1

1.1 Standard Notation and Place Valuedigit – one of the numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9number – may have several digits, for example 367

1 , 2 3 4 , 5 6 7 , 8 9 0

1 , 2 3 4 , 5 6 7 , 8 9 0

Ex a For the number above what digit names the hundred thousands place?

Ex b What digit names the number of tens?

Ex c What does the digit “2” represent in the number above?

Ex d What does the digit “7” represent in the number above?

Ex e Write 5,620,487 in word form.

Ex f Write 5,620,478 in expanded form.

Ex g Write in standard notation: Thirty-two million

2

1.2 Addition

Most problems are added vertically, even if they are originally written horizontally.

Ex a 34 + 2413 + 222 Ex b 782 + 4365 + 28

The result of an addition problem is called a ___________

Perimeter – distance around the outside of an object.

Ex c Find the perimeter of each of the objects below:

Practice Problems

3

1.3 Subtraction

Subtraction is the inverse of addition.

Subtraction is also carried out vertically, even if the original problem is written horizontally.

For each example, subtract and check by adding:

Ex a 85 – 32 Ex b 425 Ex c 3000 - 1471- 86

The result of a subtraction problem is called a _____________________

Practice Problems

1) 30,008 – 52 2) 5923 - 769

4

1.4 Multiplication

8 X 3 = 24 8 and 3 are called ____________, 24 is called the _______________

If you haven’t memorized the products of single digits (times tables), you should do so.

There are many ways to write products:

5 X 8 5⋅8 5(8) (5)(8) (5)8 (5)X8

When there is no operator shown, the operation which is understood is _____________

The purpose of parentheses is _______________________

(5) + (8) = (5)(+8) = 5(+8) = 5 + (8) =

Special Products

Multiplying by zero: 0 X 17 = 29(0) =

Multiplying by one: 1 X 392 = (53)(1) =

Multi-Digit MultiplicationEx a Multiply 2 5 9 Ex b Multiply 4 5 2 7

X 7 X 3 1

Ex c Multiply 6 5 9 X 4 0 3

Sneaky Multiplication Tricks

Ex d Multiply 1000 X 7834 Ex e Multiply 6 8 2 4X 3 0 0

5

Area

Ex f Find the area of the rectangle:

6

1.5 Division

Division is the inverse of multiplication:

12÷6 = can be rewritten as

Ex a Divide and check:

455 21÷3 6 |30

Special Quotients

Dividing by 1: 19÷1 =397

1

Dividing by itself: 19÷19 =397397

Dividing 0 by a number: 0÷42 =0

20

Dividing a number by 0: 79÷0 =110

Long Division

Ex b Divide 296÷4 407

5 23 |7 2 5 8

7

1.6 Rounding and Estimating

Ex a Round 29 to the nearest 10

Round 22 to the nearest 10

Round 25 to the nearest 10

Rounding Whole Numbers Procedure – for a specific place1. Find the digit in the specified place.2. Look at the digit AFTER that place

3. If the digit ___________________________

If the digit ___________________________

4. Replace the rounded digits with __________

Ex b Round 3,682,357 to the nearest:

million ten thousand hundred ten

Estimating

Ex c Estimate the following amounts for easier calculations:Restaurant bill: $ 43.58 Truck: $27,875 House: $239,995

Ex d Estimate the sum by rounding each number to the nearest ten: 58 + 91 + 37

Ex e Estimate the difference by rounding each number to the nearest hundred:564 – 238

Ex f Estimate the product by rounding to the nearest hundred: 287 X 726

Ex g Estimate the quotient by rounding to the nearest ten: 476 ¿ 59

8

1.7 Solving Equations

Ex a My husband’s brother is 4 years older than he is. If his brother is 59, how old is he?

Solve by Trial (guess and check)

Ex b Solve: x – 9 = 33 Solve: 4x = 36

Solving by Opposite (Inverse) Operations

The opposite (inverse) of addition is ______________________

The opposite (inverse) of multiplication is __________________

We want to isolate the variable.

