixed number.
Write the mixed numbers as improper
fractions. 1 = 2 = 7 4 __ 13
6 __ 3
4 __ 1
6 __
3
Course 1
5-2 Multiplying Mixed Numbers
Try This: Example 2B Find each product. Write the answer in simplest form.
B. 1 • 3 2 9
__ 3 5
__
11 9 __ 18
5 __ •
11 • 2 1 • 5
______
Use the GCF to simply before multiplying.
22 5 __
Multiply numerators. Multiply denominators.
Write the mixed numbers as improper
fractions. 1 = . 3 = 11 9 __ 18
5 __ 2
9 __ 3
5 __
Simplify. 2 5 __ 4
11 9 __ 18
5 __ •
2
1
Course 1
5-2 Multiplying Mixed Numbers
Try This: Example 2C Find each product. Write the answer in simplest form.
C. 3 • 6 1 5
__
Use the Distributive Property.
Add.
3 5 __ 18 +
3 • 6 1 5 __
3 • (6 + ) 1 5 __
(3 • 6) + ( 3 • )
3 1 __
Multiply.
(3 • 6) + ( • ) 1 5 __
1 5 __
3 5 __ 18
Lesson Quiz
Multiply. Write each answer in simplest form.
1. 3 2. 1
Find each product. Write the answer in simplest form.
3. 2 • 1 4. 5 • 3
5. A nurse gave a patient 3 tablets of a medication. If each
tablet contained grain of the medication, how much
medication did the patient receive?
2
Insert Lesson Title Here
Course 1
5-2 Multiplying Mixed Numbers
5 8 __ 1
5 __ •
5 7 __ 7
10 __
•
1 4 __ 1
5 __ 1
5 __ 16
1 2 __
1 20 __
1 3 14 __
2 3 4 __
grain 7 40 __
5-3 Dividing Fractions and Mixed Numbers
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Multiply. Write each answer in simplest form.
1. 2 • 2 2. 1 • 3 3. 3 • 3 4. 2 • 1
5
or 5
10
Course 1
5-3 Dividing Fractions and Mixed Numbers
1 5 __
1 2 __
1 2 __
1 2 __
1 3 __
1 4 __
21 4 __ 1
4 __
or 2 11 4 __ 3
4 __
Problem of the Day
Rachel is building a doghouse. She needs to cut a 12-foot-long board into lengths of 65 in. How many lengths can she cut, and will there be any wood left? If so, how much? 2 cuts; yes, 14 in.
Course 1
5-3 Dividing Fractions and Mixed Numbers
Learn to divide fractions and mixed numbers.
Course 1
5-3 Dividing Fractions and Mixed Numbers
Vocabulary reciprocal
Insert Lesson Title Here
Course 1
5-3 Dividing Fractions and Mixed Numbers
Course 1
5-3 Dividing Fractions and Mixed Numbers
Reciprocals can help you divide by fractions. Two numbers are reciprocals if their product is 1.
Course 1
5-3 Dividing Fractions and Mixed Numbers
Additional Example 1A: Finding Reciprocals
Find the reciprocal.
A. 1 9
__
1 9 __ • = 1 Think: of what number is 1?
1 9 __ 1
9 __
1 9 __ • 9 = 1 1 9 __ of is 1. 1
9 __ 9
1 __
The reciprocal of is 9. 1 9 __
Course 1
5-3 Dividing Fractions and Mixed Numbers
Additional Example 1B: Finding Reciprocals
Find the reciprocal.
B. 2 3
__
2 3 __ • = 1 Think: of what number is 1?
2 3 __
3 2 __ 2
3 __ • = = 1 6
6 __ of is 1. 2
3 __ 3
2 __
The reciprocal of is . 2 3 __ 3
2 __
Course 1
5-3 Dividing Fractions and Mixed Numbers
Additional Example 1C: Finding Reciprocals
Find the reciprocal.
C. 3 1 5
__
16 5 __ • = 1 Write 3 as . 1
5 __
5 16 __ 16
5 __ • = = 1 80
80 __ of is 1. 16
5 __ 5
16 __
The reciprocal of is . 16 5 __ 5
16 __
16 5 __
Course 1
5-3 Dividing Fractions and Mixed Numbers
Try This: Example 1A
Find the reciprocal.
A. 1 4
__
1 4 __ • = 1 Think: of what number is 1?
1 4 __
1 4 __ • 4 = 1 of is 1. 1
4 __ 4
1 __
The reciprocal of is 4. 1 4 __
Course 1
5-3 Dividing Fractions and Mixed Numbers
Try This: Example 1B
Find the reciprocal.
B. 4 5
__
4 5 __ • = 1 Think: of what number is 1?
4 5 __
5 4 __ 4
5 __ • = = 1 20
20 __ of is 1. 4
5 __ 5
4 __
The reciprocal of is . 4 5 __ 5
4 __
Course 1
5-3 Dividing Fractions and Mixed Numbers
Try This: Example 1C
Find the reciprocal.
C. 4 1 8
__
33 8 __ • = 1 Write 4 as . 1
8 __
8 33 __ 33
8 __ • = = 1 264
264 ___ of is 1. 33
8 __ 8
33 __
The reciprocal of is . 33 8 __ 8
33 __
33 8 __
Course 1
5-3 Dividing Fractions and Mixed Numbers
Look at the relationship between the
fractions and . If you switch the
numerator and denominator of a fraction,
you will find its reciprocal. Dividing by a
number is the same as multiplying by its
reciprocal.
3 4 __ 4
3 __
24 ÷ 4 = 6 24 • = 6 1 4 __
Course 1
5-3 Dividing Fractions and Mixed Numbers
Additional Example 2A: Using Reciprocals to Divide Fractions and Mixed Numbers
Divide. Write each answer in simplest form.
A. ÷ 7 8 7
__
8 7 __ ÷ 7 = •
Rewrite as multiplication using the
reciprocal of 7, . 1 7 __
Multiply by the reciprocal.
8 7 __ 1
7 __
8 • 1 7 • 7 ____ =
The answer is in simplest form. = 8 49 __
Course 1
5-3 Dividing Fractions and Mixed Numbers
Additional Example 2B: Using Reciprocals to Divide Fractions and Mixed Numbers
B. ÷ 5 6
__
5 6 __ ÷ = •
Rewrite as multiplication using the
reciprocal of , . 3 2 __
Simplify before multiplying.
5 6 __ 3
2 __
5 • 3 6 • 2 ____ =
You can write the answer as a mixed number.
= 5 4 __
2 3
__
2 3 __ 2
3 __
1
2
Multiply.
= 1 1 4 __
Course 1
5-3 Dividing Fractions and Mixed Numbers
Additional Example 2C: Using Reciprocals to Divide Fractions and Mixed Numbers
C. 2 ÷ 1 3 4
__
Write the mixed numbers as
improper fractions. 2 =
and 1 = 13 12 __
Rewrite as multiplication.
11 • 12 4 • 13 ______
=
You can write the answer as a mixed number.
=
33 13 __
1 12 __
3 4 __
1
3 Simplify before multiplying.
= 2 7 13 __
2 ÷ 1 = ÷ 3 4 __ 1
12 __ 11
4 __ 13
12 __ 11
4 __
1 12 __
11 4 __ 12
13 __ •
Multiply. =
Course 1
5-3 Dividing Fractions and Mixed Numbers
Try This: Example 2A
Divide. Write each answer in simplest form.
A. ÷ 3 2 3
__
2 3 __ ÷ 3 = •
Rewrite as multiplication using the
reciprocal of 3, . 1 3 __
Multiply by the reciprocal.
2 3 __ 1
3 __
2 • 1 3 • 3 ____ =
The answer is in simplest form. = 2 9 __
Course 1
5-3 Dividing Fractions and Mixed Numbers
Try This: Example 2B
B. ÷ 7 10 __
7 10 __ ÷ = •
Rewrite as multiplication using the
reciprocal of , . 5 1 __
Simplify before multiplying.
7 10 __ 5
1 __
7 • 5 10 • 1 ____ =
You can write the answer as a mixed number.
= 7 2 __
1 5
__
1 5 __ 1
5 __
1
2
Multiply.
= 3 1 2 __
Course 1
5-3 Dividing Fractions and Mixed Numbers
Try This: Example 2C
C. 3 ÷ 1 2 3
__
Write the mixed numbers as
improper fractions. 3 =
and 1 = . 10 9 __
Rewrite as multiplication.
11 • 9 3 • 10 ______
=
You can write the answer as a mixed number.
=
33 10 __
1 9
__
2 3 __
1
3 Simplify before multiplying.
= 3 3 10 __
3 ÷ 1 = ÷ 2 3 __ 1
9 __ 11
3 __ 10
9 __ 11
3 __
1 9 __
11 3 __ 9
10 __ •
Multiply. =
Lesson Quiz
Find the reciprocal.
1. 2.
Divide. Write each answer in simplest form.
3. ÷ 20 4. 3 ÷ 2
5. Rhonda put 2 pounds of pecans into a -pound bag. How many bags did Rhonda fill?
Insert Lesson Title Here
Course 1
5-3 Dividing Fractions and Mixed Numbers
1 11 __ 8
13 __
4 7 __ 1
2 __ 1
2 __
3 4 __ 1
4 __
11 1 __ 13
8 __
1 35 __ 7
5 __ 2
5 __ , or 1
11 bags
5-4 Solving Fraction Equations: Multiplication and Division
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Solve.
1. x – 5 = 17 2. 5x = 125 3. x + 12 = 86 4. 9x = 108
x = 22
x = 25
x = 74
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
x = 12
Problem of the Day
Stephen forgot his locker number, but he remembered that the sum of the digits is 11 and that the digits are all odd numbers. The locker numbers are from 1 to 120. What is Stephen’s locker number? 119
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Learn to solve equations by multiplying and dividing fractions.
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Dividing by a number is the same as multiplying by its reciprocal.
Remember!
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Additional Example 1A: Solving Equations by Multiplying and Dividing
Solve the equation. Write the answer in simplest form.
A. j = 25 3 5
__
j ÷ = 25 ÷ 3 5 __ 3
5 __ 3
5 __
j • = 25 • 3 5 __ 5
3 __ 5
3 __
j = 25 • 5 1 • 3
_____
j = 25 • 5 3 __
j = , or 41 125 3
___ 2 3 __
Divide both sides of the equation by . 3 5 __
Multiply by , the reciprocal of . 3 5 __ 5
3 __
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Additional Example 1B: Solving Equations by Multiplying and Dividing
Solve the equation. Write the answer in simplest form.
B. 7x = 2 5
__
7x 1 __ 2
5 __ 1
7 __ 1
7 __ • = •
x = 2 • 1 5 • 7 ____
x = 2 35 __
Multiply both sides by the reciprocal of 7.
The answer is in simplest form.
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Additional Example 1C: Solving Equations by Multiplying and Dividing
Solve the equation. Write the answer in simplest form.
C. = 6 5y 8
__
5y 8 __ 6
1 __ 5
8 __ 5
8 __ ÷ = ÷
y = , or 9 48 5 __
Divide both sides by . 5 8 __
5y 8 __ 6
1 __ 8
5 __ 8
5 __ • = •
3 5 __
5 8 Multiply by the reciprocal of . __
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Try This: Example 1A Solve the equation. Write the answer in simplest form.
A. j = 19 3 4
__
j ÷ = 19 ÷ 3 4 __ 3
4 __ 3
4 __
j • = 19 • __ 3 4 __ 4
3 4 3 __
j = 19 • 4 1 • 3
_____
4 j = 19 • 3 __
76 ___ 1 3
j = , or 25 3 __
Divide both sides of the equation by . 3 4 __
Multiply by , the reciprocal of . 3 4 __ 4
3 __
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Try This: Example 1B Solve the equation. Write the answer in simplest form.
B. 3x = 1 7
__
3x 1 __ 1
7 __ 1
3 __ 1
3 __ • = •
x = 1 • 1 7 • 3 ____
x = 1 21 __
Multiply both sides by the reciprocal of 3.
The answer is in simplest form.
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Try This: Example 1C Solve the equation. Write the answer in simplest form.
C. = 4 6y 7
__
6y 7 __ 4
1 __ 6
7 __ 6
7 __ ÷ = ÷
y = , or 4 28 6 __
Divide both sides by .
Multiply by the reciprocal of .
6 7 __
6y 7 __ 4
1 __ 7
6 __ 7
6 __ • = •
2 3 __
6 7 __
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Additional Example 2: Problem Solving Application
2 3
__
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Dexter makes of a recipe, and he uses 12 cups of powdered milk. How many cups of powdered milk are in the recipe?
Additional Example 2 Continued
1 Understand the Problem
The answer will be the number of cups of powdered milk in the recipe.
List the important information:
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
2 3 __ • He makes of the recipe.
• He uses 12 cups of powdered milk.
2 Make a Plan
You can write and solve an equation. Let x represent the number of cups in the recipe.
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
He uses 12 cups, which is two-thirds of the amount of the recipe.
2 3 __ 12 = x
Additional Example 2 Continued
Solve 3 2 3 __ 12 = x
12 • = x • 3 2 __ 2
3 __ 3
2 __
12 1 __ 3
2 __ • = x
6
1 18 = x
There are 18 cups of powdered milk in the recipe.
Multiply both sides by , the
reciprocal of .
3 2 __
2 3 __
Simplify. Then multiply.
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Additional Example 2 Continued
Look Back 4
Check 12 = x 2 3 __
2 3 __ 12 = ? (18)
36 3 __ 12 = ?
12 = 12 ?
Substitute 18 for x.
Multiply and simplify.
18 is the solution.
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Additional Example 2 Continued
12 1
Try This: Example 2
2 5
__
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Rich makes of a recipe, and he uses 8 cups of wheat flour. How many cups of wheat flour are in the recipe?
Try This: Example 2 Continued
1 Understand the Problem
The answer will be the number of cups of wheat flour in the recipe.
2 5 __
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
List the important information:
• He makes of the recipe.
• He uses 8 cups of powdered milk.
2 Make a Plan
You can write and solve an equation. Let y represent the number of cups in the recipe.
2 5 __
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
He uses 8 cups, which is two-fifths of the amount of the recipe.
8 = y
Try This: Example 2 Continued
Solve 3 2 5 __ 8 = y
8 • = y • 5 2 __ 2
5 __ 5
2 __
8 1 __ 5
2 __ • = y
4
1 20 = y
There are 20 cups of wheat flour in the recipe.
Multiply both sides by , the
reciprocal of .
5 2 __
2 5 __
Simplify. Then multiply.
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Try This: Example 2 Continued
Look Back 4
Check 8 = y 2 5 __
2 5 __ 8 = ? (20)
40 5 __ 8 = ?
8 = 8 ?
Substitute 20 for y.
Multiply and simplify.
20 is the solution.
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
Try This: Example 2 Continued
8 1
Lesson Quiz Solve each equation. Write the answer in simplest form.
1. 3x = 2. x = 4
3. x = 14 4. = 9
5. Rebecca used 3 pt of paint to paint of the trim in her bedroom. How many pints will Rebecca use for the trim in the entire bedroom?
Insert Lesson Title Here
1 8 __ 1
4 __
3 7 __ y
7 __
1 4 __
x = 16 1 24 __ x =
98 3 __ 2
3 __ x = or 32 y = 63
12
Course 1
5-4 Solving Fraction Equations: Multiplication and Division
5-5 Least Common Multiple
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Write the first five multiples of each number.
1. 5
2. 6
3. 10
4. 12
5, 10, 15, 20, 25
6, 12, 18, 24, 30
10, 20, 30, 40, 50
Course 1
5-5 Least Common Multiple
12, 24, 36, 48, 60
Problem of the Day
Greg, Sam and Mary all work at the same high school. One of them is a principal, one of them is a teacher, and one of them is a janitor. Sam is older than Mary. Mary does not live in the same town as the principal. The teacher, the oldest of the three, often plays golf with Greg. What is each person’s job. Greg, principal; Sam, teacher; Mary, janitor
Course 1
5-5 Least Common Multiple
Learn to find the least common multiple (LCM) of a group of numbers.
Course 1
5-5 Least Common Multiple
Vocabulary
least common multiple (LCM)
Insert Lesson Title Here
Course 1
5-5 Least Common Multiple
Course 1
5-5 Least Common Multiple
Additional Example 1: Consumer Application English muffins come in packs of 8, and eggs in cartons of 12. If there are 24 students, what is the least number of packs and cartons needed so that each student has a muffin sandwich with one egg and there are no muffins left over?
There are 24 English muffins and 24 eggs.
So 3 packs of English muffins and 2 cartons of eggs are needed.
Draw muffins in groups of 8. Draw eggs in groups of 12. Stop when you have drawn the same number of each.
Course 1
5-5 Least Common Multiple Try This: Example 1
There are 18 dog cookies and 18 bones.
So 3 packages of dog cookies and 2 bags of bones are needed.
Dog cookies come in packages of 6, and bones in bags of 9. If there are 18 dogs, what is the least number of packages and bags needed so that each dog has a treat box with one bone and one cookie and there are no bones or cookies left over?
Draw cookies in groups of 6. Draw bones in groups of 9. Stop when you have drawn the same number of each.
Course 1
5-5 Least Common Multiple
The smallest number that is a multiple of two or more numbers is the least common multiple (LCM). In Additional Example 1, the LCM of 8 and 12 is 24.
Course 1
5-5 Least Common Multiple
Additional Example 2A: Using Multiples to Find the LCM
Find the least common multiple (LCM).
Method 1: Use a number line. A. 3 and 4
Use a number line to skip count by 3 and 4.
0 2 4 6 8 10 12
The least common multiple (LCM) of 3 and 4 is 12.
Course 1
5-5 Least Common Multiple
Additional Example 2B: Using Multiples to Find the LCM
Find the least common multiple (LCM).
Method 2: Use a list.
4: 4, 8, 12 , 16, 20, 24, 28, 32, 36, 40, 44, . . .
5: 5, 10, 15, 20, 25, 30, 35, 40, 45, . . .
8: 8, 16, 24, 32, 40, 48, . . .
LCM: 40
List multiples of 4, 5, and 8.
Find the smallest number that is in all the lists.
B. 4, 5, and 8
Course 1
5-5 Least Common Multiple
The prime factorization of a number is the number written as a product of its prime factorization.
Remember!
Course 1
5-5 Least Common Multiple
Additional Example 2C: Using Multiples to Find the LCM
Find the least common multiple (LCM).
Method 3: Use prime factorization. C. 6 and 20
6 = 2 • 3 20 = 2 • 2 • 5
Write the prime factorization of each number in exponential form.
Line up the common factors. 2 • 3 • 2 • 5
2 • 3 • 2 • 5 = 60 To find the LCM, multiply one number from each column.
LCM: 60
Course 1
5-5 Least Common Multiple
Additional Example 2D: Using Multiples to Find the LCM
Find the least common multiple (LCM).
D. 15, 6, and 4
15 = 3 • 5
4 = 2
Write the prime factorization of each number in exponential form.
2
3 • 5 • 2 2
3 • 5 • 2 = 60 2
To find the LCM, multiply each prime factor once with the greatest exponent used in any of the prime factorizations.
6 = 3 • 2
LCM: 60
Course 1
5-5 Least Common Multiple
Try This: Example 2A Find the least common multiple (LCM).
Method 1: Use a number line. A. 2 and 3
Use a number line to skip count by 2 and 3.
0 1 2 3 4 5 6
The least common multiple (LCM) of 2 and 3 is 6.
Course 1
5-5 Least Common Multiple
Try This: Example 2B Find the least common multiple (LCM).
Method 2: Use a list. B. 3, 4, and 9
3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, . . .
4: 4, 8, 12, 16, 20, 24, 28, 32, 36, …
9: 9, 18, 27, 36, 45, . . .
The least common multiple of 3, 4, and 9 is 36.
List multiples of 3, 4, and 9.
Find the smallest number that is in all the lists.
Course 1
5-5 Least Common Multiple
Try This: Example 2C Find the least common multiple (LCM).
Method 3: Use prime factorization. C. 4 and 10
4 = 2 • 2 10 = 2 • 5
Write the prime factorization of each number in exponential form.
Line up the common factors. 2 • 2 • 5
2 • 2 • 5 = 20 To find the LCM, multiply one number from each column.
LCM: 20
Course 1
5-5 Least Common Multiple
Try This: Example 2D Find the least common multiple (LCM).
D. 12, 6, and 8
12 = 22 • 3
6 = 2 • 3 Write the prime factorization of each number in exponential form.
23 • 3
23 • 3 = 24
To find the LCM, multiply each prime factor once with the greatest exponent used in any of the prime factorizations.
8 = 23
LCM: 24
Course 1
5-5 Least Common Multiple
Lesson Quiz Find the least common multiple (LCM).
1. 6, 14 2. 9, 12
3. 5, 6, 10 4. 12, 16, 24, 36
5. Two students in Mrs. Albring’s preschool class are stacking blocks, one on top of the other. Reece’s blocks are 4 cm high and Maddy’s blocks are 9 cm high. How tall will their stacks be when they are the same height for the first time?
36 cm
42 36
30 144
5-6 Estimating Fraction Sums and Differences
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Write each sum or difference in simplest form.
1. + 2. – 3. 2 + 1 4. 5 –1
Course 1
5-6 Estimating Fraction Sums and Differences
3 8 __ 5
8 __ 9
10 __ 3
10 __
1 4 __ 1
4 __ 8
9 __ 2
9 __
1 3 5 __
1 2 __ 3
2 3 __ 4
Problem of the Day
Students at a school dance formed equal teams to play a game. When they formed teams of 3, 4, 5, or 6, there was always one person left. What is the smallest number of people that there could have been at the dance? 61, the LCM of all the numbers, plus 1
Course 1
5-6 Estimating Fraction Sums and Differences
Learn to estimate sums and differences of fractions and mixed numbers.
Course 1
5-6 Estimating Fraction Sums and Differences
You can estimate fractions by rounding to 0, , or 1. 1 2 __
1
5 3 1 1
3
7
1 2 __
8 __
8 __
4 __
8 __
4 __
8 __
0 •
The fraction rounds to 1. 3 4 __
Course 1
5-6 Estimating Fraction Sums and Differences
closer to 0
Each numerator is much less than half the denominator, so the fractions are close to 0.
closer to 1
Each numerator is about the same as the denominator, so the fractions are close to 1.
• •
1 2 __
0 1
1 5 __ 2
11 __ 2
15 __ 4
7 __ 5
11 __ 6
7 __ 9
10 __ 16
19 __ 9
20 __
1 2 __
You can round fractions by comparing the numerator and denominator.
Course 1
5-6 Estimating Fraction Sums and Differences
closer to
Each numerator is about half the denominator, so the fractions are close to .
1 2 __
Additional Example 1A: Estimating Fractions Estimate each sum or difference by rounding to 0, , or 1.
A. +
1 2
__
6 7
__ 3 8
__
+ 6 7 __ 3
8 __
1 + = 1 1 2 __
6 7 __ 3
8 __ + is about 1 .
1 2 __
Think: rounds to 1 and rounds to . 6 7 __ 3
8 __ 1
2 __
Course 1
5-6 Estimating Fraction Sums and Differences
1 2 __
Additional Example 1B: Estimating Fractions Estimate each sum or difference by rounding to 0, , or 1.
B. –
1 2
__
9 10 __ 7
8 __
– 9 10 __ 7
8 __
1 – 1 = 0
9 10 __ 7
8 __ – is about 0.
Think: rounds to 1 and rounds to 1. 9 10 __ 7
8 __
Course 1
5-6 Estimating Fraction Sums and Differences
Try This: Example 1A Estimate each sum or difference by rounding to 0, , or 1.
A. +
1 2
__
5 6
__ 3 7
__
+ 5 6 __ 3
7 __
1 + = 1 1 2 __
5 6 __ 3
7 __ + is about 1 .
1 2 __
Think: rounds to 1 and rounds to . 5 6 __ 3
7 __ 1
2 __
Course 1
5-6 Estimating Fraction Sums and Differences
1 2 __
Try This: Example 1B Estimate each sum or difference by rounding to 0, , or 1.
B. –
1 2
__
13 19 __ 2
11 __
1 – 0 = 1
13 19 __ 2
11 __ – is about 1.
Think: rounds to 1 and rounds to 0. 13 19 __ 2
11 __
Course 1
5-6 Estimating Fraction Sums and Differences
– 2 11 __ 13
19 __
You can also estimate by rounding mixed numbers. You compare each mixed number to the
two nearest whole numbers and the nearest .
Does 10 round to 10, 10 , or 11?
1 2 __
3 10 __ 1
2 __
10 10 10 10 10 10 10 6 10 __ 2
10 __ 1
10 __ 4
10 __ 7
10 __ 9
10 __ 8
10 __
10 10 10 11 3 10 __ 1
2 __
•
The mixed number 10 rounds to 10 . 3 10 __ 1
2 __
Course 1
5-6 Estimating Fraction Sums and Differences
Additional Example 2: Sports Application The table shows the distances Tosha walked.
A. About how far did Tosha
walk on Tuesday and Thursday? Day
Tuesday
Thursday
Saturday
Sunday
Tosha’s Walking Distances
Distance (mi)
5
4
8
6
1 10 __
7 8 __
3 7 __
9 10 __
1 10 __ 7
8 __ 5 + 4
5 + 5 = 10
She walked about 10 miles on Tuesday and Thursday.
Course 1
5-6 Estimating Fraction Sums and Differences
Additional Example 2: Sports Application The table shows the distances Tosha walked.
B. About how much farther did
Tosha walk on Sunday than
Thursday?
Day
Tuesday
Thursday
Saturday
Sunday
Tosha’s Walking Distances
Distance (mi)
5
4
8
6
1 10 __
7 8 __
3 7 __
9 10 __
9 10 __ 7
8 __ 8 – 4
9 – 5 = 4
She walked about 4 miles farther on Sunday than on Thursday.
Course 1
5-6 Estimating Fraction Sums and Differences
Additional Example 2: Sports Application
The table shows the distances Tosha walked.
C. Estimate the total distance
Tosha walked on Thursday,
Saturday, and Sunday.
1 2 __
7 8 __ 4 + 6 + 8
She walked about 20 miles on Thursday, Saturday, and Sunday.
9 10 __ 3
7 __
1 2 __ 5 + 6 + 9 = 20
1 2 __
Course 1
5-6 Estimating Fraction Sums and Differences
Day
Tuesday
Thursday
Saturday
Sunday
Tosha’s Walking Distances
Distance (mi)
5
4
8
6
1 10 __
7 8 __
3 7 __
9 10 __
Try This: Example 2A The table shows the distances Jerry roller skated.
A. About how far did Jerry
skate on Tuesday and Sunday?
1 5 __ 6
7 __ 3 + 2
3 + 3 = 6
He roller skated about 6 miles on Tuesday and Sunday.
Course 1
5-6 Estimating Fraction Sums and Differences
Jerry’s Roller Skating Distances
Distance (mi)
3
6
2
8
1 5 __
3 7 __
1 7 __
6 7 __
Day
Tuesday
Thursday
Saturday
Sunday
Try This: Example 2B The table shows the distances Jerry roller skated.
B. About how much farther did
Jerry skate on Saturday than
Tuesday? 1 7 __ 1
5 __ 8 – 3
8 – 3 = 5
He roller skated about 5 miles farther on Saturday than Thursday.
Course 1
5-6 Estimating Fraction Sums and Differences
Jerry’s Roller Skating Distances
Distance (mi)
3
6
2
8
1 5 __
3 7 __
1 7 __
6 7 __
Day
Tuesday
Thursday
Saturday
Sunday
Try This: Example 2C
The table shows the distances Jerry roller skated.
C. Estimate the total distance
Tosha walked on Tuesday,
Thursday, and Sunday.
1 2 __
He roller skated about 12 miles on Tuesday, Thursday, and Sunday.
1 5 __ 3 + 6 + 2 6
7 __ 3
7 __
1 2 __ 3 + 6 + 3 = 12
1 2 __
Course 1
5-6 Estimating Fraction Sums and Differences
Jerry’s Roller Skating Distances
Distance (mi)
3
6
2
8
1 5 __
3 7 __
1 7 __
6 7 __
Day
Tuesday
Thursday
Saturday
Sunday
Lesson Quiz: Part 1
1. – 2. +
3. – 4. +
Insert Lesson Title Here
9 10 __ 2
5 __ 3
8 __ 8
9 __
10 11 __ 8
9 __ 1
4 __ 8
15 __
1 2 __ 1
2 __ 1
0 1
Course 1
5-6 Estimating Fraction Sums and Differences
Estimate each sum or difference by rounding to 0, , or 1.
1 2
__
Lesson Quiz: Part 2
Insert Lesson Title Here
Week Pounds Picked Up
1 18
2 16
3 20
1 2 __
1 3 __
9 10 __
5. The conservation club picked up trash along the road for three weeks. The table shows the number of pounds of trash they collected. About how many pounds did they collect in weeks 2 and 3?
37 pounds
Course 1
5-6 Estimating Fraction Sums and Differences
5-7 Adding and Subtracting with Unlike Denominators
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Course 1
5-7 Adding and Subtracting with Unlike Denominators
Warm Up Add. Write each answer in simplest form.
1. + 2. + 3. + 4. +
1 7 __ 1
7 __ 8
15 __ 3
15 __
7 9 __ 2
9 __ 11
20 __ 4
20 __
2 7 __ 11
15 __
1 3 4 __
Problem of the Day
If it takes 12 minutes to cut a pipe into three pieces, how long would it take to cut the pipe into four pieces. 18 minutes
Course 1
5-7 Adding and Subtracting with Unlike Denominators
Learn to add and subtract fractions with unlike denominators.
Course 1
5-7 Adding and Subtracting with Unlike Denominators
Vocabulary least common denominator (LCD)
Insert Lesson Title Here
Course 1
5-7 Adding and Subtracting with Unlike Denominators
The Pacific Ocean covers of Earth’s surface. The Atlantic
Ocean covers of Earth’s surface. To find the fraction of the
Earth’s surface that is covered by both oceans, you can add
and , which are unlike fractions.
1 3 __
1 3 __
1 5 __
1 5 __
Course 1
5-7 Adding and Subtracting with Unlike Denominators
Fractions that represent the same value are equivalent.
Remember!
Course 1
5-7 Adding and Subtracting with Unlike Denominators
To add or subtract unlike fractions, first rewrite them as equivalent fractions with a common denominator.
Additional Example 1: Cooking Application Mark made a pizza with pepperoni covering of the
pizza and onions covering another . What fraction of
the pizza is covered by pepperoni or onions?
1 4
__
1 3
__
1
__
1 4 __
1 3 __
_____ + Find a common denominator for 4 and 3.
Course 1
5-7 Adding and Subtracting with Unlike Denominators
?
1 4
__ 1 3
__ 1 4
__ 1 3
__ Add +
Additional Example 1 Continued
Write equivalent fractions with 12 as the common denominator.
1 4 __
1 3 __
_______ +
3 12 __
4 12 __
_____ Add the numerators. Keep the common denominator.
1 4
__ 1 3
__ 7 12 __
1 12
1 12
1 12
1 12
1 12
1 12
1 12
The pepperoni or onions cover of the pizza. 7 12 __
Course 1
5-7 Adding and Subtracting with Unlike Denominators
Try This: Example 1 Tori made a pizza with peppers covering of the
pizza and ham covering another . What
fraction of the pizza is covered by ham or peppers?
1 3
__
1 2
__
1 3
__ 1 2
__
? 1 3 __
1 2 __
_____ + Find a common denominator for 3 and 2.
Course 1
5-7 Adding and Subtracting with Unlike Denominators
1 3
__ 1 2
__ Add +
Try This: Example 1 Continued
Write equivalent fractions with 6 as the common denominator.
1 3 __
1 2 __
_______ +
2 6 __
3 6 __
_____ Add the numerators. Keep the common denominator.
5 6 __
1 3
__ 1 2
__
1 6
1 6
1 6
1 6
1 6
The peppers or ham cover of the pizza. 5 6 __
Course 1
5-7 Adding and Subtracting with Unlike Denominators
You can use any common denominator or the least common denominator to add and subtract unlike fractions. The least common denominator (LCD) is the least common multiple of the denominators.
Course 1
5-7 Adding and Subtracting with Unlike Denominators
Additional Example 2A: Adding and Subtracting Unlike Fractions
Add or Subtract. Write each answer in simplest form.
Method 1: Multiplying denominators.
A. – 7 10 __ 3
8 __
– 7 10 __ 3
8 __
– 56 80 __ 30
80 __
Multiply the denominators. 10 • 8 = 80
Write equivalent fractions.
Subtract.
Write the answer in simplest form.
26 80 __
13 40 __
Course 1
5-7 Adding and Subtracting with Unlike Denominators
Additional Example 2B: Adding and Subtracting Unlike Fractions
Add or Subtract. Write each answer in simplest form.
Method 2: Use the LCD.
B. – 11 12 __ 3
8 __
– 11 12 __ 3
8 __
– 22 24 __ 9
24 __
The LCD is 24.
Write equivalent fractions.
Subtract. 13 24 __
Course 1
5-7 Adding and Subtracting with Unlike Denominators
Additional Example 2C: Adding and Subtracting Unlike Fractions
Add or Subtract. Write each answer in simplest form.
Method 3: Use mental math.
C. + 5 16 __ 1
8 __
+ 5 16 __ 1
8 __
+ 5 16 __ 2
16 __
Think: 16 is a multiple of 8, so the LCD is 16. Rewrite with a denominator of 16.
Add. 7 16 __
1 8 __
Course 1
5-7 Adding and Subtracting with Unlike Denominators
Additional Example 2D: Adding and Subtracting Unlike Fractions
Add or Subtract. Write each answer in simplest form.
Method 3: Use mental math.
D. – 5 16 __ 1
8 __
– 5 16 __ 1
8 __
– 5 16 __ 2
16 __
Think: 16 is a multiple of 8, so the LCD is 16. Rewrite with a denominator of 16.
Subtract. 3 16 __
Course 1
5-7 Adding and Subtracting with Unlike Denominators
1 8 __
Try This: Example 2A Add or Subtract. Write each answer in simplest form.
Method 1: Multiplying denominators.
A. – 8 10 __ 2
6 __
– 8 10 __ 2
6 __
– 48 60 __ 20
60 __
Multiply the denominators. 10 • 6 = 60
Write equivalent fractions.
Subtract.
Write the answer in simplest form.
28 60 __
7 15 __
Course 1
5-7 Adding and Subtracting with Unlike Denominators
Try This: Example 2B Add or Subtract. Write each answer in simplest form.
Method 2: Use the LCD.
B. – 4 7
__ 1 3
__
– 4 7 __ 1
3 __
– 12 21 __ 7
21 __
The LCD is 21.
Write equivalent fractions.
Subtract. 5 21 __
Course 1
5-7 Adding and Subtracting with Unlike Denominators
Try This: Example 2C Add or Subtract. Write each answer in simplest form.
Method 3: Use mental math.
C. + 3 12 __ 1
6 __
+ 3 12 __ 1
6 __
+ 3 12 __ 2
12 __
Think: 12 is a multiple of 6, so the LCD is 12. Rewrite with a denominator of 12.
Add. 5 12 __
1 6 __
Course 1
5-7 Adding and Subtracting with Unlike Denominators
Try This: Example 2D Add or Subtract. Write each answer in simplest form.
Method 3: Use mental math.
D. – 3 12 __ 1
6 __
– 3 12 __ 1
6 __
– 3 12 __ 2
12 __
Think: 12 is a multiple of 6, so the LCD is 12. Rewrite with a denominator of 12.
Subtract. 1 12 __
1 6 __
Course 1
5-7 Adding and Subtracting with Unlike Denominators
Lesson Quiz Add or Subtract. Write each answer in simplest form.
1. + 2. –
3. – 4. +
5. Bonnie is making oatmeal bars and the recipes calls for cup of brown sugar. If she has cup of brown sugar, how much more does she need?
Insert Lesson Title Here
3 8 __ 1
6 __ 1
2 __ 1
8 __
5 12 __ 1
6 __ 1
6 __ 1
8 __
3 4 __
1 3 __
13 24 __ 3
8 __
1 4 __ 7
24 __
5 12 __ cup
Course 1
5-7 Adding and Subtracting with Unlike Denominators
5-8 Adding and Subtracting Mixed Numbers
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Course 1
5-8 Adding and Subtracting Mixed Numbers
Warm Up Add. Write each answer in simplest form.
1. – 2. + 3. + 4. –
7 12 __ 1
2 __ 3
11 __ 2
3 __
1 6 __ 3
4 __ 11
12 __ 1
3 __
1 12 __ 31
33 __
11 12 __ 7
12 __
Course 1
5-8 Adding and Subtracting Mixed Numbers
Problem of the Day
The sum of every row, column and diagonal is the same. Complete the magic square.
1
3 4 __ 11
12 ___ 1 1 3
__
1 1 6 __
12 1 1 __
5 6 __
2 3 __ 1
4 __ 1
Learn to add and subtract mixed numbers with unlike denominators.
Course 1
5-8 Adding and Subtracting Mixed Numbers
Course 1
5-8 Adding and Subtracting Mixed Numbers
Additional Example 1A: Adding and Subtracting Mixed Numbers
Find the sum. Write the answer in simplest form.
1 8
__ 5 6
__
3 1 8 __ 3 6
48 __ Multiply the denominators.
8 · 6 = 48
+1 5 6 __ + 1 40
48 __
______ ______ Write equivalent fractions with a denominator of 48.
4 = 4 46 48 __ 23
24 __ Add the fractions and then
whole numbers, and simplify.
A. 3 + 1
Course 1
5-8 Adding and Subtracting Mixed Numbers
Additional Example 1B: Adding and Subtracting Mixed Numbers
Find the difference. Write the answer in simplest form.
2 3
__ 1 4
__
5 2 3 __ 5 8
12 __ The LCD of the denominators is 12.
–1 1 4 __ – 1 3
12 __
______ ______ Write equivalent fractions with a denominator of 12.
4 5 12 __ Subtract the fractions and
then the whole numbers.
B. 5 – 1
Course 1
5-8 Adding and Subtracting Mixed Numbers
Additional Example 1C: Adding and Subtracting Mixed Numbers
Find the sum. Write the answer in simplest form.
1 2
__ 4 5
__
2 1 2 __ 2 5
10 __ The LCD of the denominators
is 10.
+4 4 5 __ + 4 8
10 __
______ ______ Write equivalent fractions with a denominator of 10.
6 = 7 13 10 __ 3
10 __ Add the fractions and then
the whole numbers.
6 = 6 + 1 13 10 __ 3
10 __
C. 2 + 4
Course 1
5-8 Adding and Subtracting Mixed Numbers
Additional Example 1D: Adding and Subtracting Mixed Numbers
Find the difference. Write the answer in simplest form.
2 3
__ 2 9
__
6 2 3 __ 6 6
9 __ Think: 9 is a multiple of 3, so 9
is the LCD. –3 2
9 __ – 3 2
9 __
______ ______ Write equivalent fractions with a denominator of 9.
3 4 9 __ Subtract the fractions and
then the whole numbers.
D. 6 – 3
Course 1
5-8 Adding and Subtracting Mixed Numbers
Try This: Example 1A Find the sum. Write the answer in simplest form.
1 4
__ 1 6
__
2 1 4 __ 2 6
24 __ Multiply the denominators.
4 · 6 = 24
+3 1 6 __ + 3 4
24 __
______ ______ Write equivalent fractions with a denominator of 24.
5 = 5 10 24 __ 5
12 __ Add the fractions and then
whole numbers, and simplify.
A. 2 + 3
Course 1
5-8 Adding and Subtracting Mixed Numbers
Try This: Example 1B Find the difference. Write the answer in simplest form.
1 2
__ 3 7
__
4 1 2 __ 4 7
14 __ The LCD of the denominators is 14.
–1 3 7 __ – 1 6
14 __
______ ______ Write equivalent fractions with a denominator of 14.
3 1 14 __ Subtract the fractions and
then the whole numbers.
B. 4 – 1
Course 1
5-8 Adding and Subtracting Mixed Numbers
Try This: Example 1C Find the sum. Write the answer in simplest form.
7 8
__ 2 3
__
2 7 8 __ 2 21
24 __ The LCD of the denominators is 24.
+4 2 3 __ + 4 16
24 __
______ ______ Write equivalent fractions with a denominator of 24.
6 = 7 37 24 __ 13
24 __ Add the fractions and then the
whole numbers. 6 = 6 + 1 37 24 __ 13
24 __
C. 2 + 4
Course 1
5-8 Adding and Subtracting Mixed Numbers
Try This: Example 1D Find the difference. Write the answer in simplest form.
3 4
__ 1 8
__
7 3 4 __ 7 6
8 __ Think: 8 is a multiple of 4, so 8
is the LCD.
–2 1 8 __ – 2 1
8 __
______ ______
Write equivalent fractions with a denominator of 8.
5 5 8 __ Subtract the fractions and
then the whole numbers.
D. 7 – 2
Course 1
5-8 Adding and Subtracting Mixed Numbers
Additional Example 2: Measurement Application The length of Jen’s kitten’s body is 10 inches. Its tail is 5
inches long. What is the total length of its body and tail?
1 4
__ 1 8
__
1 4
__ 1 8
__
10 1 4 __ 10 2
8 __ Find a common denominator. Write
equivalent fractions with the LCD, 8, as the denominator.
+5 1 8 __ + 5 1
8 __
______ ______ Add the fractions and then the whole numbers. 15
3 8 __
The total length of the kitten’s body and tail is 15 inches.
You can use mental math to find an LCD. Think: 8 is a multiple of 4 and 8. Helpful Hint
3 8 __
Add 10 + 5
Course 1
5-8 Adding and Subtracting Mixed Numbers
Try This: Example 2 The length of Regina’s mouse’s body is 2 inches. Its tail is 2
inches long. What is the total length of its body and tail?
2 3
__ 1 6
__
2 3
__ 1 6
__
2 2 3 __ 2 4
6 __ Find a common denominator. Write
equivalent fractions with the LCD, 6, as the denominator.
+2 1 6 __ + 2 1
6 __
______ ______
Add the fractions and then the whole numbers.
4 5 6 __
The total length of the mouse’s body and tail is 4 inches. 5 6 __
Add 2 +2
Lesson Quiz
Find each sum or difference. Write the answer in simplest form.
1. 7 – 4 2. 9 + 7
3. 6 + 1 4. 10 – 4
5. Miles worked 5 hours on Friday and 8 hours on
Saturday. How many total hours did he work?
11 12 __ 2
3 __ 1
6 __ 3
8 __
7 8 __ 3
10 __ 2
5 __ 1
6 __
2 3 __ 3
4 __
3 1 4 __ 16 13
24 __
8 7 40 __ 6 7
30 __
14 hours 5 12 __
Course 1
5-8 Adding and Subtracting Mixed Numbers
5-9 Renaming to Subtract Mixed Numbers
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Course 1
5-9 Renaming to Subtract Mixed Numbers
Warm Up Add or Subtract.
1. 3 + 1 2. 7 + 3. 8 – 2 4. 6 – 1
2 5 __ 3
5 __ 9
11 __ 11
11 __
4 5 __ 1
4 __ 1
3 __ 1
9 __
5 8
6 5
9 11 __
11 20 __ 2
9 __
Problem of the Day
Complete the magic square so that every row, column, and diagonal, has the same sum.
Course 1
5-9 Renaming to Subtract Mixed Numbers
1 2 1 1
1 1 2
7 8 __
5 8 __
5 8 __ 1
2 __
1 2 __ 1 7
8 __
3 4 __ 1 1
8 __ 1
1 4 __
3 8 __
3 8 __ 1
8 __ 3
4 __
1 4 __
2
Learn to rename mixed numbers to subtract.
Course 1
5-9 Renaming to Subtract Mixed Numbers
Course 1
5-9 Renaming to Subtract Mixed Numbers
Jimmy and his family planted a tree when it was 1 ft
tall. Now the tree is 2 ft tall. How much has the tree
grown since it was planted?
3 4 __
1 4 __
1 4 __ 3
4 __ The difference in the heights is 2 – 1 .