Ex c Solve each equation, then check your answer:

14 + x = 52 18 =2⋅y 22⋅6 = p

9

1.8 ApplicationsProcedure1. Familiarize – understand what is asked for, what numbers are important2. Translate – make an equation3. Solve4. Check5. State – Answer the question

Graph from Basic College Mathematics, 12/e, by Bittinger/Beecher/Johnson

Ex a (Problem 1) How much taller would the Aeropolis 2001 have been than the Nakeel Tower?

Ex b (Problem 3) The Willis Tower (formerly Sears Tower) is the tallest building in Chicago. If the Miglin-Beitler Skyneedle had been built, it would have been 551 ft. higher than the Willis Tower. What is the height of the Willis Tower?

Ex c There are 520 seats in an auditorium. If all rows have the same number of seats, and there are 20 rows, how many seats are in each row?

10

1.9 Exponential Notation and Order of Operations

24

33

52

Ex a Write in exponent form: 7⋅7⋅7 10⋅10⋅10⋅10⋅10⋅10

Ex b Evaluate: 7⋅7⋅7 10⋅10⋅10⋅10⋅10⋅10

Simplifying Expressions (Order of Operations for several operations)1. Parentheses (and grouping symbols like { } or [ ])2. Evaluate all exponential expressions3. Multiplication and Division, in order from left to right4. Addition and Subtraction, in order from left to right

Ex a 100 – (58 – 21) (100 – 58) – 21

Ex b 5⋅22 (5⋅2 )2

Ex c 60 - 36÷3⋅4 (60 - 36 )÷3⋅4

Average - Add the numbers, divide by “how many”Ex d Find the average of the test scores: 82, 72, 83

11

2.1 Factorizations

For the product a⋅b , a and b are called _____________________

Dividing

Qd , d is a factor of Q if the remainder is _____ .

If d is a factor of Q, Q is a _______________ of d, and Q is _________________ by d.

Ex a List all the factors of 24.

Ex b List all the factors of 23.

Ex c List the first 5 multiples of 13.

Ex d Show that 52 is divisible by 4.

Prime and Composite Numbers

1 is ___________________________

A _____________________has only 1 and itself (2 different factors) as factors

A _____________________ can be “broken down” into other factors besides 1 and itself

Ex e Which of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are prime?

Ex f Is 128 divisible by 7? Ex g Is 128 divisible by 8?

12

2.2 Divisibility

A number is divisible by 2 if its ones digit is ___________________________

Ex a Which numbers are divisible by 2?

17 4,201,122 3801 50,000

A number is divisible by 3 if ____________________________ is divisible by 3

Ex b Which numbers are divisible by 3?

29 4,201,122 3801 50,000

A number is divisible by 6 if it is ____________________________

Ex c Which numbers are divisible by 6?

29 4,201,122 3801 50,000

A number is divisible by 9 if ____________________________ is divisible by 9

Ex d Which numbers are divisible by 9?

387 4,201,122

A number is divisible by 10 if ____________________________

A number is divisible by 5 if ____________________________

Ex f Which numbers are divisible by 10? Which are divisible by 5?

295 3,729,231 1620

A number is divisible by 4 if ____________________________

A number is divisible by 8 if ____________________________

Ex g Which numbers are divisible by 4? Which are divisible by 8?

9024 387,231 420

13

2.3 Fractions and Fraction Notation

Ex a Shade the portions that represent 2 3

8

Ex b What fraction is represented by the shaded portions?

Ex c Find 1 7

8 and 3 3

16

Ratio – a quotient of 2 quantities (can be written as a fraction) The 2 quantities are often separated by “to”

Ex d A job opening has 97 applicants, and 4 people are hired.1) Write the ratio of people hired to applicants.

2) Write the ratio of people hired to people not hired.

14

2.4 Multiplying Fractions

Ex a Multiply

Multiplying a Fraction by a Fraction1. Multiply the 2 numerators -

2. Multiply the 2 denominators –

Do the same rules work for addition (add 2 numerators & keep, add 2 denominators & keep)?

Multiply a Fraction by Whole Number

Ex b Multiply

ApplicationsEx c For a training program, 20 out of 71 applicants are accepted. Of the accepted students, 4/5 of the students are hired. What fraction of all applicants are hired?

15

2.5 Simplifying (reducing)

Fractions that reduce to 1:

Multiplicative Identity (Multiplying identity) - Using it gives the same value (no change)

a ¿ _____ = a

Equivalent fractions – have the same value:

We can change fractions to have a new denominator, but the same value

Ex a Find a name for

27 with a denominator of 21.

Ex b Create an equivalent fraction with the new denominator.

34= ?

36

Simplifying Fraction Notation (Reducing)Simplest fractions have NO COMMON FACTOR in the numerator and denominator. To get simplest form, remove fractions that equal 1 (common factors)

Ex c Simplify:

1824

16

Practice ProblemsSimplify:

1.

1230

2.

1854

3.

11701200

17

2.6 Multiplying, Simplifying, and ApplicationsIt’s important to simplify a product before actually multiplying out the numbersEx a Simplify and multiply35⋅10

9

Procedure1. Put numerator and denominator factors together in the num. & denom., but don’t actually

multiply out the numbers2. Factor the numerator and denominator3. Remove factor fractions that equal 1, if possible.4. Multiply out the products to get a single number in numerator & denominator.

Ex b Multiply (reminder before multiplying:

Applications

Ex c The pitch of a screw is

116 inch (this is how far it moves with every full turn). How far

into a piece of wood will it go when makes 6 full turns?

Ex d Financial aid covers

35 of a student’s expenses. If expenses are $4500, how much is

covered by financial aid?

18

2.7 Division and Applications

Reciprocals - Pairs of fractions whose product = 1. We find a reciprocal by________

Ex a Find the reciprocal of

Dividing Fractions

Ex b Divide:

23÷8

9

Solving Equations

Ex c Solve

25x=10

7 Solve

32x=600

ApplicationsEx b How many 3/4 ounce servings of chips can be made from a 12 ounce bag?

19

Practice Problems

1. 9⋅( 5

12 )2.

57⋅ 110 3.

( 425 )⋅(15

16 )

4.

314

÷67 5.

6⋅( 43 )

6. Solve:

54x=3

8

20

3.1 Least Common Multiples

Ex a Find the least common multiple of 9 and 12 by making a list of multiples.

12:

9:

Some common multiples are:

The Least Common Multiple (LCM) is:

Finding LCM’s by Listing Multiples (Method 1)a) Is the largest number a multiple of the other numbers?

b) If not, list multiples of the largest number until you find one that is a multiple of the other numbers.

Ex b Ex c Find the LCM of 4, 10, and 20

Ex c Find the LCM of 4, 6, and 10:

10:

6:

4:

Prime Factorization - Breaking down numbers to the smallest possible factors

Tree method: Divide Up Method:

21

Finding the LCM by Prime Factorization (Method 2)a) Write the prime factorization of each numberb) Create a product – for each factor, use the greatest number of repeats in ONE

number

Ex c Find the LCM of 9 and 12

Ex d Find the LCM of 8, 18, and 12 using prime factorization and exponents

Practice Problem: Find the LCM of 25 and 35

We will skip Method 3 (p. 150).

22

3.2 Addition and Applications

Like Denominators1. Add numerators2. Keep same denominator3. Reduce if possible

Ex a Add:

112

+ 712 = It doesn’t say reduce – should we?

Different Denominators1. Get LCD (LCM of denominators)2. Multiply top & bottom by the needed factor.3. Add as above.

Ex b Add:

14+ 3

8 = How is this different from multiplying?

Ex c Add:

59+ 2

15 =

Ex d Add:

13+ 1

8 =

23

Applications

Ex e A tile is

14 in. thick and is glued to a board

78 in. thick. The glue is

116 . How thick is the

result?