1 4 __
3 4 __ 1
4 __
You will need to rename 2 because the fraction
in 1 is greater than .
1 4 __ Divide one whole of 2 into fourths.
Course 1
5-9 Renaming to Subtract Mixed Numbers
2 1 4 __ 1 5
4 __
–1
1 4 __
– 1 ______ ______
= 2 4 __ 1
2 __
3 4 __
Rename 2 as 1 . 3 4 __
5 4 __
The tree has grown ft since it was planted. 1 2 __
1 4 1 1
4 1 4
1 4
1 4
1 1 4
1 4
1 4
1 4
1
1
1 4 1 4
Course 1
5-9 Renaming to Subtract Mixed Numbers
Additional Example 1A: Renaming Mixed Numbers Subtract. Write each answer in simplest form.
1 6
__ 5 6
__
7 1 6 __ 6 7
6 __ Rename 7 as 6 + 1 = 6 + + .
–2 5 6 __ –2 5
6 __
______ ______
4 2 6 __
Subtract the fractions and then the whole numbers.
1 6 __ 1
6 __ 6
6 __ 1
6 __
Write the answer in simplest form. = 4 1 3 __
A. 7 – 2
Course 1
5-9 Renaming to Subtract Mixed Numbers
Additional Example 1B: Renaming Mixed Numbers Subtract. Write each answer in simplest form.
2 5
__ 7 10 __
8 4 10 __ 7 14
10 __
Rename 8 as 7 + 1 = 7 + + . –6 7 10 __ –6 7
10 __
______ ______
1 7 10 __ Subtract the fractions and then the
whole numbers.
4 10 __ 4
10 __ 10
10 __ 4
10 __
10 is a multiple of 5, so 10 is a common denominator.
B. 8 – 6
Course 1
5-9 Renaming to Subtract Mixed Numbers
Additional Example 1C: Renaming Mixed Numbers Subtract. Write each answer in simplest form.
2 3
__
6 5 3 3 __
–3 2 3 __ –3 2
3 __
______ ______
2
Subtract the fractions and then the whole numbers.
Write 6 as a mixed number with a denominator of 3. Rename 6 as 5 + . 3
3 __
1 3 __
C. 6 – 3
Course 1
5-9 Renaming to Subtract Mixed Numbers
Additional Example 1D: Renaming Mixed Numbers Subtract. Write each answer in simplest form.
1 3
__ 3 4
__
5 4 12 __ 4 16
12 __
–2 9 12 __ –2 9
12 __
______ ______
2 7 12 __ Subtract the fractions and then the
whole numbers.
Rename 5 as 4 + 1 = 4 + + . 4 12 __ 4
12 __ 12
12 __ 4
12 __
The LCM of 4 and 3 is 12.
D. 5 – 2
Course 1
5-9 Renaming to Subtract Mixed Numbers
Try This: Example 1A Subtract. Write each answer in simplest form.
1 8
__ 5 8
__
5 1 8 __ 4 9
8 __ Rename 5 as 4 + 1 = 4 + + .
–2 5 8 __ –2 5
8 __
______ ______
2 4 8 __
Subtract the fractions and then the whole numbers.
1 8 __ 1
8 __ 4
4 __ 1
8 __
Write the answer in simplest form. = 2 1 2 __
A. 5 – 2
Course 1
5-9 Renaming to Subtract Mixed Numbers
Try This: Example 1B Subtract. Write each answer in simplest form.
1 6
__ 7 12 __
7 2 12 __ 6 14
12 __
Rename 7 as 6 + 1 = 6 + + . –3 7 12 __ –3 7
12 __
______ ______
3 7 12 __ Subtract the fractions and then the
whole numbers.
2 12 __ 2
12 __ 12
12 __ 2
12 __
12 is a multiple of 6, so 12 is a common denominator.
B. 7 – 3
Course 1
5-9 Renaming to Subtract Mixed Numbers
Try This: Example 1C Subtract. Write each answer in simplest form.
3 4
__
5 4 4 4 __ Write 5 as a mixed number with a
denominator of 4. Rename 5 as 4 + .
–2 3 4 __ –2 3
4 __
______ ______
2
Subtract the fractions and then the whole numbers.
4 4 __
1 4 __
C. 5 – 2
Course 1
5-9 Renaming to Subtract Mixed Numbers
Try This: Example 1D Subtract. Write each answer in simplest form.
1 2
__ 2 3
__
8 3 6 __ 7 9
6 __
Rename 8 as 7 + 1 = 7 + + . –4 4 6 __ –4 4
6 __
______ ______
3 5 6 __ Subtract the fractions and then the
whole numbers.
3 6 __ 3
6 __ 6
6 __ 3
6 __
The LCM of 2 and 3 is 6.
D. 8 – 4
Course 1
5-9 Renaming to Subtract Mixed Numbers
Additional Example 2A: Measurement Application Li is making a quilt. She needs 15 yards of fabric.
A. Li has 2 yards of fabric. How many more yards does she need?
3 4
__
15 14 4 4 __ Write 15 as a mixed number with a
denominator of 4. Rename 15 as 14 + .
–2 3 4 __ –2 3
4 __
______ ______
12
Subtract the fractions and then the whole numbers.
4 4 __
1 4 __
Li needs another 12 yards of fabric. 1 4 __
Course 1
5-9 Renaming to Subtract Mixed Numbers
Additional Example 2B: Measurement Application
B. If Li uses 11 yards of fabric, how much of the 15
yards will she have left?
1 6
__
15 14 6 6 __ Write 15 as a mixed number with a
denominator of 6. Rename 15 as 14 + .
–11 1 6 __ –11
1 6 __
______ ______
3
Subtract the fractions and then the whole numbers.
6 6 __
5 6 __
Li will have 3 yards of fabric. 5 6 __
Course 1
5-9 Renaming to Subtract Mixed Numbers
Try This: Example 2A Peggy is making curtains. She needs 13 feet of fabric.
A. Peggy has 4 yards of fabric. How many more yards does she need?
5 6
__
13 12 6 6 __ Write 13 as a mixed number with a
denominator of 6. Rename 13 as 12 + .
–4 5 6 __ –4 5
6 __
______ ______
8
Subtract the fractions and then the whole numbers.
6 6 __
1 6 __
Peggy needs another 8 yards of fabric. 1 6 __
Course 1
5-9 Renaming to Subtract Mixed Numbers
Try This: Example 2B
B. If Peggy uses 9 yards of fabric, how much of the 13
yards will she have left?
1 4
__
13 12 4 4 __ Write 13 as a mixed number with a
denominator of 4. Rename 13 as 12 + .
–9 1 4 __ –9 1
4 __
______ ______
3
Subtract the fractions and then the whole numbers.
4 4 __
3 4 __
Peggy will have 3 yards of fabric left. 3 4 __
Lesson Quiz
Insert Lesson Title Here
Course 1
5-9 Renaming to Subtract Mixed Numbers
Subtract. Write each answer in simplest form.
1. 7 – 3 2. 8 – 3
3. 7 – 5 4. 10 – 4
5. Irena put 10 cups of food in the bird feeder. The birds ate 3 cups. How many cups are left?
4 7 __ 6
7 __ 7
10 __ 4
5 __
2 9 __ 3
8 __ 9
16 __
3 8 __
3 5 7 __ 4 9
10 __
1 7 9 __
5 13 16 __
6 cups 5 8 __
5-10 Solving Fraction Equations: Addition and Subtraction
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Solve.
1. x – 15 = 9
2. x + 21 = 34
3. 17 = x – 11
4. 22 = x – 34
x = 24 x = 13
x = 28
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
x = 56
Problem of the Day
If a newborn baby weighs two pounds plus three-fourths its own weight, how much does it weigh? 8 pounds
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
Learn to solve equations by adding and subtracting fractions.
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
Additional Example 1A: Solving Equations by Adding and Subtracting
Solve the equation. Write the solution in simplest form.
3 5
__
x + 5 = 14 3 5 __
3 5 __ – 5 – 5 3
5 __
______ ______
x = 13 – 5 3 5 __ 5
5 __
x = 8 2 5 __
Subtract 5 from both sides to undo addition.
3 5 __
Rename 14 as 13 . 5 5 __
Subtract.
A. x + 5 = 14
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
Additional Example 1B: Solving Equations by Adding and Subtracting
Solve the equation. Write the solution in simplest form.
2 9
__
3 = x – 4 2 9 __
1 3 __ + 4 + 4 1
3 __
______ ________
3 + 4 = x 3 9 __ 2
9 __
7 = x 5 9 __
Add 4 to both sides to undo the
subtraction.
1 3 __
Find a common denominator.
Add.
1 3
__
1 3 __
B. 3 = x – 4
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
Additional Example 1C: Solving Equations by Adding and Subtracting
Solve the equation. Write the solution in simplest form.
1 6
__
6 = m + 1 6 __
7 12 __ – – 7
12 __
______ ______
6 – = m 7 12 __ 2
12 __
5 = m 7 12 __
Subtract from both sides to undo the
addition.
7 12 __
Find a common denominator.
Subtract.
7 12 __
7 12 __
5 – = m 7 12 __ 14
12 __ Rename 6 as 5 + .
2 12 __ 12
12 __ 2
12 __
C. 6 = m +
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
Additional Example 1D: Solving Equations by Adding and Subtracting
Solve the equation. Write the solution in simplest form.
4 5
__
w – = 3 4 5 __
4 5 __ + + 4
5 __
_____ _____
w = 3 + 4 5 __ 3
10 __
w = 3 + 3 10 __
Add to both sides to undo addition. 4 5 __
Find a common denominator.
Add.
3 10 __
3 10 __
8 10 __
w = 3
w = 4
11 10 __
1 10 __
3 = 3 + 1 . 11 10 __ 1
10 __
D. w – = 3
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
Try This: Example 1A Solve the equation. Write the solution in simplest form.
5 8
__
x + 3 = 12 5 8 __
5 8 __ – 3 – 3 5
8 __
______ ______
x = 11 – 3 5 8 __ 8
8 __
x = 8 3 8 __
Subtract 3 from both sides to undo addition.
5 8 __
Rename 12 as 11 . 8 8 __
Subtract.
A. x + 3 = 12
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
Try This: Example 1B Solve the equation. Write the solution in simplest form.
3 8
__
2 = x – 3 3 8 __
1 4 __ + 3 + 3 1
4 __
______ ______
2 + 3 = x 2 8 __ 3
8 __
5 = x 5 8 __
Add 3 to both sides to undo the
subtraction.
1 4 __
Find a common denominator.
Add.
1 4
__
1 4 __
B. 2 = x – 3
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
Try This: Example 1C Solve the equation. Write the solution in simplest form.
2 5
__
5 = m + 2 5 __
7 10 __ – – 7
10 __
______ ______
5 – = m 7 10 __ 4
10 __
4 = m 7 10 __
Subtract from both sides to undo the
addition.
7 10 __
Find a common denominator.
Subtract.
7 10 __
7 10 __
4 - = m 7 10 __ 14
10 __ Rename 5 as 4 + .
4 10 __ 10
10 __ 4
10 __
C. 5 = m +
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
Try This: Example 1D Solve the equation. Write the solution in simplest form.
1 3
__
w – = 5 1 3 __
1 3 __ + + 1
3 __
_____ _____
w = 5 + 1 3 __ 5
6 __
w = 5 + 5 6 __
Add to both sides to undo addition. 1 3 __
Find a common denominator.
Add.
5 6
__
5 6 __
2 6 __
w = 5
w = 6
7 6 __
1 6 __
5 = 5 + 1 . 7 6 __ 1
6 __
D. w – = 5
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
Additional Example 2: Application 1 4
__ 1 2
__
d – 17 = 85 1 2 __
1 2 __ + 17 + 17
1 2 __
________ ________
1 4 __
d = 102 3 4 __
Let d represent the weight of Ian’s dog.
Add 17 to both sides to undo the
subtraction.
1 2 __
Linda’s dog weighs 85 pounds. If Linda’s dog weighs 17 pounds less than Ian’s dog, how much does Ian’s dog weigh?
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
Additional Example 2 Continued
Check
d – 17 = 85 1 2 __ 1
4 __
3 4 __ 102 – 17 = 85
1 2 __ 1
4 __ ?
3 4 __ 102 – 17 = 85
2 4 __ 1
4 __ ?
85 = 85 ? 1 4 __ 1
4 __
Substitute 102 for d. 3 4 __
Find a common denominator.
102 is the solution. 3 4 __
Ian’s dog weighs 102 pounds. 3 4 __
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
Try This: Example 2 2 3
__
1 6
__
c – 4 = 13 1 6 __
1 6 __ + 4 + 4
1 6 __
_______ ________
2 3 __
c = 17 5 6 __
Let c represent the weight of Vicki’s cat.
Add 4 to both sides to undo the
subtraction.
1 6 __
Jimmy’s cat weighs 13 pounds. If Jimmy’s cat weighs
4 pounds less than Vicki’s cat, how much does Vicki’s
cat weigh?
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
Try This: Example 2
Check
c – 4 = 13 1 6 __ 2
3 __
5 6 __ 17 – 4 = 13
1 6 __ 2
3 __ ?
5 6 __ 17 – 4 = 13
1 6 __ 4
6 __ ?
13 = 13 ? 4 6 __ 4
6 __
Substitute 17 for c. 5 6 __
Find a common denominator.
17 is the solution. 5 6 __
Vicki’s cat weighs 17 pounds. 5 6 __
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
Solve each equation. Write the solution in simplest form.
1. x – 3 = 2 2. x + 6 = 11
3. 1 = x – 2 4. 5 = x –
5. Marco needs to work 40 hours per week. He has worked 31 hours so far this week. How many hours does he need to work on Friday to meet his 40-hour requirement?
Lesson Quiz
3 4 __ 5
12 __ x = 6 1
6 __ 5
8 __
7 10 __ 1
2 __ 3
4 __
5 8 __
x = 4 3 8 __
x = 4 1 2 __ x = 6 1
4 __
8 hours 3 8 __
4 5 __
Course 1
5-10 Solving Fraction Equations: Addition and Subtraction
6-1 Make a Table
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Write the values in simplest form. 1. + 2. – 3. ÷ 4. 5 · 2
Course 1
6-1 Make a Table
13 58
58
13
78
14
56
5 12
23 24
5 24 5 8 2
1 8 13
Problem of the Day
If February 1 falls on a Tuesday, then March 1 falls on what day of the week? Tuesday or Wednesday, depending on whether or not it is a leap year.
Course 1
6-1 Make a Table
Learn to use tables to record and organize data.
Course 1
6-1 Make a Table
Course 1
6-1 Make a Table
Additional Example 1: Application Use the audience data to make a table. Then use your table to describe how attendance has changed over time.
Date People in Audience May 1 May 2 May 3
From the table you can see that the number of people in the audience increased from May 1 to May 3.
Make a table. Write the dates in order so that you can see how the attendance changed over time.
On May 1, there were 275 people in the audience at the school play. On May 2, there were 302 people. On May 3 there were 322 people.
275 302 322
Course 1
6-1 Make a Table
Try This: Example 1 Use the audience data to make a table. Then use your table to describe how attendance has changed over time.
Date People in Audience April 1 May 1 June 1
From the table you can see that the number of people in the audience decreased from April 1 to June 1.
Make a table. Write the dates in order so that you can see how the attendance changed over time.
On April 1, there were 212 people at the symphony. On May 1, there were 189 people. On June 1 there were 172 people.
212 189 172
Course 1
6-1 Make a Table
Additional Example 2: Organizing Data in a Table Use the temperature data to make a table. Then use your table to find a pattern in the data and draw a conclusion.
At 3 A.M., the temperature was 53°F. At 5 A.M., it was 52°F. At 7 A.M., it was 50°F. At 9 A.M., it was 53°F. At 11 A.M., it was 57°F.
53
50 52
Time Temperature (°F) 3 A.M.
5 A.M.
7 A.M.
9 A.M.
11 A.M. 53 57
The temperature dropped until 7 A.M., then it rose. One conclusion is that the low temperature on this day was 50°F.
Course 1
6-1 Make a Table
Try This: Example 2 Use the temperature data to make a table. Then use your table to find a pattern in the data and draw a conclusion.
The temperature dropped until 6 A.M., then it rose. One conclusion is that the low temperature on this day was 44° F.
At 2 A.M., the temperature was 48°F. At 4 A.M., it was 46°F. At 6 A.M., it was 44°F. At 8 A.M., it was 47°F. At 10 A.M., it was 51°F.
48
44 46
Time Temperature (°F) 2 A.M.
4 A.M.
6 A.M.
8 A.M.
10 A.M.
47 51
Lesson Quiz: Part 1
1. Humans have the following approximate heart rates at the ages given: newborn, 135 beats per minute (bpm); 2 years old, 110 bpm; 6 years old, 95 bpm; 10 years old, 87 bpm; 20 years old, 71 bpm; 40 years old, 72 bpm; and 60 years old, 74 bpm. Use this data to make a table.
Insert Lesson Title Here
Course 1
6-1 Make a Table
Age Heart rate newborn 135 bpm
2 110 bpm 6 95 bpm 10 87 bpm 20 71 bpm 40 72 bpm 60 74 bpm
Lesson Quiz: Part 2
2. Use the data from problem 1 to estimate how many times per minute an 8-year-old’s heart beats. 91
Insert Lesson Title Here
Course 1
6-1 Make a Table
Age Heart rate newborn 135 bpm
2 110 bpm 6 95 bpm 10 87 bpm 20 71 bpm 40 72 bpm 60 74 bpm
6-2 Range, Mean, Median, and Mode
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Order the numbers from least to greatest. 1. 8, 6, 25, 7, 4, 12 2. 60, 11, 27, 45, 32 Divide 3. 720 ÷ 4 4. 760 ÷ 10
4, 6, 7, 8, 12, 25
11, 27, 32, 45, 60
180
Course 1
6-2 Range, Mean, Median, and Mode
76
Problem of the Day
Ms. Red, Ms. Blue, and Ms. Green attended a holiday party. They each wore a different-colored dress, red, blue, or green. Ms. Red said to the lady wearing the blue dress, “Did you notice that none of us is wearing a dress with the color that corresponds to our name?” Who wore which color dress? Ms. Red: green; Ms. Blue: red; Ms. Green: blue
Course 1
6-2 Range, Mean, Median, and Mode
Learn to find the range, mean, median, and mode of a data set.
Course 1
6-2 Range, Mean, Median, and Mode
Vocabulary range mean median mode
Insert Lesson Title Here
Course 1
6-2 Range, Mean, Median, and Mode
Course 1
6-2 Range, Mean, Median, and Mode Some descriptions of a set of data are called the range, mean, median, and mode.
•The range is the difference between the least and greatest values in the set.
•The mean is the sum of all the items, divided by the number of items in the set. (The mean is sometimes called the average.)
•The median is the middle value when the data are in numerical order, or the mean of the two middle values if there are an even number of items.
•The mode is the value or values that occur most often. There may be more than one mode for a data set. When all values occur an equal number of times, the data set has no mode.
Course 1
6-2 Range, Mean, Median, and Mode
Additional Example 1A: Finding the Range, Mean, Median, and Mode of a Data Set
Depths of Puddles (in.) 5 8 3 5 4 2 1
Start by writing the data in numerical order. 1, 2, 3, 4, 5, 5, 8 range: 8 – 1 = 7
mean: 1 + 2 + 3 + 4 + 5 + 5 + 8 = 28
28 ÷ 7 = 4 median: 4 mode: 5
The range is 7 in.; the mean is 4 in.; the median is 4 in.; and the mode is 5 in.
Subtract least value from greatest value.
Add all values. Divide the sum by the number of items.
There are an odd number of items, find the middle value. 5 occurs most often.
Find the range, mean, median, and mode of each data set.
A.
Course 1
6-2 Range, Mean, Median, and Mode
Additional Example 1B: Finding the Range, Mean, Median, and Mode of a Data Set
Number of Points Scored 96 75 84 7
Write the data in numerical order. 7, 75, 84, 96 range: 96 – 7 = 89 mean: 96 + 75 + 84 + 7
4 = 65.5
median: 7, 75, 84, 96
74 + 84 2
= 79.5
mode: none
The range is 89 points; the mean is 65.5 points; the median is 79.5 points; and there is no mode.
There are an even number of items, so find the mean of the two middle values.
B.
Course 1
6-2 Range, Mean, Median, and Mode
Try This: Example 1A Find the range, mean, median, and mode of each data set.
A. Rainfall per month (in.) 1 2 10 2 5 6 9
Start by writing the data in numerical order. 1, 2, 2, 5, 6, 9, 10 range: 10 – 1 = 9
mean: 1 + 2 + 2 + 5 + 6 + 9 + 10 = 35
35 ÷ 7 = 5 median: 5 mode: 2
The range is 9 in.; the mean is 5 in.; the median is 5 in.; and the mode is 2 in.
Subtract least value from greatest value.
Add all values.
Divide the sum by the number of items. There are an odd number of items, find the middle value. 2 occurs most often.
Course 1
6-2 Range, Mean, Median, and Mode
Try This: Example 1B
B. Number of Points Scored 53 26 47 12
Write the data in numerical order. 12, 26, 47, 53 range: 53 – 12 = 41 mean: 53 + 26 + 47 + 12
4 = 34.5
median: 12, 26, 47, 54
26 + 47 2
= 36.5
mode: none
The range is 41 points; the mean is 34.5 points; the median is 36.5 points; and there is no mode.
There are an even number of items, so find the mean of the two middle values.
Lesson Quiz
Use the following data set: 18, 20, 56, 47, 30, 18, 21.
1. Find the range. 2. Find the mean.
3. Find the median. 4. Find the mode.
5. Bonnie ran a mile in 8 minutes, 8 minutes, 7 minutes, 9 minutes, and 8 minutes. What was her mean time?
38
Insert Lesson Title Here
Course 1
6-2 Range, Mean, Median, and Mode
30
21 18
8 minutes
6-3 Additional Data and Outliers
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Use the numbers to answer the questions. 146, 161, 114, 178, 150, 134, 172, 131, 128 1. What is the greatest number? 2. What is the least number? 3. How can you find the median?
178
114
Order the numbers and find the middle value.
Course 1
6-3 Additional Data and Outliers
Problem of the Day
Ms. Green has 6 red gloves and 10 blue gloves in a box. She closes her eyes and picks some gloves. What is the least number of gloves Ms. Green will have to pick to ensure 2 gloves of the same color? 3
Course 1
6-3 Additional Data and Outliers
Learn the effect of additional data and outliers.
Course 1
6-3 Additional Data and Outliers
Vocabulary outlier
Insert Lesson Title Here
Course 1
6-3 Additional Data and Outliers
Course 1
6-3 Additional Data and Outliers
The mean, median, and mode may change when you add data to a data set.
Course 1
6-3 Additional Data and Outliers
Additional Example 1A & 1B: Sports Application A. Find the mean, median, and mode of the data in the table.
EMS Football Games Won Year 1998 1999 2000 2001 2002
Games 11 5 7 5 7
mean = 7 modes = 5, 7 median = 7
B. EMS also won 13 games in 1997 and 8 games in 1996. Add this data to the data in the table and find the mean, median, and mode.
mean = 8 modes = 5,7 median = 7
The mean increased by 1, the modes remained the same, and the median remained the same.
Course 1
6-3 Additional Data and Outliers
Try This: Example 1A & 1B A. Find the mean, median, and mode of the data in the table.
MA Basketball Games Won Year 1998 1999 2000 2001 2002
Games 13 6 4 6 11
mean = 8 mode = 6 median = 6
B. MA also won 15 games in 1997 and 8 games in 1996. Add this data to the data in the table and find the mean, median, and mode.
mean = 9 mode = 6 median = 8
The mean increased by 1, the mode remained the same, and the median increased by 2.
Course 1
6-3 Additional Data and Outliers
An outlier is a value in a set that is very different from the other values.
Course 1
6-3 Additional Data and Outliers
Additional Example 2: Application Ms. Gray is 25 years old. She took a class with students who were 55, 52, 59, 61, 63, and 58 years old. Find the mean, median, and mode with and without Ms. Gray’s age.
mean ≈ 53.3 no mode median = 58
mean = 58 no mode median = 58.5
When you add Ms. Gray’s age, the mean decreases by about 4.7, the mode stays the same, and the median decreases by 0.5. The mean is the most affected by the outlier. The median is closer to most of the students’ ages.
Data with Ms. Gray’s age:
Data without Ms. Gray’s age:
Course 1
6-3 Additional Data and Outliers
Try This: Example 2 Ms. Pink is 56 years old. She volunteered to work with people who were 25, 22, 27, 24, 26, and 23 years old. Find the mean, median, and mode with and without Ms. Pink’s age.
mean = 29 no mode median = 25
mean = 24.5 no mode median = 24.5
When you add Ms. Gray’s age, the mean increases by 4.5, the mode stays the same, and the median increases by 0.5. The mean is the most affected by the outlier. The median is closer to most of the students’ ages.
Data with Ms. Pink’s age:
Data without Ms. Gray’s age:
Course 1
6-3 Additional Data and Outliers
Additional Example 3: Describing a Data Set The Yorks are shopping for skates. They found 8 pairs of skates with the following prices:
$35, $42, $75, $40, $47, $34, $45, $40
What are the mean, median, and mode of this data set? Which statistic best describes the data set?
mean = $44.75 mode = $40 median = $41
The median price is the best description of the prices. Most of the skates cost about $41.
The mean is higher than most of the prices because of the $75 skates.
Course 1
6-3 Additional Data and Outliers
Try This: Example 3 The Oswalds are shopping for gloves. They found 8 pairs of gloves with the following prices:
$17, $15, $39, $12, $13, $16, $19, $15
What are the mean, median, and mode of this data set? Which statistic best describes the data set?
mean = $18.25 mode = $15 median = $15.50
The median price is the best description of the prices. Most of the gloves cost about $15.50.
The mean is higher than most of the prices because of the $39 gloves.
Course 1
6-3 Additional Data and Outliers
Some data sets do not contain numbers. For example, the circle graph shows the result of a survey to find people’s favorite color.
When it does not contain numbers the only way to describe the data set is with the mode. You cannot find a mean or a median for a set of colors.
The mode for this data set is blue. Most people in this survey chose blue as their favorite color.
Lesson Quiz
At the college bookstore, your brother buys 6 textbooks at the following prices: $21, $58, $68, $125, $36, and $140.
1. Find the mean.
2. Find the median.
3. Find the mode.
4. Your brother signs up for an additional class, and the textbook costs $225. Recalculate the mean, including the extra book.
$63
$74.67
Insert Lesson Title Here
none
$96.14
Course 1
6-3 Additional Data and Outliers
6-4 Bar Graphs
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Use the following data set. 45 55 58 63 63 37 76 46 34
1. What is the mean of the data? 2. What is the median of the data? 3. What is the mode of the data?
53 55
63
Course 1
6-4 Bar Graphs
Problem of the Day
The distance around the bases is 4 × 90 feet. How many runs does a baseball team need to score before the scoring base runners have covered a mile? (1 mile = 5,280 feet) 15 runs
Course 1
6-4 Bar Graphs
Learn to display and analyze data in bar graphs.
Course 1
6-4 Bar Graphs
Vocabulary bar graph double-bar graph
Insert Lesson Title Here
Course 1
6-4 Bar Graphs
Course 1
6-4 Bar Graphs
A bar graph can be used to display and compare data. A bar graph displays data with vertical or horizontal bars.
Course 1
6-4 Bar Graphs
Additional Example 1A: Reading a Bar Graph
Use the bar graph to answer each question. A. Which biome in the
graph has the lowest average summer temperature?
Find the lowest bar.
The coniferous forest has the least average summer temperature.
Course 1
6-4 Bar Graphs
Additional Example 1B: Reading a Bar Graph
Use the bar graph to answer each question. B. Which biomes in the
graph have an average summer temperature of 30°C or greater? Find the bar or bars whose heights measure 30 or more than 30. The grassland and the rain forest have average summer temperatures of 30°C or greater.
Course 1
6-4 Bar Graphs
Try This: Example 1A
Use the bar graph to answer each question.
A. Which biome in the graph has the highest average summer temperature? Find the highest bar.
The rain forest has the highest average summer temperature.
Course 1
6-4 Bar Graphs
Try This: Example 1B
Use the bar graph to answer each question.
B. Which biomes in the graph have an average summer temperature of 25°C or greater? Find the bar or bars whose heights measure 25 or more than 25. The deciduous forest, the grassland, and the rain forest have average summer temperatures of 25°C or greater.
Course 1
6-4 Bar Graphs
Additional Example 2: Making a Bar Graph
Use the given data to make a bar graph. Magazine Subscriptions Sold Grade 6 Grade 7 Grade 8
258 597 374
Step 1:Find an appropriate scale and interval. The scale must include all of the data values. The interval separates the scale into equal parts.
Step 2:Use the data to determine the lengths of the bars. Draw bars of equal width. The bars cannot touch.
Step 3: Title the graph and label the axes.
Course 1
6-4 Bar Graphs
Try This: Example 2
Use the given data to make a bar graph. Tickets Sold
Grade 6 Grade 7 Grade 8 310 215 285
Step 1:Find an appropriate scale and interval. The scale must include all of the data values. The interval separates the scale into equal parts.
Step 2:Use the data to determine the lengths of the bars. Draw bars of equal width. The bars cannot touch.
Step 3: Title the graph and label the axes.
Grade 6
Grade 7
Grade 8
0
50
100
150
200
250
300
350Tickets Sold
Grade
Tick
ets
Course 1
6-4 Bar Graphs
A double-bar graph shows two sets of related data.
Course 1
6-4 Bar Graphs Additional Example 3: Problem Solving Application
Make a double-bar graph to compare the data in the table. Club Memberships
Club Art Music Science Boys 12 6 16 Girls 8 14 4
1 Understand the Problem You are asked to use a graph to compare the data given in the table. You will need to use all of the information given.
2 Make a Plan You can make a double-bar graph to display the two sets of data.
Course 1
6-4 Bar Graphs
Additional Example 3 Continued
Solve 3 Determine appropriate scales for both sets of data.
Use the data to determine the lengths of the bars. Draw bars of equal width. Bars should be in pairs. Use a different color for boy memberships and girl memberships. Title the graph and label both axes.
Include a key to show what each bar represents.
Look Back 4 You could make two separate graphs, one of boy memberships and one of girl memberships. However, it is easier to compare the two data sets when they are on the same graph.
Course 1
6-4 Bar Graphs Try This: Example 3
Make a double-bar graph to compare the data in the table.
Club Memberships Club Band Chess Year Book Boys 9 14 16 Girls 11 7 15
1 Understand the Problem You are asked to use a graph to compare the data given in the table. You will need to use all of the information given.
2 Make a Plan You can make a double-bar graph to display the two sets of data.
Course 1
6-4 Bar Graphs
Try This: Example 3 Continued
Solve 3 Determine appropriate scales for both sets of data.
Use the data to determine the lengths of the bars. Draw bars of equal width. Bars should be in pairs. Use a different color for boy memberships and girl memberships. Title the graph and label both axes.
Include a key to show what each bar represents.
Look Back 4 You could make two separate graphs, one of boy memberships and one of girl memberships. However, it is easier to compare the two data sets when they are on the same graph.
02468
10121416
Chess Band Year Book
BoysGirls
Club
Club Memberships
Mem
bers
hips
Lesson Quiz: Part 1
Use the bar graph to answer each question.
1. Which animal was least popular among students?
2. Which pet was more popular to twice as many students as rabbits were?
dog
bird
Insert Lesson Title Here
Course 1
6-4 Bar Graphs
Student Pet Survey
Lesson Quiz: Part 2
3. Make a bar graph of this data.
Insert Lesson Title Here
Course 1
6-4 Bar Graphs
Number of Daily Servings
Grains = 6
Fruit = 2
Meat = 2
Milk = 3
Vegetables = 3
Number of Daily Servings
01234567
Grains
Fruit
Meat
Milk
Vegeta
bles
Daily
Ser
ving
s
GrainsFruitMeatMilkVegetables
6-5 Frequency Tables and Histograms
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Create a bar graph of the data.
Course 1
6-5 Frequency Tables and Histograms
Favorite rides at fair: Ferris wheel = 5, loop the loop = 4, merry-go-round = 3, bumper cars = 7, sit and spin = 9
Favorite Rides at Fair
0
2
4
6
8
10
Ferris Wheel Loop the loop Merry-go-round
Bumper cars Sit and spin
Problem of the Day
A set of 7 numbers has a mean of 36, a median of 37, a mode of 37, and a range of 6. What could the 7 numbers be?
Possible answer: 33, 33, 36, 37, 37, 37, 39
Course 1
6-5 Frequency Tables and Histograms
Learn to organize data in frequency tables and histograms.
Course 1
6-5 Frequency Tables and Histograms
Vocabulary frequency table cumulative frequency histogram
Insert Lesson Title Here
Course 1
6-5 Frequency Tables and Histograms
Course 1
6-5 Frequency Tables and Histograms Additional Example 1: Making a Tally Table
Students in Mr. Ray’s class recorded their fingerprint patterns. Which type of pattern do most students in Mr. Ray’s class have?
whorl loop whorl loop
arch arch loop whorl
loop arch whorl arch
arch whorl arch loop
Make a tally table to organize the data.
Course 1
6-5 Frequency Tables and Histograms Additional Example 1 Continued
Students in Mr. Ray’s class recorded their fingerprint patterns. Which type of pattern do more students in Mr. Ray’s class have?
whorl loop whorl loop
arch arch loop whorl
loop arch whorl arch
arch whorl arch loop
Step 1: Make a column for each fingerprint pattern.
Course 1
6-5 Frequency Tables and Histograms Additional Example 1 Continued
Students in Mr. Ray’s class recorded their fingerprint patterns. Which type of pattern do more students in Mr. Ray’s class have?
whorl loop whorl loop
arch arch loop whorl
loop arch whorl arch
arch whorl arch loop
Step 1: Make a column for each fingerprint pattern.
Step 2: For each fingerprint, make a tally mark in the appropriate column.
Course 1
6-5 Frequency Tables and Histograms Additional Example 1 Continued
Students in Mr. Ray’s class recorded their fingerprint patterns. Which type of pattern do more students in Mr. Ray’s class have?
whorl loop whorl loop
arch arch loop whorl
loop arch whorl arch
arch whorl arch loop
Step 1: Make a column for each fingerprint pattern.
Step 2: For each fingerprint, make a tally mark in the appropriate column.
Number of Fingerprint Patterns Whorl Arch Loop l l l l l l l l l l l l l
Course 1
6-5 Frequency Tables and Histograms Additional Example 1 Continued
Students in Mr. Ray’s class recorded their fingerprint patterns. Which type of pattern do more students in Mr. Ray’s class have?
whorl loop whorl loop
arch arch loop whorl
loop arch whorl arch
arch whorl arch loop
Number of Fingerprint Patterns Whorl Arch Loop l l l l l l l l l l l l l
Most students in Mr. Ray’s class have an arch fingerprint.
Course 1
6-5 Frequency Tables and Histograms
A group of four tally marks with a line through it means five. llll = 5 llll llll = 10
Reading Math
Course 1
6-5 Frequency Tables and Histograms Try This: Example 1
Students in Ms. Gracie’s class recorded their fingerprint patterns. Which type of pattern do more students in Ms. Gracie’s class have?
Make a tally table to organize the data.
whorl loop whorl loop
arch whorl loop whorl
loop whorl whorl arch
arch whorl arch loop
Course 1
6-5 Frequency Tables and Histograms Try This: Example 1 Continued
Students in Ms. Gracie’s class recorded their fingerprint patterns. Which type of pattern do more students in Ms. Gracie’s class have?
Step 1: Make a column for each fingerprint pattern.
whorl loop whorl loop
arch whorl loop whorl
loop whorl whorl arch
arch whorl arch loop
Course 1
6-5 Frequency Tables and Histograms Try This: Example 1 Continued
Students in Ms. Gracie’s class recorded their fingerprint patterns. Which type of pattern do more students in Ms. Gracie’s class have?
Step 1: Make a column for each fingerprint pattern.
whorl loop whorl loop
arch whorl loop whorl
loop whorl whorl arch
arch whorl arch loop
Step 2: For each fingerprint, make a tally mark in the appropriate column.
Course 1
6-5 Frequency Tables and Histograms Try This: Example 1 Continued
Students in Ms. Gracie’s class recorded their fingerprint patterns. Which type of pattern do more students in Ms. Gracie’s class have?
Step 1: Make a column for each fingerprint pattern.
whorl loop whorl loop
arch whorl loop whorl
loop whorl whorl arch
arch whorl arch loop
Step 2: For each fingerprint, make a tally mark in the appropriate column.
Number of Fingerprint Patterns Whorl Arch Loop l l l l l l l l l l l l l l
Course 1
6-5 Frequency Tables and Histograms Try This: Example 1 Continued
Students in Ms. Gracie’s class recorded their fingerprint patterns. Which type of pattern do more students in Ms. Gracie’s class have?
whorl loop whorl loop
arch whorl loop whorl
loop whorl whorl arch
arch whorl arch loop
Number of Fingerprint Patterns Whorl Arch Loop l l l l l l l l l l l l l l
Most students in Ms. Gracie’s class have a whorl fingerprint.
Course 1
6-5 Frequency Tables and Histograms
A frequency table tells the number of times an event, category, or group occurs. The cumulative frequency column shows a running total of all frequencies.
Course 1
6-5 Frequency Tables and Histograms Additional Example 2: Making a Cumulative
Frequency Table
Step 1: Make a row for each pattern.
Step 2: The frequency is how many times each pattern occurred.
Number of Fingerprint Patterns Fingerprint Pattern Frequency Cumulative Frequency
Whorl Arch Loop
Step 3: Find the cumulative frequency for each row by adding all frequency values above or in that row.
6 5
11 16
5 5
Use the tally table in Additional Example 1 to make a cumulative frequency table.
Course 1
6-5 Frequency Tables and Histograms Try This: Additional Example 2
Step 1: Make a row for each pattern.
Step 2: The frequency is how many times each pattern occurred.
Number of Fingerprint Patterns Fingerprint Pattern Frequency Cumulative Frequency
Whorl Arch Loop
Step 3: Find the cumulative frequency for each row by adding all frequency values above or in that row.
4 5
11 16
7 7
Use the tally table in Try This: Example 1 to make a cumulative frequency table.
Course 1
6-5 Frequency Tables and Histograms
Additional Example 3: Making a Frequency Table with Intervals
Use the data in the table to make a frequency table with intervals.
Number of Pages Read per Student Last Weekend
12 15 40 19 7 5 22 34 37 18
Step 1: Choose equal intervals.
Step 2: Find the number of data values in each interval. Write these numbers in the “Frequency” row.
Course 1
6-5 Frequency Tables and Histograms
Additional Example 3 Continued
Number of Pages Read per Student Last Weekend Number 1–10 11–20 21–30 31–40
Frequency
This table shows that 2 students read between 1 and 10 pages, 4 students read between 11 and 20 pages, and so on.
2 4 1 3
Number of Pages Read per Student Last Weekend
12 15 40 19 7 5 22 34 37 18
Course 1
6-5 Frequency Tables and Histograms
Try This: Example 3
Use the data in the table to make a frequency table with intervals.
Number of Pages Read per Student Last Weekend
17 29 9 19 7 5 27 34 21 38
Step 1: Choose equal intervals.
Step 2: Find the number of data values in each interval. Write these numbers in the “Frequency” row.
Course 1
6-5 Frequency Tables and Histograms
Try This: Example 3 Continued
Number of Pages Read per Student Last Weekend
Number 1–10 11–20 21–30 31–40 Frequency
This table shows that 3 students read between 1 and 10 pages, 2 students read between 11 and 20 pages, and so on.
3 2 3 2
Number of Pages Read per Student Last Weekend
17 29 9 19 7 5 27 34 21 38
Course 1
6-5 Frequency Tables and Histograms
A histogram is a bar graph that shows the number of data items that occur within each interval.
Course 1
6-5 Frequency Tables and Histograms
Additional Example 4: Making a Histogram Use the frequency table in Additional Example 3 to make a histogram.
Step 1: Choose an appropriate scale and interval.
Step 2: Draw a bar for the number of students in each interval. The bars should touch but not overlap. Step 3: Title the graph and label the axes.
Course 1
6-5 Frequency Tables and Histograms Try This: Example 4
Use the frequency table in Try This: Example 3 to make a histogram.
Step 1: Choose an appropriate scale and interval.
Step 2: Draw a bar for the number of students in each interval. The bars should touch but not overlap. Step 3: Title the graph and label the axes.
0
1
2
3
4
1- 10 11- 20 21- 30 31- 40
Number of Pages Read per Student Last Weekend
Number of Pages
Stu
den
ts
Lesson Quiz: Part 1
1. Students listed the number of days they spent on vacation in one year. Make a tally table with intervals of 5.
2, 18, 5, 15, 7, 10, 1, 10, 4
16, 7, 11, 17, 3, 8, 14, 13, 10
Insert Lesson Title Here
Course 1
6-5 Frequency Tables and Histograms
Number of Days Spent on Vacation 1–5 6–10 11–15 16–20 |||| |||| | |||| |||
Lesson Quiz
2. Use your tally table from problem 1 to make a frequency and cumulative frequency table.
Insert Lesson Title Here
Course 1
6-5 Frequency Tables and Histograms
Number of Days Spent on Vacation
Number of Days Frequency Cumulative Frequency
1–5 5 5 6–10 6 11 11–15 4 15 16–20 3 18
6-6 Ordered Pairs
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up 24 35 47 67 36 22 80 41 Use the set of data above to find the following:
1. the mean 2. the median 3. the mode 4. the range
44 38 none
Course 1
6-6 Ordered Pairs
47
Problem of the Day
What is the largest 6-digit number with each digit different and no digit a prime number? 986,410
Course 1
6-6 Ordered Pairs
Learn to graph ordered pairs on a coordinate grid.
Course 1
6-6 Ordered Pairs
Vocabulary coordinate grid ordered pair
Insert Lesson Title Here
Course 1
6-6 Ordered Pairs
Course 1
6-6 Ordered Pairs
Cities, towns, and neighborhoods are often laid out on a grid. This makes it easier to map and find locations.
A coordinate grid is formed by horizontal and vertical lines and is used to locate points.
Each point on a coordinate grid can be located by using an ordered pair of numbers, such as (4, 6). The starting point is (0, 0).
• The first number tells how far to move horizontally from (0, 0).
• The second number tells how far to move vertically.
Course 1
6-6 Ordered Pairs
Additional Example 1A: Identifying Ordered Pairs Name the ordered pair for the location.
0 1 2 3 4 5
1 2 3 4
Gym
Theater Park
A. Gym
Start at (0, 0). Move right 2 units and then up 3 units.
The gym is located at (2, 3).
Course 1
6-6 Ordered Pairs
Additional Example 1B: Identifying Ordered Pairs Name the ordered pair for the location.
B. Theater Start at (0, 0). Move right 3 units and then up 1 unit.
The theater is located at (3, 1).
0 1 2 3 4 5
1 2 3 4
Gym
Theater Park
Course 1
6-6 Ordered Pairs
Additional Example 1C: Identifying Ordered Pairs Name the ordered pair for the location.
C. Park Start at (0, 0). Move right 4 units and then up 2 units.
The park is located at (4, 2).
0 1 2 3 4 5
1 2 3 4
Gym
Theater Park
Course 1
6-6 Ordered Pairs
Try This: Example 1A Name the ordered pair for each location.
0 1 2 3 4 5
1 2 3 4 Library
Mall School
A. Library
Start at (0, 0). Move right 1 unit and then up 4 units. The library is located at (1, 4).
Course 1
6-6 Ordered Pairs
Try This: Example 1B Name the ordered pair for the location.
B. Mall Start at (0, 0). Move right 4 units and then up 1 unit.
The mall is located at (4, 1).
0 1 2 3 4 5
1 2 3 4 Library
Mall School
Course 1
6-6 Ordered Pairs
Try This: Example 1C Name the ordered pair for the location.
C. School Start at (0, 0). Move right 2 units and then up 2 units.
The school is located at (2, 2).
0 1 2 3 4 5
1 2 3 4 Library
Mall School
Course 1
6-6 Ordered Pairs
Additional Example 2: Graphing Ordered Pairs
Graph and label each point on a coordinate grid.
A. L (3, 5) Start at (0, 0).
1 2 3 4 5 6 0 1 2 3 4 5 6
Move right 3 units. Move up 5 units.
L
B. M (4, 0) Start at (0, 0). Move right 4 units. Move up 0 units.
M
Course 1
6-6 Ordered Pairs
Try This: Example 2
Graph and label each point on a coordinate grid.
A. T (2, 6) Start at (0, 0).
1 2 3 4 5 6 0 1 2 3 4 5 6
Move right 2 units. Move up 6 units.
T
B. V (5, 0) Start at (0, 0). Move right 5 units. Move up 0 units.
V
Lesson Quiz
Give the ordered pair for each point.
1. A
2. B
3. C
4. D
Graph and label each point on a coordinate grid.
5. F(7, 2) 6. G(1, 7)
(6, 1)
(4, 6)
Insert Lesson Title Here
(1, 4)
(2, 1)
Course 1
6-6 Ordered Pairs
F
G
6-7 Line Graphs
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Describe how to graph each point on a coordinate grid.
1. (4, 5) 2. (0, 2) 3. (3, 0)
right 4, up 5 up 2
right 3
Course 1
6-7 Line Graphs
Problem of the Day
Study the first two columns to determine a pattern to help fill in the blank square at the bottom.
6
Course 1
6-7 Line Graphs
4 49 6 11 134 12 7 85
Learn to display and analyze data in line graphs.
Course 1
6-7 Line Graphs
Vocabulary line graph double-line graph
Insert Lesson Title Here
Course 1
6-7 Line Graphs
Course 1
6-7 Line Graphs
Data that shows change over time is best displayed in a line graph. A line graph displays a set of data using line segments.
Course 1
6-7 Line Graphs
Additional Example 1: Making a Line Graph
Use the data in the table to make a line graph.
Population of New Hampshire
Year Population
1650 1,300
1670 1,800
1690 4,200
1700 5,000
Course 1
6-7 Line Graphs
Because time passes whether or not the population changes, time is independent of population. Always put the independent quantity on the horizontal axis.
Helpful Hint
Course 1
6-7 Line Graphs
Additional Example 1 Continued
Step 1: Place year on the horizontal axis and population on the vertical axis. Label the axes.
Step 2: Determine an appropriate scale and interval for each axis.
Step 3: Mark a point for each data value. Connect the points with straight lines.
Step 4: Title the graph. 0 1650 1670 1690 1700
1,000 2,000 3,000 4,000
5,000 6,000
Pop
ula
tion
Year
Population of New Hampshire
Course 1
6-7 Line Graphs
Try This: Example 1
Use the data in the table to make a line graph.
School District Enrollment
Year Population
1996 2,300
1998 2,800
2000 5,200
2002 6,000
Course 1
6-7 Line Graphs
Try This: Example 1 Continued
Step 1: Place year on the horizontal axis and number of students on the vertical axis. Label the axes.
Step 2: Determine an appropriate scale and interval for each axis.
Step 3: Mark a point for each data value. Connect the points with straight lines.
Step 4: Title the graph. 0 1996 1998 2000 2002
1,000 2,000 3,000 4,000
5,000 6,000
Nu
mb
er o
f S
tud
ents
Year
School District Enrollment
Course 1
6-7 Line Graphs
Additional Example 2: Reading a Line Graph
Use the line graph to answer each question. A. In which year did
CDs cost the most? 2002
B. About how much did CDs cost in 2000? $15
C. Did CD prices increase or decrease from 1999 through 2002? They increased.
Course 1
6-7 Line Graphs
Try This: Example 2
Use the line graph to answer each question. A. In which year did
CDs cost the least? 1999
B. About how much did CDs cost in 1999? $13
C. Did CD prices increase or decrease from 2001 to 2002? They increased.
Course 1
6-7 Line Graphs
Line graphs that display two sets of data are called double-line graphs.
Course 1
6-7 Line Graphs
Additional Example 3: Making a Double-Line Graph
Use the data in the table to make a double-line graph.
Stock Prices 1985 1990 1995 2000
Corporation A $16 $20 $34 $33 Corporation B $38 $35 $31 $21
Use different colors of lines to connect the stock values so you will easily be able to tell the data apart.
Helpful Hint
Course 1
6-7 Line Graphs
Additional Example 3 Continued
Step 1: Determine an appropriate scale and interval.
Step 2: Mark a point for each Corporation A value and connect the points.
Step 3: Mark a point for each Corporation B value and connect the points.
Step 4: Title the graph and label both axes. Include a key.
$0 1985 1990 1995 2000
$10 $20 $30 $40
Pri
ce o
f S
tock
Year
Stock Prices
Corp. A
Corp. B
Course 1
6-7 Line Graphs
Try This: Example 3
Use the data in the table to make a double-line graph.
Stock Prices 1985 1990 1995 2000
Corporation C $8 $16 $20 $28 Corporation D $35 $22 $14 $7
Course 1
6-7 Line Graphs
Try This: Example 3 Continued
Step 1: Determine an appropriate scale and interval.
Step 2: Mark a point for each Corporation C value and connect the points.
Step 3: Mark a point for each Corporation D value and connect the points.
Step 4: Title the graph and label both axes. Include a key.
$0 1985 1990 1995 2000
$10 $20 $30 $40
Pri
ce o
f S
tock
Year
Stock Prices
Corp. C
Corp. D
Lesson Quiz: Part 1 1. Use the data to make a line graph.
Insert Lesson Title Here
Course 1
6-7 Line Graphs
Number of Aluminum Cans Collected Mon Tue Wed Thu Fri 100 150 200 125 175
0
50 100
150 200
250
Nu
mb
er o
f C
ans
Aluminum Cans Collected
M T W Th F
Lesson Quiz: Part 2 Use the line graph to answer each question.
2. Which plant was taller on Tuesday?
3. Which plant grew more between Thursday and Friday?
4. Which plant grew the most in one week?
Each grew the same amount.
A
Insert Lesson Title Here
A
Course 1
6-7 Line Graphs
6-8 Misleading Graphs
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Use the data below to answer each question. 20 21 23 24 27 33 34 35 36 38 40 41 42 43 46 52 53
1. What is the median? 2. What is the mode? 3. What is the range?
36 none 33
Course 1
6-8 Misleading Graphs
Problem of the Day
Nine students in a group found that their mean score was 86 for the first math test. On the next test, each student in the group scored 7 points higher than on the first test. What was their mean score for the two tests? 89.5
Course 1
6-8 Misleading Graphs
Learn to recognize misleading graphs.
Course 1
6-8 Misleading Graphs
Course 1
6-8 Misleading Graphs
Additional Example 1A: Misleading Bar Graphs
A. Why is this bar graph misleading? Because the lower part of the vertical axis is missing, the differences in prices are exaggerated.
Course 1
6-8 Misleading Graphs
Additional Example 1B: Misleading Bar Graphs
B. What might people believe from the misleading graph?
People might believe that Cars B and C cost 1 – 2
times as much as Car A. In reality, Cars B and C are only a few thousand dollars more than Car A.
1 2 __ 1
2 __
Course 1
6-8 Misleading Graphs
Try This: Example 1A
A. Why is this bar graph misleading?
The vertical axis begins at 530 rather than 0.
600
550 540 530
580 570 560
Dol
lars
Money Raised
4th graders 5th graders 6th graders
Course 1
6-8 Misleading Graphs
Try This: Example 1B
B. What might people believe from the misleading graph? That the 5th graders have raised twice as much money as the 4th graders.
600
550 540 530
580 570 560
Dol
lars
Money Raised
4th graders 5th graders 6th graders
Course 1
6-8 Misleading Graphs
Additional Example 2A: Misleading Line Graphs
A. Why are these line graphs misleading?
If you look at the scale for each graph, you will notice that the April graph goes from 54° to 66° and the May graph goes from 68° to 80°.
Course 1
6-8 Misleading Graphs
Additional Example 2B: Misleading Line Graphs
B. What might people believe from these misleading graphs? People might believe that the temperatures in May were about the same as the temperatures in April. In reality, the temperatures in April were about 15 degrees lower.
Course 1
6-8 Misleading Graphs
Additional Example 2C: Misleading Line Graphs
C. Why is this line graph misleading?
The scale goes from $0 to $80, and then increases by $5.
Course 1
6-8 Misleading Graphs
Try This: Example 2A
A. Why are these line graphs misleading?
If you look at the scale for each graph, you will notice that the September Graph goes from 85° to 70° and the November graph goes from 65° to 50°.
0102030405060708090
1 2 3 40
10
20304050
6070
1 2 3 4Week Week
Tem
pera
ture
(°F
)
Tem
pera
ture
(°F
) September November
Course 1
6-8 Misleading Graphs
Try This: Example 2B
B. What might people believe from these misleading graphs?
People might believe that the temperatures in September were about the same as the temperatures in November. In reality, the temperatures in September were about 20 degrees higher.
0102030405060708090
1 2 3 40
10
20304050
6070
1 2 3 4Week Week
Tem
pera
ture
(°F
)
Tem
pera
ture
(°F
) September November
Course 1
6-8 Misleading Graphs
Try This: Example 2C
C. Why is this line graph misleading?
The scale goes from $0 to $50, and then increases by $10.
0
50
60
70 80
1985 1990 1995 2000
Stock prices
Year
Pric
e of
sto
ck ($
) Corp. C
Corp. D
Lesson Quiz
1. Why might this line graph be misleading?
2. What might people believe from the graph?
Possible answer: that there were hardly any visitors on Monday
The scale does not start at zero.
Insert Lesson Title Here
Course 1
6-8 Misleading Graphs
6-9 Stem-and-Leaf Plots
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up A set of data ranges from 12 to 86. What intervals would you use to display this data in a histogram with four intervals?
Possible answer: 10–29, 30–49, 50–69, 70–89
Course 1
6-9 Stem-and-Leaf Plots
Problem of the Day
What is the least number that can be divided evenly by each of the numbers 1 through 12? 27,720
Course 1
6-9 Stem-and-Leaf Plots
Learn to make and analyze stem-and-leaf plots.
Course 1
6-9 Stem-and-Leaf Plots
Vocabulary stem-and-leaf plot
Insert Lesson Title Here
Course 1
6-9 Stem-and-Leaf Plots
Course 1
6-9 Stem-and-Leaf Plots
A stem-and-leaf plot shows data arranged by place value. You can use a stem-and-leaf plot when you want to display data in an organized way that allows you to see each value.
Course 1
6-9 Stem-and-Leaf Plots
Additional Example 1: Creating Stem-and-Leaf Plots
Use the data in the table to make a stem-and-leaf plot.
Test Scores 75 86 83 91 94 88 84 99 79 86
Step 1: Group the data by tens digits.
Step 2: Order the data from least to greatest.
75 79 83 84 86 86 88 91 94 99
Course 1
6-9 Stem-and-Leaf Plots
To write 42 in a stem-and-leaf plot, write each digit in a separate column.
4 2
Helpful Hint
Stem Leaf
Course 1
6-9 Stem-and-Leaf Plots
Additional Example 1 Continued Step 3: List the tens digits of
the data in order from least to greatest. Write these in the “stems” column.
Step 4: For each tens digit, record the ones digits of each data value in order from least to greatest. Write these in the “leaves” column.
Step 5: Title the graph and add a key.
Stems Test Scores
Key: 7 5 means 75
Leaves
7 8 9
5 9 3 4 6 6 8 1 4 9
75 79 83 84 86 86 88 91 94 99
Course 1
6-9 Stem-and-Leaf Plots
Try This: Example 1
Use the data in the table to make a stem-and-leaf plot.
Test Scores 72 88 64 79 61 84 83 76 74 67
Step 1: Group the data by tens digits.
Step 2: Order the data from least to greatest.
61 64 67 72 74 76 79 83 84 88
Course 1
6-9 Stem-and-Leaf Plots
Try This: Example 1 Cont.
Step 3: List the tens digits of the data in order from least to greatest. Write these in the “stems” column.
Step 4: For each tens digit, record the ones digits of each data value in order from least to greatest. Write these in the “leaves” column.
Step 5: Title the graph and add a key.
Stems Test Scores
Key: 6 1 means 61
Leaves
6 7 8
1 4 7 2 4 6 9 3 4 8
61 64 67 72 74 76 79 83 84 88
Course 1
6-9 Stem-and-Leaf Plots Additional Example 2: Reading Stem-and-Leaf
Plots
Find the least value, greatest value, mean, median, mode, and range of the data.
Stems
Leaves 4 5 6
0 0 1 5 7 1 1 2 4 3 3 3 5 9 9
7 8 9
0 4 4 3 6 7 1 4
The least stem and least leaf give the least value, 40.
The greatest stem and greatest leaf give the greatest value, 94.
Use the data values to find the mean (40 + … + 94) ÷ 23 = 64.
Key: 4 0 means 40
Course 1
6-9 Stem-and-Leaf Plots
Additional Example 2 Continued
The median is the middle value in the table, 63.
To find the mode, look for the number that occurs most often in a row of leaves. Then identify its stem. The mode is 63.
The range is the difference between the greatest and the least value. 94 – 40 = 54.
Stems
Leaves 4 5 6
0 0 1 5 7 1 1 2 4 3 3 3 5 9 9
7 8 9
0 4 4 3 6 7 1 4
Key: 4 0 means 40
Course 1
6-9 Stem-and-Leaf Plots
Try This: Example 2
Find the least value, greatest value, mean, median, mode, and range of the data.
Stems
Leaves 3 4 5
0 2 5 6 8 1 1 3 4 4 5 6 9 9 9
6 7 8
1 2 4 5 6 9 1 5
The least stem and least leaf give the least value, 30.
The greatest stem and greatest leaf give the greatest value, 85.
Use the data values to find the mean (30 + … + 85) ÷ 23 = 55.
Key: 3 0 means 30
Course 1
6-9 Stem-and-Leaf Plots
Try This: Example 2 Continued
The median is the middle value in the table, 56.
To find the mode, look for the number that occurs most often in a row of leaves. Then identify its stem. The mode is 59.
The range is the difference between the greatest and the least value. 85 – 30 = 55.
Stems
Leaves 3 4 5
0 2 5 6 8 1 1 3 4 4 5 6 9 9 9
6 7 8
1 2 4 5 6 9 1 5
Key: 3 0 means 30
Lesson Quiz: Part 1
1. Make a stem-and-leaf plot of the data.
42 36 40 31 29 49 21 28 52 27 22 35 30 46 34 34
Insert Lesson Title Here
Course 1
6-9 Stem-and-Leaf Plots
2 1 2 7 8 9
3 0 1 4 4 5 6
4 0 2 6 9
5 2
Stems
Leaves
Key: 3 | 0 means 30
Lesson Quiz: Part 2
Find each value using the stem-and-leaf plot.
2. What is the least value?
3. What is the mean?
4. What is the median?
5. What is the mode?
34.75
21
Insert Lesson Title Here
34
34
Course 1
6-9 Stem-and-Leaf Plots
7-1 Points, Lines, and Planes
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up What geometry term might you associate with each object?
1. a string on a guitar
2. a window
3. the tip of a pencil
4. a sheet of paper
line segment
plane or rectangle
point
Course 1
7-1 Points, Lines, and Planes
plane or rectangle
Problem of the Day
Draw a clock face that includes the numerals 1–12. Draw two lines that do not intersect and that separate the clock face into three parts so that the sums of the numbers on each part are the same.
Course 1
7-1 Points, Lines, and Planes
Learn to describe figures by using the terms of geometry.
Course 1
7-1 Points, Lines, and Planes
Vocabulary point line plane line segment ray
Insert Lesson Title Here
Course 1
7-1 Points, Lines, and Planes
Course 1
7-1 Points, Lines, and Planes
The building blocks of geometry are points, lines, and planes.
A plane is a flat surface that plane LMN, extends without end in all plane MLN, directions. plane NLM
A plane is named by three points on the plane that are not on the same line.
M
N
L
A point is an exact location. P point P, P
A point is named by a capital letter.
A line is a straight path that line AB, AB, extends without end in A B line BA, BA opposite directions.
A line is named by two points on the line.
Course 1
7-1 Points, Lines, and Planes
Additional Example 1A &1B: Identifying Points, Lines, and Planes
Use the diagram to name each geometric figure.
A. three points M, N, and P Five points are labeled: points M, N, P, Q, and R.
B. two lines PR and NR You can also write RP and RN.
Q
N
P R M
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7-1 Points, Lines, and Planes
Additional Example 1C & 1D: Identifying Points, Lines, and Planes
Use the diagram to name each geometric figure.
C. a point shared by two lines point R
D. a plane plane QRM Use any three points in the plane that are not on the same line. Write the three points in any order.
Point R is a point on PR and NR.
Q
N
P R M
Course 1
7-1 Points, Lines, and Planes
Try This: Example 1A &1B Use the diagram to name each geometric figure.
A. three points S, T, and U Five points are labeled: points S, T, U, V, and W.
B. two lines UW and SW You can also write WU and WS.
V
S
U W T
Course 1
7-1 Points, Lines, and Planes
Try This: Example 1C & 1D Use the diagram to name each geometric figure.
C. a point shared by two lines point W
D. a plane plane VUT Use any three points in the plane that are not on the same line. Write the three points in any order.
Point W is a point on WS and WU. V
S
U W T
Course 1
7-1 Points, Lines, and Planes
A line segment is a line segment XY, XY, made of two endpoints Y line segment YX, YX and all the points between X the endpoints.
A line segment is named by its endpoints.
A ray has one endpoint. ray JK, JK From the endpoint, the ray J K extends without end in one direction only.
A ray is named by its endpoint first followed by another point on the ray.
Course 1
7-1 Points, Lines, and Planes
Additional Example 2A & 2B: Identifying Line Segments and Rays
Use the diagram to give a possible name to each figure.
A. three different line segments
B. three ways to name the line
You can also write BA, CB, and CA.
AB, BC, and AC You can also write BA, CB, and CA.
AB, BC, and AC C
A B
Course 1
7-1 Points, Lines, and Planes
Additional Example 2C & 2D: Identifying Line Segments and Rays
Use the diagram to give a possible name to each figure. C. six different rays
D. another name for ray AB
A is still the endpoint. C is another point on the ray.
AB, AC, BC, CB,
CA, and BA
AC
C
A B
Course 1
7-1 Points, Lines, and Planes
Try This: Example 2A & 2B Use the diagram to give a possible name to each figure.
A. three different line segments
B. three ways to name the line
You can also write ED, FE, and FD.
DE, EF, and DF You can also write ED, FE, and FD.
DE, EF, and DF F
D E
Course 1
7-1 Points, Lines, and Planes
Try This: Example 2C & 2D Use the diagram to give a possible name to each figure.
C. six different rays
D. another name for ray DE
D is still the endpoint. F is another point on the ray.
DE, EF, DF, FE,
FD, and ED
DF
F
D E
Lesson Quiz
Use the diagram to name each geometric figure.
1. three points
2. two lines
3. a point shared by a line and a ray
4. a plane
Possible answer: A, B, C
Insert Lesson Title Here
Possible answer: F
Possible answer: plane AEC
Course 1
7-1 Points, Lines, and Planes
Possible answer: AD, CE
7-2 Angles
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up
1. Draw two points. Label one point A and the other point B. 2. Draw a line through points A and B. 3. Draw a ray with A as an endpoint and C as a point on the ray. 4. Name all the rays in your drawing.
Course 1
7-2 Angles
AB, BA, and AC
A B
C
Problem of the Day
The measure of Jack’s angle is twice that of Amy’s and half that of Nate’s. The sum of the measures of Amy’s and Trisha’s angles is equal to the sum of the measures of Jack’s and Nate’s angles. The sum of the measures of all the angles is equal to 180°. What is the measure of each student’s angle? Jack’s angle: 30°; Nate’s angle: 60°; Amy’s angle: 15°; Trisha’s angle: 75°
Course 1
7-2 Angles
Learn to name, measure, classify, estimate, and draw angles.
Course 1
7-2 Angles
Vocabulary angle vertex acute angle right angle obtuse angle straight angle
Insert Lesson Title Here
Course 1
7-2 Angles
Course 1
7-2 Angles
An angle is formed by two rays with a common endpoint, called the vertex. An angle can be named by its vertex or by its vertex and a point from each ray. The middle point in the name should always be the vertex.
Angles are measured in degrees. The number of degrees determines the type of angle. Use the symbol ° to show degrees: 90° means “90 degrees.”
Course 1
7-2 Angles
An acute angle measures less than 90°.
A right angle measures exactly 90°.
Course 1
7-2 Angles
An obtuse angle measures more than 90° and less than 180°.
A straight angle measures exactly 180°.
Course 1
7-2 Angles Additional Example 1: Measuring an Angle with a
Protractor Use a protractor to measure the angle. Tell what type of angle it is.
• Place the center point of the protractor on the vertex of the angle.
G H
F
Course 1
7-2 Angles Additional Example 1 Continued
Use a protractor to measure the angle. Tell what type of angle it is.
• Place the protractor so that ray GH passes through the 0° mark.
G H
F
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7-2 Angles
• Using the scale that starts with 0° along ray GH, read the measure where ray GF crosses.
Additional Example 1 Continued Use a protractor to measure the angle. Tell what type of angle it is.
G H
F
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7-2 Angles
• The measure of FGH is 120°. Write this as m FGH = 120°.
Additional Example 1 Continued Use a protractor to measure the angle. Tell what type of angle it is.
G H
F
Course 1
7-2 Angles
• Since 120° > 90° and 120° < 180°, the angle is obtuse.
Additional Example 1 Continued Use a protractor to measure the angle. Tell what type of angle it is.
G H
F
Course 1
7-2 Angles Try This: Example 1
Use a protractor to measure the angle. Tell what type of angle it is.
H I
G
• Place the center point of the protractor on the vertex of the angle.
Course 1
7-2 Angles Try This: Example 1 Continued
Use a protractor to measure the angle. Tell what type of angle it is.
• Place the protractor so that ray HI passes through the 0° mark.
H I
G
Course 1
7-2 Angles Try This: Example 1 Continued
Use a protractor to measure the angle. Tell what type of angle it is.
H I
G
• Using the scale that starts with 0° along ray HI, read the measure where ray HI crosses.
Course 1
7-2 Angles Try This: Example 1 Continued
Use a protractor to measure the angle. Tell what type of angle it is.
H I
G
• The measure of GHI is 70°. Write this as m GHI = 70°.
Course 1
7-2 Angles Try This: Example 1 Continued
Use a protractor to measure the angle. Tell what type of angle it is.
H I
G
• Since 70° < 90°, the angle is acute.
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7-2 Angles
Additional Example 2: Drawing an Angle with a Protractor
Use a protractor to draw an angle that measures 80°.
• Draw a ray on a sheet of paper.
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7-2 Angles
Additional Example 2 Continued Use a protractor to draw an angle that measures 80°.
• Place the center point of the protractor on the endpoint of the ray. • Place the protractor so that the ray passes through the 0° mark.
Course 1
7-2 Angles
Additional Example 2 Continued Use a protractor to draw an angle that measures 80°.
• Make a mark at 80° above the scale on the protractor.
• Use a straightedge to draw a ray from the endpoint of the first ray through the mark you make at 80°.
Course 1
7-2 Angles
Try This: Example 2 Use a protractor to draw an angle that measures 45°.
• Draw a ray on a sheet of paper.
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7-2 Angles
• Place the center point of the protractor on the endpoint of the ray. • Place the protractor so that the ray passes through the 0° mark.
Try This: Example 2 Continued Use a protractor to draw an angle that measures 45°.
Course 1
7-2 Angles
• Make a mark at 45° above the scale on the protractor.
• Use a straightedge to draw a ray from the endpoint of the first ray through the mark you make at 45°.
Try This: Example 2 Continued Use a protractor to draw an angle that measures 45°.
Course 1
7-2 Angles
To estimate the measure of an angle, compare it with an angle whose measure you already know. A right angle has half the measure of a straight angle. A 45° angle has half the measure of a right angle.
Course 1
7-2 Angles
Additional Example 3: Estimating Angle Measures
Estimate the measure of the angle, and then use a protractor to check the reasonableness of your estimate.
Think: The measure of the angle is about halfway between 90° and 180°. A good estimate would be 135°.
The angle measures 131°, so the estimate is reasonable.
Course 1
7-2 Angles
Try This: Example 3 Estimate the measure of the angle, and then use a protractor to check the reasonableness of your estimate.
Think: The measure of the angle is a little more than halfway between 0° and 90°. A good estimate would be 50°.
The angle measures 52°, so the estimate is reasonable.
Lesson Quiz
Use a protractor to draw an angle with the given measure. Tell what type of angle it is.
1. 140°
2. 20°
3. Draw a right angle.
4. Is the angle shown closer to 30° or 120°?
acute
obtuse
Insert Lesson Title Here
30°
Course 1
7-2 Angles
7-3 Angle Relationships
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Identify the type of angle.
1. 70°
2. 90°
3. 140°
4. 180°
acute
right
obtuse
Course 1
7-3 Angle Relationships
straight
Problem of the Day
A line forms an angle of 57° with the vertical axis. What angle does the line form with the horizontal axis? 33° or 147°
Course 1
7-3 Angle Relationships
Learn to understand relationships of angles.
Course 1
7-3 Angle Relationships
Vocabulary congruent vertical angles adjacent angles complementary angles supplementary angles
Insert Lesson Title Here
Course 1
7-3 Angle Relationships
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7-3 Angle Relationships
When angles have the same measure, they are said to be congruent.
Vertical angles are formed opposite each other when two lines intersect. Vertical angles have the same measure, so they are always congruent. MRP and NRQ are vertical angles.
MRN and PRQ are vertical angles
M N 160°
160°
20° 20° R
P Q
Course 1
7-3 Angle Relationships Adjacent angles are side by side and have a common vertex and ray. Adjacent angles may or may not be congruent.
MRN and NRQ are adjacent angles. They share vertex R and RN.
NRQ and QRP are adjacent angles. They share vertex R and RQ.
M N 160°
160°
20° 20° R
P Q
Course 1
7-3 Angle Relationships
Additional Example 1A: Identifying Types of Angle Pairs
Identify the type of each angle pair shown.
A.
5 and 6 are opposite each other and are formed by two intersecting lines.
They are vertical angles.
5 6
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7-3 Angle Relationships
Additional Example 1B: Identifying Types of Angle Pairs
Identify the type of each angle pair shown.
B. 7 and 8 are side by side and
have a common vertex and ray.
They are adjacent angles.
7 8
Course 1
7-3 Angle Relationships
Try This: Additional Example 1A
Identify the type of each angle pair shown.
A.
3 and 4 are side by side and have a common vertex and ray.
They are adjacent angles.
3 4
Course 1
7-3 Angle Relationships
Try This: Additional Example 1B
Identify the type of each angle pair shown.
B.
7 and 8 are opposite each other and are formed by two intersecting lines.
They are vertical angles.
7
8
Course 1
7-3 Angle Relationships
65° + 25° = 90°
LMN and NMP are complementary.
Complementary angles are two angles whose measures have a sum of 90°.
P
N
M
L
25° 65°
Course 1
7-3 Angle Relationships
Supplementary angles are two angles whose measures have a sum of 180°.
65° + 115° = 180°
GHK and KHJ are supplementary.
J
K
H 115° 65°
G
Course 1
7-3 Angle Relationships
Additional Example 1A: Identifying an Unknown Angle Measure
Find each unknown angle measure.
71° + a = 90° –71° –71° a = 19°
The sum of the measures is 90°.
a
71°
A. The angles are complementary.
Course 1
7-3 Angle Relationships
Additional Example 1B: Identifying an Unknown Angle Measure
Find each unknown angle measure.
125° + b = 180° –125° –125° b = 55°
The sum of the measures is 180°.
b 125°
B. The angles are supplementary.
Course 1
7-3 Angle Relationships
Additional Example 1C: Identifying an Unknown Angle Measure
Find each unknown angle measure.
c = 82° Vertical angles are congruent.
c 82°
C. The angles are vertical angles.
Course 1
7-3 Angle Relationships
Additional Example 1D: Identifying an Unknown Angle Measure
Find each unknown angle measure.
x + y + 80° = 180° –80° –80°
x + y = 100°
The sum of the measures is 180°.
D. JKL and MKN are congruent.
x = 50° and y = 50° Each angle measures half of 100°.
x 80°
K
M L
N J y
Course 1
7-3 Angle Relationships
Try This: Example 1A
Find each unknown angle measure.
65° + d = 90° –65° –65° d = 25°
The sum of the measures is 90°.
d
65°
A. The angles are complementary.
Course 1
7-3 Angle Relationships
Try This: Example 1B
Find each unknown angle measure.
145° + s = 180° –145° –145° s = 35°
The sum of the measures is 180°.
s 145°
B. The angles are supplementary.
Course 1
7-3 Angle Relationships
Try This: Example 1C
Find each unknown angle measure.
t = 32° Vertical angles are congruent.
t
32°
C. The angles are vertical angles.
Course 1
7-3 Angle Relationships
Try This: Example 1D
Find each unknown angle measure.
x + y + 50° = 180° –50° –50°
x + y = 130°
The sum of the measures is 180°.
D. ABC and DBE are congruent.
x = 65° and y = 65° Each angle measures half of 130°.
x 50°
B
D C
E A y
Lesson Quiz
Give the complement of each angle.
1. 70° 2. 42°
Give the supplement of each angle.
3. 120° 4. 17°
5. Identify the type of angle pair shown.
48° 20°
Insert Lesson Title Here
60° 163°
Course 1
7-3 Angle Relationships
adjacent
7-4 Classifying Lines
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Give the complement and supplement of each angle.
1. 80°
2. 64°
3. 15°
10°, 100°
26°, 116°
75°, 165°
Course 1
7-4 Classifying Lines
Problem of the Day
Draw three points that are not in a straight line. Label them A, B, and C. How many different lines can you draw that contains two of the points? Name the lines.
Course 1
7-4 Classifying Lines
3; AB, AC, BC
Learn to classify the different types of lines.
Course 1
7-4 Classifying Lines
Vocabulary parallel lines perpendicular lines skew lines
Insert Lesson Title Here
Course 1
7-4 Classifying Lines
Course 1
7-4 Classifying Lines
Intersecting lines are lines that cross at one common point.
Parallel lines are lines in the same plane that never intersect.
Line YZ intersects line WX.
YZ intersects WX.
Line AB is parallel to line ML.
AB ML.
Y W
Z X
B
A
M
L
Course 1
7-4 Classifying Lines
Perpendicular lines intersect to form 90° angles, or right angles.
Line RS is perpendicular to line TU.
RS TU. S
R
T U
The square inside a right angle shows that the rays of the angle are perpendicular.
Writing Math
Course 1
7-4 Classifying Lines
Skew lines are lines that lie in different planes. They are neither parallel nor intersecting.
Line AB and line ML are skew.
AB and ML are skew.
M
L
A B
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7-4 Classifying Lines
Additional Example 1A: Classifying Pairs of Lines
Classify each pair of lines.
A.
The lines intersect to form right angles.
They are perpendicular.
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7-4 Classifying Lines
Additional Example 1B: Classifying Pairs of Lines
Classify each pair of lines.
B.
The lines are in different planes and are not parallel or intersecting.
They are skew.
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7-4 Classifying Lines
Additional Example 1C: Classifying Pairs of Lines
Classify each pair of lines.
C.
The lines are in the same plane. They do not appear to intersect.
They are parallel.
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7-4 Classifying Lines
Additional Example 1D: Classifying Pairs of Lines
Classify each pair of lines.
D.
The lines cross at one common point.
They are intersecting.
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7-4 Classifying Lines
Try This: Example 1A
Classify each pair of lines.
A.
The lines are in the same plane. They do not appear to intersect.
They are parallel.
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7-4 Classifying Lines
Try This: Example 1B
Classify each pair of lines.
B.
The lines intersect to form right angles.
They are perpendicular.
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7-4 Classifying Lines
Try This: Additional Example 1C
Classify each pair of lines.
C.
The lines are in different planes and are not parallel or intersecting.
They are skew.
Course 1
7-4 Classifying Lines
Try This: Additional Example 1D
Classify each pair of lines.
D.
The lines cross at one common point.
They are intersecting.
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7-4 Classifying Lines
Additional Example 2: Application
The handrails on an escalator are in the same plane. What type of line relationship do they represent?
The handrails are in the same plane and do not intersect.
The lines are parallel.
Course 1
7-4 Classifying Lines
Try This: Example 2
The roads are in the same plane. What type of line relationship do they represent?
The lines cross at one common point.
The lines are intersecting.
Lesson Quiz
1. Sketch a pair of perpendicular lines.
2. Sketch a pair of parallel lines.
Use the figure to classify the lines.
3. AD and BC
4. AD and CF
5. AB and BG
parallel
Insert Lesson Title Here
skew
perpendicular
Course 1
7-4 Classifying Lines
7-5 Triangles
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up
1. What are two angles whose sum is 90°? 2. What are two angles whose sum is 180°? 3. A part of a line between two points is called a _________. 4. Two lines that intersect at 90° are ______________.
complementary angles
supplementary angles
segment
Course 1
7-5 Triangles
perpendicular
Problem of the Day
Find the total number of shaded triangles in each figure.
3 6 10
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7-5 Triangles
Learn to classify triangles and solve problems involving angle and side measures of triangles.
Course 1
7-5 Triangles
Vocabulary acute triangle obtuse triangle right triangle scalene triangle isosceles triangle equilateral triangle
Insert Lesson Title Here
Course 1
7-5 Triangles
Course 1
7-5 Triangles
A triangle is a closed figure with three line segments and three angles. Triangles can be classified by the measures of their angles. An acute triangle has only acute angles. An obtuse triangle has one obtuse angle. A right triangle has one right angle.
Acute triangle Obtuse triangle Right triangle
Course 1
7-5 Triangles
To decide whether a triangle is acute, obtuse, or right, you need to know the measures of its angles.
The sum of the measures of the angles in any triangle is 180°. You can see this if you tear the corners from a triangle and arrange them around a point on a line.
By knowing the sum of the measures of the angles in a triangle, you can find unknown angle measures.
Course 1
7-5 Triangles
Additional Example 1: Application
D
E
F
To classify the triangle, find the measure of D on the trophy.
So the measure of D is 90°. Because DEF has one right angle, the trophy is a right triangle.
Subtract the sum of the known angle measures from 180°
m D = 180° – (38° + 52°)
m D = 180° – 90°
m D = 90°
Sara designed this triangular trophy. The measure of E is 38°, and the measure of F is 52°. Classify the triangle.
Course 1
7-5 Triangles
Try This: Example 1
D
E
F
To classify the triangle, find the measure of D on the trophy.
So the measure of D is 136°. Because DEF has one obtuse angle, the trophy is an obtuse triangle.
Subtract the sum of the known angle measures from 180°
m D = 180° – (22° + 22°)
m D = 180° – 44°
m D = 136°
Sara designed this triangular trophy. The measure of E is 22°, and the measure of F is 22°. Classify the triangle.
Course 1
7-5 Triangles
You can use what you know about vertical, adjacent, complementary, and supplementary angles to find the measures of missing angles.
Course 1
7-5 Triangles
Additional Example 2A: Using Properties of Angles to Label Triangles
Use the diagram to find the measure of each indicated angle.
QTR and STR are supplementary angles, so the sum of m QTR and m STR is 180°.
m QTR = 180° – 68°
= 112°
P
S R
Q
T
68°
55°
A. QTR
Course 1
7-5 Triangles
Additional Example 2B: Using Properties of Angles to Label Triangles
QRT and SRT are complementary angles, so the sum of m QRT and m SRT is 90°. m SRT = 180° – (68° + 55°)
= 180° – 123° = 57°
m QRT = 90° – 57° = 33°
P
S R
Q
T
68°
55°
B. QRT
Course 1
7-5 Triangles
Try This: Example 2A
Use the diagram to find the measure of each indicated angle.
MNO and PNO are supplementary angles, so the sum of m MNO and m PNO is 180°.
m MNO = 180° – 44°
= 136°
L
P O
M
N
44°
60°
A. MNO
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7-5 Triangles
Try This: Example 2B
MON and PON are complementary angles, so the sum of m MON and m PON is 90°.
m PON = 180° – (44° + 60°) = 180° – 104° = 76°
m MON = 90° – 76° = 14°
L
P O
M
N
44°
60°
B. MON
Course 1
7-5 Triangles
Triangles can be classified by the lengths of their sides. A scalene triangle has no congruent sides. An isosceles triangle has at least two congruent sides. An equilateral triangle has three congruent sides.
Course 1
7-5 Triangles
Additional Example 3: Classifying Triangles by Lengths of Sides
Classify the triangle. The sum of the lengths of the sides is 19.5 in.
c = 6.5
6.5 in.
N L c
M
c + (6.5 + 6.5) = 19.5 c + 13 = 19.5
c + 13 – 13 = 19.5 – 13
6.5 in.
Side c is 6.5 inches long. Because LMN has three congruent sides, it is equilateral.
Course 1
7-5 Triangles
Try This: Example 3
Classify the triangle. The sum of the lengths of the sides is 21.6 in.
d = 7.2
7.2 in.
C A d
B
d + (7.2 + 7.2) = 21.6 d + 14.4 = 21.6
d + 14.4 – 14.4 = 21.6 – 14.4
7.2 in.
Side d is 7.2 inches long. Because ABC has three congruent sides, it is equilateral.
Lesson Quiz
If the angles can form a triangle, classify the triangle as acute, obtuse, or right.
1. 37°, 53°, 90° 2. 65°, 110°, 25°
3. 61°, 78°, 41° 4. 115°, 25°, 40°
The lengths of three sides of a triangle are given. Classify the triangle.
5. 12, 16, 25 6. 10, 10, 15
not a triangle right
Insert Lesson Title Here
acute obtuse
Course 1
7-5 Triangles
scalene isosceles
7-6 Quadrilaterals
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up The lengths of three sides of a triangle are given. Classify the triangle.
1. 12, 12, 12
2. 18, 10, 14
3. 15, 15, 26
4. 23, 36, 16
equilateral
scalene
isosceles
Course 1
7-6 Quadrilaterals
scalene
Problem of the Day
How many different rectangles are in the figure.
10; if the rectangles are marked 1-5 clockwise from upper left, they are 1; 2; 3; 4; 5; 1-2; 3-4; 4-5; 3-5; 1-2-3-4-5.
Course 1
7-6 Quadrilaterals
Learn to identify, classify, and compare quadrilaterals.
Course 1
7-6 Quadrilaterals
Vocabulary quadrilateral parallelogram rectangle rhombus square trapezoid
Insert Lesson Title Here
Course 1
7-6 Quadrilaterals
Course 1
7-6 Quadrilaterals
A quadrilateral is a plane figure with four sides and four angles.
Five special types of quadrilaterals and their properties will be shown in this lesson. The same mark on two or more sides of a figure indicates that the sides are congruent.
Course 1
7-6 Quadrilaterals
Parallelogram
Opposite sides are parallel and congruent. Opposite angles are congruent.
Rectangle Parallelogram with four right angles.
Course 1
7-6 Quadrilaterals
Rhombus Parallelogram with four congruent sides.
Square Rectangle with four congruent sides.
Course 1
7-6 Quadrilaterals
Trapezoid Quadrilateral with exactly two parallel sides. May have two right angles.
Course 1
7-6 Quadrilaterals
Additional Example 1A: Naming Quadrilaterals
Give the most descriptive name for each figure.
The figure is a quadrilateral and a trapezoid.
Trapezoid is the most descriptive name.
A.
Course 1
7-6 Quadrilaterals
Additional Example 1B: Naming Quadrilaterals
Give the most descriptive name for each figure.
B.
The figure is a quadrilateral, a parallelogram, and a rectangle.
Rectangle is the most descriptive name.
Course 1
7-6 Quadrilaterals
Additional Example 1C: Naming Quadrilaterals
Give the most descriptive name for each figure.
C. The figure is a plane figure with three congruent sides.
Equilateral triangle is the most descriptive name.
Course 1
7-6 Quadrilaterals
Additional Example 1D: Naming Quadrilaterals
Give the most descriptive name for each figure.
D. The figure is a quadrilateral, parallelogram, and rhombus.
Rhombus is the most descriptive name.
Course 1
7-6 Quadrilaterals
Try This: Additional Example 1A
Give the most descriptive name for each figure.
A. The figure is a quadrilateral and a trapezoid.
Trapezoid is the most descriptive name.
Course 1
7-6 Quadrilaterals
Try This: Additional Example 1B
Give the most descriptive name for each figure.
B.
The figure is a quadrilateral.
Parallelogram is the most descriptive name.
Course 1
7-6 Quadrilaterals
Try This: Additional Example 1C
Give the most descriptive name for each figure.
C. The figure is a plane figure, but it has more than 4 sides.
This figure is not a quadrilateral. It is a hexagon.
Course 1
7-6 Quadrilaterals
Try This: Additional Example 1D
Give the most descriptive name for each figure.
D. The figure is a quadrilateral and a parallelogram.
Parallelogram is the most descriptive name.
Course 1
7-6 Quadrilaterals
You can draw a diagram to classify quadrilaterals based on their properties.
Course 1
7-6 Quadrilaterals
Additional Example 2A & 2B: Classifying Quadrilaterals
Complete each statement.
A. A rectangle can also be called a ________.
B. A parallelogram cannot be a __________.
A rectangle has opposite sides that are parallel; it can be called a parallelogram.
?
?
A parallelogram has opposite sides that are parallel; it cannot be called a trapezoid.
Course 1
7-6 Quadrilaterals
Try This: Example 2A & 2B
Complete each statement.
A. A rhombus with four right angles is a ________.
B. A trapezoid has exactly _______ parallel sides, and may have _____ right angles.
A rhombus has four congruent sides, and the opposite sides are parallel. If it has four right angles, then it is a square.
?
A trapezoid is a quadrilateral with exactly two parallel sides and it may have two right angles.
? ?
Lesson Quiz Complete each statement.
1. A quadrilateral with four right angles is a _________.
2. A parallelogram with four right angles and four congruent sides is a _______.
3. A figure with 4 sides and 4 angles is a ______.
4. Give the most descriptive name for this quadrilateral.
square
square or rectangle
Insert Lesson Title Here
quadrilateral
trapezoid
Course 1
7-6 Quadrilaterals
?
?
?
7-7 Polygons
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up True or false?
1. Some trapezoids are parallelograms. 2. Some figures with 4 right angles are
squares.
false
true
Course 1
7-7 Polygons
Problem of the Day
Four square tables pushed together can seat either 8 or 10 people. How many people could 12 square tables pushed together seat?
14, 16, 18, or 26 people
Course 1
7-7 Polygons
Learn to identify regular and not regular polygons and to find the angle measures of regular polygons.
Course 1
7-7 Polygons
Vocabulary polygon regular polygon
Insert Lesson Title Here
Course 1
7-7 Polygons
Course 1
7-7 Polygons
Triangles and quadrilaterals are examples of polygons. A polygon is a closed plane figure formed by three or more line segments. A regular polygon is a polygon in which all sides are congruent and all angles are congruent.
Polygons are named by the number of their sides and angles.
Course 1
7-7 Polygons
Course 1
7-7 Polygons
Additional Example 1A: Identifying Polygons
Tell whether each shape is a polygon. If so, give its name and tell whether it appears to be regular or not regular.
A. The shape is a closed plane figure formed by three or more line segments. polygon
There are five sides and five angles. pentagon
All 5 sides do not appear to be congruent. Not regular
Course 1
7-7 Polygons
Additional Example 1B: Identifying Polygons
Tell whether each shape is a polygon. If so, give its name and tell whether it appears to be regular or not regular.
B.
There are eight sides and eight angles. octagon The sides and angles appear to be congruent. regular
The shape is a closed plane figure formed by three or more line segments. polygon
Course 1
7-7 Polygons
Try This: Example 1A
Tell whether each shape is a polygon. If so, give its name and tell whether it appears to be regular or not regular.