Practice Problems

1. Add:

34+ 2

5 =

2. Add:

215

+ 49+ 1

6 =

24

3.3 Subtraction, Order, Applications

Like Denominators1. Subtract numerators2. Keep same denominator3. Reduce if possible

Ex a Add:

910

− 310 =

Different Denominators1. Get LCD 2. Multiply top & bottom by the needed factor.3. Subtract as above.

Ex b Subtract:

53−1

4 =

Ex c Subtract:

75− 1

14 =

Order (which is bigger?) - Use < or >

Ex e Use < or > to write a true statement.

35

25

34

57

Ex f I have ½ lb. butter, and use 1/3 lb. How much is left?

25

3.4 Mixed Numerals (also called mixed numbers)What fraction is represented below?

Ex a Write as a mixed numeral: 11 +

34 = 9 +

35 =

Ex b Convert to fraction notation (commonly called _________________________ )

4

35 5

23

Fraction notation (improper fractions) and mixed numbers.

Ex c Convert the improper fractions to mixed numbers:

957

1583

26

3.5 Add & Subtract Using Mixed Numerals; Applications

Addition

Ex a 3 1

4 Ex b 1 3

5

+11 38

+4 23

Subtraction

Ex c 8 5

6 Ex d 13 1

7

- 2 1

12

−7 5

7

Ex e (# 38) A plumber uses 2 pipes, each of length 51 5

16 , and one pipe of length 34 3

4 . How much pipe was used in all?

27

3.6 Multiplication and Division with Mixed Numbers

Convert mixed numbers to fraction notation (improper fractions)

Ex a (1 3

5 )(2 1

4 )

Ex b (12)(3 5

6 )

Ex c 2 1

2÷1 3

5

Ex d 8÷1 1

3

Ex e A space shuttle orbits the earth in 1 1

2 hr. How many orbits are made in 24 hours?

28

3.7 Order of Operations/Complex Fractions/Estimation

Ex a Simplify ( 2

3 )2− 1

3⋅(2 1

2 )

Ex b Simplify:

3106

35

Ex c Find the average of

34

and 58

Ex d Estimate, rounding to the nearest whole number

Compare the numerator to ______________________________

58

13

4 311

12 78

Ex d Estimate each term to the nearest whole number, then perform the operations:

7 319

+5 45−231

37

29

4.1 Decimal Notation, Order, and Rounding

Decimal ValuesEx a Write the value of $178.95 in expanded form.

Place Value Chart (for 1.73205)

Decimal Notation and Word Names – decimal words are similar to the fraction they represent

1. Number left of decimal point:

2. Point:

3. Number right of point:

Ex b Give the word name of 1.73205

Ex c The median age in CA is 35.2 -- write the word name.

30

Converting Decimals to Fractions:1. Count the number of decimal places2. Write that number of zeroes in the denominator, with 1 in front3. Write the digits in the numerator

Ex Convert 0.357 0.0182 23.41

Note: Whole number parts on the left of the decimal point make ______________________

Converting Fractions to Decimals (“simple” denominators with powers of 10)1. Count the number of zeroes2. Use that number of place values to make the numerator smaller (move left).

Ex

Order (state which is larger using > or <)How to make equivalent decimals:

31

Comparing numbers in decimal notation1. If needed, tack on zeroes to make the decimals equal length2. Compare digits beginning starting immediately after the point (if needed, tack on zeroes to

make numbers the same length)3. When digits differ, the larger number gives the larger amount

Ex

Rounding1. Look at the specified digit2. Look at the next place value (immediately after the specified one)3. If the next digit is

0 – 4, keep the desired digit5 – 9, round up

Ex Round to the nearest

a) thousandth

b) hundredth

c) whole number (unit)

d) ten

e) hundred

Ex Round to the nearest

a) hundredth

b) tenth

c) whole number (unit)

d) ten

e) hundred

32

4.2 Addition & Subtraction

Procedure:1. Line up decimal points! (most important)2. Fill in zeroes at the end of decimals if needed