A. There are four sides and four angles. quadrilateral
The sides and angles appear to be congruent. regular
Course 1
7-7 Polygons
Try This: Example 1B
Tell whether each shape is a polygon. If so, give its name and tell whether it appears to be regular or not regular.
B. There are four sides and four angles.
quadrilateral
The sides and angles appear to be congruent.
regular
Course 1
7-7 Polygons
The sum of the interior angle measures in a triangle is 180°, so the sum of the interior angle measures in a quadrilateral is 360°.
Course 1
7-7 Polygons
Additional Example 2: Problem Solving Application
Malcolm designed a wall hanging that was a regular 9-sided polygon (called a nonagon). What is the measure of each angle of the nonagon?
1 Understand the Problem The answer will be the measure of each angle in a nonagon.
List the important information:
• A regular nonagon has 9 congruent sides and 9 congruent angles.
Course 1
7-7 Polygons
2 Make a Plan Make a table to look for a pattern using regular polygons.
Solve 3 Draw some regular polygons and divide each into triangles.
Additional Example 2 Continued
Course 1
7-7 Polygons
Additional Example 2 Continued
720°
Course 1
7-7 Polygons
Additional Example 2 Continued
The number of triangles is always 2 fewer than the number of sides.
A nonagon can be divided into 9 – 2 = 7 triangles.
The sum of the interior angle measures in a nonagon is 7 × 180° = 1,260°.
So the measure of each angle is 1,260° ÷ 9 = 140°.
Course 1
7-7 Polygons Additional Example 2 Continued
Look Back 4
Each angle in a nonagon is obtuse. 140° is a reasonable answer, because an obtuse angle is between 90° and 180°.
Course 1
7-7 Polygons
Try This: Additional Example 2
Sara designed a picture that was a regular 6-sided polygon (called a hexagon). What is the measure of each angle of the hexagon?
1 Understand the Problem The answer will be the measure of each angle in a hexagon.
List the important information:
• A regular hexagon has 6 congruent sides and 6 congruent angles.
Course 1
7-7 Polygons
2 Make a Plan Make a table to look for a pattern using regular polygons.
Solve 3 Draw some regular polygons and divide each into triangles.
Try This: Example 2 Continued
Course 1
7-7 Polygons
Try This: Example 2 Continued
Course 1
7-7 Polygons
The number of triangles is always 2 fewer than the number of sides. A hexagon can be divided into 6 – 2 = 4 triangles.
The sum of the interior angles in a octagon is 4 × 180° = 720°.
So the measure of each angle is 720° ÷ 6 = 120°.
Try This: Example 2 Continued
Course 1
7-7 Polygons
Look Back 4
Each angle in a hexagon is obtuse. 120° is a reasonable answer, because an obtuse angle is between 90° and 180°.
Try This: Example 2 Continued
Lesson Quiz
1. Name each polygon and tell whether it appears to be regular or not regular.
2. What is the measure of each angle in a regular dodecagon (12-sided figure)?
150°
nonagon, regular; octagon, not regular
Insert Lesson Title Here
Course 1
7-7 Polygons
7-8 Geometric Pattern
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Divide.
1. What is the sum of the angle measures in a quadrilateral?
2. What is the sum of the angle measures
in a hexagon? 3. What is the measure of each angle in a
regular octagon?
360°
720°
135°
Course 1
7-8 Geometric Patterns
Problem of the Day
Which three letters come next in the following series: W, T, L, C, N, I, . . .
T, F, S; the letters are the initial letters of the words in the question.
Course 1
7-8 Geometric Patterns
Learn to recognize, describe, and extend geometric patterns.
Course 1
7-8 Geometric Patterns
Course 1
7-8 Geometric Patterns
Additional Example 1A: Extending Geometric Patterns
Identify a possible pattern. Use the pattern to draw the next figure.
A.
Each circle has one more dot than the one to its left. The dots are positioned around the circle from top to bottom and right to left.
So the next figure might be .
Course 1
7-8 Geometric Patterns
Additional Example 1B: Extending Geometric Patterns
Identify a possible pattern. Use the pattern to draw the next figure.
B. Each figure has one or two more circles than the figure to its left.
So the next figure might be .
Course 1
7-8 Geometric Patterns
Try This: Example 1A
Identify a possible pattern. Use the pattern to draw the next figure.
A. Each square has one more dot than the one to its left. The dots are positioned around the square from top to bottom and right to left.
So the next figure might be .
Course 1
7-8 Geometric Patterns
Try This: Example 1B
Identify a possible pattern. Use the pattern to draw the next figure.
B. Each figure has two more triangles than the figure before it. The triangles within the figure are shaded one at a time from left to right.
So the next figure might be .
Course 1
7-8 Geometric Patterns
Additional Example 2A: Completing Geometric Patterns
Identify a possible pattern. Use the pattern to draw the missing figure.
A. The first figure has 1 right triangle. The second figure has 2 right triangles arranged counterclockwise. The fourth figure has 4 right triangles.
So the missing figure might be .
Course 1
7-8 Geometric Patterns
Additional Example 2B: Completing Geometric Patterns
Identify a possible pattern. Use the pattern to draw the missing figure.
B.
The first figure has 1 dot. The second figure is double the first figure, vertically. The third figure is double the second figure, horizontally. The fourth figure could be double the third figure, vertically, and then the fifth figure will be double the fourth figure, horizontally.
So the missing figure might be
.
Course 1
7-8 Geometric Patterns
Try This: Example 2A
Identify a possible pattern. Use the pattern to draw the missing figure.
A. Each figure has 1 more square than the previous figure.
So the missing figure might be .
?
Course 1
7-8 Geometric Patterns
Try This: Example 2B
Identify a possible pattern. Use the pattern to draw the missing figure.
B. Each figure is a right triangle. The first figure has 2 red triangles along the base. The third figure has 4 red triangles, and the last figure has 5.
?
So the missing figure might be .
Course 1
7-8 Geometric Patterns
Additional Example 3: Art Application
Travis is painting a platter. Identify a pattern that Travis is using and draw what the finished platter might look like.
The pattern from inside to outside is narrow stripe, narrow stripe, wide stripe, narrow stripe, narrow stripe, wide strip. The color pattern from inside to outside is brown, green, brown, green.
Following this pattern, the finished platter might look like this platter.
Course 1
7-8 Geometric Patterns
Try This: Example 3
Nancy is designing a plate. Identify a pattern that Nancy is using and draw what the finished plate might look like.
The pattern from inside to outside is narrow stripe, wide stripe, narrow stripe, wide stripe. The color pattern from inside to outside is dark blue, light blue, dark blue, light blue.
Following this pattern, the finished plate might look like this plate.
Lesson Quiz
Identify a possible pattern. Use the pattern to draw the next figure.
Insert Lesson Title Here
Course 1
7-8 Geometric Patterns
Possible answer:
7-9 Congruence
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up
1. A straight angle has what degree measure? 2. How do you name a line?
180°
Name any two points on a line.
Course 1
7-9 Congruence
Problem of the Day
The sum of two decimals is 9.3; their difference is 4.3, and their product is 17.00. What are they? 2.5, 6.8
Course 1
7-9 Congruence
Learn to identify congruent figures and to use congruence to solve problems.
Course 1
7-9 Congruence
Course 1
7-9 Congruence Additional Example 1A: Identifying Congruent
Figures
These figures do not have the same shape and they are not the same size.
These figures are not congruent.
Decide whether the figures in each pair are congruent. If not, explain.
A.
Course 1
7-9 Congruence
Additional Example 1B: Identifying Congruent Figures
Decide whether the figures in each pair are congruent. If not, explain.
B. These figures have the same shape size and size.
These figures are congruent.
Course 1
7-9 Congruence
Additional Example 1C: Identifying Congruent Figures
Decide whether the figures in each pair are congruent. If not, explain.
C. Each figure is a trapezoid. The corresponding sides are 5.7 in., 7.5 in., 5 in., and 10 in.
These figures are congruent.
Course 1
7-9 Congruence
Additional Example 1D: Identifying Congruent Figures
Decide whether the figures in each pair are congruent. If not, explain.
D. Each figure is a rectangle. Each short side of each rectangle measure 4 cm. Each long side of each rectangle measures 9 cm.
These figures are congruent.
Course 1
7-9 Congruence
Try This: Example 1A
Decide whether the figures in each pair are congruent. If not, explain.
A. These figures do not have the same shape and they are not the same size.
These figures are not congruent.
Course 1
7-9 Congruence
Try This: Example 1B
Decide whether the figures in each pair are congruent. If not, explain.
B.
These figures have the same shape size and size.
These figures are congruent.
Course 1
7-9 Congruence
Try This: Example 1C
Decide whether the figures in each pair are congruent. If not, explain.
C. Each figure is a triangle. The corresponding sides are 3 cm, 4 cm, and 5 cm.
These figures are congruent.
3 cm
3 cm
4 cm
4 cm 5 cm
5 cm
Course 1
7-9 Congruence
Try This: Example 1D
Decide whether the figures in each pair are congruent. If not, explain.
D.
Each figure is a rectangle. Each short side of each rectangle measures 5 in. Each long side of each rectangle measures 12 in. These figures are congruent.
12 in. 12 in.
5 in.
5 in.
Course 1
7-9 Congruence Additional Example 2: Consumer Application Jodi needs a sleeping pad that is congruent to her sleeping bag. Which sleeping pad should she buy?
Both sleeping pads are trapezoids. Only sleeping pad B is the same size as the sleeping bag.
Which sleeping pad is the same size and shape as the sleeping bag?
Sleeping pad B is congruent to the sleeping bag.
Course 1
7-9 Congruence
Try This: Example 2
Jodi needs a frame that is congruent to her flag. Which frame should she buy?
Both frames are triangles. Only Frame B is the same size as the flag.
Which frame is the same size and shape as the flag?
Frame B is congruent to the flag.
11.2 in.
7.8 in.
4 in. 10 in. 11.2 in.
7.8 in. 9.2 in.
4 in. 4 in.
Flag Frame A Frame B
Lesson Quiz True or false?
1. If two figures have sides of the same length, they are congruent.
2. Congruent figures have angle measures that are equal.
3. Decide whether the figures are congruent. If not, explain.
true
false
Insert Lesson Title Here
No, both are quadrilaterals but they are not the same size or shape.
Course 1
7-9 Congruence
7-10 Transformations
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Tell whether the figures described are congruent.
1. two triangles, each with sides measuring 24 cm, 32 cm, and 40 cm 2. a square with sides of length 22 cm and a rectangle with side lengths 12 cm and 11 cm
congruent
not congruent
Course 1
7-10 Transformations
Problem of the Day
Imagine that each figure is folded so that point A lies on point B. What figures would be formed?
trapezoid, right triangle
Course 1
7-10 Transformations
Learn to use translations, reflections, and rotations to transform geometric shapes.
Course 1
7-10 Transformations
Vocabulary transformation translation rotation reflection line of reflection
Insert Lesson Title Here
Course 1
7-10 Transformations
A rigid transformation moves a figure without changing its size or shape. So the original figure and the transformed figure are always congruent.
The illustrations of the alien will show three transformations: a translation, a rotation, and a reflection. Notice the transformed alien does not change in size or shape.
Course 1
7-10 Transformations
A translation is the movement of a figure along a straight line.
Only the location of the figure changes with a translation.
Course 1
7-10 Transformations
A rotation is the movement of a figure around a point. A point of rotation can be on or outside a figure.
Course 1
7-10 Transformations
The location and position of a figure can change with a rotation.
When a figure flips over a line, creating a mirror image, it is called a reflection. The line the figure is flipped over is called line of reflection.
The location and position of a figure change with a reflection.
Course 1
7-10 Transformations
Additional Example 1A: Identifying Transformations
Tell whether each is a translation, rotation, or reflection.
A.
The figure is flipped over a line.
It is a reflection.
Course 1
7-10 Transformations
Additional Example 1B: Identifying Transformations
Tell whether each is a translation, rotation, or reflection.
The figure is moved along a line.
It is a translation.
Course 1
7-10 Transformations
B.
Additional Example 1C: Identifying Transformations
Tell whether each is a translation, rotation, or reflection.
C. The figure moves around a point.
It is a rotation.
Course 1
7-10 Transformations
Try This: Example 1A
Tell whether each is a translation, rotation, or reflection.
A.
The figure is flipped over a line.
It is a reflection.
Course 1
7-10 Transformations
Try This: Example 1B
Tell whether each is a translation, rotation, or reflection.
B.
The figure moves around a point.
It is a rotation.
Course 1
7-10 Transformations
Try This: Example 1C
Tell whether each is a translation, rotation, or reflection.
C.
The figure is moved along a line.
It is a translation.
Course 1
7-10 Transformations
A full turn is a 360°
rotation. So a turn
is 90°, and a turn
is 180°.
1 2 __
1 4 __
90°
180°
360°
Course 1
7-10 Transformations
Additional Example 2A: Drawing Transformations
Draw each transformation.
A. Draw a 180° rotation about the point shown.
Trace the figure and the point of rotation.
Place your pencil on the point of rotation.
Rotate the figure 180°.
Trace the figure in its new location.
Course 1
7-10 Transformations
Additional Example 2B: Drawing Transformations
Draw each transformation.
B. Draw a horizontal reflection.
Trace the figure and the line of reflection.
Fold along the line of reflection.
Trace the figure in its new location.
Course 1
7-10 Transformations
Try This: Example 2A
Draw each transformation.
A. Draw a 180° rotation about the point shown.
Trace the figure and the point of rotation.
Place your pencil on the point of rotation.
Rotate the figure 180°.
Trace the figure in its new location.
Course 1
7-10 Transformations
Try This: Example 2B
Draw each transformation.
B. Draw a horizontal reflection.
Trace the figure and the line of reflection.
Fold along the line of reflection.
Trace the figure in its new location.
Course 1
7-10 Transformations
Lesson Quiz
1. Tell whether the figure is translated, rotated, or reflected.
2. Draw a vertical reflection of the first figure in problem 1.
rotated
Insert Lesson Title Here
Course 1
7-10 Transformations
7-11 Symmetry
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up True or false?
1. In a reflection the original figure is flipped over a line of symmetry.
2. In a translation the original figure slides
along a straight line and is flipped. 3. In a rotation a figure is moved around a
point.
true
false
true
Course 1
7-11 Symmetry
Problem of the Day
Name four plane figures that can be rotated 180° around a center point and look the same after the rotation as they did before. Possible answer: rectangle, square, parallelogram, regular hexagon
Course 1
7-11 Symmetry
Learn to identify line symmetry.
Course 1
7-11 Symmetry
Vocabulary line symmetry line of symmetry
Insert Lesson Title Here
Course 1
7-11 Symmetry
A figure has line symmetry if it can be folded or reflected so that the two parts of the figure match, or are congruent. The line of reflection is called the line of symmetry.
Course 1
7-11 Symmetry
Additional Example 1A: Identifying Lines of Symmetry
Determine whether each dashed line appears to be a line of symmetry.
A. The two parts of the figure appear to match exactly when folded or reflected across the line.
The line appears to be a line of symmetry.
Course 1
7-11 Symmetry
Additional Example 1B: Identifying Lines of Symmetry
Determine whether each dashed line appears to be a line of symmetry.
B. The two parts of the figure do not appear congruent.
The line does not appear to be a line of symmetry.
Course 1
7-11 Symmetry
Try This: Example 1A
Determine whether each dashed line appears to be a line of symmetry.
A.
The two parts of the figure do not appear congruent.
The line does not appear to be a line of symmetry.
Course 1
7-11 Symmetry
Try This: Example 1B
Determine whether each dashed line appears to be a line of symmetry.
B. The two parts of the figure appear to match exactly when folded or reflected across the line.
The line appears to be a line of symmetry.
Course 1
7-11 Symmetry
Some figures have more than one line of symmetry.
Course 1
7-11 Symmetry
Additional Example 2A: Finding Multiple Lines of Symmetry
Find all of the lines of symmetry in the regular polygon.
A. Trace the figure and cut it out.
Fold the figure in half in different ways.
Count the lines of symmetry.
5 lines of symmetry
Course 1
7-11 Symmetry
Additional Example 2B: Finding Multiple Lines of Symmetry
Find all of the lines of symmetry in the regular polygon.
B.
Count the lines of symmetry.
8 lines of symmetry
Course 1
7-11 Symmetry
Additional Example 2C: Finding Multiple Lines of Symmetry
Find all of the lines of symmetry in the regular polygon.
C.
Count the lines of symmetry.
4 lines of symmetry
Course 1
7-11 Symmetry
Try This: Example 2A
Find all of the lines of symmetry in the regular polygon.
A.
1 line of symmetry
Count the lines of symmetry.
Course 1
7-11 Symmetry
Try This: Example 2B
Find all of the lines of symmetry in the regular polygon.
B.
Count the lines of symmetry.
4 lines of symmetry
Course 1
7-11 Symmetry
Try This: Example 2C
Find all of the lines of symmetry in the regular polygon.
C. Count the lines of symmetry.
4 lines of symmetry
Course 1
7-11 Symmetry
Additional Example 3A: Social Studies Application
Find all of the lines of symmetry in the flag.
A. Alaska
There are no lines of symmetry.
Course 1
7-11 Symmetry
Additional Example 3B: Social Studies Application
Find all of the lines of symmetry in the flag.
B. Arizona
1 line of symmetry Course 1
7-11 Symmetry
Additional Example 3C: Social Studies Application
Find all of the lines of symmetry in the flag.
C. Colorado
1 line of symmetry Course 1
7-11 Symmetry
Additional Example 3D: Social Studies Application
Find all of the lines of symmetry in the flag.
D. New Mexico
2 lines of symmetry Course 1
7-11 Symmetry
Try This: Example 3A
2 lines of symmetry
Course 1
7-11 Symmetry
Find all of the lines of symmetry in each design.
There are no lines of symmetry. Course 1
7-11 Symmetry
Try This: Example 3B
Find all of the lines of symmetry in each design.
1 line of symmetry Course 1
7-11 Symmetry
Try This: Example 3C
Find all of the lines of symmetry in each design.
2 lines of symmetry Course 1
7-11 Symmetry
Try This: Example 3D
Find all of the lines of symmetry in each design.
Lesson Quiz
Does the described figure have line symmetry?
1. right isosceles triangle
2. rectangle
Do the following capital letters of the alphabet have symmetry? If so, is the line of symmetry horizontal or vertical?
3. H 4. R
yes
yes
Insert Lesson Title Here
yes, vertical and horizontal no
Course 1
7-11 Symmetry
7-12 Tessellations
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Do the following letters of the alphabet have symmetry, and if so, is the line of symmetry horizontal or vertical?
1. X 2. P 3. C 4. T
yes; both no yes; horizontal
Course 1
7-12 Tessellations
yes; vertical
Problem of the Day
If either diagonal is drawn in a figure, the figure is divided into two congruent triangles. What is the most general classification of this figure? parallelogram
Course 1
7-12 Tessellations
Learn to identify tessellations and shapes that can tessellate.
Course 1
7-12 Tessellations
Vocabulary tessellation
Insert Lesson Title Here
Course 1
7-12 Tessellations
A tessellation is a repeating arrangement of one or more shapes that completely covers a plane with no gaps and no overlaps.
Although most tessellations are made by humans, a few occur in nature. Honeycombs are naturally occurring tessellations of regular hexagons.
Course 1
7-12 Tessellations
Additional Example 1A: Identifying Polygons that Tessellate the Plane
Identify whether the polygon can tessellate the plane. Make a drawing to show your answer.
A. The trapezoids cover the plane without any gaps or overlaps.
The trapezoid can tessellate the plane.
Course 1
7-12 Tessellations
Additional Example 1B: Identifying Polygons that Tessellate the Plane
Identify whether the polygon can tessellate the plane. Make a drawing to show your answer.
B. There are gaps between the pentagons.
The pentagon cannot tessellate the plane.
Course 1
7-12 Tessellations
Try This: Example 1A
Identify whether the polygon can tessellate the plane. Make a drawing to show your answer.
A. The triangles cover the plane without any gaps or overlaps.
The triangle can tessellate the plane.
Course 1
7-12 Tessellations
Try This: Example 1B
Identify whether the polygon can tessellate the plane. Make a drawing to show your answer.
B.
There are no gaps between the parallelograms.
The parallelogram can tessellate the plane.
Course 1
7-12 Tessellations
Additional Example 2A: Identifying Nonpolygons That Tessellate the Plane
Identify whether the shape can tessellate the plane. Make a drawing to show your answer.
The shapes cover the plane without any gaps or overlaps.
This shape can tessellate the plane.
Course 1
7-12 Tessellations
A.
Additional Example 2B: Identifying Nonpolygons That Tessellate the Plane
Identify whether the shape can tessellate the plane. Make a drawing to show your answer.
B.
The shapes cover the plane without any gaps or overlaps.
This shape can tessellate the plane.
Course 1
7-12 Tessellations
Try This: Example 2A
Identify whether the shape can tessellate the plane. Make a drawing to show your answer.
A. There are gaps between the shapes.
This shape cannot tessellate the plane.
Course 1
7-12 Tessellations
Try This: Example 2B
Identify whether the shape can tessellate the plane. Make a drawing to show your answer.
B.
The shapes cover the plane with any gaps or overlaps.
This shape can tessellate the plane.
Course 1
7-12 Tessellations
Lesson Quiz
True or false?
1. A regular hexagon can tessellate the plane.
2. A regular octagon can tessellate the plane.
3. A square can tessellate the plane.
4. A regular pentagon can tessellate the plane.
false
true
Insert Lesson Title Here
true
false
Course 1
7-12 Tessellations
8-1 Ratios and Rates
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Write each fraction in simplest form.
1. 2. 3. 4.
Course 1
8-1 Ratios and Rates
2 6 __ 1
3 __ 4
16 __ 1
4 __
25 70 ___ 5
14 __ 6
52 __ 3
26 __
Problem of the Day
What three consecutive odd numbers are factors of 105?
3, 5, 7
Course 1
8-1 Ratios and Rates
Learn to write ratios and rates and to find unit rates.
Course 1
8-1 Ratios and Rates
Vocabulary ratios equivalent ratios rate unit rate
Insert Lesson Title Here
Course 1
8-1 Ratios and Rates
Course 1
8-1 Ratios and Rates
You can compare the different groups by using ratios. A ratio is a comparison of two quantities using division.
For a time, the Boston Symphony Orchestra was made up of 95 musicians.
Violins 29 Violas 12 Cellos 10 Basses 9 Flutes 5 Trumpets 3 Double reeds 8 Percussion 5 Clarinets 4 Harp 1 Horns 6 Trombones 3
Course 1
8-1 Ratios and Rates
For example, you can use a ratio to compare the number of violins (29) with the number of violas (12). This ratio can be written in three ways.
Notice that the ratio of violins to violas, is
different from the ratio of violas to violins, . The order of the terms is important.
29 12 ___
29 12 ___
12 29 ___
29 to 12 29:12 Terms
Ratios can be written to compare a part to a part, a part to the whole, or the whole to a part.
Course 1
8-1 Ratios and Rates
Read the ratio as “twenty-nine to twelve.”
Reading Math 29
12 ___
Course 1
8-1 Ratios and Rates
Additional Example 1A: Writing Ratios
Use the table to write the ratio.
A. cats to rabbits
Animals at the Vet Cats 5 Dogs 7 Rabbits 2
or 5 to 2 or 5:2 5 2 __ Part to part
Course 1
8-1 Ratios and Rates
Additional Example 1B: Writing Ratios
Use the table to write the ratio.
B. dogs to total number of pets
Animals at the Vet Cats 5 Dogs 7
Rabbits 2
or 7 to 14 or 7:14 7 14
__ Part to whole
Course 1
8-1 Ratios and Rates
Additional Example 1C: Writing Ratios
Use the table to write the ratio.
C. total number of pets to cats
Animals at the Vet Cats 5 Dogs 7
Rabbits 2
or 14 to 5 or 14:5 14 5
__ Whole to part
Course 1
8-1 Ratios and Rates
Try This: Example 1A
Use the table to write the ratio.
A. birds to total number of pets
Animals at the Vet Birds 6
Hamsters 9 Snakes 3
or 6 to 18 or 6:18 6 18
__ Part to whole
Course 1
8-1 Ratios and Rates
Try This: Example 1B
Use the table to write the ratio.
B. snakes to birds
or 3 to 6 or 3:6 3 6
__ Part to part
Animals at the Vet Birds 6
Hamsters 9 Snakes 3
Course 1
8-1 Ratios and Rates
Try This: Example 1C
Use the table to write the ratio.
C. total number of pets to hamsters
or 18 to 9 or 18:9 18 9
__ Whole to part
Animals at the Vet Birds 6
Hamsters 9 Snakes 3
Course 1
8-1 Ratios and Rates
Equivalent ratios are ratios that name the same comparison. You can find an equivalent ratio by multiplying or dividing both terms of a ratio by the same number.
Course 1
8-1 Ratios and Rates
Additional Example 2: Writing Equivalent Ratios
Write three equivalent ratios to compare the number of diamonds to the number of spades in the pattern.
There are 3 diamonds and 6 spades.
number of diamonds number of spades =
3 6 __
There is 1 diamond for every 2 spades.
3 6 __ =
3 ÷ 3 6 ÷ 3 ____ =
1 2 __
3 6 __ = 9
18 __ =
If you triple the pattern, there will be 9 diamonds for 18 spades.
So , , and are equivalent ratios. 3 6 __ 1
2 __ 9
18 __
3 • 3 6 • 3
Course 1
8-1 Ratios and Rates Try This: Example 2
Write three equivalent ratios to compare the number of triangles to the number of hearts in the pattern.
There are 3 triangles and 9 hearts.
number of triangles number of hearts =
3 9 __
There is 1 triangle for every 3 hearts.
3 9 __ =
3 ÷ 3 9 ÷ 3 ____ =
1 3 __
3 9 __ = 9
27 __ =
If you triple the pattern, there will be 9 triangles for 27 hearts.
So , , and are equivalent ratios. 3 9 __ 1
3 __ 9
27 __
3 • 3 9 • 3
Course 1
8-1 Ratios and Rates
A rate compares two quantities that have different units of measure. Suppose a 2-liter bottle of soda costs $1.98.
When the comparison is to one unit, the rate is called a unit rate. Divide both terms by the second term to find the unit rate.
When the prices of two or more items are compared, the item with the lowest unit rate is the best deal.
rate = price number of liters _____________ $1.98
2 liters ________ = $1.98 for 2 liters
unit rate = $1.98 2 _____ $1.98 ÷ 2
2 ÷ 2 ________ = $0.99 for 1 liter = $0.99
1 _____
Course 1
8-1 Ratios and Rates
Additional Example 3: Consumer Application
A 3-roll pack of paper towels costs $2.79. A 6-roll pack of the same paper towels costs $5.46. Which is the better deal?
Write the rate. $2.79 3 rolls _____
$2.79 ÷ 3 3 rolls ÷ 3 _________ Divide both
terms by 3.
$0.93 1 roll _____ $0.93 for 1 roll.
Write the rate. $5.46 6 rolls _____
$5.46 ÷ 6 6 rolls ÷ 6 _________ Divide both
terms by 6.
$0.91 1 roll _____ $0.91 for 1 roll.
The 6-roll pack of paper towels is the better deal.
Course 1
8-1 Ratios and Rates
Try This: Example 3
A 3-pack of juice boxes costs $2.10. A 9-pack of the same juice boxes costs $5.58. Which is the better deal?
Write the rate. $2.10 3 pack _____
$2.10 ÷ 3 3 pack ÷ 3 _________ Divide both
terms by 3.
$0.70 1 box _____ $0.70 for 1 juice
box.
Write the rate. $5.58 9 pack _____
$5.58 ÷ 9 9 pack ÷ 9 _________ Divide both
terms by 9.
$0.62 1 box _____ $0.62 for 1 juice
box.
The 9-pack of juice boxes is the better deal.
Lesson Quiz
Use the table to
write each ratio.
1. tulips to daffodils
2. crocuses to total number of bulbs
3. total bulbs to lilies
4. A dozen eggs cost $1.25 at one market. At a competing market, 1 dozen eggs cost $2.00. Which is the better buy?
20:51
9:17
Insert Lesson Title Here
51:5
eggs from the first market
Course 1
8-1 Ratios and Rates
1 2 __
8-2 Proportions
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Use the table to write each ratio.
1. giraffes to monkeys 2. polar bears to all bears 3. monkeys to all animals 4. all animals to all bears
2:17 4:7
17:26
Course 1
8-2 Proportions
Brown bears 3 Giraffes 2 Monkeys 17 Polar Bears 4
26:7
Problem of the Day
A carpenter can build one doghouse in one day. How many doghouses can 12 carpenters build in 20 days? 240
Course 1
8-2 Proportions
Learn to write and solve proportions.
Course 1
8-2 Proportions
Vocabulary proportion
Insert Lesson Title Here
Course 1
8-2 Proportions
Course 1
8-2 Proportions
A proportion is an equation that shows two equivalent ratios.
Read the proportion = as “two is to
one as four is to two.”
2 1 __ 4
2 __ = 4
2 __ 8
4 __ = 2
1 __ 6
3 __ =
2 1 __ 4
2 __
Course 1
8-2 Proportions
Additional Example 1: Modeling Proportions
Write a proportion for the model.
number of hearts number of stars
4 8 __ =
First write the ratio of hearts to stars.
Next separate the hearts and stars into two equal groups.
Course 1
8-2 Proportions
Additional Example 1 Continued
Now write the ratio of hearts to stars in each group.
number of hearts number of stars
2 4 __ =
A proportion shown by the model is = . 4 8 __ 2
4 __
Course 1
8-2 Proportions
Try This: Example 1
Write a proportion for the model.
number of faces number of moons
2 6 __ =
First write the ratio of faces to moons.
Next separate the faces and moons into two equal groups.
Course 1
8-2 Proportions
Try This: Example 1
Now write the ratio of hearts to stars in each group.
number of faces number of moons
1 3 __ =
A proportion shown by the model is = . 2 6 __ 1
3 __
Course 1
8-2 Proportions
CROSS PRODUCTS
Cross products in proportions are equal 3 5 __ 9
15 ___ = 4
8 __ 2
4 __ = 9
6 __ 3
2 __ = 14
7 __ 2
1 __ =
8 • 2 = 4 • 4 5 • 9 = 3 • 15
16 = 16 45 = 45
6 • 3 = 9 • 2
18 = 18
7 • 2 = 14 • 1
14 = 14
Course 1
8-2 Proportions
Additional Example 2: Using Cross Products to Complete Proportions
Find the missing value in the proportion.
6n 6
___
5 6 __ n
18 __ = Find the cross products.
6 • n = 5 • 18 The cross products are equal.
6n = 90 n is multiplied by 6.
90 6
___ = Divide both sides by 6 to undo the multiplication.
n = 15
Course 1
8-2 Proportions
Try This: Example 2
Find the missing value in the proportion.
5n 5
___
3 5 __ n
15 __ = Find the cross products.
5 • n = 3 • 15 The cross products are equal.
5n = 45 n is multiplied by 5.
45 5
___ = Divide both sides by 5 to undo the multiplication.
n = 9
Course 1
8-2 Proportions
In a proportion, the units must be in the same order in both ratios.
Helpful Hint
tsp lb
___ tsp lb
___ =
lb tsp
___ lb tsp
___ = or
Course 1
8-2 Proportions Additional Example 3: Measurement Application
According to the label, 1 tablespoon of plant fertilizer should be used per 6 gallons of water. How many tablespoons of fertilizer would you use for 4 gallons of water?
f 4 gal
____ 1 tsp 6 gal
_____ = Let f be the amount of fertilizer for 4 gallons of water.
f 4 gal
____ 1 tsp 6 gal
_____ = Write a proportion.
6 • f = 1 • 4 The cross products are equal.
Course 1
8-2 Proportions
Additional Example 3 Continued
f is multiplied by 6.
Divide both sides by 6 to undo the multiplication.
Write your answer in simplest form.
6f = 4
6f 6
___ 4 6
__ =
f = tbsp 2 3
__
You would use tbsp of fertilizer for 4 gallons of water.
2 3
__
Course 1
8-2 Proportions
Try This: Example 3
According to the label, 3 tablespoon of plant fertilizer should be used per 9 gallons of water. How many tablespoons of fertilizer would you use for 2 gallons of water?
f 2 gal
____ 3 tsp 9 gal
_____ = Let f be the amount of fertilizer for 2 gallons of water.
f 2 gal
____ 3 tsp 9 gal
_____ = Write a proportion.
9 • f = 3 • 2 The cross products are equal.
Course 1
8-2 Proportions
Try This: Example 3 Continued
f is multiplied by 9.
Divide both side by 9 to undo the multiplication.
Write your answer in simplest form.
9f = 6
9f 9
___ 6 9
__ =
f = tbsp 2 3
__
You would use tbsp of fertilizer for 2 gallons of water.
2 3
__
Lesson Quiz
1. Write a proportion for the model.
Find the missing value in each proportion.
2. = 3. =
4. The label on a bottle of salad dressing states that there are 3 grams of fat per tablespoon. If you use 3 tablespoons, how many grams of fat would you be getting?
p = 30 x = 25
Insert Lesson Title Here
9 g
Course 1
8-2 Proportions
2 6
__ 1 3
__ =
9 5
__ 45 x
__ p 36
__ 5 6
__
8-3 Proportions and Customary Measurement
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Find the missing value in each proportion.
1. = 2. = 3. = 4. =
x = 6
x = 25
x = 35
Course 1
8-3 Proportions and Customary Measurement
4 1
__ 24 x
__
x 30
__ 5 6
__
14 16
__ x 40
__
8 x
__ 12 18
__ x = 12
Problem of the Day
There are 4 ounces in a gill. There are 4 gills in a pint. There are 8 pints in a gallon. How many ounces are the same as the total of 3 gallons, 3 pints, 3 gills, and 3 ounces? 447
Course 1
8-3 Proportions and Customary Measurement
Learn to use proportions to make conversions within the customary system.
Course 1
8-3 Proportions and Customary Measurement
Course 1
8-3 Proportions and Customary Measurement
Additional Example 1A: Using Proportions to Convert Measurements
A. Sonja went hiking for 4 hours. For how many minutes did she go hiking?
4 hr x min
____ 1 hr 60 min
_____ = 1 hour is 60 minutes. Write a proportion. Use a variable for the value you are trying to find. The cross products are equal. 60 • 4 = 1 • x
240 = x
Sonja hiked for 240 minutes.
Course 1
8-3 Proportions and Customary Measurement
Additional Example 1B: Using Proportions to Convert Measurements
B. Mr. Lee is 72 inches tall. Find his height in feet.
x ft 72 in.
____ 1 ft 12 in
____ = 1 foot is 12 inches. Write a proportion. Use a variable for the value you are trying to find. The cross products are equal. 12 • x = 1 • 72 x is multiplied by 12. 12x = 72
Mr. Lee is 6 feet tall.
12x 12
___ 72 12
___ = Divide both sides by 12 to undo the multiplication.
x = 6
Course 1
8-3 Proportions and Customary Measurement
Additional Example 1C: Using Proportions to Convert Measurements
C. Hunter used 56 cups of water to wash his car. How many gallons of water did he use?
x gal 56 cups
______ 1 gal 16 cups
_______ = 1 gallon is 16 cups. Write a proportion. Use a variable for the value you are trying to find. The cross products are equal. 16 • x = 1 • 56 x is multiplied by 16. 16x = 56
Hunter used 3.5 gallons of water to wash his car.
16x 16
___ 56 16
___ = Divide both sides by 16 to undo the multiplication.
x = 3.5
Course 1
8-3 Proportions and Customary Measurement
Try This: Example 1A
A. Helga went biking for 3 hours. For how many minutes did she go biking?
3 hr x min
____ 1 hr 60 min
_____ = 1 hour is 60 minutes. Write a proportion. Use a variable for the value you are trying to find.
The cross products are equal. 60 • 3 = 1 • x
180 = x
Helga biked for 180 minutes.
Course 1
8-3 Proportions and Customary Measurement
Try This: Example 1B
B. Ms. Hagan is 60 inches tall. Find her height in feet.
x ft 60 in.
____ 1 ft 12 in.
____ = 1 foot is 12 inches. Write a proportion. Use a variable for the value you are trying to find.
The cross products are equal. 12 • x = 1 • 60
x is multiplied by 12. 12x = 60
Ms. Hagan is 5 feet tall.
12x 12
___ 60 12
___ = Divide both sides by 12 to undo the multiplication.
x = 5
Course 1
8-3 Proportions and Customary Measurement
Try This: Example 1C
C. Seymour used 40 cups of water to wash his dog. How many gallons of water did he use?
x gal 40 cups
______ 1 gal 16 cups
_______ = 1 gallon is 16 cups. Write a proportion. Use a variable for the value you are trying to find.
The cross products are equal. 16 • x = 1 • 40
x is multiplied by 16. 16x = 40
Seymour used 2.5 gallons of water to wash his dog.
16x 16
___ 40 16
___ = Divide both sides by 16 to undo the multiplication.
x = 2.5
Course 1
8-3 Proportions and Customary Measurement
Lesson Quiz
Find the missing value.
1. in. = 6 yd
2. 24 pt = gal
Compare. Write <, >, or =.
3. 42 oz 2 lb.
4. 7,920 ft 1.5 mi
5. If your family is going to take a 3-week vacation, how many days will you be gone?
3
216
Insert Lesson Title Here
>
=
1 2
__
21 days Course 1
8-3 Proportions and Customary Measurement
8-4 Similar Figures
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Fill in the missing value.
1. c = 2 qt
2. 180 in. = yd
3. 3 tons = lb
4. min = 2,760 s
8
5
6,000
Course 1
8-4 Similar Figures
45
Problem of the Day
How many 8 in. by 10 in. rectangular tiles would be needed to cover a 16 ft by 20 ft floor? 576
Course 1
8-4 Similar Figures
Learn to use ratios to identify similar figures.
Course 1
8-4 Similar Figures
Vocabulary similar corresponding sides corresponding angles
Insert Lesson Title Here
Course 1
8-4 Similar Figures
Course 1
8-4 Similar Figures
Two or more figures are similar if they have exactly the same shape. Similar figures may be different sizes. Similar figures have corresponding sides and corresponding angles.
• Corresponding sides have lengths that are proportional.
• Corresponding angles are congruent.
Course 1
8-4 Similar Figures
A D
B C
3 cm
2 cm 2 cm
3 cm
W Z
X Y
9 cm
9 cm
6 cm 6 cm
Corresponding sides: Corresponding angles:
AB corresponds to WX.
AD corresponds to WZ. CD corresponds to YZ. BC corresponds to XY.
A corresponds to W. B corresponds to X. C corresponds to Y. D corresponds to Z.
Course 1
8-4 Similar Figures
A D
B C
3 cm
2 cm 2 cm
3 cm
W Z
X Y
9 cm
9 cm
6 cm 6 cm
In the rectangles above, one proportion is
= , or = . AB WX
AD WZ
2 6
3 9
If you cannot use corresponding side lengths to write a proportion, or if corresponding angles are not congruent, then the figures are not similar.
Course 1
8-4 Similar Figures Additional Example 1: Finding Missing
Measures in Similar Figures
111 y
___ 100 200
____ = Write a proportion using corresponding side lengths. The cross products are equal. 200 • 111 = 100 • y
The two triangles are similar. Find the missing length y and the measure of D.
Course 1
8-4 Similar Figures
Additional Example 1 Continued
y is multiplied by 100. 22,200 = 100y
22,200 100
______ 100y 100
____ = Divide both sides by 100 to undo the multiplication.
222 mm = y
Angle D is congruent to angle C, and m C = 70°.
m D = 70°
The two triangles are similar. Find the missing length y and the measure of D.
Course 1
8-4 Similar Figures
Try This: Example 1
52 y
___ 50 100
____ = Write a proportion using corresponding side lengths.
The cross products are equal. 100 • 52 = 50 • y
A B 60 m 120 m
50 m 100 m
y 52 m
65°
45°
The two triangles are similar. Find the missing length y and the measure of B.
Course 1
8-4 Similar Figures
Try This: Example 1 Continued
y is multiplied by 50. 5,200 = 50y
5,200 50
_____ 50y 50
___ = Divide both sides by 50 to undo the multiplication.
104 m = y
Angle B is congruent to angle A, and m A = 65°.
m B = 65°
The two triangles are similar. Find the missing length y and the measure of B.
Course 1
8-4 Similar Figures
Additional Example 2: Problem Solving Application This reduction is similar to a picture that Katie painted. The height of the actual painting is 54 centimeters. What is the width of the actual painting?
1 Understand the Problem The answer will be the width of the actual painting. List the important information: • The actual painting and the reduction above are similar. • The reduced painting is 2 cm tall and 3 cm wide. • The actual painting is 54 cm tall.
Course 1
8-4 Similar Figures
Additional Example 2 Continued
Draw a diagram to represent the situation. Use the corresponding sides to write a proportion.
2 Make a Plan
Reduced Actual
2 54
3 w
Course 1
8-4 Similar Figures
Additional Example 2 Continued
Solve 3
54 • 3 = 2 • w
162 = 2w
162 2
____ 2w 2
___ =
81 = w
The width of the actual painting is 81 cm.
Write a proportion.
The cross products are equal.
w is multiplied by 2.
Divide both sides by 2 to undo the multiplication.
3 cm w cm
2 cm 54 cm
_____ =
Course 1
8-4 Similar Figures
Additional Example 2 Continued
Look Back 4 Estimate to check your answer. The ratio of the heights is about 2:50 or 1:25. The ratio of the widths is about 3:90, or 1:30. Since these ratios are close to each other, 81 cm is a reasonable answer.
Course 1
8-4 Similar Figures
Try This: Example 2
This reduction is similar to a picture that Marty painted. The height of the actual painting is 39 inches. What is the width of the actual painting?
1 Understand the Problem The answer will be the width of the actual painting. List the important information: • The actual painting and the reduction above are similar. • The reduced painting is 3 in. tall and 4 in. wide. • The actual painting is 39 in. tall.
4 in.
3 in.
Course 1
8-4 Similar Figures
Try This: Example 2 Continued
Draw a diagram to represent the situation. Use the corresponding sides to write a proportion.
2 Make a Plan
Reduced Actual
3 39
4 w
Course 1
8-4 Similar Figures
Try This: Example 2 Continued
Solve 3 4 in w in
____ 3 in 39 in
_____ =
39 • 4 = 3 • w
156 = 3w
156 3
____ 3w 3
___ =
52 = w
The width of the actual painting is 52 inches.
Write a proportion.
The cross products are equal.
w is multiplied by 3.
Divide both sides by 3 to undo the multiplication.
Course 1
8-4 Similar Figures
Try This: Example 2 Continued
Look Back 4 Estimate to check your answer. The ratio of the heights is about 4:40, or 1:10. The ratio of the widths is about 5:50, or 1:10. Since these ratios are the same, 52 inches is a reasonable answer.
Lesson Quiz These two triangles are similar.
1. Find the missing length x.
2. Find the measure of J.
3. Find the missing length y.
4. Find the measure of P.
5. Susan is making a wood deck from plans for an 8 ft by 10 ft deck. However, she is going to increase its size proportionally. If the length is to be 15 ft, what will the width be?
36.9°
30 in.
Insert Lesson Title Here
4 in.
90°
Course 1
8-4 Similar Figures
12 ft
8-5 Indirect Measurement
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Find the missing value in each proportion.
1. = 2. = 3. = 4. =
t = 15
k = 5
n = 56
Course 1
8-5 Indirect Measurement
6 t
__ 18 45
__
k 19
__ 20 76
__
6 8
__ 42 n
__
21 11
__ x 44
__ x = 84
Problem of the Day
Bryce, Kate, and Annie have drawn rectangles. Each side of Bryce’s rectangle is twice the size of one side of Kate’s. The same side of Kate’s rectangle is congruent to one side of Annie’s. One side of Annie’s rectangle is congruent to one side of Bryce’s. Which two rectangles could be congruent? Kate’s and Annie’s
Course 1
8-5 Indirect Measurement
Learn to use proportions and similar figures to find unknown measures.