Ex a Add: 2.68 + 11.3 + 0.009

Ex b Subtract: 6 – 4.27

Ex c Solve: x + 3.7 = 9.431

Practice Problems - Perform the operations or solve

1) 7 – 2.381 2) 14.843 + 0.34 + 1.9 + 10 3) Solve: x – 42.87 = 19.4

33

4.3 MultiplicationDecimals have fractional equivalents:

2.35 X 0.4

Procedure:1. Multiply digits as if they were whole numbers2. Move the point the number of decimal places after all points3. Fill in zeroes if needed

Ex a Multiply: (6.7)(0.038)

Multiply by 0.1, 0.01, 0.001, etc. (small numbers)Ex b Multiply 18.47 X 0.001

Multiply by 10, 100, 1000, etc. (large numbersEx c Multiply 18.47 X 1000

Large Number NamesEx d Convert $14.5 million to digits

Dollars and Cents $1 = 100¢ and 1¢ = 0.01$

Ex e Convert 89 cents to dollars Ex f Convert $22.51 to cents

34

Practice Problems

1) 4.6 X 0.9 2) 0.01 X 821.37

3) Convert 530,792¢ to dollars 4) Convert 192.5 thousand to standard form

4.4 Division

Divide decimals by whole numbers – similar to whole number long division, but put the decimal point in the quotient

Ex a 15|25 .5

Decimal divisors (denominators) - make fraction and move point to get a whole number in denominator.

Ex b 2.732¿ 0.04

35

Divide by 10, 100, 1000, etc. (large numbers)

Ex c Divide 128.54 ¿ 1000

Divide by 0.1, 0.01, 0.001, etc. (small numbers)

Ex d Divide

0 .0630 .001

SolvingEx e Solve: 2.5t = 300

Practice Problems

1)

14 . 310 .01

2) 11.2 ¿ 4 3) Solve: 0.3y = 1.38

36

4.5 Converting Fractions to Decimals - Use Long Division

Write in decimal notation:

Ex a Ex b Ex c

58

2 13

311

For each of the decimals above, round to the nearest tenth, hundredth, and thousandth.

tenth:

hundredth

thousandth

Practice Problems

Write in decimal notation and round to the nearest tenth

1)

134 2)

56

37

4.6 – 4.7 Estimating and Applications

Estimating Sums and Differences – operations are - Round to the same place value(s) then add or subtract.

Ex a On a shopping trip, Mia buys items costing $38.95, $129.99 and $9.77. Estimate the cost by rounding to the nearest ten.

Ex b A $491.79 tablet is discounted by $109.21. Estimate the final price.

Estimating Products and Quotients – operations are - Round to one non-zero digit OR round to “easy” digits.

Ex b Dan is paid $892.12 for 11 days. Estimate his daily pay, then calculate the exact amount to the nearest cent.

Practice Problems - Perform the operations or solve

1) Estimate, rounding to the nearest tenth:1.4368 + 0.1724 – 0.0913

2) Coffee costs $3.61 (including tax). How much is spent in a 30-day month?

3) Cole earned $620.80 working 40 hours in a week. What is his hourly wage?

4) A shipment of seafood costs $88.65, and there are 6.245 lb. Estimate each number, then divide the estimates to approximate the cost/lb.

38

5.1 Intro to Ratios

ratio – a ___________________ of 2 quantities

There are several ways to write ratios. For example for a TV screen 16 inches wide and 9 inches tall, the width to height ratio can be written as:

Ex a Write the ratios in 2 other formats (without reducing)

3 to 5 14.7:100 8½ to 11

Ex b Find the ratio of length to width:

Ex c For the triangle, find the ratios listed and reduce.

height to base ratio hypotenuse to base ratio height to hypotenuse ratio

39

5.2 Rates and Unit Prices

rate – a ratio whose numerator and denominator have different units.

Ex a My car travels 500 miles on 15 gallons of gas. What is the rate of miles per gallon (also known as gas mileage)?

Ex b Al earns $30,000 in a year. What is his rate of pay in dollars per month?

Unit rate – ratio where the denominator number is 1

Unit price – ratio of price to number of units, where the number of units is reduced to ____________

To find unit rates (including unit price), use ______________________________

Ex c Find the unit cost of the following jars of peanut butter. Which is the better buy?