Course 1
8-5 Indirect Measurement
Vocabulary indirect measurement
Insert Lesson Title Here
Course 1
8-5 Indirect Measurement
Course 1
8-5 Indirect Measurement
One way to find a height that you cannot measure directly is to use similar figures and proportions. This method is called indirect measurement.
Course 1
8-5 Indirect Measurement
Additional Example 1: Using Indirect Measurement
Use the similar triangles to find the height of the tree.
2 7
__ 6 h
__ =
h • 2 = 6 • 7
2h = 42
2h 2
___ 42 2
___ =
h = 21
Write a proportion using corresponding sides.
The cross products are equal.
h is multiplied by 2.
Divide both sides by 2 to undo multiplication.
The tree is 21 feet tall.
Course 1
8-5 Indirect Measurement
Try This: Example 1
Use the similar triangles to find the height of the tree.
3 9
__ 6 h
__ =
h • 3 = 6 • 9
3h = 54
3h 3
___ 54 3
___ =
h = 18
Write a proportion using corresponding sides.
The cross products are equal.
h is multiplied by 3.
Divide both sides by 3 to undo multiplication.
The tree is 18 feet tall.
3 ft. 9 ft.
6 ft. h
Course 1
8-5 Indirect Measurement
Additional Example 2: Measurement Application
A rocket casts a shadow that is 91.5 feet long. A 4-foot model rocket casts a shadow that is 3 feet long. How tall is the rocket?
91.5 3
____ h 4
__ =
4 • 91.5 = h • 3
366 = 3h
366 3
___ 3h 3
___ =
122 = h
Write a proportion using corresponding sides.
The cross products are equal.
h is multiplied by 3.
Divide both sides by 3 to undo multiplication.
The rocket is 122 feet tall.
Course 1
8-5 Indirect Measurement
Try This: Example 2
A building casts a shadow that is 72.5 feet long when a 4-foot model building casts a shadow that is 2 feet long. How tall is the building?
72.5 2
____ h 4
__ =
4 • 72.5 = h • 2
290 = 2h
290 2
___ 2h 2
___ =
145 = h
Write a proportion using corresponding sides. The cross products are equal.
h is multiplied by 2. Divide both sides by 2 to undo multiplication.
The building is 145 feet tall.
h
72.5 ft 2 ft
4 ft
Lesson Quiz
1. On a sunny day, a telephone pole casts a shadow 21 ft long. A 5-foot-tall mailbox next to the pole casts a shadow 3 ft long. How tall is the pole?
2. On a sunny afternoon, a goalpost casts a 75 ft shadow. A 6.5 ft football player next to the goal post has a shadow 19.5 ft long. How tall is the goalpost?
25 feet
35 feet
Insert Lesson Title Here
Course 1
8-5 Indirect Measurement
8-6 Scale Drawings and Maps
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Find the unknown heights.
1. A tower casts a 56 ft shadow. A 5 ft girl next to it casts a 3.5 ft shadow. How tall is the tower?
2. On a sunny day, a 50 ft silo casts a 10 ft
shadow. The barn next to the silo casts a shadow that is 4 ft long. How tall is the barn?
80 ft
20 ft
Course 1
8-6 Scale Drawings and Maps
Problem of the Day
Hal runs 4 miles in 32 minutes. Julie runs 5 miles more than Hal runs. If Julie runs at the same rate as Hal, for how many minutes will Julie run? 72 minutes
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8-6 Scale Drawings and Maps
Learn to read and use map scales and scale drawings.
Course 1
8-6 Scale Drawings and Maps
Vocabulary scale drawing scale
Insert Lesson Title Here
Course 1
8-6 Scale Drawings and Maps
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8-6 Scale Drawings and Maps The map shown is a scale drawing. A scale drawing is a drawing of a real object that is proportionally smaller or larger than the real object. In other words, measurements on a scale drawing are in proportion to the measurements of the real object.
A scale is a ratio between two sets of measurements. In the map above, the scale is 1 in:100 mi. This ratio means that 1 inch on the map represents 100 miles.
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8-6 Scale Drawings and Maps Additional Example 1: Finding Actual Distances The scale on a map is 4 in: 1 mi. On the map, the distance between two towns is 20 in. What is the actual distance?
20 in. x mi
_____ 4 in. 1 mi
____ =
1 • 20 = 4 • x 20 = 4x 20 4
___ 4x 4
___ =
5 = x
Write a proportion using the scale. Let x be the actual number of miles between the two towns. The cross products are equal. x is multiplied by 4. Divide both sides by 4 to undo multiplication.
The actual distance between the two towns is 5 miles.
Course 1
8-6 Scale Drawings and Maps
In Additional Example 1, think “4 inches is 1 mile, so 20 inches is how many miles?” This approach will help you set up proportions in similar problems.
Helpful Hint
Course 1
8-6 Scale Drawings and Maps
Try This: Example 1
18 in. x mi
_____ 3 in. 1 mi
____ =
1 • 18 = 3 • x 18 = 3x 18 3
___ 3x 3
___ =
6 = x
Write a proportion using the scale. Let x be the actual number of miles between the two cities. The cross products are equal. x is multiplied by 3. Divide both sides by 3 to undo multiplication.
The actual distance between the two cities is 6 miles.
The scale on a map is 3 in: 1 mi. On the map, the distance between two cities is 18 in. What is the actual distance?
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8-6 Scale Drawings and Maps Additional Example 2A: Astronomy Application A. If a drawing of the planets were made using the scale 1 in:30 million km, the distance from Mars to Jupiter on the drawing would be about 18.3 in. What is the actual distance between Mars to Jupiter?
18.3 in. x million km
_________ 1 in. 30 million km
___________ =
30 • 18.3 = 1 • x
549 = x
Write a proportion. Let x be the actual distance from Mars to Jupiter.
The cross products are equal.
The actual distance from Mars to Jupiter is about 549 million km.
Course 1
8-6 Scale Drawings and Maps Additional Example 2B: Astronomy Application B. The actual distance from Earth to Mars is about 78 million kilometers. How far apart should Earth and Mars be drawn?
x in. 78 million km
__________ 1 in. 30 million km
___________ =
30 • x = 1 • 78
x = 2
Write a proportion. Let x be the distance from Earth to Mars on the drawing. The cross products are equal.
Earth and Mars should be drawn 2 inches apart.
30x = 78 30x 30
___ 78 30
___ =
x is multiplied by 30. Divide both sides by 30 to undo multiplication.
3 5 __
3 5 __
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8-6 Scale Drawings and Maps Try This: Additional Example 2A A. If a drawing of the planets were made using the scale 1 in:15 million km, the distance from Mars to Venus on the drawing would be about 8 in. What is the actual distance from Mars to Venus?
8 in. x million km
_________ 1 in. 15 million km
___________ =
15 • 8 = 1 • x
120 = x
Write a proportion. Let x be the distance from Mars to Venus.
The cross products are equal.
The actual distance from Mars to Venus is about 120 million km.
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8-6 Scale Drawings and Maps Try This: Example 2B B. The distance from Earth to the Sun is about 150 million kilometers. How far apart should Earth and the Sun be drawn?
x in. 150 mil km
________ 1 in. 15 mil km
________ =
15 • x = 1 • 150
x = 10
Write a proportion. Let x be the distance from Earth to the Sun on the drawing. The cross products are equal.
Earth and the Sun should be drawn 10 inches apart.
15x = 150 15x 15
___ 150 15
____ =
x is multiplied by 15.
Divide both sides by 15 to undo multiplication.
Lesson Quiz
On a map of the Great Lakes, 2 cm = 45 km. Find the actual distance of the following, given their distances on the map.
1. Detroit to Cleveland = 12 cm
2. Duluth to Nipigon = 20 cm
3. Buffalo to Syracuse = 10 cm
4. Sault Ste. Marie to Toronto = 33 cm
450 km
270 km
Insert Lesson Title Here
225 km
742.5 km
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8-6 Scale Drawings and Maps
8-7 Percents
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Warm Up
Lesson Presentation
Problem of the Day
Warm Up Write each fraction as a decimal.
1. 2. Write each decimal as a fraction. 3. 0.375 4. 0.05
0.75 0.9
Course 1
8-7 Percents
3 4 __ 9
10 __
3 8 __ 1
20 __
Problem of the Day
Wally wanted to change the scale of a drawing from 1 in. = 2 ft to 1 in. = 10 ft. The scale height of a building in the first drawing is 25 in. How high is the building in the new drawing? 5 in.
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8-7 Percents
Learn to write percents as decimals and as fractions.
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8-7 Percents
Vocabulary percent
Insert Lesson Title Here
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8-7 Percents
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8-7 Percents
Most states charge sales tax on items you purchase. Sales tax is a percent of the item’s price. A percent is a ratio of a number to 100.
You can remember that percent means “per hundred.” For example, 8% means “8 per hundred,” or “8 out of 100.”
Course 1
8-7 Percents
If a sales tax rate is 8%, the following statements are true:
• For every $1.00 you spend, you pay $0.08 in sales tax.
• For every $10.00 you spend, you pay $0.80 in sales tax.
• For every $100 you spend, you pay $8 in sales tax.
Because percent means “per hundred,” 100% means “100 out of 100.” This is why 100% is often used to mean “all” or “the whole thing.”
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8-7 Percents Additional Example 1: Modeling Percent
Use a 10-by-10-square grid to model 17%.
A 10-by-10 square grid has 100 squares.
17% means “17 out of 100”
or .
Shade 17 squares out of 100 squares.
17 100
___
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8-7 Percents Try This: Example 1
Use a 10-by-10-square grid to model 26%.
A 10-by-10 square grid has 100 squares.
26% means “26 out of 100”
or .
Shade 26 squares out of 100 squares.
26 100
___
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8-7 Percents Additional Example 2: Writing Percents as
Fractions
Write 35% as a fraction in simplest form.
Write the percent as a fraction with a denominator of 100.
Write the fraction in simplest form.
35% = 35 100
___
35 ÷ 5 100 ÷ 5
_______ = 7 20 __
Written as a fraction, 35% is . 7 20 __
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8-7 Percents Try This: Example 2
Write 65% as a fraction in simplest form.
Write the percent as a fraction with a denominator of 100.
Write the fraction in simplest form.
65% = 65 100
___
65 ÷ 5 100 ÷ 5
_______ = 13 20 __
Written as a fraction, 65% is . 13 20 __
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8-7 Percents Additional Example 3: Life Science Application
Janell is an ice skater with 20% body fat. Write 20% as a fraction in simplest form.
Write the percent as a fraction with a denominator of 100.
Write the fraction in simplest form.
20% = 20 100
___
20 ÷ 20 100 ÷ 20
_______ = 1 5 __
Written as a fraction, 20% is . 1 5 __
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8-7 Percents Try This: Example 3
Timmy is a football player with 10% body fat. Write 10% as a fraction in simplest form .
Write the percent as a fraction with a denominator of 100.
Write the fraction in simplest form.
10% = 10 100
___
10 ÷ 10 100 ÷ 10
_______ = 1 10
__
Written as a fraction, 10% is . 1 10 __
Course 1
8-7 Percents
Additional Example 4: Writing Percents as Decimals
Write 56% as a decimal. Write the percent as a fraction with a denominator of 100.
Write the fraction as a decimal.
56% = 56 100
___
Written as a fraction, 56% is 0.56.
100 56.00 0.56
–500 600
–600 0
To divide by 100, move the decimal point two places to the left. 56 ÷ 100 = 0.56
Remember!
Course 1
8-7 Percents Try This: Example 4
Write 32% as a decimal.
Write the percent as a fraction with a denominator of 100.
Write the fraction as a decimal.
32% = 32 100
___
Written as a fraction, 32% is 0.32.
100 32.00 0.32
–300 200
–200 0
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8-7 Percents Additional Example 5: Application
Water made up 85% of the fluids that Kirk drank yesterday. Write 85% as a decimal.
Write the percent as a fraction with a denominator of 100.
Write the fraction as a decimal.
85% = 85 100
___
85 ÷ 100 = 0.85
Written as a decimal, 85% is 0.85.
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8-7 Percents Try This: Example 5
Water made up 95% of the fluids that Lisa drank yesterday. Write 95% as a decimal.
Write the percent as a fraction with a denominator of 100.
Write the fraction as a decimal.
95% = 95 100
___
95 ÷ 100 = 0.95
Written as a decimal, 95% is 0.95.
Lesson Quiz
Write each percent as a fraction in simplest form.
1. 52% 2. 29%
Write each percent as a decimal.
3. 17% 4. 86%
5. A store clerk has an 8% sales increase. Write the increase as a fraction in simplest form and as a decimal.
Insert Lesson Title Here
, 0.08
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8-7 Percents
13 25 __ 29
100 ___
0.17 0.86
2 25 __
8-8 Percents, Decimals, and Fractions
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Warm Up
Lesson Presentation
Problem of the Day
Warm Up Write each percent as a fraction in simplest form.
1. 37% 2. 78% Write each percent as a decimal. 3. 59% 4. 7% 0.59 0.07
Course 1
8-8 Percents, Decimals, and Fractions
37 100
___ 39 50
___
Problem of the Day
Jennifer gave Karen 50% of her comic books. Karen gave Jack 50% of the books she got from Jennifer. Jack gave Lucas 50% of the books he got from Karen. Lucas got 4 comic books. What percent of Jennifer’s comic books does Lucas have? 12.5%
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8-8 Percents, Decimals, and Fractions
Learn to write decimals and fractions as percents.
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8-8 Percents, Decimals, and Fractions
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8-8 Percents, Decimals, and Fractions Additional Example 1A: Writing Decimals as
Percents
Write each decimal as a percent.
Method 1: Use place value.
A. 0.7
Write the decimal as a fraction.
Write an equivalent fraction with 100 as the denominator.
0.7 = 7 10
__
7 • 10 10 • 10
_______ = 70 100
___
70 100
___ = 70% Write the numerator with a percent symbol.
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8-8 Percents, Decimals, and Fractions Additional Example 1B: Writing Decimals as
Percents
Write each decimal as a percent.
Method 1: Use place value.
B. 0.16
Write the decimal as a fraction. 0.16 = 16 100
__
16 100
___ = 16% Write the numerator with a percent symbol.
Course 1
8-8 Percents, Decimals, and Fractions Additional Example 1C: Writing Decimals as
Percents
Write each decimal as a percent.
Method 2: Multiply by 100.
C. 0.4118
Multiply by 100. 0.4118 • 100
41.18% Add the percent symbol.
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8-8 Percents, Decimals, and Fractions Additional Example 1D: Writing Decimals as
Percents
Write each decimal as a percent.
Method 2: Multiply by 100.
D. 0.067
Multiply by 100. 0.067 • 100
6.7% Add the percent symbol.
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8-8 Percents, Decimals, and Fractions Try This: Example 1A
Write each decimal as a percent.
Method 1: Use place value.
A. 0.9
Write the decimal as a fraction.
Write an equivalent fraction with 100 as the denominator.
0.9 = 9 10
__
9 • 10 10 • 10
_______ = 90 100
___
90 100
___ = 90% Write the numerator with a percent symbol.
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8-8 Percents, Decimals, and Fractions Try This: Example 1B
Write each decimal as a percent.
Method 1: Use place value.
B. 0.31
Write the decimal as a fraction. 0.31 =
31 100
___ = 31% Write the numerator with a percent symbol.
31 100
Course 1
8-8 Percents, Decimals, and Fractions Try This: Example 1C
Write each decimal as a percent.
Method 2: Multiply by 100.
C. 0.1225
Multiply by 100. 0.1225 • 100
12.25% Add the percent symbol.
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8-8 Percents, Decimals, and Fractions Try This: Example 1D
Write each decimal as a percent.
Method 2: Multiply by 100.
D. 0.023
Multiply by 100. 0.023 • 100
2.3% Add the percent symbol.
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8-8 Percents, Decimals, and Fractions
When the denominator is a factor of 100, it is often easier to use method 1. When the denominator is not a factor of 100, it is usually easier to use method 2.
Helpful Hint
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8-8 Percents, Decimals, and Fractions Additional Example 2A: Writing Fractions as
Percents
Write an equivalent fraction with 100 as the denominator.
9 • 4 25 • 4
______ = 36 100
___
36 100
___ = 36% Write the numerator with a percent symbol.
9 25
__
Write each fraction as a percent.
Method 1: Write an equivalent fraction with a denominator of 100.
A.
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8-8 Percents, Decimals, and Fractions Additional Example 2B: Writing Fractions as
Percents
Divide the numerator by the denominator.
0.15 = 15% Multiply by 100 by moving the decimal point right two places. Add the percent symbol.
3 20
__
100
20 3.00 0.15
–20
–100 0
Method 2: Use division to write the fraction as a decimal.
B.
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8-8 Percents, Decimals, and Fractions Try This: Example 2A
Write each fraction as a percent.
Method 1: Write an equivalent fraction with a denominator of 100.
A.
Write an equivalent fraction with 100 as the denominator.
7 • 2 50 • 2
______ = 14 100
___
14 100
___ = 14% Write the numerator with a percent symbol.
7 50
__
Course 1
8-8 Percents, Decimals, and Fractions
Try This: Example 2B
Divide the numerator by the denominator.
0.16 = 16% Multiply by 100 by moving the decimal point right by two places. Add the percent symbol.
8 50
__
50 8.00 0.16
Method 2: Use division to write the fraction as a decimal.
B.
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8-8 Percents, Decimals, and Fractions
Additional Example 3: Application
One year, of people with home offices
were self-employed. What percent of people with home offices were self employed?
Write an equivalent fraction with a denominator of 100.
7 • 4 25 • 4
______ = 28 100
___
28 100
___ = 28%
7 25
__
7 25
__
28% of people with home offices were self-employed.
Course 1
8-8 Percents, Decimals, and Fractions Try This: Example 3
One year, of people who bought new cars
bought cars that were yellow. What percent of people that bought new cars bought yellow cars?
Write an equivalent fraction with a denominator of 100.
3 • 2 50 • 2
______ = 6 100
___
6 100
___ = 6%
3 50
__
3 50
__
6% of people with new cars bought yellow cars.
Course 1
8-8 Percents, Decimals, and Fractions
Lesson Quiz
Write each decimal as a percent.
1. 0.26 2. 0.419
Write each fraction as a percent.
3. 4.
5. About of all the students at a local high
school own their own car. What percent is this?
41.9% 26%
Insert Lesson Title Here
20% 56.25%
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8-8 Percents, Decimals, and Fractions
1 5
__ 9 16
__
1 16
__
6.25%
8-9 Percent Problems
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Write each decimal as a percent and fraction.
1. 0.38 2. 0.06 3. 0.2
38%,
6%,
20%,
Course 1
8-9 Percent Problems
19 50
__
3 50
__
1 5
__
Problem of the Day
Lucky Jim won $16,000,000 in a lottery. Every year for 10 years he spent 50% of what was left. How much did Lucky Jim have after 10 years? $15,625
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8-9 Percent Problems
Learn to find the missing value in a percent problem.
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8-9 Percent Problems
Course 1
8-9 Percent Problems
To find the percent of one number is of another, use this proportion:
% 100 = is
of
If you are looking for 45% of 420, 45 replaces the percent sign and 420 replaces “of.” The first denominator, 100, always stays the same. The “is” part is what you have been asked to find.
Course 1
8-9 Percent Problems
Additional Example 1: Application
First estimate your answer. Think: 35% = ,
which is close to , and 560 is close to 600. So
about of the students participate in after-school sports.
35 100
___
1 3
__
1 3
__
1 3
__ This is the estimate. • 600 = 200
Think: “35 out of 100 is how many out of 560?” Helpful Hint
There are 560 students in Ella’s school. If 35% of the students participate in after-school sports, how many students participate in after-school sports?
Course 1
8-9 Percent Problems
Additional Example 1 Continued Now solve:
s 560
___ 35 100
___ =
100 • s = 35 • 560 100s = 19,600 100s 100
____ 19,600 100
_____ =
s = 196
Let s represent the number of students who participate in after-school sports. The cross products are equal. s is multiplied by 100. Divide both sides of the equation by 100 to undo multiplication.
Since 196 is close to your estimate of 200, 196 is a reasonable answer.
196 students participate in after-school sports.
Course 1
8-9 Percent Problems
Try This: Example 1
There are 480 students in Tisha’s school. If 70% of the students participate in the fundraising program, how many students participate in the fundraising program?
First estimate your answer. Think: 70% = ,
which is close to , and 480 is close to 500. So
about of the students participate in after
school sports.
70 100
___
3 4
__
3 4
__
This is the estimate. 3 4
__ • 500 = 375
Course 1
8-9 Percent Problems
Try This: Example 1 Continued Now solve:
s 480
___ 70 100
___ =
100 • s = 70 • 480 100s = 33,600 100s 100
____ 33,600 100
_____ =
s = 336
Let s represent the number of students who participate in after-school sports. The cross products are equal. s is multiplied by 100. Divide both sides of the equation by 100 to undo multiplication.
Since 336 is close to your estimate of 375, 336 is a reasonable answer.
336 students participate in the fundraising program.
Course 1
8-9 Percent Problems
Additional Example 2: Application
Johan is 25% of the way through his exercises. If he has exercised for 20 minutes so far, how much longer does he have to work out?
is of
__ % 100
___ =
100 • 20 = 25 • m
25% of the exercises are completed, so 20 minutes is 25% of the total time needed.
The cross products are equal.
20 m
__ 25 100
___ = Set up a proportion. The “of” part is what you have been asked to find.
Course 1
8-9 Percent Problems
Additional Example 2 Continued
2,000 = 25m
2,000 25
_____ 25m 25
____ =
80 = m
m is multiplied by 25.
Divide both sides by 25 to undo multiplication.
The time needed for the exercises is 80 min. So far, the exercises have taken 20 min. Because 80 – 20 = 60, Johan will be finished in 60 min.
Course 1
8-9 Percent Problems
Try This: Example 2
Phil is 30% of the way through his homework. If he has worked for 15 minutes so far, how much longer does he have to work?
is of
__ % 100
___ =
100 • 15 = 30 • m
30% of the exercises are completed, so 15 minutes is 30% of the total time needed.
The cross products are equal.
15 m
__ 30 100
___ = Set up a proportion. The “of” part is what you have been asked to find.
Course 1
8-9 Percent Problems
Try This: Example 2 Continued
1,500 = 30m
1,500 30
_____ 30m 30
____ =
50 = m
m is multiplied by 30.
Divide both sides by 30 to undo multiplication.
The time needed for the homework is 50 min. So far, the homework has taken 15 min. Because 50 – 15 = 35, Phil will be finished in 35 min.
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8-9 Percent Problems
Instead of using proportions, you can also multiply to find a percent of a number.
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8-9 Percent Problems Additional Example 3: Multiplying to Find a
Percent of a Number
36% = 0.36 0.36 • 50
Write the percent as a decimal. Multiply using the decimal.
18 So 18 is 36% of 50.
17% = 0.17 0.17 • 95
Write the percent as a decimal. Multiply using the decimal.
16.15 So 16.15 is 17% of 95.
A. Find 36% of 50.
B. Find 17% of 95.
Course 1
8-9 Percent Problems Try This: Example 3
20% = 0.20 0.20 • 70
Write the percent as a decimal. Multiply using the decimal.
14 So 14 is 20% of 70.
12% = 0.12 0.12 • 83
Write the percent as a decimal. Multiply using the decimal.
9.96 So 9.96 is 12% of 83.
A. Find 20% of 70.
B. Find 12% of 83.
Lesson Quiz
1. Find 28% of 310.
2. Find 70% of 542.
3. Martha is taking a 100-question test. She has completed 60% of the test in 45 minutes. How much longer will it take her to finish the test?
4. Crystal has a collection of 72 pennies. If 25% of them are Canadian, how many Canadian pennies does she have?
379.4
86.8
Insert Lesson Title Here
30 min
18
Course 1
8-9 Percent Problems
8-10 Using Percents
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Find the percent of each number.
1. 75% of 300 2. 93% of 56 3. 32% of 128 4. 9% of 60
225 52.08 40.96
Course 1
8-10 Using Percents
5.4
Problem of the Day
A chessboard is 8 squares wide by 8 squares long. Each player has 8 pawns, 1 king, and 7 other pieces. At the start of a game, all the pieces are on the board, 1 piece per square. What percent of the total number of squares have a chess piece? 50%
Course 1
8-10 Using Percents
Learn to solve percent problems that involve discounts, tips, and sales tax.
Course 1
8-10 Using Percents
Vocabulary discount tip sales tax
Insert Lesson Title Here
Course 1
8-10 Using Percents
Common Uses of Percents Discounts A discount is an amount that is
subtracted from the regular price of an item. discount = price • discount rate total cost = price – discount
Tips A tip is an amount added to a bill for service. tip = bill • tip rate total cost = bill + tip
Sales tax Sales tax is an amount added to the price of an item.
sales tax = price • sales tax rate total cost = price + sales tax
Course 1
8-10 Using Percents
Additional Example 1: Finding Discounts A clothing store is having a 10% off sale. If Angela wants to buy a sweater whose regular price is $19.95, about how much will she pay for the sweater after the discount? Step 1: First round $19.95 to $20. Step 2: Find 10% of $20 by multiplying 0.10 • 20. (Hint: Moving the decimal point one place left is a shortcut.) 10% of 20 = 0.10 • $20 = $2.00
The approximate discount is $2.00. Subtract this amount from $20.00 to estimate the cost of the sweater. $20.00 – $2.00 = $18.00 Angela will pay about $18.00 for the sweater.
Course 1
8-10 Using Percents
To multiply by 0.10, move the decimal point one place left.
Remember!
Course 1
8-10 Using Percents
Try This: Example 1 A fishing store is having a 10% off sale. If Gerald wants to buy a fishing pole whose regular price is $39.95, about how much will he pay for the pole after the discount? Step 1: First round $39.95 to $40. Step 2: Find 10% of $40 by multiplying 0.10 • 40.
10% of 40 = 0.10 • $40 = $4.00
The approximate discount is $4.00. Subtract this amount from $40.00 to estimate the cost of the pole. $40.00 - $4.00 = $36.00
Gerald will pay about $36.00 for the fishing pole.
Course 1
8-10 Using Percents
When estimating percents, use percents that you can calculate mentally.
• You can find 10% of a number by moving the decimal point one place to the left.
• You can find 1% of a number by moving the decimal point two places to the left.
• You can find 5% of a number by finding one-half of 10% of the number.
Course 1
8-10 Using Percents
Additional Example 2: Finding Tips
Ben’s dinner bill is $7.85. He wants to leave a tip that is 15% of the bill. About how much should his tip be? Step 1: First round $7.85 to $8. Step 2: Think: 15% = 10% + 5%
Step 3: 5% = 10% ÷ 2 = $0.80 ÷ 2 = $0.40
Step 4: 15% = 10% + 5%.
Ben should leave about $1.20 as a tip.
= $0.80 + $0.40 = $1.20
10% of $8 = 0.10 • $8 = $0.80
Course 1
8-10 Using Percents
Try This: Example 2
Lita’s dinner bill is $11.95. She wants to leave a tip that is 15% of the bill. About how much should her tip be? Step 1: First round $11.95 to $12. Step 2: Think: 15% = 10% + 5%
Step 3: 5% = 10% ÷ 2 = $1.20 ÷ 2 = $0.60
Step 4: 15% = 10% + 5%.
Lita should leave about $1.80 as a tip.
= $1.20 + $0.60 = $1.80
10% of $12 = 0.10 • $12 = $1.20
Course 1
8-10 Using Percents
Additional Example 3: Finding Sales Tax Ann is buying a dog bed for $29.75. The sales tax rate is 7%. About how much will the total cost of the dog bed be? Step 1: First round $29.75 to $30. Step 2: Think: 7% = 7 • 1%
1% of $30 = 0.01 • $30 = $0.30 Step 3: 7% = 7 • 1%.
= 7 • $0.30 = $2.10. The approximate sales tax is $2.10. Add this amount to $30 to estimate the total cost of the dog bed.
$30 + $2.10 = $32.10
Ann will pay about $32.10 for the dog bed. Course 1
8-10 Using Percents
Try This: Example 3 Erik is buying a blanket for $19.83. The sales tax rate is 8%. About how much will the total cost of the blanket be? Step 1: First round $19.83 to $20. Step 2: Think: 8% = 8 • 1%
1% of $20 = 0.01 • $20 = $0.20 Step 3: 8% = 8 • 1%
= 8 • $0.20 = $1.60. The approximate sales tax is $1.60. Add this amount to $20 to estimate the total cost of the blanket.
$20 + $1.60 = $21.60
Erik will pay about $21.60 for the blanket. Course 1
8-10 Using Percents
Lesson Quiz
1. Sean’s new jeans are priced at $29.97, but the sale sign reads, “Take 15% off.” About how much will the jeans cost after the discount?
2. The bill for a family dinner comes to $56.78. About how much would a 20% tip be?
3. The price on a book is $12.99. If sales tax is 4%, about how much will its total cost be?
4. Megan wants a new bike. She is happy to see a sign that reads, “All bikes 10% off.” If the original price of the bike was $159.90 and sales tax is 6%, about how much will the total cost of the bike be?
About $11.50
About $25.50
Insert Lesson Title Here
About $13.50
About $150 Course 1
8-10 Using Percents
9-1 Understanding Integers
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Add or subtract.
1. 16 + 25 2. 84 – 12 3. Graph the even numbers from 1 to 10 on
a number line.
41 72
Course 1
9-1 Understanding Integers
0 1 2 3 4 5 6 7 8 9 10
Problem of the Day
Carlo uses a double-pan balance and three different weights to weigh bird seed. If his weights are 1 lb, 2 lb, and 5 lb, what whole pound amounts is he able to weigh? 1, 2, 3, 5, 6, 7, and 8 lb
Course 1
9-1 Understanding Integers
Learn to identify and graph integers, find opposites, and find the absolute value of an integer.
Course 1
9-1 Understanding Integers
Vocabulary positive number negative number opposites integer absolute value
Insert Lesson Title Here
Course 1
9-1 Understanding Integers
Course 1
9-1 Understanding Integers
Positive numbers are greater than 0. They may be written with a positive sign (+), but they are usually written without it.
Negative numbers are less than 0. They are always written with a negative sign (–).
Course 1
9-1 Understanding Integers
Additional Example 1A & 1B: Identifying Positive and Negative Numbers in the Real World
Name a positive or negative number to represent each situation.
A. a jet climbing to an altitude of 20,000 feet
B. taking $15 out of the bank
Positive numbers can represent climbing or rising.
+20,000
Negative numbers can represent taking out or withdrawing. –15
Course 1
9-1 Understanding Integers
Additional Example 1 Continued
Name a positive or negative number to represent each situation.
C. 7 degrees below zero
Negative numbers can represent values below or less than a certain value. –7
Course 1
9-1 Understanding Integers
Try This: Additional Example 1A & 1B
Name a positive or negative number to represent each situation.
A. 300 feet below sea level
B. a hiker hiking to an altitude of 4,000 feet
Negative numbers can represent values below or less than a certain value. –300
Positive numbers can represent climbing or rising. +4,000
Course 1
9-1 Understanding Integers
Try This: Example 1C
Name a positive or negative number to represent each situation?
C. spending $34
Negative numbers can represent losses or decreases. –34
Course 1
9-1 Understanding Integers
You can graph positive and negative numbers on a number line. On a number line, opposites are the same distance from 0 but on different sides of 0. Integers are the set of all whole numbers and their opposites.
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
Opposites
Positive Integers Negative Integers
0 is neither negative nor positive.
Course 1
9-1 Understanding Integers
The set of whole numbers includes zero and the counting numbers.
{0, 1, 2, 3, 4, …}
Remember!
Course 1
9-1 Understanding Integers
Additional Example 2A & 2B: Graphing Integers
Graph each integer and its opposite on a number line.
A. +2
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
B. –3
–2 is the same distance from 0 as +2.
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
+3 is the same distance from 0 as –3.
Course 1
9-1 Understanding Integers
Additional Example 2C: Graphing Integers
Graph each integer and its opposite on a number line.
C. +1
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
–1 is the same distance from 0 as +1.
Course 1
9-1 Understanding Integers
Try This: Example 2A & 2B
Graph each integer and its opposite on a number line.
A. +5
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
B. –4
–5 is the same distance from 0 as +5.
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
+4 is the same distance from 0 as –4.
Course 1
9-1 Understanding Integers
Try This: Example 2C
Graph each integer and its opposite on a number line.
C. +4
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
–4 is the same distance from 0 as +4.
Course 1
9-1 Understanding Integers
|–3| = 3 |3| = 3
The absolute value of an integer is its distance from 0 on a number line. The symbol for absolute value is | |.
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
3 units 3 units
Course 1
9-1 Understanding Integers
• Absolute values are never negative.
• Opposite integers have the same absolute value.
• |0| = 0
Read |3| as “the absolute value of 3.”
Read |–3| as “the absolute value of negative 3.”
Reading Math
Course 1
9-1 Understanding Integers
Additional Example 3A & 3B: Finding Absolute Value
Use the number line to find the absolute value of each integer.
A. |–4|
B. |2|
–4 is 4 units from 0, so |–4| = 4. 4
2 is 2 units from 0, so |2| = 2. 2
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
Course 1
9-1 Understanding Integers
Try This: Example 3A & 3B
Use the number line to find the absolute value of each integer.
A. |–3|
B. |1|
–3 is 3 units from 0, so |–3| = 3. 3
1 is 1 unit from 0, so |1| = 1. 1
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
Lesson Quiz Name a positive or negative number to represent each situation.
1. saving $15
2. 12 feet below sea level
3. What is the opposite of –6?
4. What is the absolute value of –12?
5. When the Swanton Bulldogs football team passed the football, they gained 25 yards. Write an integer to represent this situation.
–12
+15
Insert Lesson Title Here
6
12
Course 1
9-1 Understanding Integers
+25
9-2 Comparing and Ordering Integers
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Compare. Write <, >, or =.
1. 8,426 8,246
2. 9,625 6,852
3. 2,071 2,171
4. 2,250 2,250
>
>
<
Course 1
9-2 Comparing and Ordering Integers
=
Problem of the Day
Four friends are waiting in line at the amusement park. Jenna is in front of Kyle. Kyle is behind Gary and in front of Maggie. Gary is first. In what order are they waiting?
Gary, Jenna, Kyle, Maggie
Course 1
9-2 Comparing and Ordering Integers
Learn to compare and order integers.
Course 1
9-2 Comparing and Ordering Integers
Course 1
9-2 Comparing and Ordering Integers
Numbers on a number line increase in value as you move from left to right.
Remember!
Course 1
9-2 Comparing and Ordering Integers
Additional Example 1: Comparing Integers Use the number line to compare each pair of integers. Write < or >.
A. –2 2
B. 3 –5
C. –1 –4
–5 –4 –3 –2 –1 0 1 2 3 4 5
–2 is to the left of 2 on the number line. –2 < 2
3 > –5 3 is to the right of –5 on the number line.
–1 is to the right of –4 on the number line. –1 > –4
Course 1
9-2 Comparing and Ordering Integers
Try This: Example 1
Use the number line to compare each pair of integers. Write < or >.
A. –2 1
B. 2 –3
C. –3 –4
–5 –4 –3 –2 –1 0 1 2 3 4 5
–2 is to the left of 1 on the number line. –2 < 1
2 > –3 2 is to the right of –3 on the number line.
–3 is to the right of –4 on the number line. –3 > –4
Course 1
9-2 Comparing and Ordering Integers Additional Example 2: Ordering Integers
Order the integers in each set from least to greatest.
A. –2, 3, –1
B. 4, –3, –5, 2
–3 –2 –1 0 1 2 3
Graph the integers on the same number line.
Then read the numbers from left to right: –2, –1, 3.
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6
Graph the integers on the same number line.
Then read the numbers from left to right: –5, –3, 2, 4.
Course 1
9-2 Comparing and Ordering Integers Try This: Example 2
Order the integers in each set from least to greatest.
A. –2, 2, –3
B. 6, –2, 5, –3
–3 –2 –1 0 1 2 3
Graph the integers on the same number line.
Then read the numbers from left to right: –3, –2, 2.
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6
Graph the integers on the same number line.
Then read the numbers from left to right: –3, –2, 5, 6.
Course 1
9-2 Comparing and Ordering Integers
Additional Example 3: Problem Solving Application
1 Understand the Problem
The answer will be the player with the lowest score. List the important information:
• Craig scored +2.
• Cameron scored +3.
• Rob scored –1.
In a golf match, Craig scored +2, Cameron scored +3, and Rob scored –1. Who won the golf match?
Course 1
9-2 Comparing and Ordering Integers
Additional Example 3 Continued
2 Make a Plan
You can draw a diagram to order the scores from least to greatest.
Solve 3 Draw a number line and graph each player’s score on it.
–3 –2 –1 0 1 2 3 • • •
Rob’s score, –1, is farthest to the left, so it is the lowest score. Rob won the golf match.
Course 1
9-2 Comparing and Ordering Integers
Additional Example 3 Continued
Negative integers are always less than positive integers, so neither Craig nor Cameron won the golf match.
Look Back 4
Course 1
9-2 Comparing and Ordering Integers
Try This: Example 3
1 Understand the Problem
The answer will be the player with the lowest score. List the important information:
• Melissa scored +6.
• Trista scored –3.
• Alyssa scored –1.
In a golf match, Melissa scored +6, Trista scored –3, and Alyssa scored –1. Who won the golf match?
Course 1
9-2 Comparing and Ordering Integers
Try This: Example 3 Continued
2 Make a Plan
You can draw a diagram to order the scores from least to greatest.
Solve 3 Draw a number line and graph each player’s score on it.
• • •
Trista’s score, –3, is farthest to the left, so it is the lowest score. Trista won the golf match.
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6
Course 1
9-2 Comparing and Ordering Integers
Negative integers are always less positive integers, so Melissa cannot be the winner. Since Trista’s score of -3 is less than Alyssa’s score of -1, Trista won.
Look Back 4
Try This: Example 3 Continued
Lesson Quiz Order the integers in each set from least to greatest.
1. –3, 7, 4
2. –11, 2, 5, –15
Compare. Write <, >, or =.
3. –3 4 4. –12 –10
5. A location in Carlsbad Caverns is 752 ft below sea level, and another location is 910 ft below sea level. Which location is closer to sea level?
–15, –11, 2, 5
–3, 4, 7
Insert Lesson Title Here
< <
Course 1
9-2 Comparing and Ordering Integers
the location at –752 feet
9-3 The Coordinate Plane
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Use the number line to compare each pair of integers. Write < or >.
1. 7 –7 2. –8 –3 3. 0 –4 4. –2 –5
> < >
Course 1
9-3 The Coordinate Plane
>
Problem of the Day
While delivering pizza, Christian drove 4 miles south, 6 miles west, 2 miles north, 8 miles east, and then 2 miles north. How far is Christian from where he started? 2 miles
Course 1
9-3 The Coordinate Plane
Learn to locate and graph points on the coordinate plane.
Course 1
9-3 The Coordinate Plane
Vocabulary coordinate plane axes x-axis y-axis quadrants origin coordinates x-coordinate y-coordinate
Insert Lesson Title Here
Course 1
9-3 The Coordinate Plane
Course 1
9-3 The Coordinate Plane
A coordinate plane is formed by two number lines in a plane that intersect at right angles. The point of intersection is the zero on each number line.
• The two number lines are called the axes.
Course 1
9-3 The Coordinate Plane
A coordinate plane is formed by two number lines in a plane that intersect at right angles. The point of intersection is the zero on each number line.
• The horizontal axis is called the x-axis.
Course 1
9-3 The Coordinate Plane
A coordinate plane is formed by two number lines in a plane that intersect at right angles. The point of intersection is the zero on each number line.
• The vertical axis is called the y-axis.
Course 1
9-3 The Coordinate Plane
A coordinate plane is formed by two number lines in a plane that intersect at right angles. The point of intersection is the zero on each number line.
• The two axes divide the coordinate plane into four quadrants.
Course 1
9-3 The Coordinate Plane
A coordinate plane is formed by two number lines in a plane that intersect at right angles. The point of intersection is the zero on each number line.
• The point where the axes intersect is called the origin.
Course 1
9-3 The Coordinate Plane
Additional Example 1: Identifying Quadrants
Name the quadrant where each point is located.
A. X
B. Y
C. S
Quadrant IV
Quadrant III
Quadrant II
Course 1
9-3 The Coordinate Plane
Try This: Example 1
Name the quadrant where each point is located.
A. V
B. Z
C. T
Quadrant I
y-axis
Quadrant IV
Points on the axes are not in any quadrant. Helpful Hint
no quadrant
Insert Lesson Title Here
Course 1
9-3 The Coordinate Plane
An ordered pair gives the location of a point on a coordinate plane. The first number tells how far to move right (positive) or left (negative) from the origin. The second number tells how far to move up (positive) or down (negative).
The numbers in an ordered pair are called coordinates. The first number is called the x-coordinate. The second number is called the y-coordinate.
The ordered pair for the origin is (0,0).
Course 1
9-3 The Coordinate Plane
Additional Example 2A: Locating Points on a Coordinate Plane
Give the coordinates of each point.
A. X
From the origin, X is 4 units right and 1 unit down. (4, –1)
Course 1
9-3 The Coordinate Plane
Additional Example 2B: Locating Points on a Coordinate Plane
Give the coordinates of each point.
B. Y
(–2, –3)
From the origin, Y is 2 units left, and 3 units down.
Course 1
9-3 The Coordinate Plane
Additional Example 2C: Locating Points on a Coordinate Plane
Give the coordinates of each point.
C. S
From the origin, S is 3 units left, and 3 units up. (–3, 3)
Course 1
9-3 The Coordinate Plane
Try This: Example 2A
Give the coordinates of each point.
A. V
From the origin, V is 4 units right and 2 units up. (4, 2)
Course 1
9-3 The Coordinate Plane
Try This: Example 2B
Give the coordinates of each point.
B. Z
(0, 4)
From the origin, Z is 0 units right, and 4 units up.
Course 1
9-3 The Coordinate Plane
Try This: Example 2C
Give the coordinates of each point.
C. T
From the origin, T is 1 unit right, and 3 units down. (1, –3)
Course 1
9-3 The Coordinate Plane
Additional Example 3A & 3B: Graphing Points on a Coordinate Plane
Graph each point on a coordinate plane.
A. M(4, 3)
B. Q (–4, 1)
x
y
–4 –2 0 2 4
4
2
–2
–4
From the origin, move 4 units right, and 3 units up.
M
Q
From the origin, move 4 units left, and 1 unit up.
Course 1
9-3 The Coordinate Plane
Additional Example 3C: Graphing Points on a Coordinate Plane
Graph each point on a coordinate plane.
C. R(0, –3)
From the origin, move 3 units down.
x
y
–4 –2 0 2 4
4
2
–2
–4 R
Course 1
9-3 The Coordinate Plane
Try This: Example 3A & 3B
Graph each point on a coordinate plane.
A. L(3, 4)
B. M (–3, –3)
x
y
–4 –2 0 2 4
4
2
–2
–4
From the origin, move 3 units right, and 4 units up.
L
M From the origin, move 3 units left, and 3 units down.
Course 1
9-3 The Coordinate Plane
Try This: Example 3C
Graph each point on a coordinate plane.
C. P(1, –2)
From the origin, move 1 unit right and 2 units down.
x
y
–4 –2 0 2 4
4
2
–2
–4
P
Lesson Quiz
Name the quadrant where each ordered pair is located.
1. (3, –5) 2. (–4, –2)
3. (6, 2) 4. (–7, 9)
Give the coordinates of each point.
5. A 6. B
7. C 8. D
III IV
Insert Lesson Title Here
I II
Course 1
9-3 The Coordinate Plane
(–2, 3) (1, 2)
(0, –2) (–2, –3)
9-4 Adding Integers
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Add.
1. 2 + 3 2. 7 + 4 3. 8 + 5 4. 17 + 12 Name the integer that corresponds to each point. 5. A 6. B
5 11
–3
Course 1
9-4 Adding Integers
29 13
5
Problem of the Day
Karen earned 40 points on part I of a test, 26 points on part II, and 25 points on part III. What number of points did Karen earn on parts I and III combined?
65
Course 1
9-4 Adding Integers
Learn to add integers.
Course 1
9-4 Adding Integers
Course 1
9-4 Adding Integers
Adding Integers on a Number Line
Move right on a number line to add a positive integer. Move left on a number line to add a negative integer.
Course 1
9-4 Adding Integers
Parentheses are used to separate addition, subtraction, multiplication, and division signs from negative integers.
–2 + (–5) = –7
Writing Math
Course 1
9-4 Adding Integers
Additional Example 1A & 1B: Writing Integer Addition
Write the addition modeled on each number line.
A.
B.
–1 0 1 2 3 4 5 6
5 + (–4)
The addition modeled is 5 + (–4) = 1.
–5 –4 –3 –2 –1 0 1 2 3 4
(–4) +7
The addition modeled is –4 + 7 = 3.
Course 1
9-4 Adding Integers
Additional Example 1C: Writing Integer Addition
Write the addition modeled on each number line.
C.
3 + (–5)
The addition modeled is 3 + (–5) = –2.
–4 –3 –2 –1 0 1 2 3 4
Course 1
9-4 Adding Integers
Try This: Example 1A & 1B
Write the addition modeled on each number line.
A.
B.
6 + (–3)
The addition modeled is 6 + (–3) = 3.
(–2) +3
The addition modeled is –2 + 3 = 1.
–1 0 1 2 3 4 5 6
–5 –4 –3 –2 –1 0 1 2 3 4
Course 1
9-4 Adding Integers
Try This: Example 1C
Write the addition modeled on each number line.
C.
4 + (–7)
The addition modeled is 4 + (–7) = –3.
–4 –3 –2 –1 0 1 2 3 4
Course 1
9-4 Adding Integers
Additional Example 2A: Adding Integers
Find the sum.
A. –3 + (–2)
–6 –5 –4 –3 –2 –1 0 1
–3 + (–2)
–3 + (–2) = –5
Think:
Course 1
9-4 Adding Integers
Additional Example 2B: Adding Integers
Find the sum.
B. 6 + (–8)
6 + (–8) = –2
+(–8) 6
–3 –2 –1 0 1 2 3 4 5 6 7
Think:
Course 1
9-4 Adding Integers
Try This: Example 2A
Find the sum.
A. –2 + (–4)
–6 –5 –4 –3 –2 –1 0 1
–2 + (–4)
–2 + (–4) = –6
Think:
Course 1
9-4 Adding Integers
Try This: Example 2B
Find the sum.
B. 3 + (–6)
3 + (–6) = –3
+ (–6) 3
–3 –2 –1 0 1 2 3 4 5 6 7
Think:
Course 1
9-4 Adding Integers
Additional Example 3: Evaluating Integer Expressions
Evaluate y + (–2) for each value of y.
A. y = 7
B. y = 1
y + (–2) Write the expression. 7 + (–2) Substitute 7 for y.
5 Add.
y + (–2) Write the expression. 1 + (–2) Substitute 1 for y.
–1 Add.
Course 1
9-4 Adding Integers
Try This: Example 3
Evaluate z + (–4) for each value of z.
A. z = 2
B. z = 3
z + (–4) Write the expression. 2 + (–4) Substitute 2 for z. –2 Add.
z + (–4) Write the expression. 3 + (–4) Substitute 3 for z. –1 Add.
Course 1
9-4 Adding Integers
Additional Example 4: Application
A sunken ship is 12 m below the sea level. A search plane flies 35 m above the sunken ship. How far above the sea is the plane?
The ship is 12 m below the sea level and the plane is 35 m above the ship.
–12 + 35
23
The plane is 23 m above the sea.
Course 1
9-4 Adding Integers
Try This: Example 4
A sunken ship is 33 m below the sea level. A search plane flies 54 m above the sunken ship. How far above the sea is the plane?
The ship is 33 m below the sea level and the plane is 54 m above the ship.
–33 + 54
21
The plane is 21 m above the sea.
Lesson Quiz Find each sum.
1. 7 + (–3)
2. –5 + 2
3. –8 + (–4)
4. Evaluate x + 5 for x = –4.
5. At midnight on a winter night, the temperature was –12°F. By 10 A.M. the temperature had risen 42°F. What was the new temperature?
–3
4
Insert Lesson Title Here
–12
1
Course 1
9-4 Adding Integers
30°F
9-5 Subtracting Integers
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Find each sum.
1. –4 + 6 2. 5 + 12 3. –3 + 3 4. –8 + 9 Give the opposite of each number. 5. 7 6. –8 7. –3 8. 1
2 17 0
Course 1
9-5 Subtracting Integers
1
–7 8 3 –1
Problem of the Day Jamal knows his average score for 3 science tests. If he adds the average and the amount that each score differs from the average, what will his sum be?
Course 1
9-5 Subtracting Integers
The average
Learn to subtract integers.
Course 1
9-5 Subtracting Integers
Course 1
9-5 Subtracting Integers
Subtracting Integers on a Number Line Move left on a number line to subtract a positive integer. Move right on a number line to subtract a negative integer.
Course 1
9-5 Subtracting Integers
Additional Example 1: Writing Integer Subtraction
Write the subtraction modeled on each number line.
A.
B.
–8 –7 – 6 –5 –4 –3 –2 –1 0 1
–3 –4
The subtraction modeled is –3 – 4 = –7.
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5
(–5) – (–9)
The addition modeled is –5 – (–9) = 4.
Course 1
9-5 Subtracting Integers
Try This: Example 1
Write the subtraction modeled on each number line.
A.
B.
–5 –2
The subtraction modeled is –5 – 2 = –7.
(–3) –(–5)
The addition modeled is –3 – (–5) = 2.
–8 –7 – 6 –5 –4 –3 –2 –1 0 1
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5
Course 1
9-5 Subtracting Integers
Additional Example 2A: Subtracting Integers
Find the difference.
A. 4 – 6
4 – 6 = –2
(–6) 4
–3 –2 –1 0 1 2 3 4 5
Course 1
9-5 Subtracting Integers
Additional Example 2B: Subtracting Integers
Find the difference.
B. 3 – (–3)
–(–3) 3
3 – (–3) = 6
–1 0 1 2 3 4 5 6 7
Course 1
9-5 Subtracting Integers
Try This: Example 2A
Find the difference.
A. 2 – (–5)
–1 0 1 2 3 4 5 6 7
– (–5) 2
2 – (–5) = 7
Course 1
9-5 Subtracting Integers
Try This: Example 2B
Find the difference.
B. 3 – 6
3 – 6 = –3
(–6) 3
–3 –2 –1 0 1 2 3 4 5
Course 1
9-5 Subtracting Integers
Additional Example 3: Evaluating Integer Expressions
Evaluate a – 4 for each value of a.
A. a = 2
B. a = 8
a – 4 Write the expression. 2 – 4 Substitute 2 for a. –2 Subtract.
a – 4 Write the expression. 8 – 4 Substitute 8 for a. 4 Subtract.
Course 1
9-5 Subtracting Integers
Try This: Example 3
Evaluate b – 6 for each value of b.
A. b = 8
B. b = 12
b – 6 Write the expression. 8 – 6 Substitute 8 for b. 2 Subtract.
b – 6 Write the expression. 12 – 6 Substitute 12 for b.
6 Subtract.
Lesson Quiz
Find each difference.
1. 8 – 5 2. –9 – (–3)
Evaluate x – (–7) for each value of x.
3. x = –2 4. x = 5
5. The peak of Mt. Everest is approximately 26,544 ft above sea level. The shore of the Dead Sea is about 1,200 ft below sea level. How much higher is the peak of Mt. Everest than the shore of the Dead Sea?
–6 3
Insert Lesson Title Here
5
27,744 ft Course 1
9-5 Subtracting Integers
12
9-6 Multiplying Integers
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Find each product.
1. 8 • 4 2. 7 • 12 3. 3 • 9 4. 6 • 5 5. 80 • 6 6. 50 • 6 7. 40 • 90 8. 20 • 700
32 84 27
Course 1
9-6 Multiplying Integers
30 480 300 3,600 14,000
Problem of the Day
Catherine has $14.00 and earns $12.00 for each lawn she mows. If she mows 4 lawns and buys 5 DVDs that cost $11.95 each, including tax, how much money does she have left?. $2.25
Course 1
9-6 Multiplying Integers
Learn to multiply integers.
Course 1
9-6 Multiplying Integers
Course 1
9-6 Multiply Integers
Numbers 3 • 2 –3 • 2
Words 3 groups of 2 the opposite of 3 groups of 2
Addition 2 + 2 + 2 –(2 + 2 + 2)
Product 6 –6
Course 1
9-6 Multiply Integers
Numbers 3 • (–2) –3 • (–2)
Words 3 groups of –2 the opposite of 3 groups of –2
Addition (–2) + (–2) + (–2) –[(–2) + (–2) + (–2)]
Product –6 6
Course 1
9-6 Multiply Integers
Additional Example 1A & 1B: Multiplying Integers
Find each product.
A. 5 • 2
B. 4 • (–5)
5 • 2 = 10 Think: 5 groups of 2.
4 • (–5) = –20 Think: 4 groups of –5.
To find the opposite of a number, change the sign. The opposite of 6 is –6. The opposite of –4 is 4.
Remember!
Course 1
9-6 Multiply Integers
Additional Example 1C & 1D: Multiplying Integers
Find each product.
C. –3 • 2
D. –2 • (–4)
–3 • 2 = –6 Think: the opposite of 3 groups of 2.
–2 • (–4) = 8 Think: the opposite of 2 groups of –4.
Course 1
9-6 Multiply Integers
Try This: Example 1A & 1B
Find each product.
A. 3 • 4
B. 2 • (–7)
3 • 4 = 12 Think: 3 groups of 4.
2 • (–7) = –14 Think: 2 groups of –7.
Course 1
9-6 Multiply Integers
Try This: Example 1C & 1D
Find each product.
C. –5 • 3
D. –4 • (–6)
–5 • 3 = –15 Think: the opposite of 5 groups of 3.
–4 • (–6) = 24 Think: the opposite of 4 groups of –6.
Course 1
9-6 Multiplying Integers
MULTIPLYING INTEGERS If the signs are the same, the product is positive. 4 • 3 = 12 –6 • (–3) = 18
If the signs are different, the product is negative. –2 • 5 = –10 7 • (–8) = –56
The product of any number and 0 is 0.
0 • 9 = 0 (–12) • 0 = 0
Course 1
9-6 Multiplying Integers
Additional Example 2: Evaluating Integer Expressions
Evaluate –7x for each value of x.
A. x = –3
B. x = 5
–7x Write the expression. –7 • (–3) Substitute –3 for x. 21 The signs are the same, so the answer
is positive.
–7x Write the expression. –7 • 5 Substitute 5 for x. –35 The signs are different,
so the answer is negative. –7x means –7 • x.
Remember!
Course 1
9-6 Multiplying Integers
Try This: Example 2
Evaluate –4y for each value of y.
A. y = – 2
B. y = 7
–4y Write the expression. –4 • (–2) Substitute –2 for y.
8 The signs are the same, so the answer is positive.
–4y Write the expression. –4 • 7 Substitute 7 for y. –28 The signs are different, so the answer
is negative.
Lesson Quiz Find each product.
1. 6 • (4) 2. 3 • (–2)
3. –9 • (–2) 4. –6 • 5
5. Evaluate 3y for y = –7.
6. During a football game, Raymond’s team lost 6 yards on each of 3 plays and gained 8 yards on each of two plays. What integer represents the total change in the team’s position?
–6 24
Insert Lesson Title Here
18 –30
Course 1
9-6 Multiplying Integers
–21
–2
9-7 Dividing Integers
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Find each quotient.
1. 18 ÷ 2 2. 42 ÷ 7
3. 56 ÷ 8 4. 24 ÷ 6
5. 3,600 ÷ 4 6. 540 ÷ 60
9 6 7
Course 1
9-7 Dividing Integers
4
900 9
Problem of the Day
Hank wanted to record the number of marbles he lost to his friend Marcus each day for 5 days. He forgot to record one day, but for the other days he wrote 8, 2, 3, and 4. The average number he lost was 4. What number did he forget to write? 3
Course 1
9-7 Dividing Integers
Learn to divide integers.
Course 1
9-7 Dividing Integers
Course 1
9-7 Dividing Integers
Multiplication and division are inverse operations. To solve a division problem, think of the related multiplication.
Course 1
9-6 Dividing Integers
Additional Example 1: Dividing Integers
Find each quotient.
A. –30 ÷ 6
B. –42 ÷ (–7)
Think: What number times 6 equals –30?
–5 • 6 = –30, so –30 ÷ 6 = –5.
Think: What number times –7 equals –42?
6 • (–7) = –42, so –42 ÷ (–7) = 6.
Course 1
9-6 Dividing Integers
Try This: Example 1
Find each quotient.
A. –15 ÷ 5
B. –36 ÷ (–6)
Think: What number times 5 equals –15?
–3 • 5 = –15, so –15 ÷ 5 = –3.
Think: What number times -6 equals –36?
6 • (–6) = –36, so –36 ÷ (–6) = 6.
Course 1
9-7 Dividing Integers
Dividing Integers If the signs are the same, the product is positive. 24 ÷ 3 = 8 –6 ÷ (–3) = 2
If the signs are different, the quotient is negative. –20 ÷ 5 = –4 72 ÷ (–8) = –9
Zero divided by any integer equals 0.
= 0 = 0 0 14 __ 0
–11 __
You cannot divide any integer by 0.
Because division is the inverse of multiplication, the rules for dividing integers are the same as the rules for multiplying integers.
Course 1
9-7 Dividing Integers
Additional Example 2A: Evaluating Integer Expressions
Evaluate for each value of d.
A. d = 16
Write the expression.
= 16 ÷ 4 Substitute 16 for d.
= 4 The signs are the same, so the answer is positive.
d 4 __
d 4 __
16 4 __
Course 1
9-7 Dividing Integers
Additional Example 2B: Evaluating Integer Expressions
Evaluate for each value of d.
B. d = –24
Write the expression.
= –24 ÷ 4 Substitute –24 for d.
= –6 The signs are different, so the answer is negative.
d 4 __
d 4 __
-24 4 ___
Course 1
9-7 Dividing Integers
Additional Example 2C: Evaluating Integer Expressions
Evaluate for each value of d.
C. d = –12
Write the expression.
= –12 ÷ 4 Substitute –12 for d.
= –3 The signs are different, so the answer is negative.
d 4 __
d 4 __
-12 4 ___
Course 1
9-7 Dividing Integers
Try This: Example 2A
Evaluate for each value of c.
B. c = –15
Write the expression.
= –15 ÷ 3 Substitute –15 for c.
= –5 The signs are different, so the answer is negative.
c 3 __
c 3 __
-15 3 ___
Course 1
9-7 Dividing Integers
Try This: Example 2B
Evaluate for each value of c.
B. c = 12
Write the expression.
= 12 ÷ 3 Substitute 12 for c.
= 4
c 3 __
c 3 __
12 3 ___
The signs are the same, so the answer is positive.
Course 1
9-7 Dividing Integers
Try This: Example 2C
Evaluate for each value of c.
C. c = 27
Write the expression.
= 27 ÷ 3 Substitute 27 for c.
= 9 The signs are the same, so the answer is positive.
c 3 __
c 3 __
27 3 __
Lesson Quiz Find each quotient.
1. 18 ÷ (–3) 2. –36 ÷ (–6)
3. –64 ÷ (–8) 4. –45 ÷ 3
Evaluate for each value of x.
5. x = –12 6. x = 20
7. Cameron’s mom has $3,200 in her bank account. If she uses the money to make monthly rent payments of $375, how many whole payments can she make?
6 –6
Insert Lesson Title Here
8 –15
Course 1
9-7 Dividing Integers
–3 5
8
x 4 __
9-8 Solving Integer Equation
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Use mental math to find each solution.
1. 6 + x = 12 2. 3x = 15 3. = 4 4. x – 8 = 12
x = 6 x = 5
x = 20
Course 1
9-8 Solving Integer Equations
x = 20 x 5 __
Problem of the Day
Marie spent $15 at the fruit stand, buying peaches that cost $2 per container and strawberries that cost $3 per container. She bought the same number of containers of each fruit. How many containers each of peaches and strawberries did she buy? 3
Course 1
9-8 Solving Integer Equations
Learn to solve equations containing integers.
Course 1
9-8 Solving Integer Equations
Course 1
9-8 Solving Integer Equations
A. Solve –8 + y = –13. Check your answer.
–8 + y = –13 –8 is added to y.
+ 8 + 8 Subtracting –8 from both sides to undo the addition is the same as adding +8.
y = –5 Check –8 + y = –13 Write the equation.
Substitute –5 for y. –5 is a solution.
–8 + (–5) = –13 ?
–13 = –13 ?
Additional Example 1A: Adding and Subtracting to Solve Equations
Course 1
9-8 Solving Integer Equations
To solve the equation on the previous slide using algebra tiles, you can add four tiles to both sides and then remove pairs of red and yellow tiles. This is because subtracting a number is the same as adding its opposite.
Helpful Hint
Course 1
9-8 Solving Integer Equations
Additional Example 1B: Adding and Subtracting to Solve Equations
B. n – 2 = –8. n – 2 = –8 2 is subtracted from n.
+ 2 + 2 Add 2 to both sides to undo the subtraction.
n = –6
Check n – 2 = –8 Write the equation.
Substitute –6 for n. –6 is a solution.
–6 – 2 = –8 ?
–8 = –8 ?
Course 1
9-8 Solving Integer Equations
Try This: Example 1A
A. Solve –2 + y = –7. Check your answer.
–2 + y = –7 –2 is added to y.
+ 2 + 2 Subtracting –2 from both sides to undo the addition is the same as adding +2.
y = –5
Check –2 + y = –7 Write the equation.
Substitute –5 for y. –5 is a solution.
–2 + (–5) = –7 ?
-7 = -7 ?
Course 1
9-8 Solving Integer Equations
Try This: Example 1B
B. n – 6 = –14. n – 6 = –14 –14 is added to n.
+ 6 + 6 Add 6 to both sides to undo the subtraction.
n = –8
Check
n – 6 = –14 Write the equation. Substitute –8 for n. –8 is a solution.
–8 – 6 = –14 ?
–14= –14 ?
Course 1
9-8 Solving Integer Equations
Additional Example 2A: Multiplying and Dividing to Solve Equations
A. 4m = –20
= m is multiplied by 4. Divide both sides by 4 to undo the multiplication.
m = –5
Check 4m = –20 Write the equation.
Substitute –5 for m. –5 is a solution.
4(–5) = –20 ?
–20 = –20 ?
4m 4 ___ -20
4 ___
Course 1
9-8 Solving Integer Equations
3 • = 3 • (–7) x is divided by 3. Multiply both sides by 3 to undo the division.
x = –21
Check
= –7 Write the equation.
Substitute –21 for x.
–21 is a solution.
–21 ÷ 3 = –7 ?
–7 = –7 ?
x 3 __
x 3 __
Additional Example 2B: Multiplying and Dividing to Solve Equations
B. = –7 x 3
Course 1
9-8 Solving Integer Equations
Try This: Example 2A
A. 3n = –15
= n is multiplied by 3. Divide both sides by 3 to undo the multiplication.
n = –5
Check
3n = –15 Write the equation.
Substitute –5 for n. –5 is a solution.
3(–5) = –15 ?
–15 = –15 ?
3n 3 ___ -15
3 ___
Course 1
9-8 Solving Integer Equations
Try This: Example 2B
B. = –6
4 • = 4 • (–6) y is divided by 4. Multiply both sides by 4 to undo the division.
y = –24
Check
= –6 Write the equation.
Substitute –24 for y.
–24 is a solution.
–24 ÷ 4 = –6 ?
–6 = –6 ?
y 4 __
y 4 __
y 4 __
Lesson Quiz
Solving each equation.
1. 5 + x = –6 2. y – 9 = –7
3. –6a = 24 4. = –9
5. A submarine captain sets the following diving course: dive 300 ft, stop, and then dive another 300 ft. If this pattern is continued, how many dives will be necessary to reach a location 3,000 ft below sea level?
a = –4
x = –11
Insert Lesson Title Here
x = –45
10
Course 1
9-8 Solving Integer Equations
y = 2 x
5 __
10-1 Finding Perimeter
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up
1. What figure has four equal sides and four right angles? 2. What figure has eight sides? 3. What figure has five sides?
square octagon
pentagon
Course 1
10-1 Finding Perimeter
Problem of the Day
Find the greatest perimeter possible when 6 identical unit squares are arranged to form a closed figure. Adjacent squares must share an entire side. 14 units
Course 1
10-1 Finding Perimeter
Learn to find the perimeter and missing side lengths of a polygon.
Course 1
10-1 Finding Perimeter
Vocabulary perimeter
Insert Lesson Title Here
Course 1
10-1 Finding Perimeter
The perimeter of a figure is the distance around it. To find the perimeter you can add the lengths of the sides.
Course 1
10-1 Finding Perimeter
Additional Example 1: Finding the Perimeter of a Polygon
Find the perimeter of the figure.
2.8 + 3.6 + 3.5 + 3 + 4.3
Add all the side lengths.
The perimeter is 17.2 in.
Course 1
10-1 Finding Perimeter
Try This: Example 1
Find the perimeter of the figure.
3.3 + 3.3 + 4.2 + 4.2 + 3
Add all the side lengths. 3.3 in. 3.3 in.
4.2 in. 4.2 in.
3 in.
The perimeter is 18 in.
Course 1
10-1 Finding Perimeter
Course 1
10-1 Finding Perimeter
Additional Example 2: Using a Formula to Find Perimeter
Find the perimeter P of the rectangle.
P = 2l+ 2w Substitute 12 for l and 3 for w. P = (2 • 12) + (2 • 3)
P = 24 + 6 P = 30
Multiply. Add.
The perimeter is 30 cm.
12 cm
3 cm
Course 1
10-1 Finding Perimeter
Try This: Example 2
Find the perimeter P of the rectangle.
P = 2l + 2w Substitute 15 for l and 2 for w. P = (2 • 15) + (2 • 2)
P = 30 + 4 P = 34
Multiply. Add.
The perimeter is 34 cm.
15 cm
2 cm
Course 1
10-1 Finding Perimeter
A. What is the length of side a if the perimeter equals 1,471 mm?
P = sum of side lengths
Use the values you know. 1,471 = 416 + 272 + 201 + 289 + a
1,471 = 1,178 + a
1,471 – 1,178 = 1,178 + a – 1,178
Add the known lengths.
Subtract 1,178 from both sides. 293 = a
Side a is 293 mm long.
Additional Example 3A: Finding Unknown Side Lengths and the Perimeter of a Polygon
Find each unknown measure.
Course 1
10-1 Finding Perimeter
Additional Example 3B: Finding Unknown Side Lengths and the Perimeter of a Polygon
Find each unknown measure.
B. What is the perimeter of the polygon?
First find the unknown side length.
Find the sides opposite side b.
The length of side b = 50 + 22.
Side b is 72 cm long.
P = 72 + 55 + 50 + 33 + 22 + 22 P = 254
Find the perimeter. The perimeter of the polygon is 254 cm.
Course 1
10-1 Finding Perimeter
72 cm
l = 4w Find the length.
l = (4 • 19)
l = 76
Substitute 19 for w.
Multiply.
Additional Example 3C: Finding Unknown Side Lengths and the Perimeter of a Polygon
C. The width of a rectangle is 19 cm. What is the perimeter of the rectangle if the length is 4 times the width? 4w
19 cm
Course 1
10-1 Finding Perimeter
Additional Example 3C Continued
P = 2l + 2w
Substitute 76 and 19. P = 2(76) + 2 (19)
P = 152 + 38
P = 190
Multiply.
Add.
Use the formula for the perimeter of a rectangle.
The perimeter of the rectangle is 190 cm.
4w
19 cm
Course 1
10-1 Finding Perimeter
Try This: Example 3A
Find each unknown measure.
A. What is the length of side a if the perimeter equals 1,302 mm?
P = sum of side lengths
Use the values you know. 1,302 = 212 + 280 + 250 + 240 + a
1,302 = 982 + a
1,302 – 982 = 982 + a – 982
Add the known lengths.
Subtract 982 from both sides. 320 = a
Side a is 320 mm long.
212 mm 280 mm
a 250 mm
240 mm
Course 1
10-1 Finding Perimeter
First find the unknown side length. Find the sides opposite side b. The length of side b = 12 + 4.
Side b is 16 m long.
P = 14 + 12 + 8 + 4 + 6 + 16 P = 60
Find the perimeter.
The perimeter of the polygon is 60 m.
Try This: Example 3B
Find each unknown measure.
B. What is the perimeter of the polygon? 12 m
14 m
b m
8 m
6 m 4 m
16 m
Course 1
10-1 Finding Perimeter
l = 5w Find the length. l = (5 • 13)
l = 65
Substitute 13 for w.
Multiply.
Try This: Example 3C
C. The width of a rectangle is 13 cm. What is the perimeter of the rectangle if the length is 5 times the width?
5w
13 cm
Course 1
10-1 Finding Perimeter
Try This: Example 3C Continued
65 cm
13 cm
P = 2l + 2w
Substitute 65 and 13. P = 2(65) + 2 (13)
P = 130 + 26
P = 156
Multiply.
Add.
Use the formula for the perimeter of a rectangle.
The perimeter of the rectangle is 156 cm.
Course 1
10-1 Finding Perimeter
Lesson Quiz
Find each perimeter.
1. 2.
3. What is the perimeter of a polygon with side lengths of 15 cm, 18 cm, 21 cm, 32 cm, and 26 cm?
4. What is the perimeter of a rectangle with length 22 cm and width 8 cm?
9 cm 4 ft
Insert Lesson Title Here
112 cm
60 cm
5 6 __
Course 1
10-1 Finding Perimeter
10-2 Estimating and Finding Area
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up
1. What is the perimeter of a square with side lengths of 15 in.? 2. What is the perimeter of a rectangle with length 16 cm and width 11 cm?
60 in.
54 cm
Course 1
10-2 Estimating and Finding Area
Problem of the Day
Two wibbles equal four wabbles, and eight wabbles equal two bibbles. If a square has a perimeter of 16 wibbles, what is its area in square bibbles? 4 square bibbles
Course 1
10-2 Estimating and Finding Area
Learn to estimate the area of irregular figures and to find the area of rectangles, triangles, and parallelograms.
Course 1
10-2 Estimating and Finding Area
Vocabulary area
Insert Lesson Title Here
Course 1
10-2 Estimating and Finding Area
The area of a figure is the amount of surface it covers. We measure area in square units.
Course 1
10-2 Estimating and Finding Area
Additional Example 1: Finding the Area of an Irregular Figure
Find the area of the rectangle. Count full squares: 19 red squares. Count almost-full squares: 9 blue squares. Count squares that are about half-full: 4 green squares = 2 full squares. Do not count almost empty yellow squares. Add. 19 + 9 + 2 = 30
The area of the figure is about 30 mi . 2
Course 1
10-2 Estimating and Finding Area
Try This: Example 1
Estimate the are of the figure.
Count full squares: 11 red squares. Count almost-full squares: 5 green squares. Count squares that are about half-full: 4 blue squares = 2 full squares. Do not count almost empty yellow squares. Add. 11 + 5 + 2 = 18
The area of the figure is about 18 mi . 2
= mi2
Course 1
10-2 Estimating and Finding Area
Course 1
10-2 Estimating and Finding Area
Additional Example 2: Estimating the Area of a Rectangle
Find the area of the figure.
Write the formula.
Substitute 15 for l. Substitute 9 for w.
A = lw
A = 15 • 9
A = 135
The area is about 135in2.
15 in.
9 in.
Course 1
10-2 Estimating and Finding Area
Try This: Example 2
Find the area of the figure.
Write the formula.
Substitute 13 for l. Substitute 7 for w.
A = lw
A = 13 • 7
A = 91
The area is about 91 in2.
13 in.
7 in.
Course 1
10-2 Estimating and Finding Area
You can use the formula for the area of a rectangle to write a formula for the area of a parallelogram. Imagine cutting off the triangle drawn in the parallelogram and sliding it to the right to form a rectangle.
The area of a parallelogram = bh. The area of a rectangle = lw.
The base of the parallelogram is the length of the rectangle. The height of the parallelogram is the width of the rectangle.
Course 1
10-2 Estimating and Finding Area
Additional Example 3: Finding the Area of a Parallelogram
Find the area of the parallelogram.
Write the formula.
Multiply.
A = bh
2 ft 1 2
A = 2 • 1 1 2 __ 1
4 __ Substitute 2 for b
and 1 for h.
1 2 __
1 4 __
A = • 5 2 __ 5
4 __
A = or 3 25 8
___ 1 8 __
The area is 3 ft2. 1 8 __
1 4 1 ft
Course 1
10-2 Estimating and Finding Area
Try This: Example 3
Find the area of the parallelogram.
Write the formula.
Multiply.
A = bh
A = 1 • 3 1 2 __ 1
2 __ Substitute 1 for b
and 3 for h.
1 2 __
1 2 __
A = • 3 2 __ 7
2 __
A = or 5 21 4
___ 1 4 __
The area is 5 ft2. 1 4 __
1 1 ft 2
1 3 ft 2
Course 1
10-2 Estimating and Finding Area
You can make a parallelogram out of two congruent triangles.
The area of each triangle is half the area of the parallelogram, so the formula for the area of a
triangle is A = bh. 1 2 __
Course 1
10-2 Estimating and Finding Area
Additional Example 4: Finding the Area of a Triangle
Find the area of the triangle.
Write the formula.
Substitute 9.6 for b. Substitute 2.9 for h.
Multiply.
The area is 13.92 cm2.
A = 13.92
A = bh 1 2 __
A = (9.6 • 2.9) 1 2 __
A = (27.84) 1 2 __
Course 1
10-2 Estimating and Finding Area
Try This: Example 4
Find the area of the triangle.
Write the formula.
Substitute 8.5 for b. Substitute 3.4 for h.
Multiply.
The area is 14.45 cm2.
3.4 cm
8.5 cm
A = 14.45
A = bh 1 2 __
A = (8.5 • 3.4) 1 2 __
A = (28.9) 1 2 __
Course 1
10-2 Estimating and Finding Area
Lesson Quiz
Find the area of each figure.
1. 2.
3. What is the area of a parallelogram with base 16 in. and height 10 in.?
4. What is the area of a triangle with base 10 in. and height 7 in.?
Insert Lesson Title Here
4.5 m 9.2 m
41.4 m2 24 ft2
160 in2
35 in2
Course 1
10-2 Estimating and Finding Area
10-3 Break into Simpler Parts
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up
1. What is the area of a rectangle with length 10 cm and width 4 cm?
2. What is the area of a parallelogram with
base 18 ft and height 12 ft? 3. What is the area of a triangle with base
16 cm and height 8 cm?
40 cm2
216 ft2
64 cm2
Course 1
10-3 Break into Simpler Parts
Problem of the Day
Four squares are stacked in a tower. The bottom square is 12 inches on a side. The perimeter of each of the other squares is half of the one below it. What is the perimeter of the combined figure? 69 in.
Course 1
10-3 Break into Simpler Parts
Learn to break a polygon into simpler parts to find its area.
Course 1
10-3 Break into Simpler Parts
Additional Example 1A: Finding Areas of Composite Figures
Find the area of the polygon.
A.
Think: Break the polygon apart into rectangles. Find the area of each rectangle.
1.7 cm
4.9 cm 1.3 cm
2.1 cm
Course 1
10-3 Break into Simpler Parts
Additional Example 1A Continued
A = lw A = lw A = 4.9 • 1.7 A = 2.1 • 1.3
Write the formula for the area of a rectangle. A = 8.33 A = 2.73
8.33 + 2.73 = 11.06 Add to find the total area. The area of the polygon is 11.06 cm2.
1.7 cm
4.9 cm
1.3 cm
2.1 cm
Course 1
10-3 Break into Simpler Parts
Additional Example 1B Continued
Find the area of the polygon.
B.
Think: Break the figure apart into a rectangle and a triangle.
Find the area of each polygon. Course 1
10-3 Break into Simpler Parts
Additional Example 1B Continued
A = lw A = 28 • 24 A = 672 A = 168
672 + 168 = 840 Add to find the total area of the polygon.
The area of the polygon is 840 ft2.
A = bh 1 2 __
A = 28 • 12 1 2 __
Course 1
10-3 Break into Simpler Parts
Try This: Example 1A
Find the area of the polygon.
A.
Think: Break the polygon apart into rectangles. Find the area of each rectangle.
1.9 cm
5.5 cm 1.5 cm 2 cm
1.9 cm
1.5 cm
2 cm
3.4 cm
5.5 cm
Course 1
10-3 Break into Simpler Parts
A = lw A = lw A = 5.5 • 1.9 A = 2 • 1.5
Write the formula for the area of a rectangle. A = 10.45 A = 3
10.45 + 3 = 13.45 Add to find the total area. The area of the polygon is 13.45 cm2.
Try This: Example 1A Continued 1.9 cm
5.5 cm 1.5 cm 2 cm
Course 1
10-3 Break into Simpler Parts
Try This: Example 1B
Find the area of the polygon.
B.
Think: Break the figure apart into a rectangle and a triangle.
Find the area of each polygon.
36 ft
22 ft
20 ft
20 ft
22 ft
22 ft
16 ft
Course 1
10-3 Break into Simpler Parts
A = lw A = 22 • 20 A = 440 A = 176
440 + 176 = 616 Add to find the total area of the polygon.
The area of the polygon is 616 ft2.
A = bh 1 2 __
A = 22 • 16 1 2 __
Try This: Example 1B Continued
20 ft 22 ft
22 ft
16 ft
Course 1
10-3 Break into Simpler Parts
Additional Example 2: Art Application
Patrick made a design. All the sides are 5 inches long, except for two longer sides that are each 20 inches. All the angles are right angles. What is the area of the quilt design?
Think: Divide the design into 3 rectangles. Find the area of one rectangle that has a length of 20 in and a width of 5 in. Write the formula. A = lw
A = 20 • 5 = 100 3 • 100 = 300 Multiply to find the area of the 3
rectangles. The area of the design is 300 in2.
20 in.
20 in.
5 in.
Course 1
10-3 Break into Simpler Parts
You can also use the formula A = s2 , where s is the length of a side, to find the area of a square.
Helpful Hint
Course 1
10-3 Break into Simpler Parts
Try This: Example 2
Yvonne made quilt design. All the sides are 4 inches long, except for the two longer sides that are each 16 inches. All the angles are right angles. What is the area of the quilt design?
Think: Divide the quilt design into 10 squares. Find the area of one square that has a side length of 4 in.
Write the formula. A = lw A = 4 • 4 = 16 10 • 16 = 160
Multiply to find the area of the 10 squares.
The area of the quilt design is 160 in2.
4 in. 16 in.
16 in.
Course 1
10-3 Break into Simpler Parts
Lesson Quiz
Find the area of the figure shown.
Insert Lesson Title Here
220 units2
Course 1
10-3 Break into Simpler Parts
10-4 Comparing Perimeter and Area
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up
1. What is the area of a figure made up of a rectangle with length 12 cm and height 4 cm and a parallelogram with length 12 cm and height 6 cm?
2. What is the area of a figure consisting of
a triangle sitting on top of a rectangle? The triangle has a base of 12 in. and height of 9 in., and the rectangle has a base of 12 in. and height of 5 in.
Course 1
10-4 Comparing Perimeter and Area
120 cm2
114 in2
Problem of the Day
If sixteen people sit, evenly spaced, in a circle for story time, who sits directly across from person 5? person 13
Course 1
10-4 Comparing Perimeter and Area
Learn to make a model to explore how area and perimeter are affected by changes in the dimensions of a figure.
Course 1
10-4 Comparing Perimeter and Area
Additional Example 1: Changing Dimensions
Find how the perimeter and the area of the figure change when its dimensions change.
P = 29.2 in. A = 35 in.2
P = 14.6 in. A = 8.75 in.2
When the dimensions of the triangle are divided by 2, the perimeter is divided by 2, and the area is divided by 4, or 22.
Course 1
10-4 Comparing Perimeter and Area
Divide each dimension by 2.
Try This: Example 1
Find how the perimeter and the area of the figure change when its dimensions change.
Multiply each dimension by 2.
4 in. 2 in.
8 in.
4 in.
P = 12 in. A = 8 in.2
P = 24 in. A = 32 in.2
When the dimensions of the rectangle are multiplied by 2, the perimeter is multiplied by 2, and the area is multiplied by 4, or 22.
Course 1
10-4 Comparing Perimeter and Area
Additional Example 2: Application
Draw a rectangle whose dimensions are 4 times as large as the given rectangle. How do the perimeter and area change?
Multiply each dimension by 4. P = 10 cm
A = 6 cm2 P = 40 cm
A = 96 cm2 When the dimensions of the rectangle are multiplied by 4, the perimeter is multiplied by 4, and the area is multiplied by 16, or 42.
3 cm 2 cm 8 cm
12 cm
Course 1
10-4 Comparing Perimeter and Area
Try This: Example 2
Draw a rectangle whose dimensions are 2 times as large as the given rectangle. How do the perimeter and area change?
Multiply each dimension by 2. P = 16 cm
A = 15 cm2 P = 32 cm
A = 60 cm2 When the dimensions of the rectangle are multiplied by 2, the perimeter is multiplied by 2, and the area is multiplied by 4, or 22.
5 cm 3 cm 6 cm
10 cm
Course 1
10-4 Comparing Perimeter and Area
Lesson Quiz
Find how the perimeter and area of the triangle change when its dimensions change.
The perimeter is multiplied by 2, and the area is multiplied by 4; perimeter = 24, area = 24; perimeter = 48, area = 96.
Insert Lesson Title Here
Course 1
10-4 Comparing Perimeter and Area
10-5 Circles
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up The length and width of a rectangle are each multiplied by 5. Find how the perimeter and area of the rectangle change.
The perimeter is multiplied by 5, and the area is multiplied by 25.
Course 1
10-5 Circles
Problem of the Day
When using a calculator to find the height of a rectangle whose length one knew, a student accidentally multiplied by 20 when she should have divided by 20. The answer displayed was 520. What is the correct height? 1.3
Course 1
10-5 Circles
Learn to identify the parts of a circle and to find the circumference and area of a circle.
Course 1
10-5 Circles
Vocabulary circle center radius (radii) diameter circumference pi
Insert Lesson Title Here
Course 1
10-5 Circles
A circle is the set of all points in a plane that are the same distance from a given point, called the center.
Center
Course 1
10-5 Circles
A line segment with one endpoint at the center of the circle and the other endpoint on the circle is a radius (plural: radii).
Center Radius
Course 1
10-5 Circles
A chord is a line segment with both endpoints on a circle. A diameter is a chord that passes through the center of the circle. The length of the diameter is twice the length of the radius.
Center Radius
Diameter
Course 1
10-5 Circles
Additional Example 1: Naming Parts of a Circle
Name the circle, a diameter, and three radii.
N The circle is circle Z.
LM is a diameter.
ZL, ZM, and ZN are radii.
M
Z L
Course 1
10-5 Circles
Try This: Example 1
Name the circle, a diameter, and three radii.
The circle is circle D.
IG is a diameter.
DI, DG, and DH are radii.
G
H
D I
Course 1
10-5 Circles
The distance around a circle is called the circumference.
Center Radius
Diameter
Circumference
Course 1
10-5 Circles
The ratio of the circumference to the diameter, , is the same for any circle. This
ratio is represented by the Greek letter π, which is read “pi.”
C d
C d
= π
Course 1
10-5 Circles
The formula for the circumference of a circle is C = πd, or C = 2πr.
The decimal representation of pi starts with 3.14159265 . . . and goes on forever without repeating. We estimate pi using either 3.14
or . 22 7
Course 1
10-5 Circles
Additional Example 2A: Using the Formula for the Circumference of a Circle
Find the missing value to the nearest hundredth. Use 3.14 for pi.
A. d = 11 ft; C = ?
C = πd
C ≈ 3.14 • 11
C ≈ 34.54 ft
Write the formula.
Replace π with 3.14 and d with 11.
11 ft
Course 1
10-5 Circles
Additional Example 2B: Using the Formula for the Circumference of a Circle
Find each missing value to the nearest hundredth. Use 3.14 for pi.
B. r = 5 cm; C = ?
C = 2πr
C ≈ 2 • 3.14 • 5
C ≈ 31.4 cm
Write the formula.
Replace π with 3.14 and r with 5.
5 cm
Course 1
10-5 Circles
Additional Example 2C: Using the Formula for the Circumference of a Circle
Find each missing value to the nearest hundredth. Use 3.14 for pi.
C. C = 21.98 cm; d = ?
C = πd
21.98 ≈ 3.14d
7.00 cm ≈ d
Write the formula.
Replace C with 21.98 and π with 3.14.
21.98 3.14d _______ _______
3.14 3.14 ≈ Divide both sides by 3.14.
Course 1
10-5 Circles
Try This: Example 2A
Find the missing value to the nearest hundredth. Use 3.14 for pi.
A. d = 9 ft; C = ?
C = πd
C ≈ 3.14 • 9
C ≈ 28.26 ft
Write the formula.
Replace π with 3.14 and d with 9.
9 ft
Course 1
10-5 Circles
Try This: Example 2B
Find each missing value to the nearest hundredth. Use 3.14 for pi.
B. r = 6 cm; C = ?
C = 2πr
C ≈ 2 • 3.14 • 6
C ≈ 37.68 cm
Write the formula.
Replace π with 3.14 and r with 6.
6 cm
Course 1
10-5 Circles
Try This: Example 2C
Find each missing value to the nearest hundredth. Use 3.14 for pi.
C. C = 18.84 cm; d = ?
C = πd
18.84 ≈ 3.14d
6.00 cm ≈ d
Write the formula.
Replace C with 18.84 and π with 3.14.
18.84 3.14d _______ _______
3.14 3.14 ≈ Divide both sides by 3.14.
Course 1
10-5 Circles
The formula for the area of a circle is A = πr2.
Course 1
10-5 Circles
Additional Example 3: Using the Formula for the Area of a Circle
Find the area of the circle. Use for pi.
d = 42 cm; A = ?
Write the formula to find the area. A = πr2
r = d ÷ 2 r = 42 ÷ 2 = 21
The length of the diameter is twice the length of the radius.
Replace π with and r with 21. 22 7 __
A ≈ • 441 22 7 __ Use the GCF to simplify. 63
A ≈ 1,386 cm2 Multiply.
22 7
A ≈ • 212 22 7
1
42 cm
Course 1
10-5 Circles
Write the formula to find the area. A = πr2
r = d ÷ 2 r = 28 ÷ 2 = 14
The length of the diameter is twice the length of the radius.
Replace π with and r with 14. 22 7 __
A ≈ • 196 22 7 __ Use the GCF to simplify. 28
A ≈ 616 cm2 Multiply.
Try This: Example 3
Find the area of the circle. Use for pi.
d = 28 cm; A = ?
22 7
A ≈ • 142 22 7
1
28 cm
Course 1
10-5 Circles
Lesson Quiz
Find the circumference and area of each circle. Use 3.14 for π.
1. 2.
3. Find the area of a circle with a diameter of 20 feet. Use 3.14 for π.
C = 25.12 in.
Insert Lesson Title Here
C = 18.84 in.
8 in.
314 ft2
A = 50.24 in2 A = 28.26 in2
3 in.
Course 1
10-5 Circles
10-6 Solid Figures
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Solve. Use 3.14 for π.
37.68 in.
56.52 cm
Course 1
10-6 Solid Figures
452.16 ft2
1. The diameter of a circle is 12 in. What is the circumference?
2. The radius of a circle is 9 cm. What is its circumference?
3. Find the area of a circle with a 12 ft radius.
Problem of the Day
To measure the perimeter of her square patio, Becky used an old bicycle wheel with a 22 in. diameter. She rolled the wheel from one corner of the patio along the edge to the next. The wheel made 6.75 revolutions. What is the perimeter in feet of the patio? Use 3.14 for π. 155.43 ft
Course 1
10-6 Solid Figures
Learn to name solid figures.