Brand A is 40 oz. and costs $5.00 Brand B is 28 oz. and costs $3.00

Ex d I drive 390 miles in 6 hours. What is the unit rate in miles/hour?

40

5.3 ProportionsProportion – 2 ratios that equal each other:

The pairs 1, 2 and 3, 6 can be used to form a ratio:

We can test if 2 proportions are equal if their cross products are _____________________

Ex a Are the pairs or ratios proportional?

1)

47

511 2) 3,5 and 21, 35

3)

120

0 . 040 .8 4) 2½ , 4½ and 10, 18

Solving Proportions – set cross products ____________________________, then solve for x

Ex b Solve:

x4=9

6x5= 0

210 .5y

=1 .53 .5

41

Practice Problems:

1) Write fraction notation and reduce:

8 to 12 2.4 to 6

2) Find each rate:

65 meters to 5 seconds 243 miles per 4 hours

3) Are the pairs proportional:

3, 7 and 15, 45 2.4, 1.5 and 0.16, 0.1

4) Solve

27=10x

p1. 2

=1. 10. 6

42

5.4 Applications of Proportions

When to use proportions? You have 2 quantities that are related. One quantity changes, and you want to find the changed value of the second quantity.

Ex a Ravi makes $315 working 21 hours. How much would he make if he works 40 hours?

Ex b Two cities on a map are 2½ inches apart, which represents 300 miles. How far apart are two cities if they are 6¼ inches apart on the map?

Ex c In 2015, 1 US dollar is worth 16.8 pesos. How many dollars is 400 pesos worth, to the nearest dollar?

43

6.1 Percent Notation

The Butte fire was 15% contained (as of Sept. 12, 2015). What does that mean?

Percent notation:

Fraction notation: or ratio:

Decimal notation:

Percent of a Quantity

What is

35 of 40?

What is 60% of 40?

Proper fractions (less than 1) and “common” percentages (less than 100%) are similar

Amount = Percent of Base (If you have “Percent of”, the next is always base)

Ex b A discount is 20% of the original price. If the item is marked $30, what is the discount?

44

Converting percent to decimal – Replace % with ___________ or ______________

This causes you to remove _______, make number ____________Ex a Convert to decimal:

58% 7.2% 150% 0.03%

Ex b Convert 5 2

3%

to decimal:

Ex c Write decimal values for each of the percentages listed.

Monthly Expenses

If monthly income is $1000, how much spent on transportation?

Convert decimal to percent – multiply by ___________ Does this change the value?

Ex d Write percent notation for

0.27 0.735 0.4

2.7 0.0009

45

6.2 Percents and Fractions

Converting a fraction to percent

1. First, convert fraction to decimal (From 4.5, use ________________________

2. Next, convert decimal to percent (From 6.1, multiply by ______________________

Ex a Convert 7/8 to a percent

Ex b Convert 7 2

3 to a percent

Shortcut - Only works when denominator is a factor of 1001. Multiply top and bottom to build the denominator to 100.2. Change /100 to %

Ex c Convert to a percent:

1320

725

310

4750

Converting Percent to a Fraction (postpone repeating decimal to Math 20)1. Replace % with __________________2. Reduce

Ex d Convert to a fraction and simplify

46

70% 12.5% 0.4% 16 2

3%

Practice Problems

1. A lawn requires 300 gallons of water for every 500 square feet. How much water does a lawn which is 1800 square feet require?

2. Sal burned 200 calories in ¾ hour of walking. How many calories would be burned in 1 ¾ hours of walking?

3 Find percent notation for:

0.7 0.3891

512

725

4. Find decimal notation for:

57% 1.5% 22 ½ % 240%

5. Find fraction notation for:

57% 1.5% 22 ½ % 240%

47

48

6.3 Solving Percent Problems - Percent Equations

Translating to Equations

of multiply

is equals

% convert number to decimal or a fraction using 1/100

What

Ex a What is 7% of 45?