Course 1
10-6 Solid Figures
Vocabulary polyhedron face edge vertex prism base pyramid cylinder cone
Insert Lesson Title Here
Course 1
10-6 Solid Figures
A polyhedron is a three-dimensional object, or solid figure, with flat surfaces, called faces, that are polygons.
When two faces of a solid figure share a side, they form an edge. On a solid figure, a point at which three or more edges meet is a vertex (plural: vertices).
Course 1
10-6 Solid Figures
Additional Example 1: Identifying Faces, Edges, and Vertices
Identify the number of faces, edges, and vertices on each solid figure.
A.
B.
5 faces 8 edges
5 vertices
7 faces 15 edges
10 vertices
Course 1
10-6 Solid Figures
Try This: Example 1
Identify the number of faces, edges, and vertices on each solid figure.
A.
B.
6 faces 12 edges
8 vertices
5 faces 9 edges
6 vertices
Course 1
10-6 Solid Figures
A prism is a polyhedron with two congruent, parallel bases, and other faces that are all parallelograms. A prism is named for the shape of its bases. A cylinder also has two congruent, parallel bases, but bases of a cylinder are circular. Cylinders are not polyhedra because not every surface is a polygon.
Course 1
10-6 Solid Figures
A pyramid has one polygon shaped base, and the other faces are triangles that come to a point. A pyramid is named for the shape of its base. A cone has a circular base and a curved surface that comes to a point. Cones are not polyhedra because not every surface is a polygon.
Course 1
10-6 Solid Figures
The point of a cone is called its vertex.
Helpful Hint
Course 1
10-6 Solid Figures
Additional Example 2A: Naming Solid Figures
Name the solid figures represented by the object.
A.
The figure is not a polyhedron.
There is a curved surface.
The figure represents a cylinder.
There are two congruent, parallel bases.
The bases are circles.
Course 1
10-6 Solid Figures
Additional Example 2B: Naming Solid Figures
Name the solid figures represented by the object.
B.
The figure is a polyhedron.
All the faces are flat and are polygons.
The figure is a triangular pyramid.
There is one base and the other faces are triangles that meet at a point, so the figure is a pyramid. The base is a triangle.
Course 1
10-6 Solid Figures
Additional Example 2C: Naming Solid Figures
Name the solid figures represented by the object.
C.
The figure is a polyhedron.
All the faces are flat and are polygons.
The figure is a rectangular prism.
There are two congruent, parallel bases, so the figure is a prism. The bases are rectangles.
Course 1
10-6 Solid Figures
Try This: Example 2A
Name the solid figures represented by the object.
A.
The figure is a polyhedron.
All the faces are flat and are polygons.
The figure is a square pyramid.
There is one base and the other faces are triangles that meet at a point, so the figure is a pyramid. The base is a square.
Course 1
10-6 Solid Figures
Try This: Example 2B
Name the solid figures represented by the object.
B.
The figure is a polyhedron.
All the faces are flat and are polygons.
The figure is a rectangular prism.
There are two congruent, parallel bases, so the figure is a prism. The bases are rectangles.
Course 1
10-6 Solid Figures
Try This: Example 2C
Name the solid figures represented by the object.
C.
The figure is not a polyhedron.
There is a curved surface.
The figure represents a cylinder.
There are two congruent, parallel bases.
The bases are circles.
Course 1
10-6 Solid Figures
Lesson Quiz
1. Identify the number of faces, edges, and vertices in the figure shown.
Identify the figure described
2. two congruent circular faces connected by a curved surface
3. one flat circular face and a curved lateral surface that comes to a point
cylinder
8 faces, 18 edges, and 12 vertices
Insert Lesson Title Here
cone Course 1
10-6 Solid Figures
10-7 Surface Area
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Identify the figure described.
1. two parallel congruent faces, with the other faces being parallelograms 2. a polyhedron that has a vertex and a face at opposite ends, with the other faces being triangles
prism
pyramid
Course 1
10-7 Surface Area
Problem of the Day
Which figure has the longer side and by how much, a square with an area of 81 ft2 or a square with perimeter of 84 ft?
A square with a perimeter of 84 ft; by 12 ft
Course 1
10-7 Surface Area
Learn to find the surface areas of prisms, pyramids, and cylinders.
Course 1
10-7 Surface Area
Vocabulary surface area net
Insert Lesson Title Here
Course 1
10-7 Surface Area
The surface area of a solid figure is the sum of the areas of its surfaces. To help you see all the surfaces of a solid figure, you can use a net. A net is the pattern made when the surface of a solid figure is layed out flat showing each face of the figure.
Course 1
10-7 Surface Area
Additional Example 1A: Finding the Surface Area of a Prism
Find the surface area S of the prism.
A. Method 1: Use a net.
Draw a net to help you see each face of the prism.
Use the formula A = lw to find the area of each face.
Course 1
10-7 Surface Area
Additional Example 1A Continued
A: A = 5 × 2 = 10
B: A = 12 × 5 = 60
C: A = 12 × 2 = 24
D: A = 12 × 5 = 60
E: A = 12 × 2 = 24
F: A = 5 × 2 = 10
S = 10 + 60 + 24 + 60 + 24 + 10 = 188 Add the areas of each face.
The surface area is 188 in2. Course 1
10-7 Surface Area
Additional Example 1B: Finding the Surface Area of a Prism
Find the surface area S of each prism.
B. Method 2: Use a three-dimensional drawing.
Find the area of the front, top, and side, and multiply each by 2 to include the opposite faces.
Course 1
10-7 Surface Area
Additional Example 1B Continued
Front: 9 × 7 = 63
Top: 9 × 5 = 45
Side: 7 × 5 = 35
63 × 2 = 126
45 × 2 = 90
35 × 2 = 70
S = 126 + 90 + 70 = 286 Add the areas of each face.
The surface area is 286 cm2.
Course 1
10-7 Surface Area
Try This: Example 1A
Find the surface area S of the prism.
A. Method 1: Use a net.
Draw a net to help you see each face of the prism.
Use the formula A = lw to find the area of each face.
3 in. 11 in.
6 in. 11 in.
6 in. 6 in. 3 in.
3 in.
3 in.
3 in.
A
B C D E F
Course 1
10-7 Surface Area
Try This: Example 1A
A: A = 6 × 3 = 18
B: A = 11 × 6 = 66
C: A = 11 × 3 = 33
D: A = 11 × 6 = 66
E: A = 11 × 3 = 33
F: A = 6 × 3 = 18
S = 18 + 66 + 33 + 66 + 33 + 18 = 234 Add the areas of each face.
The surface area is 234 in2.
11 in.
6 in. 6 in. 3 in.
3 in.
3 in.
3 in.
A
B C D E F
Course 1
10-7 Surface Area
Try This: Example 1B
Find the surface area S of each prism.
B. Method 2: Use a three-dimensional drawing.
Find the area of the front, top, and side, and multiply each by 2 to include the opposite faces.
6 cm 10 cm
8 cm
top front side
Course 1
10-7 Surface Area
Try This: Example 1B Continued
Front: 10 × 8 = 80
Top: 10 × 6 = 60
Side: 8 × 6 = 48
80 × 2 = 160
60 × 2 = 120
48 × 2 = 96
S = 160 + 120 + 96 = 376 Add the areas of each face.
The surface area is 376 cm2.
6 cm 10 cm
8 cm
top front side
Course 1
10-7 Surface Area
The surface area of a pyramid equals the sum of the area of the base and the areas of the triangular faces. To find the surface area of a pyramid, think of its net.
Course 1
10-7 Surface Area
Additional Example 2: Finding the Surface Area of a Pyramid
Find the surface area S of the pyramid. S = area of square + 4 × (area of
triangular face)
S = 49 + 4 × 28 S = 49 + 112
Substitute.
S = s2 + 4 × ( bh) 1 2 __
S = 72 + 4 × ( × 7 × 8) 1 2 __
S = 161 The surface area is 161 ft2.
Course 1
10-7 Surface Area
Try This: Example 2
Find the surface area S of the pyramid.
S = area of square + 4 × (area of triangular face)
S = 25 + 4 × 25 S = 25 + 100
Substitute.
S = s2 + 4 × ( bh) 1 2 __
S = 52 + 4 × ( × 5 × 10) 1 2 __
S = 125 The surface area is 125 ft2.
5 ft
5 ft
10 ft
10 ft 5 ft
Course 1
10-7 Surface Area
The surface area of a cylinder equals the sum of the area of its bases and the area of its curved surface.
To find the area of the curved surface of a cylinder, multiply its height by the circumference of the base.
Helpful Hint
Course 1
10-7 Surface Area
Additional Example 3: Finding the Surface Area of a Cylinder
Find the surface area S of the cylinder. Use 3.14 for π, and round to the nearest hundredth.
S = area of lateral surface + 2 × (area of each base) Substitute. S = h × (2πr) + 2 × (πr2)
S = 7 × (2 × π × 4) + 2 × (π × 42)
ft
Course 1
10-7 Surface Area
Additional Example 3 Continued
Find the surface area S of the cylinder. Use 3.14 for π, and round to the nearest hundredth.
S ≈ 7 × 8(3.14) + 2 × 16(3.14)
S ≈ 7 × 25.12 + 2 × 50.24
The surface area is about 276.32 ft2.
Use 3.14 for π.
S ≈ 175.84 + 100.48
S ≈ 276.32
S = 7 × 8π + 2 × 16π
Course 1
10-7 Surface Area
Try This: Example 3
Find the surface area S of the cylinder. Use 3.14 for π, and round to the nearest hundredth.
S = area of lateral surface + 2 × (area of each base) Substitute. S = h × (2πr) + 2 × (πr2)
S = 9 × (2 × π × 6) + 2 × (π × 62)
6 ft
9 ft
Course 1
10-7 Surface Area
Try This: Example 3 Continued
Find the surface area S of the cylinder. Use 3.14 for π, and round to the nearest hundredth.
S ≈ 9 × 12(3.14) + 2 × 36(3.14)
S ≈ 9 × 37.68 + 2 × 113.04
The surface area is about 565.2 ft2.
Use 3.14 for π.
S ≈ 339.12 + 226.08
S ≈ 565.2
S = 9 × 12π + 2 × 36π
Course 1
10-7 Surface Area
Lesson Quiz
Find the surface area of each figure. Use 3.14 for π.
1. rectangular prism with base length 6 ft, width 5 ft, and height 7 ft
2. cylinder with radius 3 ft and height 7 ft
3. Find the surface area of the figure shown.
Insert Lesson Title Here
Course 1
10-7 Surface Area
214 ft2
188.4 ft2
208 ft2
10-8 Finding Volume
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Find the surface area of each figure. Use 3.14 for π.
Course 1
10-8 Finding Volume
432 in2
226.08 ft2
1. rectangular prism with base length 8 in., width 6 in., and height 12 in.
2. cylinder with diameter 8 ft and height 5 ft
Problem of the Day
A rectangular park is bordered by a 3-foot-wide sidewalk. The park, including the sidewalk, measures 125 ft by 180 ft. What is the area of the park, not including the sidewalk? 20,706 ft2
Course 1
10-8 Finding Volume
Learn to estimate and find the volumes of rectangular prisms and triangular prisms.
Course 1
10-8 Finding Volume
Vocabulary volume
Insert Lesson Title Here
Course 1
10-8 Finding Volume
Volume is the number of cubic units needed to fill a space.
Course 1
10-8 Finding Volume
It takes 10, or 5 · 2, centimeter cubes to cover the bottom layer of this rectangular prism.
There are 3 layers of 10 cubes each. It takes 30, or 5 · 2 · 3, cubes to fill the prism.
The volume of the prism is 5 cm · 2 cm · 3 cm = 30 cm3.
Course 1
10-8 Finding Volume
Additional Example 1: Finding the Volume of a Rectangular Prism
Find the volume of the rectangular prism.
V = lwh Write the formula.
V = 26 • 11 • 13 l = 26; w = 11; h = 13
Multiply. V = 3,718 in3
13 in.
26 in. 11 in.
Course 1
10-8 Finding Volume
Try This: Example 1
Find the volume of the rectangular prism.
V = lwh Write the formula.
V = 29 • 12 • 16 l = 29; w = 12; h = 16
Multiply. V = 5,568 in3
16 in.
29 in. 12 in.
Course 1
10-8 Finding Volume
To find the volume of any prism, you can use the formula V= Bh, where B is the area of the base, and h is the prism’s height. So, to find the volume of a triangular prism, B is the area of the triangular base and h is the height of the prism.
Course 1
10-8 Finding Volume
Additional Example 2A: Finding the Volume of a Triangular Prism
Find the volume of each triangular prism.
A.
V = Bh Write the formula. V = ( • 3.9 • 1.3) • 4 1
2 __ B = • 3.9 • 1.3; h = 4. 1
2 __
Multiply. V = 10.14 m3
Course 1
10-8 Finding Volume
Additional Example 2B: Finding the Volume of a Triangular Prism
Find the volume of the triangular prism.
B.
V = Bh Write the formula. V = ( • 6.5 • 7) • 6 1
2 __ B = • 6.5 • 7; h = 6. 1
2 __
Multiply. V = 136.5 ft 3
Course 1
10-8 Finding Volume
Try This: Example 2A
Find the volume of each triangular prism.
A.
V = Bh Write the formula. V = ( • 4.2 • 1.6) • 7 1
2 __ B = • 4.2 • 1.6; h = 7. 1
2 __
Multiply. V = 23.52 m3
1.6 m 7 m 4.2 m
Course 1
10-8 Finding Volume
Try This: Example 2B
Find the volume of each triangular prism.
B.
V = Bh Write the formula. V = ( • 4.5 • 9) • 5 1
2 __ B = • 4.5 • 9; h = 5. 1
2 __
Multiply. V = 101.25 ft3
4.5 ft 5 ft
9 ft
Course 1
10-8 Finding Volume
Additional Example 3: Problem Solving Application
Suppose a facial tissue company ships 16 cubic tissue boxes in each case. What are the possible dimensions for a case of tissue boxes?
1 Understand the Problem
The answer will be all possible dimensions for a case of 16 cubic boxes.
List the important information:
• There are 16 tissue boxes in a case.
• The boxes are cubic, or square prisms.
Course 1
10-8 Finding Volume
You can make models using cubes to find the possible dimensions for a case of 16 tissue boxes.
2 Make a Plan
Additional Example 3 Continued
Course 1
10-8 Finding Volume
Solve 3
You can make models using cubes to find the possible dimensions for a case of 16 cubes.
Additional Example 3 Continued
The possible dimensions for a case of 16 cubic tissue boxes are the following: 1 • 1 • 16, 1 • 2 • 8, 1 • 4 • 4, and 2 • 2 • 4 .
Course 1
10-8 Finding Volume
Notice that each dimension is a factor of 16. Also, the product of the dimensions (length • width • height) is 16, showing that the volume of each case is 16 cubes.
Look Back 4
Additional Example 3 Continued
Course 1
10-8 Finding Volume
Try This: Example 3
Suppose a paper company ships 12 cubic boxes of envelopes in each case. What are the possible dimensions for a case of envelope boxes? 1 Understand the Problem
The answer will be all possible dimensions for a case of 12 cubic boxes.
List the important information:
• There are 12 envelope boxes in a case.
• The boxes are cubic, or square prisms.
Course 1
10-8 Finding Volume
You can make models using cubes to find the possible dimensions for a case of 12 boxes of envelopes.
2 Make a Plan
Try This: Example 3 Continued
Course 1
10-8 Finding Volume
Solve 3 You can make models using cubes to find the possible dimensions for a case of 12 cubes.
The possible dimensions for a case of 24 cubic envelope boxes are the following: 1 • 1 • 12, 1 • 2 • 6, 1 • 3 • 4, and 2 • 2 • 3.
Try This: Example 3 Continued
Course 1
10-8 Finding Volume
Notice that each dimension is a factor of 12. Also, the product of the dimensions (length • width • height) is 12, showing that the volume of each case is 12 cubes.
Look Back 4
Try This: Example 3 Continued
Course 1
10-8 Finding Volume
Lesson Quiz
Find the volume of each figure.
1. rectangular prism with length 20 cm, width 15 cm, and height 12 cm
2. triangular prism with a height of 12 cm and a triangular base with base length 7.3 cm and height 3.5 cm
3. Find the volume of the figure shown.
Insert Lesson Title Here
3,600 cm3
153.3 cm3
38.13 cm3
Course 1
10-8 Finding Volume
10-9 Volume of Cylinders
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Find the volume of each figure described.
Course 1
10-9 Volume of Cylinders
359.04 cm3
1,320 cm3
1. rectangular prism with length 12 cm, width 11 cm, and height 10 cm
2. triangular prism with height 11 cm and triangular base with base length 10.2 cm and height 6.4 cm
Problem of the Day
The height of a box is half its width. The length is 12 in. longer than its width. If the volume of the box is 28 in , what are the dimensions of the box? 1 in. × 2 in. × 14 in.
3
Course 1
10-9 Volume of Cylinders
Learn to find volumes of cylinders.
Course 1
10-9 Volume of Cylinders
To find the volume of a cylinder, you can use the same method as you did for prisms: Multiply the area of the base by the height.
volume of a cylinder = area of base × height
The area of the circular base is πr2, so the formula is V = Bh = πr2h.
Course 1
10-9 Volume of Cylinders
Additional Example 1A: Finding the Volume of a Cylinder
Find the volume V of the cylinder to the nearest cubic unit.
A.
Write the formula. Replace π with 3.14, r with 4, and h with 7. Multiply. V ≈ 351.68
V = πr2h V ≈ 3.14 × 42 × 7
The volume is about 352 ft3. Course 1
10-9 Volume of Cylinders
Additional Example 1B: Finding the Volume of a Cylinder
B.
10 cm ÷ 2 = 5 cm Find the radius.
Write the formula. Replace π with 3.14, r with 5, and h with 11. Multiply. V ≈ 863.5
V = πr2h V ≈ 3.14 × 52 × 11
The volume is about 864 cm3. Course 1
10-9 Volume of Cylinders
Additional Example 1C: Finding the Volume of a Cylinder
C.
Find the radius. r = + 4 h 3 __
r = + 4 = 7 9 3 __ Substitute 9 for h.
Write the formula. Replace π with 3.14, r with 7, and h with 9. Multiply. V ≈ 1,384.74
V = πr2h V ≈ 3.14 × 72 × 9
The volume is about 1,385 in3. Course 1
10-9 Volume of Cylinders
Try This: Example 1A
Find the volume V of each cylinder to the nearest cubic unit.
A.
Multiply. V ≈ 565.2
The volume is about 565 ft3.
6 ft
5 ft
Write the formula. Replace π with 3.14, r with 6, and h with 5.
V = πr2h V ≈ 3.14 × 62 × 5
Course 1
10-9 Volume of Cylinders
Try This: Example 1B
B.
Multiply. V ≈ 301.44
8 cm ÷ 2 = 4 cm
The volume is about 301 cm3.
Find the radius.
8 cm
6 cm
Write the formula. Replace π with 3.14, r with 4, and h with 16.
V = πr2h V ≈ 3.14 × 42 × 6
Course 1
10-9 Volume of Cylinders
Try This: Example 1C
C.
Multiply. V ≈ 1230.88 The volume is about 1231 in3.
Find the radius. r = + 5 h 4 __
r = + 5 = 7 8 4 __ Substitute 8 for h.
r = + 5
h = 8 in
h 4
Write the formula. Replace π with 3.14, r with 7, and h with 8.
V = πr2h V ≈ 3.14 × 72 × 8
Course 1
10-9 Volume of Cylinders
Additional Example 2A: Application Ali has a cylinder-shaped pencil holder with a 3 in. diameter and a height of 5 in. Scott has a cylinder-shaped pencil holder with a 4 in. diameter and a height of 6 in. Estimate the volume of each cylinder to the nearest cubic inch.
A. Ali’s pencil holder
Write the formula. Replace π with 3.14, r with 1.5, and h with 5. Multiply. V ≈ 35.325
3 in. ÷ 2 = 1.5 in.
V ≈ 3.14 × 1.52 × 5
The volume of Ali’s pencil holder is about 35 in3.
Find the radius.
V = πr2h
Course 1
10-9 Volume of Cylinders
Additional Example 2B: Application
B. Scott’s pencil holder
Write the formula.
Multiply.
4 in. ÷ 2 = 2 in.
The volume of Scott’s pencil holder is about 75 in3.
Find the radius.
V = πr2h
Replace π with , r with
2, and h with 6.
22 7
__ V ≈ × 22 × 6 22
7 __
V ≈ = 75 528 7
___ 3 7
__
Course 1
10-9 Volume of Cylinders
Try This: Example 2A
Sara has a cylinder-shaped sunglasses case with a 3 in. diameter and a height of 6 in. Ulysses has a cylinder-shaped pencil holder with a 4 in. diameter and a height of 7 in. Estimate the volume of each cylinder to the nearest cubic inch.
A. Sara’s sunglasses case
Write the formula. Replace π with 3.14, r with 1.5, and h with 6. Multiply. V ≈ 42.39
3 in. ÷ 2 = 1.5 in.
V ≈ 3.14 × 1.52 × 6
The volume of Sara’s sunglasses case is about 42 in3.
Find the radius.
V = πr2h
Course 1
10-9 Volume of Cylinders
Try This: Example 2B
B. Ulysses’ pencil holder
Write the formula.
Multiply.
4 in. ÷ 2 = 2 in.
The volume of Scott’s pencil holder is about 75 in3.
Find the radius.
V = πr2h
Replace π with , r with
2, and h with 7.
22 7
__ V ≈ × 22 × 7 22
7 __
V = 88
Course 1
10-9 Volume of Cylinders
Additional Example 3A & 3B: Comparing Volumes of Cylinders
Find which cylinder has the greater volume.
Cylinder 1:
V ≈ 3.14 × 1.52 × 12 V = πr2h
V ≈ 84.78 cm3
Cylinder 2:
V ≈ 3.14 × 32 × 6 V = πr2h
V ≈ 169.56 cm3
Cylinder 2 has the greater volume because 169.56 cm3 > 84.78 cm3.
Course 1
10-9 Volume of Cylinders
Try This: Example 3A & 3B
Find which cylinder has the greater volume.
Cylinder 1:
V ≈ 3.14 × 2.52 × 10 V = πr2h
V ≈ 196.25 cm3
Cylinder 2:
V ≈ 3.14 × 22 × 4 V = πr2h
V ≈ 50.24 cm3
Cylinder 1 has the greater volume because 196.25 cm3 > 50.24 cm3.
10 cm 2.5 cm
4 cm
4 cm
Course 1
10-9 Volume of Cylinders
Lesson Quiz
Find the volume of each cylinder to the nearest cubic unit. Use 3.14 for π.
Insert Lesson Title Here
cylinder B
1,560.14 ft3
193 ft3
1017 ft3
1,181.64 ft3
Course 1
10-9 Volume of Cylinders
1. radius = 9 ft, height = 4 ft
2. radius = 3.2 ft, height = 6 ft
3. Which cylinder has a greater volume?
a. radius 5.6 ft and height 12 ft
b. radius 9.1 ft and height 6 ft
11-1 Introduction to Probability
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Write each fraction as a decimal and as a percent.
1. 2. 3. 4.
0.75; 75%
0.50; 50%
Course 1
11-1 Introduction to Probability
1 2 __
3 4 __
2 9 __
3 8 __ 0.375; 37.5%
0.2; 22.2% __
Problem of the Day
What fraction of the numbers from 0 to 99 are divisible by 3?.
or 34 100
___ 17 50
__
Course 1
11-1 Introduction to Probability
Learn to estimate the likelihood of an event and to write and compare probabilities.
Course 1
11-1 Introduction to Probability
Vocabulary probability
Insert Lesson Title Here
Course 1
11-1 Introduction to Probability
Probability is the measure of how likely an event is to occur.
Course 1
11-1 Introduction to Probability
Probabilities are written as fractions or decimals from 0 to 1 or as percents from 0% to 100%. The higher an event’s probability, the more likely that event is to happen.
• Events with a probability of 0, or 0%, never happen.
• Events with a probability of 1, or 100%, always happen.
• Events with a probability of 0.5, or 50%, have the same chance of happening as of not happening.
Course 1
11-1 Introduction to Probability
Impossible Unlikely As likely as not Likely Certain
0 0.5 1
2 1 __ 0% 100%
50%
Course 1
11-1 Introduction to Probability
Insert Lesson Title Here Additional Example 1A & 1B: Estimating the
Likelihood of an Event
Write impossible, unlikely, as likely as not, likely, or certain to describe each event.
A. You will roll an even number on a standard number cube.
B. The month of February has 28 days.
as likely as not
certain
A standard number cube is numbered from 1 to 6.
Helpful Hint
Course 1
11-1 Introduction to Probability
Insert Lesson Title Here
Additional Example 1C & 1D: Estimating the Likelihood of an Event
Write impossible, unlikely, as likely as not, likely, or certain to describe each event.
C. This spinner lands on blue.
D. This spinner lands on an odd number.
unlikely
impossible
Course 1
11-1 Introduction to Probability
Insert Lesson Title Here
Try This: Example 1A & 1B
Write impossible, unlikely, as likely as not, likely, or certain to describe each event.
A. You guess a number between 1 and 1000.
B. You will roll an odd number on a standard number cube.
unlikely
as likely as not
A standard number cube is numbered from 1 to 6.
Helpful Hint
Course 1
11-1 Introduction to Probability
Insert Lesson Title Here
Try This: Example 1C & 1D
Write impossible, unlikely, as likely as not, likely, or certain to describe each event.
C. This spinner lands on green.
D. This spinner lands on an even number.
unlikely
certain
Course 1
11-1 Introduction to Probability
Insert Lesson Title Here
Additional Example 2A & 2B: Writing Probabilities
A. The weather report gives a 75% chance of snow. Write this probability as a decimal and as a fraction.
75% = 0.75
B. The chance of being chosen is 0.8. Write this probability as a fraction and as a percent.
Write as a decimal.
75% = = 75 100 ___ 3
4 __ Write as a fraction in
simplest form.
0.8 = = 8 10 __ 4
5 __ Write as a fraction in
simplest form. 0.8 = 80% Write as a percent.
Course 1
11-1 Introduction to Probability
Insert Lesson Title Here
Additional Example 2C: Writing Probabilities
C. There is a chance of getting a ring. Write
this probability as a decimal and as a percent.
Write as a decimal.
Write as a percent.
7 50 __
= 7 ÷ 50 = 0.14 7 50 __
= = = 14% 7 • 2 50 • 2 _____ 7
50 __ 14
100 ___
In Example 2C, after you find the decimal form
of , you can use it to find the percent.
0.14 = 14%.
Helpful Hint
7 50 __
Course 1
11-1 Introduction to Probability
Insert Lesson Title Here
Try This: Example 2A & 2B
A. The weather report gives a 25% chance of snow. Write this probability as a decimal and as a fraction.
25% = 0.25
B. The chance of being chosen is 0.6. Write this probability as a fraction and as a percent.
Write as a decimal.
25% = = 25 100 ___ 1
4 __ Write as a fraction in
simplest form.
0.6 = = 6 10 __ 3
5 __ Write as a fraction in
simplest form. 0.6 = 60% Write as a percent.
Course 1
11-1 Introduction to Probability
Insert Lesson Title Here
Try This: Example 2C
C. There is a chance of getting a ring. Write
this probability as a decimal and as a percent.
Write as a decimal.
Write as a percent.
4 25 __
= 4 ÷ 25 = 0.16 4 25 __
= = = 16% 4 • 4 25 • 4 _____ 4
25 __ 16
100 ___
Course 1
11-1 Introduction to Probability
Insert Lesson Title Here
Additional Example 3A: Comparing Probabilities
A. On a standard number cube, there is a 50%
chance of rolling a multiple of 2 and a 33 %
chance of rolling a multiple of 3. Is it more likely to roll a multiple of 2 or a multiple of 3?
It is more likely to roll a multiple of 2.
Compare: 33 % < 50% 1 3 __
1 3 __
Course 1
11-1 Introduction to Probability
Insert Lesson Title Here
Additional Example 3B: Comparing Probabilities
B. When you spin a certain spinner, there is a 15% chance that it will land on yellow, a 15% chance it will land on green, and a 70% chance that it will land on purple. Is it more likely to land on purple or on green?
It is more likely to land on purple than on green.
Compare: 70% > 15%
Course 1
11-1 Introduction to Probability
Insert Lesson Title Here
Try This: Example 3A
A. When you spin a certain spinner, there is a 25% chance that it will land on purple, a 35% chance it will land on green, and a 40% chance that it will land on red. Is it more likely to land on red or on green?
It is more likely to land on red than on green.
Compare: 40% > 35%
Course 1
11-1 Introduction to Probability
Insert Lesson Title Here
Try This: Example 3B
B. When you spin a certain spinner, there is a
9% chance that it will land on yellow, a 35 %
chance it will land on brown, and a 55 %
chance that it will land on blue. Is it more likely to land on brown or on yellow?
It is more likely to land on brown than on yellow.
Compare: 9% < 35 % 1 3 __
1 3 __
2 3 __
Course 1
11-1 Introduction to Probability
Lesson Quiz Write impossible, unlikely, equally likely, likely, or certain to describe each event.
1. The sun will rise tomorrow.
2. You will roll 13 when rolling two dice.
3. There is a 0.125 chance of picking the winning ticket. Write this probability as a fraction and as a percent.
4. At Hamburger Hut, there is a 20% chance of getting a plastic dinosaur cup and a 35% chance of getting a plastic rabbit cup. It is less likely that you will receive a rabbit cup or dinosaur cup?
impossible
certain
Insert Lesson Title Here
dinosaur cup
, 12.5% 1 8 __
Course 1
11-1 Introduction to Probability
11-2 Experimental Probability
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Write impossible, unlikely, equally likely, likely, or certain to describe each event. 1. A particular person’s birthday falls on the first of a month. 2. You roll an odd number on a fair number cube. 3. There is a 0.14 probability of picking the winning ticket. Write this as a fraction and as a percent.
unlikely
equally likely
Course 1
11-2 Experimental Probability
, 14% 7 50 __
Problem of the Day
Max picks a letter out of this problem at random. What is the probability that the letter is in the first half of the alphabet?
57 101 ___
Course 1
11-2 Experimental Probability
Learn to find the experimental probability of an event.
Course 1
11-2 Experimental Probability
Vocabulary experiment outcome sample space experimental probability
Insert Lesson Title Here
Course 1
11-2 Experimental Probability
An experiment is an activity involving chance that can have different results. Flipping a coin and rolling a number cube are examples of experiments.
The different results that can occur are called outcomes of the experiment. If you are flipping a coin, heads is one possible outcome.
The sample space of an experiment is the set of all possible outcomes. You can use {} to show sample spaces. When a coin is being flipped, {heads, tails} is the sample space.
Course 1
11-2 Experimental Probability
Additional Example 1A: Identifying Outcomes and Sample Spaces
For each experiment, identify the outcome shown and the sample space.
A. Spinning two spinners
outcome shown: B1 sample space: {A1, A2, B1, B2}
Course 1
11-2 Experimental Probability
Additional Example 1B: Identifying Outcomes and Sample Spaces
For each experiment, identify the outcome shown and the sample space.
B. Spinning a spinner
outcome shown: green sample space: {red, purple, green}
Course 1
11-2 Experimental Probability
Try This: Example 1A
For each experiment, identify the outcome shown and the sample space.
A. Spinning two spinners
outcome shown: C3
sample space: {C3, C4, D3, D4}
C D 3 4
Course 1
11-2 Experimental Probability
Try This: Example 1B
For each experiment, identify the outcome shown and the sample space.
B. Spinning a spinner
outcome shown: blue
sample space: {blue, orange, green}
Course 1
11-2 Experimental Probability
Performing an experiment is one way to estimate the probability of an event. If an experiment is repeated many times, the experimental probability of an event is the ratio of the number of times the event occurs to the total number of times the experiment is performed.
Course 1
11-2 Experimental Probability
The probability of an event can be written as P(event). P(blue) means “the probability that blue will be the outcome.”
Writing Math
Course 1
11-2 Experimental Probability
Additional Example 2: Finding Experimental Probability
For one month, Mr. Crowe recorded the time at which his train arrived. He organized his results in a frequency table.
Time 6:49-6:52 6:53-6:56 6:57-7:00
Frequency 7 8 5
Course 1
11-2 Experimental Probability
Additional Example 2A Continued
= 7 + 8
20 _____
= 15 20 ___ =
3 4 __
Before 6:57 includes 6:49-6:52 and 6:53-6:56.
P(before 6:57) ≈ number of times the event occurs total number of trials
___________________________
A. Find the experimental probability that the train will arrive before 6:57.
Course 1
11-2 Experimental Probability
Additional Example 2B: Finding Experimental Probability
= 8 20 ___ =
2 5 __
P(between 6:53 and 6:56) ≈ number of times the event occurs
total number of trials ___________________________
B. Find the experimental probability that the train will arrive between 6:53 and 6:56.
Course 1
11-2 Experimental Probability
Try This: Example 2
For one month, Ms. Simons recorded the time at which her bus arrived. She organized her results in a frequency table.
Time 4:31-4:40 4:41-4:50 4:51-5:00
Frequency 4 8 12
Course 1
11-2 Experimental Probability
Try This: Example 2A
= 4 + 8
24 _____
= 12 24 ___ =
1 2 __
Before 4:51 includes 4:31-4:40 and 4:41-4:50.
P(before 4:51) ≈ number of times the event occurs total number of trials
___________________________
A. Find the experimental probability that the bus will arrive before 4:51.
Course 1
11-2 Experimental Probability
Try This: Example 2B
= 8
24 ___ =
1
3 __
P(between 4:41 and 4:50) ≈ number of times the event occurs
total number of trials ___________________________
B. Find the experimental probability that the bus will arrive between 4:41 and 4:50.
Course 1
11-2 Experimental Probability
Additional Example 3: Comparing Experimental Probabilities
Erika tossed a cylinder 30 times and recorded whether it landed on one of its bases or on its side. Based on Erika’s experiment, which way is the cylinder more likely to land?
Outcome On a base On its side
Frequency llll llll llll llll llll llll l
Find the experimental probability of each outcome.
Course 1
11-2 Experimental Probability
Additional Example 3 Continued
= 21 30 ___ P(side) ≈
number of times the event occurs total number of trials
___________________________
Compare the probabilities. 9 30 ___ <
21 30 ___
It is more likely that the cylinder will land on its side.
= 9 30 ___ P(base) ≈
number of times the event occurs total number of trials
___________________________
Course 1
11-2 Experimental Probability
Additional Example 3: Comparing Experimental Probabilities
Chad tossed a cylinder 25 times and recorded whether it landed on one of its bases or on its side. Based on Chads’s experiment, which way is the cylinder more likely to land?
Outcome On a base On its side
Frequency llll llll llll llll llll
Find the experimental probability of each outcome.
Course 1
11-2 Experimental Probability
Try This: Example 3 Continued
= 20 25 ___ P(side) ≈
number of times the event occurs total number of trials
___________________________
Compare the probabilities. 5 25 ___ <
20 25 ___
It is more likely that the cylinder will land on its side.
= 5 25 ___ P(base) ≈
number of times the event occurs total number of trials
___________________________
Course 1
11-2 Experimental Probability
Lesson Quiz: Part 1
1. The spinner below was spun. Identify the outcome shown and the sample space.
outcome: green; sample space: {red, blue, green, purple, yellow}
Insert Lesson Title Here
Course 1
11-2 Experimental Probability
Lesson Quiz: Part 2
2. Find the experimental probability that the spinner will land on blue.
3. Find the experimental probability that the spinner will land on red.
4. Based on the experiment, on which color will the spinner most likely land?
Insert Lesson Title Here
red
2 9 __
4 9 __
Sandra spun the spinner above several times and recorded the results in the table.
Course 1
11-2 Experimental Probability
11-3 Theoretical Probability
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Tim took one marble from a bag, recorded the color, and returned it to the bag. He repeated this several times and recorded the results. 1. Find the experimental probability that a marble selected from the bag will be green.
2. Find the experimental probability that a marble selected from the bag will not be yellow.
Course 1
11-3 Theoretical Probability
3 5 __
4 5 __
Problem of the Day
What is the probability that the sum of four consecutive whole numbers is divisible by 4?. 0
Course 1
11-3 Theoretical Probability
Learn to find the theoretical probability of an event.
Course 1
11-3 Theoretical Probability
Vocabulary theoretical probability equally likely fair
Insert Lesson Title Here
Course 1
11-3 Theoretical Probability
Another way to estimate probability of an event is to use theoretical probability. One situation in which you can use theoretical probability is when all outcomes have the same chance of occurring. In other words, the outcomes are equally likely.
Course 1
11-3 Theoretical Probability
An experiment with equally likely outcomes is said to be fair. You can usually assume that experiments involving items such as coins and number cubes are fair.
Course 1
11-3 Theoretical Probability
Additional Example 1A: Finding Theoretical Probability
A. What is the probability that this fair spinner will land on 3? There are three possible outcomes when spinning this spinner: 1, 2, or 3. All are equally likely because the spinner is fair.
P(3)= 3 possible outcomes _________________
There is only one way for the spinner to land on 3.
P(3)= 3 possible outcomes __________________ 1 way event can occur
= 1 3 __
Course 1
11-3 Theoretical Probability
Additional Example 1B: Finding Theoretical Probability
B. What is the probability of rolling a number greater than 4 on a fair number cube?
There are six possible outcomes when a fair number cube is rolled: 1, 2, 3, 4, 5, or 6. All are equally likely. There are 2 ways to roll a number greater than 4:5 or 6.
P(greater than 4)= 6 possible outcomes _________________
P(greater than 4)= 6 possible outcomes ____________________ 2 ways events can occur
= 2 6 __
Course 1
11-3 Theoretical Probability
Try This: Example 1A
A. What is the probability that this fair spinner will land on 1?
There are three possible outcomes when spinning this spinner: 1, 2, or 3. All are equally likely because the spinner is fair.
P(3)= 3 possible outcomes _________________
There is only one way for the spinner to land on 1.
P(3)= 3 possible outcomes __________________ 1 way event can occur
= 1 3 __
Course 1
11-3 Theoretical Probability
Try This: Example 1B
B. What is the probability of rolling a number less than 4 on a fair number cube?
There are six possible outcomes when a fair number cube is rolled: 1, 2, 3, 4, 5, or 6. All are equally likely. There are 3 ways to roll a number greater than 4:3, 2 or 1.
P(less than 4)= 6 possible outcomes _________________
P(less than 4)= 6 possible outcomes ____________________ 3 ways events can occur = 1
2 __
Course 1
11-3 Theoretical Probability
Think about a single experiment, such as tossing a coin. There are two possible outcomes, heads or tails. What is P(heads) + P(tails)?
Experimental Probability (coin tossed 10 times)
Theoretical Probability
H T
llll l llll P(heads) = P(tails) =
+
6 10 __ 4
10 __ 10
10 __ = = 1
6 10 __ 4
10 __ P(heads) = P(tails)=
1 2 __ 6
10 __
1 2 __ + 1
2 __ 2
2 __ = = 1
Course 1
11-3 Theoretical Probability
No matter how you determine the probabilities, their sum is 1.
This is true for any experiment—the probabilities of the individual outcomes add to 1 (or 100%, if the probabilities are given as percents.)
Course 1
11-3 Theoretical Probability
Additional Example 2: Finding Probabilities of Events not Happening
Suppose there is a 45% chance of snow tomorrow. What is the probability that it will not snow?
In this situation there are two possible outcomes, either it will snow or it will not snow. P(snow) + P(not snow) = 100%
45% + P(not snow) = 100% -45% -45%
P(not snow) = 55% _____ _____ Subtract 45%
from each side.
Course 1
11-3 Theoretical Probability
Course 1
11-3 Theoretical Probability
Try This: Example 2
Suppose there is a 35% chance of rain tomorrow. What is the probability that it will not rain?
In this situation there are two possible outcomes, either it will rain or it will not rain.
P(rain) + P(not rain) = 100%
35% + P(not rain) = 100%
P(not rain) = 65%
-35% -35% _____ _____ Subtract 35% from each side.
Lesson Quiz
Use the spinner shown for problems 1-3.
1. P(2)
2. P(odd number)
3. P(factor of 6)
4. Suppose there is a 2% chance of spinning the winning number at a carnival game. What is the probability of not winning?
Insert Lesson Title Here
98%
2 7 __
4 7 __
4 7 __
Course 1
11-3 Theoretical Probability
11-4 Make an Organized List
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up A game die with eight sides numbered 1 through 8 is rolled. Find each probability.
1. P(1, 2, or 3) 2. P(even number) 3. P(number greater than 9)
Course 1
11-4 Make an Organized List
3 8 __
1 2 __
0
Problem of the Day
Sam and Pam can have an apple, an orange, or a pear. What is the probability that they will pick the same snack?
Course 1
11-4 Make an Organized List
1 3 __
Learn to make an organized list to find all possible outcomes.
Course 1
11-4 Make an Organized List
Course 1
11-4 Make an Organized List
When you have to find many possibilities, one way to find them all is to make an organized list. A tree diagram is one way to organize information.
Course 1
11-4 Make an Organized List Additional Example 1: Using a Tree Diagram Matt wants to take a 3-day weekend trip to visit his grandparents. He can take either Friday or Monday off from work, and he can either fly, drive, take a train, or take a bus. How many options are available to Matt?
Friday
Monday
fly drive take a train take a bus
Friday and fly Friday and drive Friday and take a train Friday and take a bus
fly drive take a train take a bus
Monday and fly Monday and drive Monday and take a train Monday and take a bus
Course 1
11-4 Make an Organized List
Additional Example 1 Continued
Follow each branch on the tree diagram to find all of the possible outcomes. There are 8 different weekend trip combinations available to Matt.
Course 1
11-4 Make an Organized List Try This: Example 1
For her work uniform, Missy has a choice of three colors of pants—black, khaki, or navy. She has four choices for shirt colors—red, white, green, and yellow. How many different uniforms can Missy wear?
black pants
khaki pants
red shirt white shirt green shirt yellow shirt
black pants and red shirt black pants and white shirt black pants and green shirt black pants and yellow shirt
red shirt white shirt green shirt yellow shirt
khaki pants and red shirt khaki pants and white shirt khaki pants and green shirt khaki pants and yellow shirt
Course 1
11-4 Make an Organized List
Additional Example 1 Continued
Follow each branch on the tree diagram to find all of the possible outcomes. There are 12 different uniform combinations available to Missy.
navy pants
navy pants and red shirt navy pants and white shirt navy pants and green shirt navy pants and yellow shirt
red shirt white shirt green shirt yellow shirt
Course 1
11-4 Make an Organized List
Additional Example 2: Problem Solving Application
One girl and one boy will be chosen to go to the state science fair. The girl finalists are Alia, Brenda, Cathy, Deb, and Erika. The boy finalists are Frank, Greg, and Hal. How many different pairs of one girl and one boy can be formed?
Course 1
11-4 Make an Organized List
Additional Example 2 Continued
1 Understand the Problem
The answer will be the number of different pairs of one girl and one boy.
List the important information:
• There are five girls, A, B, C, D, and E.
• There are three boys, F, G, and H.
• Only one girl and one boy will be chosen.
Use each student’s first initial.
Course 1
11-4 Make an Organized List
You can make an organized list to keep track of the sequences.
2 Make a Plan
Additional Example 2 Continued
Course 1
11-4 Make an Organized List
Solve 3 • List all the pairs that begin with A. AF, AG, AH
• List all the pairs that begin with B. BF, BG, BH
• List all the pairs that begin with C. CF, CG, CH
• List all the pairs that begin with D. DF, DG, DH
• List all the pairs that begin with E. EF, EG, EH
Additional Example 2 Continued
There are 5 groups of 3 pairs.
3 + 3 + 3 + 3 + 3 = 15 There are 15 pairs of one girl and one boy.
Course 1
11-4 Make an Organized List
You could have made a list beginning with a boy’s name. There would be 3 groups of 5 pairs.
5 + 5 + 5 = 15
Look Back 4
Each list will have 15 pairs of one girl and one boy.