Ex b 28% of 30 is what?

Ex c 15 is what percent of 75?

Practice Problems1. What percent of 42 is 7?

2. 9 is 25% of what?

3. 70% of what is 35?

49

6.5 Applications

Ex a From the pie chart below:

Monthly Expenses 1) If a person makes $3000/month, how much is spent on housing?

2) If a person is spending $600/month on transportation, what is their total income?

Ex b A test has 60 questions, and Jan gets 49 correct. What percent are correct (to the nearest whole number percent)?

Percent Increase & Decrease

Ex c Rent was $750/month last month and increased to $800 this month. What is the percent increase?

Ex d A TV cost $400 last year but costs $320 this year. What is the percent decrease?

50

6.6 Sales Tax, Commission, and Discount

Sales tax and commission increase an original price.Discount decreases the original price.Rate of discount or increase is the same as percent of discount or increase.

Ex a Sales tax adds $12.74 to the price of a fire pit. If the sales tax rate is 8%, find the original price.

Ex b A real estate agent earns a 6% commission on a house valued at $240,000. How much commission does he earn?

Ex c A backpack is discounted and sells for $25. If the amount of discount is $15, find the original price and rate of discount.

Practice Problems1. A $60 meal is charged 7.35% sales tax. How much tax is charged, and what is the final

price?

2. A sweater originally costing $75 is marked down to $45. What is the rate of discount?

51

7.1 Average, Median and ModeMean (Average) – Procedure for calculating

1. Find the sum of values2. Divide by how many values

Ex a Trey’s test scores are 88, 92, 79, and 84. What is the average?

Weighted Average - If a class has a higher number of units, it “counts more”

GPA = total grade points/# of units

Ex b A student takes earns an “A” in a 3-unit English course and a “C” in a 4-unit math course. What is the GPA?

Class Units Grade

English 3 A (4.0)

Math 4 C (2.0)

Ex b My syllabus has the following weights:Category Weight Your score

Exams 50%

Homework 20%

Participation 5%

Final Exam 25%

52

Median – for an odd number of values1. Rewrite the list of values in order (smallest to largest or largest to smallest)2. Choose the middle value (the median)

Ex The list prices of neighborhood houses is below. Find the median.

180,000 250,000 176,000 220,000 1,000,000 206,000 240,000 192,000 220,000

Median – a more sophisticated approach for an even number of values1. Rewrite the list in order2. Look at the middle 2 values.3. Calculate the average of the middle 2 values.

Ex Find the median:

180,000 250,000 176,000 220,000 1,000,000 206,000 240,000 192,000

ModeThe most common value is the mode. If there are 2 (or more) most common values, there are 2 (or more) modes. If no value is more common than any other, there is no mode.

Ex Find the mode of the ages of students:

22, 20, 19, 20, 18, 35, 19, 58, 21, 19, 28

Find the mode of these ages:

9, 9, 9, 12, 15

Find the mode:

9, 9, 12, 12, 15, 15, 17, 17

53

7.2 Interpreting Data from Tables and GraphsEx a From the table in Example 1, page 4191) Which country has the smallest land area? Which has the largest land area?

2) Which country or countries had a population decrease from 2008 to 2012

3) Find the average population density of China, Japan, and India in 2012.

4) Estimate the population in the United States in 2000 and 2012 to the nearest million. Calculate the percent increase to the nearest whole number percent.

Ex b From the pictograph in Example 2, page 4211) Which continent has the greatest number of roller coasters?

2) About how many roller coasters are there in Asia?

3) About how many more roller coasters are there in South American than in Australia?

54

7.3 Bar Graphs and Line Graphs

Ex a From the bar graph in Example 1, page 4301) Which country has the highest per capital tea consumption?

2) What is the coffee consumption in Brazil?

3) In what country do people drink almost no coffee?

4) What is the difference in coffee consumption between the 2 highest coffee consumers?

5) What is the percent decrease from the largest consumer to the second largest?

Ex c Make a bar graph of class data showing favorite technology applications

App Number of users

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Line Graphs

Ex c From the line graph, Example 3 on page 432,1) What was the average price of gold in 2010?