Additional Example 2 Continued
Course 1
11-4 Make an Organized List
Try This: Example 2
One girl and one boy will be chosen to go to the movie preview. The girl finalists are Fay, Gerri, Heidi, and Ingrid. The boy finalists are Kevin, Larry, and Marc. How many different pairs of one girl and one boy can be formed?
Course 1
11-4 Make an Organized List
Try This: Example 2 Continued
1 Understand the Problem
The answer will be the number of different pairs of one girl and one boy.
List the important information:
• There are five girls, F, G, H, and I.
• There are three boys, K, L, and M.
• Only one girl and one boy will be chosen.
Use each student’s first initial.
Course 1
11-4 Make an Organized List
You can make an organized list to keep track of the sequences.
2 Make a Plan
Try This: Example 2 Continued
Course 1
11-4 Make an Organized List
Solve 3 • List all the pairs that begin with F. FK, FL, FM
• List all the pairs that begin with G. GK, GL, GM
• List all the pairs that begin with H. HK, HL, HM
• List all the pairs that begin with I. IK, IL, IM
Try This: Example 2 Continued
There are 4 groups of 3 pairs.
3 + 3 + 3 + 3 = 12
There are 12 pairs of one girl and one boy.
Course 1
11-4 Make an Organized List
You could have made a list beginning with a boy’s name. There would be 3 groups of 4 pairs.
4 + 4 + 4 = 12
Look Back 4
Each list will have 12 pairs of one girl and one boy.
Try This: Example 2 Continued
Lesson Quiz
1. A baseball coach has 4 pitchers, 3 catchers, and 2 shortstops on his team. How many different combinations of players can he use for the positions?
2. You are taking a 5-question true/false test. How many possible combinations of answers are there?
3. You are planning a small game booth at the local street fair. You have a choice of 3 games and 4 different prizes. How many combinations of games and prizes are there?
10
24
Insert Lesson Title Here
12
Course 1
11-4 Make an Organized List
11-5 Compound Events
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up A café offers a soup-and-sandwich combination lunch. You can choose tomato soup, chicken noodle soup, or clam chowder. You can choose a turkey, ham, veggie, or tuna sandwich. How many lunch combinations are there?
12
Course 1
11-5 Compound Events
Problem of the Day
Rory dropped a quarter, a nickel, a dime, and a penny. What is the probability that all four landed tails up?
1 16 __
Course 1
11-5 Compound Events
Learn to list all the outcomes and find the theoretical probability of a compound event.
Course 1
11-5 Compound Events
Vocabulary compound event
Insert Lesson Title Here
Course 1
11-5 Compound Events
A compound event consists of two or more single events. For example, the birth of one child is a single event. The births of four children make up a compound event.
Course 1
11-5 Compound Events
Additional Example 1A: Finding Probabilities of Compound Events
Jerome spins the spinner and rolls a fair number cube.
A. Find the probability that the number cube will show an even number and that the spinner will show a B.
Course 1
11-5 Compound Events
Additional Example 1A Continued
1 2 3 4 5 6 A 1, A 2, A 3, A 4, A 5, A 6, A B 1, B 2, B 3, B 4, B 5, B 6, B C 1, C 2, C 3, C 4, C 5, C 6, C
Number Cube
Spinner
First find all of the possible outcomes.
There are 18 possible outcomes, and all are equally likely.
Three of the outcomes have an even number and B: 2, B; 4, B; and 6, B.
Course 1
11-5 Compound Events
Additional Example 1A Continued
P(even, B)= 18 possible outcomes ____________________ 3 ways events can occur
= 3 18 ___
= 1 6 __ Write your answer in simplest
form.
Course 1
11-5 Compound Events
Additional Example 1B: Finding Probabilities of Compound Events
B. Find the probability that the number cube will show a 4 and that the spinner will show an A.
Only one outcome is 4, A.
P(4, A)= 18 possible outcomes ____________________ 1 way event can occur
= 1 18 ___
Course 1
11-5 Compound Events
Additional Example 1C: Finding Probabilities of Compound Events
C. In the experiment on page 575, what is the probability that the coin will show tails, the spinner will land on purple, and a green marble will be chosen?
There are 18 equally likely outcomes.
P(tails, purple, green)= 18 possible outcomes ____________________ 1 way event can occur
= 1 18 ___
Course 1
11-5 Compound Events
Try This: Example 1A
Kiki spins the spinner and rolls a fair number cube.
A. Find the probability that the number cube will show an odd number and that the spinner will show an A.
Course 1
11-5 Compound Events
Try This: Example 1A Continued
1 2 3 4 5 6 A 1, A 2, A 3, A 4, A 5, A 6, A B 1, B 2, B 3, B 4, B 5, B 6, B C 1, C 2, C 3, C 4, C 5, C 6, C
Number Cube
Spinner
First find all of the possible outcomes.
There are 18 possible outcomes, and all are equally likely.
Course 1
11-5 Compound Events
Three of the outcomes have an odd number and BA: 1, A; 3, A; and 5, A.
P(odd, A)= 18 possible outcomes ____________________ 3 ways events can occur
= 3 18 ___
= 1 6 __ Write your answer in simplest
form.
Try This: Example 1A Continued
Course 1
11-5 Compound Events
Try This: Example 1B
B. Find the probability that the number cube will show a 6 and that the spinner will show an C.
Only one outcome is 6, C.
P(6, C)= 18 possible outcomes ____________________ 1 way event can occur
= 1 18 ___
Course 1
11-5 Compound Events
Try This: Example 1C
C. In the experiment on page 575 in the student book, what is the probability that the coin will show heads, the spinner will land on orange, and a red marble will be chosen?
There are 18 equally likely outcomes .
P(heads, orange, red)= 18 possible outcomes ____________________ 1 way event can occur
= 1 18 ___
Course 1
11-5 Compound Events
Lesson Quiz An experiment involves one spin of the spinner and one flip of the coin. Find each probability.
1. What is the probability of the spinner landing on red and the coin landing tails up?
2. What is the probability of the spinner landing on an odd number and the coin landing heads up?
3. What is the probability of the spinner landing on an even number and the coin landing tails up?
Insert Lesson Title Here
1 6 __
1 3 __
1 6 __
Course 1
11-5 Compound Events
11-6 Making Predictions
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up
1. Zachary rolled a fair number cube twice. Find the probability of the number cube showing an odd number both times.
2. Larissa rolled a fair number cube twice.
Find the probability of the number cube showing the same number both times.
Course 1
11-6 Making Predictions
1 4 __
1 36 ___
Problem of the Day
The average of three numbers is 45. If the average of the first two numbers is 47, what is the third number?
41
Course 1
11-6 Making Predictions
Learn to use probability to predict events.
Course 1
11-6 Making Predictions
Vocabulary prediction
Insert Lesson Title Here
Course 1
11-6 Making Predictions
Insert Lesson Title Here
A prediction is a guess about something in the future. A way to make a prediction is to use probability.
Course 1
11-6 Making Predictions
Additional Example 1A: Using Probability to Make Prediction
A. A store claims that 78% of shoppers end up buying something. Out of 1,000 shoppers, how many would you predict will buy something?
You can write a proportion. Remember that percent means “per hundred.”
Course 1
11-6 Making Predictions
Additional Example 1A Continued
100x 100 ____ 78,000
100 ______ =
Divide both sides by 100 to undo the multiplication.
x = 780
You can predict that about 780 out of 1,000 customers will buy something.
78 100 ___ x
1000 ____ =
Think: 78 out of 100 is how many out of 1,000.
100 • x = 78 • 1,000
100x = 78,000
The cross products are equal.
x is multiplied by 100.
Course 1
11-6 Making Predictions
Additional Example 1B: Using Probability to Make Predictions
B. If you roll a number cube 30 times, how many times do you expect to roll a number greater than 2?
2 3 __ x
30 ___ =
Think: 2 out of 3 is how many out of 30.
3 • x = 2 • 30
3x = 60
The cross products are equal.
x is multiplied by 3.
P(greater than 2) = = 4 6 __ 2
3 __
Course 1
11-6 Making Predictions
Additional Example 1B Continued
Divide both sides by 3 to undo the multiplication.
x = 20
You can expect to roll a number greater than 2 about 20 times.
3x 3 __ 60
3 __ =
Course 1
11-6 Making Predictions
Try This: Example 1A
A. A store claims 62% of shoppers end up buying something. Out of 1,000 shoppers, how many would you predict will buy something?
You can write a proportion. Remember that percent means “per hundred.”
Course 1
11-6 Making Predictions
Try This: Example 1A Continued
100x 100 ____ 62,000
100 ______ =
Divide both sides by 100 to undo the multiplication.
x = 620 You can predict that about 620 out of 1,000 customers will buy something.
62 100 ___ x
1000 ____ =
Think: 62 out of 100 is how many out of 1,000.
100 • x = 62 • 1,000
100x = 62,000
The cross products are equal.
x is multiplied by 100.
Course 1
11-6 Making Predictions
Try This: Example 1B
B. If you roll a number cube 30 times, how many times do you expect to roll a number greater than 3?
1 2 __ x
30 ___ =
Think: 1 out of 2 is how many out of 30.
2 • x = 1 • 30
2x = 30
The cross products are equal.
x is multiplied by 2.
P(greater than 3) = = 3 6 __ 1
2 __
Course 1
11-6 Making Predictions
Try This: Example 1B Continued
Divide both sides by 2 to undo the multiplication.
x = 15
You can expect to roll a number greater than 3 about 15 times.
2x 2 __ 30
2 __ =
Course 1
11-6 Making Predictions
Additional Example 2: Problem Solving Application
A stadium sell yearly parking passes. If you have a parking pass, you can park at that stadium for any event during that year.
The managers of the stadium estimate that the probability that a person with a pass will attend any one event is 50%. The parking lot has 400 spaces. If the managers want the lot to be full at every event, how many passes should they sell?
Course 1
11-6 Making Predictions
1 Understand the Problem The answer will be the number of parking passes they should sell.
List the important information:
• P(person with pass attends event): = 50%
• There are 400 parking spaces
The managers want to fill all 400 spaces. But on average, only 50% of parking pass holders will attend. So 50% of pass holders must equal 400. You can write an equation to find this number.
2 Make a Plan
Course 1
11-6 Making Predictions
Solve 3
50 100 ___ 400
x ____ =
Think: 50 out of 100 is 400 out of how many?
100 • 400 = 50 • x
40,000 = 50x
The cross products are equal.
x is multiplied by 50.
40,000 50 ______ 50x
50 ___ = Divide both sides by 50 to
undo the multiplication.
800 = x
The managers should sell 800 parking passes.
Course 1
11-6 Making Predictions
Insert Lesson Title Here
If the managers sold only 400 passes, the parking lot would not usually be full because only about 50% of the people with passes will attend any one event. The managers should sell more than 400 passes, so 800 is a reasonable answer.
Look Back 4
Course 1
11-6 Making Predictions
Try This: Example 2
A stadium sells yearly parking passes. If you have a parking pass, you can park at that stadium for any event during that year.
The managers estimate that the probability that a person with a pass will attend any one event is 60%. The parking lot has 600 spaces. If the managers want the lot to be full at every event, how many passes should they sell?
Course 1
11-6 Making Predictions
1 Understand the Problem The answer will be the number of parking passes they should sell.
List the important information:
• P(person with pass attends event): = 60%
• There are 600 parking spaces
The managers want to fill all 600 spaces. But on average, only 60% of parking pass holders will attend. So 60% of pass holders must equal 600. You can write an equation to find this number.
2 Make a Plan
Course 1
11-6 Making Predictions
Solve 3
60 100 ___ 600
x ____ =
Think: 60 out of 100 is 600 out of how many?
100 • 600 = 60 • x
60,000 = 60x
The cross products are equal.
x is multiplied by 60.
60,000 60 ______ 60x
60 ___ = Divide both sides by 60 to
undo the multiplication.
1000 = x
The managers should sell 1000 parking passes.
Course 1
11-6 Making Predictions
Insert Lesson Title Here
If the managers sold only 600 passes, the parking lot would not usually be full because only about 50% of the people with passes will attend any one event. The managers should sell more than 600 passes, so 1000 is a reasonable answer.
Look Back 4
Course 1
11-6 Making Predictions
Lesson Quiz: Part 1
1. The owner of a local pizzeria estimates that 72% of his customers order pepperoni on their on their pizza. Out of 250 orders taken in one day, how many would you predict to have pepperoni? 180
Insert Lesson Title Here
Course 1
11-6 Making Predictions
Lesson Quiz: Part 2
2. A bag contains 9 red chips, 4 blue chips, and 7 yellow chips. You pick a chip from the bag, record its color, and put the chip back in the bag. If you do this 100 times, how many times do you expect to remove a yellow chip from the bag?
3. A quality-control inspector has determined that 3% of the items he checks are defective. If the company he works for produces 3,000 items per day, how many does the inspector predict will be defective?
35
Insert Lesson Title Here
90
Course 1
11-6 Making Predictions
12-1 Tables and Functions
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Evaluate each expression for the given value of the variable.
1. 4x – 1 for x = 2 2. 7y + 3 for y = 5 3. x + 2 for x = –6 4. 8y – 3 for y = –2
7 38
–1
Course 1
12-1 Tables and Functions
1 2 __
–19
Problem of the Day
Maria rented an electric car at a rate of $40 per day and $0.15 per mile. She returned the car the same day, gave the rental clerk a $100 bill, and got $21.75 back in change. How far did Maria drive the car?
255 miles
Course 1
12-1 Tables and Functions
Learn to use data in a table to write an equation for a function and to use the equation to find a missing value.
Course 1
12-1 Tables and Functions
Vocabulary function input output
Insert Lesson Title Here
Course 1
12-1 Tables and Functions
A function is a rule that relates two quantities so that each input value corresponds exactly to one output value.
Course 1
12-1 Tables and Functions
Additional Example 1A: Writing Equations from Function Tables
x 3 4 5 6 7 10 y 13 16 19 22 25
y is 3 times x + 4.
y = 3x + 4
Compare x and y to find a pattern. Use the pattern to write an equation.
y = 3(10) + 4 Substitute 10 for x.
y = 30 + 4 = 34 Use your function rule to find y when x = 10.
Write an equation for a function that gives the values in the table. Use the equation to find the value of y for the indicated value of x.
Course 1
12-1 Tables and Functions
When all the y-values are greater than the corresponding x-values, use addition and/or multiplication in your equation.
Helpful Hint
Course 1
12-1 Tables and Functions
Try This: Example 1
x 3 4 5 6 7 10 y 10 12 14 16 18
y is 2 times x + 4.
y = 2x + 4
Compare x and y to find a pattern. Use the pattern to write an equation.
y = 2(10) + 4 Substitute 10 for x.
y = 20 + 4 = 24 Use your function rule to find y when x = 10.
Write an equation for a function that gives the values in the table. Use the equation to find the value of y for the indicated value of x.
Course 1
12-1 Tables and Functions
You can write equations for functions that are described in words.
Course 1
12-1 Tables and Functions
Additional Example 2: Translating Words into Math
The height of a painting is 7 times its width.
h = height of painting Choose variables for the equation.
h = 7w Write an equation.
Write an equation for the function. Tell what each variable you use represents.
w = width of painting
Course 1
12-1 Tables and Functions
Try This: Example 2
The height of a mirror is 4 times its width.
h = height of painting Choose variables for the equation.
h = 4w Write an equation.
Write an equation for the function. Tell what each variable you use represents.
w = width of painting
Course 1
12-1 Tables and Functions
Additional Example 3: Problem Solving Application
The school choir tracked the number of tickets sold and the total amount of money received. They sold each ticket for the same price. They received $80 for 20 tickets, $88 for 22 tickets, and $108 for 27 tickets. Write an equation for the function.
1 Understand the Problem
The answer will be an equation that describes the relationship between the number of tickets sold and the money received.
Course 1
12-1 Tables and Functions
You can make a table to display the data. 2 Make a Plan
Solve 3 Let t be the number of tickets. Let m be the amount of money received.
t 20 22 27 m 80 88 108
m is equal to 4 times t. Compare t and m.
m = 4t. Write an equation.
Course 1
12-1 Tables and Functions
Substitute the t and m values in the table to check that they are solutions of the equation m = 4t.
Look Back 4
m = 4t (20,80)
80 = 4 • 20 ?
80 = 80 ?
m = 4t (22,88)
88 = 4 • 22 ?
88 = 88 ?
m = 4t (27, 108)
108 = 4 • 27 ?
108 = 108 ?
Course 1
12-1 Tables and Functions
Try This: Example 3
The school choir tracked the number of tickets sold and the total amount of money received. They sold each ticket for the same price. They received $60 for 20 tickets, $66 for 22 tickets, and $81 for 27 tickets. Write an equation for the function.
1 Understand the Problem
The answer will be an equation that describes the relationship between the number of tickets sold and the money received.
Course 1
12-1 Tables and Functions
You can make a table to display the data. 2 Make a Plan
Solve 3 Let t be the number of tickets. Let m be the amount of money received.
t 20 22 27 m 60 66 81
m is equal to 3 times t. Compare t and m.
m = 3t. Write an equation.
Course 1
12-1 Tables and Functions
Substitute the t and m values in the table to check that they are solutions of the equation m = 3t.
Look Back 4
m = 3t (20, 60)
60 = 3 • 20 ?
60 = 60 ?
m = 3t (22, 66)
88 = 3 • 22 ?
88 = 66 ?
m = 3t (27, 81)
81 = 3 • 27 ?
81 = 81 ?
Course 1
12-1 Tables and Functions
Lesson Quiz
1. Write an equation for a function that gives the values in the table below. Use the equation to find the value for y for the indicated value of x.
2. Write an equation for the function. Tell what each variable you use represents. The height of a round can is 2 times its radius.
h = 2r, where h is the height and r is the radius.
y = 3x; 21
Insert Lesson Title Here
Course 1
12-1 Tables and Functions
X 0 1 3 5 7 y 0 3 9 15
12-2 Graphing Functions
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Write an equation for each function. Tell what each variable you use represents.
1. The length of a wall is 4 ft more than three times the height. 2. The number of trading cards is 3 less than the number of buttons.
l = 3h + 4, where l is length and h is height.
c = b – 3, where c is the number of cards and b is the number of buttons.
Course 1
12-2 Graphing Functions
Problem of the Day
Steve saved $1.50 each week. How many weeks did it take him to save enough to buy a $45 skateboard?
30
Course 1
12-2 Graphing Functions
Learn to represent linear functions using ordered pairs and graphs.
Course 1
12-2 Graphing Functions
Vocabulary linear equation
Insert Lesson Title Here
Course 1
12-2 Graphing Functions
Additional Example 1: Finding Solutions of Equations with Two Variables
Use the given x-values to write solutions of the equation y = 4x + 2 as ordered pairs. x =1, 2, 3, 4. Make a function table by using the given values for x to find values for y.
x 4x + 2 y 1 4(1) + 2 6 2 4(2) + 2 10 3 4(3) + 2 14 4 4(4) + 2 18
Write these solutions as ordered pairs.
(x, y) (1, 6) (2, 10) (3, 14) (4, 18)
Course 1
12-2 Graphing Functions
Try This: Example 1
Use the given x-values to write solutions of the equation y = 3x + 2 as ordered pairs. x = 2, 3, 4, 5. Make a function table by using the given values for x to find values for y.
x 3x + 2 y 2 3(2) + 2 8 3 3(3) + 2 11 4 3(4) + 2 14 5 3(5) + 2 17
Write these solutions as ordered pairs.
(x, y) (2, 8) (3, 11) (4, 14) (5, 17)
Course 1
12-2 Graphing Functions
Insert Lesson Title Here
Check if an ordered pair is a solution of an equation by putting the x and y values into the equation to see if they make it a true statement.
Course 1
12-2 Graphing Functions
Additional Example 2: Checking Solutions of Equations with Two Variables
Determine whether the ordered pair is a solution to the given equation.
(3, 21); y = 7x
y = 7x Write the equation.
21 = 7(3) ?
21 = 21 ? Substitute 3 for x and 21 for y.
So (3, 21) is a solution to y = 7x.
Course 1
12-2 Graphing Functions
Try This: Example 2
Determine whether the ordered pair is a solution to the given equation.
(4, 20); y = 5x
y = 5x Write the equation.
20 = 5(4) ?
20 = 20 ? Substitute 4 for x and 20 for y.
So (4, 20) is a solution to y = 5x.
Course 1
12-2 Graphing Functions
Insert Lesson Title Here
You can also graph the solutions of an equation on a coordinate plane. When you graph the ordered pairs of some functions, they form a straight line. The equations that express these functions are called linear equations.
Course 1
12-2 Graphing Functions
Insert Lesson Title Here
Additional Example 3: Reading Solutions on Graphs
Use the graph of the linear function to find the value of y for the given value of x.
x = 4
When x = 4, y = 2.
Start at the origin and move 4 units right. Move up until you reach the graph. Move left to find the y-value on the y-axis.
The ordered pair is (4, 2).
x
y
-4 -2 0 2 4
4
-2
-4
2
Course 1
12-2 Graphing Functions
Insert Lesson Title Here
Try This: Example 3
Use the graph of the linear function to find the value of y for the given value of x.
x = 2
When x = 2, y = 4.
Start at the origin and move 2 units right. Move up until you reach the graph. Move left to find the y-value on the y-axis.
The ordered pair is (2, 4).
x
y
-4 -2 0 2 4
4
-2
-4
2
Course 1
12-2 Graphing Functions
Additional Example 4: Graphing Linear Functions
Graph the function described by the equation. y = –x – 2
Make a function table.
x –x – 2 y
–1 –(–1) – 2 –1
0 –(0) – 2 –2
1 –(1) – 2 –3
Write these solutions as ordered pairs.
(x, y) (–1, –1) (0, –2) (1, –3)
Course 1
12-2 Graphing Functions
Insert Lesson Title Here
Additional Example 4 Continued
x
y
Graph the ordered pairs on a coordinate plane.
-5 -4 -3 -2 -1 0 1 2 3 4 5
5 4 3 2 1
-5 -4 -3 -2 -1
Draw a line through the points to represent all the values of x you could have chosen and the corresponding values of y.
Course 1
12-2 Graphing Functions
Try This: Example 4
Graph the function described by the equation. y = –x – 4
Make a function table.
x –x – 4 y
–1 –(– 1) – 4 –3
0 –(0) – 4 –4
1 –(1) – 4 –5
Write these solutions as ordered pairs.
(x, y) (–1, –3) (0, –4) (1, –5)
Course 1
12-2 Graphing Functions
Insert Lesson Title Here
Try This: Example 4
x
y
Graph the ordered pairs on a coordinate plane.
-5 -4 -3 -2 -1 0 1 2 3 4 5
5 4 3 2 1
-5 -4 -3 -2 -1
Draw a line through the points to represent all the values of x you could have chosen and the corresponding values of y.
Course 1
12-2 Graphing Functions
Lesson Quiz
1. Use the given x-values to write solutions as ordered pairs to the equation y = –3x + 1 for x = 0, 1, 2, and 3.
2. Determine whether (4, –2) is a solution to the equation y = –5x + 3.
3. Graph the function described by the equation y = –x + 3.
No, –2 ≠ –5(4) + 3
(0, 1), (1, –2), (2, –5), (3, –8)
Insert Lesson Title Here
Course 1
12-2 Graphing Functions
12-3 Graphing Translations
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up
1. Use the given x-values to write solutions of the following equation as ordered pairs. y = 6x – 2 for x = 0, 1, 2, 3
2. Determine whether (3, –13) is a solution
to the equation y = –4x – 1.
(0, –2), (1, 4), (2, 10), (3, 16)
yes
Course 1
12-3 Graphing Translation
Problem of the Day
Samantha’s house is 3 blocks east and 5 blocks south of Tyra. If Tyra walks straight south and then straight east to Samantha’s house, does she walk more blocks east or more blocks south? How many more?
south; 2 blocks
Course 1
12-3 Graphing Translation
Learn to use translations to change the positions of figures on a coordinate plane.
Course 1
12-3 Graphing Translation
A translation is a movement of a figure along a straight line. You can translate a figure on a coordinate plane by sliding it horizontally, vertically, or diagonally.
Course 1
12-3 Graphing Translation
Additional Example 1: Translating Figures on a Coordinate Plane
Give the coordinates of the vertices of the figure after the given translation.
Translate triangle DEF 4 units left and 3 units up.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
6 5 4 3 2 1
-1 -2 -3 -4 -5 -6
F
D
E
y
x
Course 1
12-3 Graphing Translation
Additional Example 1 Continued
To move the triangle 4 units left, subtract 4 from each of the x-coordinates.
To move the triangle 3 units up, add 3 to each of the y-coordinates.
DEF D’E’F’ D(1, 4)
E(4, 2)
F(–3, –3)
D’(1 – 4, 4 + 3) D’(–3, 7)
E’(0, 5)
F’(–7, 0)
E’(4 – 4, 2 + 3)
F’(-3 – 4, –3 + 3)
Course 1
12-3 Graphing Translation
Additional Example 1 Continued
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
5 4 3
1
-2 -3
-5 -6
6
2
-1
-4
E’
F’
D’ y
x
F
D
E
Course 1
12-3 Graphing Translation
Try This: Example 1
Give the coordinates of the vertices of the figure after the given translation.
Translate triangle GHJ 3 units left and 3 units up.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
6 5 4 3 2 1
-1 -2 -3 -4 -5 -6
J
G H
y
x
Course 1
12-3 Graphing Translation
Try This: Example 1
To move the triangle 3 units left, subtract 3 from each of the x-coordinates.
To move the triangle 3 units up, add 3 to each of the y-coordinates.
GHJ G’H’J’ G(2, 4)
H(4, 4)
J(–3, –2)
G’(2 – 3, 4 + 3) G’(–1, 7)
H’(1, 7)
J’(–6, 1)
H’(4 – 3, 4 + 3)
J’(–3 – 3, –2 + 3)
Course 1
12-3 Graphing Translation
Try This: Example 1 Continued
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
5 4 3
1
-2 -3
-5 -6
6
2
-1
-4
H’
J’
G’ y
x
J
G H
x
Course 1
12-3 Graphing Translation
Additional Example 2: Music Application
Members of a marching band begin in a trapezoid formation represented by trapezoid KLMN. Then they move 4 steps right and 5 steps down. Give the coordinates of the vertices of the trapezoid after such a translation.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
6 5 4 3 2 1
-1 -2 -3 -4 -5 -6
M
K L
y
x
N
-7 -8
Course 1
12-3 Graphing Translation
Additional Example 2 Continued
To move 4 steps right, add 4 to each of the x-coordinates.
To move 5 steps down, subtract 5 from each of the y-coordinates.
KLMN K’L’M’N’ K(–2, 1)
L(1, 1)
M(3, –3)
K’(–2 + 4, 1 – 5) K’(2, –4)
L’(5, –4)
M’(7, –8)
L’(1 + 4, 1 – 5)
M’(3 + 4, –3 – 5)
N(–4, –3) N’(0, –8) M’(–4 + 4, –3 – 5)
Course 1
12-3 Graphing Translation
Additional Example 2 Continued
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
6 5 4 3 2 1
-1 -2 -3 -4 -5 -6
M
K L
y
x
N
-7 M’
K’ L’
N’ -8
x
Course 1
12-3 Graphing Translation
Try This: Example 2
Members of a flag team begin in a trapezoid formation represented by trapezoid KLMN. Then they move 3 steps right and 2 steps down. Give the coordinates of the vertices of the trapezoid after such a translation.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
6 5 4 3 2 1
-1 -2 -3 -4 -5 -6
M
K L
y
x
N
Course 1
12-3 Graphing Translation
Try This: Example 2
To move 3 steps right, add 3 to each of the x-coordinates.
To move 2 steps down, subtract 2 from each of the y-coordinates.
KLMN K’L’M’N’ K(–1, 3)
L(1, 3)
M(3, –1)
K’(–1 + 3, 3 – 2) K’(2, 1)
L’(4, 1)
M’(6, –3)
L’(1 + 3, 3 – 2)
M’(3 + 3, –1 – 2)
N(–3, –1) N’(0, –3) M’(-3 + 3, –1 – 2)
Course 1
12-3 Graphing Translation
Try This: Example 2 Continued
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
6 5 4 3 2 1
-1 -2 -3 -4 -5 -6
M
K L
y
N
x
M’
K’ L’
N’
Course 1
12-3 Graphing Translation
Lesson Quiz
Give the coordinates of the vertices of triangle ABC, with vertices A(-5, -4), B(-3, 2), and C(1, -3), after the given translations.
1. Translate triangle ABC 3 units up and 2 units right.
2. Translate triangle ABC 5 units down and 3 units left.
3. Translate triangle ABC 2 units down and 4 units right.
A’(-3, -1), B’(-1, 5), and C’(3,0)
Insert Lesson Title Here
A’(-8, -9), B’(-6, -3), and C’(-2,-8)
A’(-1, -6), B’(1, 0), and C’(5,-5)
Course 1
12-3 Graphing Translation
12-4 Graphing Reflections
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up
A parallelogram has vertices (-4, 1), (0, 1), (-3, 4), and (1, 4). What are its vertices after it has been translated 4 units right and 3 units up? (0, 4), (4, 4), (1, 7), (5, 7)
Course 1
12-4 Graphing Reflections
Problem of the Day
These are rits: 24042, 383, and 4994. These are not rits: 39239, 28, and 5505. Which of these are rits: 39883, 4040, and 101? Why?
101 is a rit because it is the same forward and backward.
Course 1
12-4 Graphing Reflections
Learn to use reflections to change the positions of figures on a coordinate plane.
Course 1
12-4 Graphing Reflections
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
Additional Example 1: Reflecting Figures on a Coordinate Plane
Give the coordinates of the figure after the given reflection. Reflect parallelogram EFGH across the y-axis.
E F
G H
Course 1
12-4 Graphing Reflections
Additional Example 1 Continued
To reflect the parallelogram across the y-axis, write the opposites of the x-coordinates.
The y-coordinates do not change.
EFGH E’F’G’H’
E(–4, 0) E’(4, 0)
F(0, 0) F’(0, 0) G(–2, –3) G’(2, –3)
H(–6, –3) H’(6, –3)
Course 1
12-4 Graphing Reflections
Additional Example 1 Continued
Reflect parallelogram EFGH across the y-axis.
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x E F
G H
E’ F’
G’ H’
Course 1
12-4 Graphing Reflections
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
Try This: Example 1
Give the coordinates of the figure after the given reflection. Reflect parallelogram EFGH across the x-axis.
E F
G H
Course 1
12-4 Graphing Reflections
Try This: Example 1 Continued
To reflect the parallelogram across the x-axis, write the opposites of the y-coordinates.
The x-coordinates do not change.
EFGH E’F’G’H’
E(–4, 0) E’(–4, 0)
F(0, 0) F’(0, 0) G(–2, –3) G’(–2, 3)
H(–6, –3) H’(–6, 3)
Course 1
12-4 Graphing Reflections
Try This: Example 1 Continued
Reflect parallelogram EFGH across the x-axis.
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x E F
G H
E’ F’
G’ H’
Course 1
12-4 Graphing Reflections
Additional Example 2: Design Application A designer is using a stencil that is shaped like the figure below. A pattern is made by reflecting the figure across the x-axis. Give the coordinates of the vertices of the figure after the reflection.
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
P
Q S
R
Course 1
12-4 Graphing Reflections
Additional Example 2 Continued
To reflect the quadrilateral across the x-axis, write the opposites of the y-coordinates.
The x-coordinates do not change.
PQRS P’Q’R’S’ P(1, 7) P’(1, –7) Q(2, 1) Q’(2, –1) R(–1, 1) R’(–1, –1) S(–3, 3) S’(–3, –3)
Course 1
12-4 Graphing Reflections
Additional Example 2 Continued
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
P
Q S
R
P’
Q’ S’
R’
Course 1
12-4 Graphing Reflections
Try This: Example 2
A designer is using a stencil that is shaped like the figure below. A pattern is made by reflecting the figure across the y-axis. Give the coordinates of the vertices of the figure after the reflection.
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
P
Q S
R
Course 1
12-4 Graphing Reflections
Try This: Example 2 Continued
To reflect the quadrilateral across the y-axis, write the opposites of the x-coordinates.
The y-coordinates do not change.
PQRS P’Q’R’S’ P(–3, 7) P’(3, 7) Q(–2, 1) Q’(2, 1) R(–5, 1) R’(5, 1) S(–7, 3) S’(7, 3)
Course 1
12-4 Graphing Reflections
Try This: Example 2 Continued
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
P
Q S
R Q’
P’
S’ R’
Course 1
12-4 Graphing Reflections
Lesson Quiz
Give the coordinates of the vertices of each figure after the given reflection.
1. Reflect triangle ABC with vertices A(5, 1), B(2, 3), and C(4, 6) across the x-axis.
2. Reflect parallelogram PQRS with vertices P(–4, –4), Q(1, –4), R(–2, 2), and S(3, 2) across the y-axis.
P’(4, –4), Q’(–1, –4), R’(2, 2), and S’(–3, 2)
A’(5, –1), B’(2, –3), C’(4, –6)
Insert Lesson Title Here
Course 1
12-4 Graphing Reflections
12-5 Graphing Rotations
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up Parallelogram ABCD has vertices (-4, 1), (0, 1), (–3, 4), and (1, 4).
1. What are the vertices of ABCD after it has been reflected across the x-axis? 2. What are the vertices after is has been reflected across the y-axis?
(–4, –1), (0, –1), (–3, –4), (1, –4)
(4, 1), (0, 1), (3, 4), (–1, 4)
Course 1
12-5 Graphing Rotations
Problem of the Day
If each of the capital letters of the alphabet is rotated a half turn around its center, which will look the same?
H, I, N, O, S, X, Z
Course 1
12-5 Graphing Rotations
Learn to use rotations to change positions of figures on a coordinate plane.
Course 1
12-5 Graphing Rotations
You can rotate a figure about the origin or another point on a coordinate plane.
Course 1
12-5 Graphing Rotations
A rotation is the movement of a figure about a point. Rotating a figure “about the origin” means that the origin is the center of rotation.
Remember!
Course 1
12-5 Graphing Rotations
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
Additional Example 1: Rotating Figures on a Coordinate Plane
E F
G H
Course 1
12-5 Graphing Rotations
Give the coordinates of the vertices of the figure after the given rotation. Rotate parallelogram EFGH clockwise 90° about the origin.
Additional Example 1 Continued
The new x-coordinates are the old y-coordinates.
The new y-coordinates are the opposites of the old x-coordinates.
EFGH E’F’G’H’
E(–4, 0) E’(0, 4)
F(0, 0) F’(0, 0) G(–2, –3) G’(–3, 2)
H(–6, –3) H’(–3, 6)
Course 1
12-5 Graphing Rotations
Additional Example 1 Continued
Rotate parallelogram EFGH 90° about the origin.
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x E F
G H
F’
E’ G’
H’
Course 1
12-5 Graphing Rotations
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
Try This: Example 1
Give the coordinates of the vertices of the figure after the given rotation. Rotate parallelogram EFGH counterclockwise 180° about the origin.
E F
G H
Course 1
12-5 Graphing Rotations
Try This: Example 1 Continued
The new x-coordinates are the opposites of the old x-coordinates.
The new y-coordinates are the opposites of the old y-coordinates.
EFGH E’F’G’H’
E(–4, 0) E’(4, 0)
F(0, 0) F’(0, 0) G(–2, –3) G’(2, 3)
H(–6, –3) H’(6, 3)
Course 1
12-5 Graphing Rotations
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
Try This: Example 1 Continued
Rotate parallelogram EFGH counterclockwise 180° about the origin.
E F
G H
E’ F’
G’ H’
Course 1
12-5 Graphing Rotations
Additional Example 2A: Art Application
An artist draws this figure. She rotates the figure without changing its size or shape. Give the coordinates of the vertices of the figure after a clockwise rotation of 90° about the origin.
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
C
D B
A E
Course 1
12-5 Graphing Rotations
ABCDE A’B’C’D’E’
A(0, 0) A’(0, 0)
B(1, 4) B’(4, –1)
C(3, 5) C’(5, –3)
D(7, 3) D’(3, –7)
Additional Example 2A Continued
E(6, 0) E’(0, –6)
The new x-coordinates are the old y-coordinates.
The new y-coordinates are the opposites of the old x-coordinates.
Course 1
12-5 Graphing Rotations
Additional Example 2A Continued
Rotate the figure after a clockwise 90° about the origin.
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
C
D B
A E
C’
D’
B’ A’
E’
Course 1
12-5 Graphing Rotations
B. Give the coordinates of the vertices of the figure after a counterclockwise rotation of 180° about the origin.
Additional Example 2B: Art Application
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
C
D B
A E
Course 1
12-5 Graphing Rotations
The new x-coordinates are the opposites of the old x-coordinates.
The new y-coordinates are the opposites of the old y-coordinates.
Additional Example 2B Continued
ABCDE A’B’C’D’E’
A(0, 0) A’(0, 0)
B(1, 4) B’(–1, –4)
C(3, 5) C’(–3, –5)
D(7, 3) D’(–7, –3)
E(6, 0) E’(–6, 0)
Course 1
12-5 Graphing Rotations
Additional Example 2B Continued
Rotate the figure after a clockwise 180° about the origin.
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
C
D B
A E
C’
D’ B’
A’ E’
Course 1
12-5 Graphing Rotations
Try This: Example 2A
An artist draws this figure. She rotates the figure without changing its size or shape. Give the coordinates of the vertices of the figure after a clockwise rotation of 90° about the origin.
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
C
D B
A E
Course 1
12-5 Graphing Rotations
ABCDE A’B’C’D’E’
A(–6, 0) A’(0, 6)
B(–5, 4) B’(4, 5)
C(–3, 5) C’(5, 3)
D(1, 3) D’(3, –1)
Try This: Example 2A Continued
E(0, 0) E’(0, 0)
The new x-coordinates are the old y-coordinates.
The new y-coordinates are the opposites of the old x-coordinates.
Course 1
12-5 Graphing Rotations
Try This: Example 2A Continued
Rotate the figure after a clockwise 90° about the origin.
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
C
D B
A E
C’
D’
B’ A’
E’
Course 1
12-5 Graphing Rotations
B. Give the coordinates of the vertices of the figure after a counterclockwise rotation of 180° about the origin.
Try This: Example 2B
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
C
D B
A E
Course 1
12-5 Graphing Rotations
The new x-coordinates are the opposites of the old x-coordinates.
The new y-coordinates are the opposites of the old y-coordinates.
Try This: Example 2B Continued
ABCDE A’B’C’D’E’
A(–6, 0) A’(6, 0)
B(–5, 4) B’(5, –4)
C(–3, 5) C’(3, –5)
D(1, 3) D’(1, –3)
E(0, 0) E’(0, 0)
Course 1
12-5 Graphing Rotations
Try This: Example 2B Continued
Rotate the figure after a clockwise 180° about the origin.
-8 -6 -4 -2 2 4 6 8
8
2
4
6
-2
-4
-6
-8
y
x
C
D B
A E
C’ D’ B’
A’ E’
Course 1
12-5 Graphing Rotations
Lesson Quiz Give the coordinates of the vertices of the trapezoid after the given rotation.
1. Rotate trapezoid ABCD counterclockwise 180° about the origin.
2. Rotate trapezoid ABCD clockwise 90° about the origin.
A’(0, 5), B’(0, 0), C’(–3, 1), D’(–3, 4)
A’(5, 0), B’(0, 0), C’(1, 3), D’(4, 3)
Insert Lesson Title Here
Course 1
12-5 Graphing Rotations
12-6 Stretching and Shrinking
Course 1
Warm Up
Lesson Presentation
Problem of the Day
Warm Up ABCD has vertices (–4, 1), (0, 1), (–3, 4), and (1, 4). What are the vertices of ABCD after it has been rotated clockwise 180° about the origin?
(4, –1), (0, –1), (3, –4), (–1, –4)
Course 1
12-6 Stretching and Shrinking
Problem of the Day
Alice was 4 feet tall. She took a bite of one side of the caterpillar’s mushroom and became 5 times as tall! Then she took a bite of the other side of the mushroom and became times as tall. She took a bite from each side two more times. How tall was Alice then?
1 4 __
7 feet 13 16 ___
Course 1
12-6 Stretching and Shrinking
Learn to visualize and show the results of stretching or shrinking a figure.
Course 1
12-6 Stretching and Shrinking
Additional Example 1A: Stretching Figures
Write the dimensions of each part of the figure. Stretch the figure as stated and give the new dimensions of each part.
A. Increase the horizontal dimensions of the face, eyes, and mouth by a factor of 3.
Course 1
12-6 Stretching and Shrinking
Additional Example 1A Continued
Original Dimensions New Dimensions Face Vertical 8
Horizontal 9 Vertical 8 Horizontal 27
Eyes Vertical 2 Horizontal 2
Vertical 2 Horizontal 6
Mouth Vertical 2 Horizontal 5
Vertical 2 Horizontal 15
Course 1
12-6 Stretching and Shrinking
Original Dimensions
New Dimensions
Face Vertical 8 Horizontal 9
Vertical 16 Horizontal 9
Eyes Vertical 2 Horizontal 2
Vertical 4 Horizontal 2
Mouth Vertical 2 Horizontal 5
Vertical 4 Horizontal 5
Additional Example 1B: Stretching Figures
B. Increase the vertical dimensions of the face, eyes, and mouth by a factor of 2.
Course 1
12-6 Stretching and Shrinking
Try This: Example 1A
Write the dimensions of each part of the figure. Stretch the figure as stated and give the new dimensions of each part.
A. Increase the horizontal dimensions of the face, eyes, and mouth by a factor of 3.
Course 1
12-6 Stretching and Shrinking
Additional Example 1A Continued
Original Dimensions New Dimensions Face Vertical 5
Horizontal 6 Vertical 5 Horizontal 18
Eyes Vertical 1 Horizontal 1
Vertical 1 Horizontal 3
Mouth Vertical 1 Horizontal 2
Vertical 1 Horizontal 6
Course 1
12-6 Stretching and Shrinking
Original Dimensions
New Dimensions
Face Vertical 5 Horizontal 6
Vertical 10 Horizontal 6
Eyes Vertical 1 Horizontal 1
Vertical 2 Horizontal 1
Mouth Vertical 1 Horizontal 2
Vertical 2 Horizontal 2
Try This: Example 1B
B. Increase the vertical dimensions of the face, eyes, and mouth by a factor of 2.
Course 1
12-6 Stretching and Shrinking
Additional Example 2A: Shrinking Figures Write the dimensions of each figure. Shrink the figure as stated and give the new dimensions.
A. Decrease the vertical dimensions by
multiplying by . 1 3 __
Original Dimensions New Dimensions Vertical 12 Vertical 4 Horizontal 10 Horizontal 10
Course 1
12-6 Stretching and Shrinking
Additional Example 2B: Shrinking Figures
B. Decrease the horizontal dimensions by
multiplying by . 1 2 __
Original Dimensions
New Dimensions
Vertical 12 Vertical 12 Horizontal 10 Horizontal 5
Course 1
12-6 Stretching and Shrinking
Try This: Example 2A
Write the dimensions of each figure. Shrink the figure as stated and give the new dimensions.
A. Decrease the vertical dimensions by
multiplying by . 1 3 __
Original Dimensions New Dimensions Vertical 9 Vertical 3 Horizontal 6 Horizontal 6
Course 1
12-6 Stretching and Shrinking
Try This: Example 2B
B. Decrease the horizontal dimensions by
multiplying by . 1 2 __
Original Dimensions
New Dimensions
Vertical 9 Vertical 9 Horizontal 6 Horizontal 3
Course 1
12-6 Stretching and Shrinking
Lesson Quiz: Part 1 Determine the vertical and horizontal dimensions of the figure. Stretch the figure as stated and give the new dimensions.
1. Increase the vertical dimensions of the shaded region by a factor of 4. Vertical 12 48; horizontal 9 no change
Insert Lesson Title Here
Course 1
12-6 Stretching and Shrinking
Lesson Quiz: Part 2
2. Decrease the horizontal dimensions of the
shaded region by multiplying by .
Vertical 12no change; horizontal 93
Insert Lesson Title Here
1 3 __
Course 1
12-6 Stretching and Shrinking