2) During which 5-year period did the price of gold remain essentially unchanged?

3) In what year was the average gold price about $600?

4) What was the change in average price from 2010 – 2012?

5) What was the percent change during that period?

Ex b Make a line graph of class data showing the number of siblings of the students in the class.

How many siblings?

Number of responses

0

1

2

3

4

5

6

7 or more

56

7.4 Circle Graphs (aka Pie Graphs)

Ex a From Example 1, page 4391) What 2 species have the largest populations?

2) What species accounts for 15% of the population of endangered whales?

3) What is the percent of right whales (from both North Atlantic and North Pacific)?

4) If there are about 300,000 endangered whales of these species, how many endangered right whales are there?

Ex b Draw a circle graph for the following data:

Age of student

< 18 2%

18-24 61%

25 – 34 20%

35 -44 9%

45+ 8%

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8.1 Linear Measures: American Units

Length Area Volume (capacity) are 3 different kinds of quantities.

Units:

Ex a Convert: 5 ½ yards = ___________ inches

Note: When converting from larger units to smaller ones:

Converting between units: Multiply by fractions that equal 1

Ex b 64 inches = _________ feet.

Ex c Convert 0.6 miles to feet

Ex d Convert 11. 37 ft to yards

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8.2 Linear Measures: Metric

Examples of metric length:

Recall: Converting large units to small units means

Ex a 13.4 kilometers = ____________________ meters

Ex b 2.98 m = _________________cm

Ex c Convert 681 mm to meters

Ex d 57.7 mm = _____________________cm

Using mental conversion

Practice Problems (8.1 and 8.2)

1. 927.1 dm = _________________km 2. 69 inches = _____________ft

3. 0.3 miles = ____________ ft 4. 578 mm = _______________ m

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8.3 Converting Between American and Metric Units

Ex a Convert 10 cm to inches

Ex b 400 m = _______________ ft.

Ex c How many miles is a 10K run (10 km)?

Ex d The Horseshoe Falls of Niagara Falls is 173 ft tall. How many meters is it (to the nearest meter)?

Practice Problems

1. 12 inches = _____________ cm.

2. 1000 meters = _______________ miles

3. Convert 40 ft. to meters.

60

8.4 Weight & Mass/ Medical Applications

Ex a Convert 100 ounces to pounds.

Ex b 3.4 tons = _____________ pounds

Ex c 56 kg = ______________pounds

Ex d 325 mg = __________ g

Medical – Micrograms

1 micrograms = 1 mcg =

11,000,000 grams; 1 mcg =

11000 mg

1,000,000 mcg = 1 g 1000 mcg = 1 mg

Ex e Convert 0.3 mg to mcg

Practice Problems

1. 2 kg = ___________ grams 2. 2 lb = ______________ oz.

3. 475 mcg = _____________ mg 4. 2 lb = ______________ g

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8.5 Capacity/Medical Applications

Capacity (volume) -- American UnitsEx a How many gallons is 32 quarts?

Ex b Convert 3 quarts to fluid ounces (more than 1 step)

Metric UnitsThe same prefixes used for length are used for capacity.Abbreviations for liters:

Also: 1 ml = 1 centimeter cubed (cc) = 1 cm3

Ex c 0.287 L = ____________________mL

Ex d How many liters is 4380 mL?

Medical ApplicationsEx e A bottle of medicine contains 300 mL. If 1 tsp = 5 mL, how many tsp. is this?

If each dose is 2 tsp., how many doses are in the bottle?

62

Ex f A doctor orders 40 mL per hour of saline ordered intravenously. How many mL should be given in a day?

If a bag contains 500 mL saline, how long will it take to empty the bag?

Practice Problems1. How many mL is 3.75 L?

2. A punch bowl contains 3 gallons of punch. How many cups is this?

3. A patient needs to receive 2.0 L of saline over a 24 hour period. a) How many liters should be given in 1 hour?

b) How many mL is this equivalent to?

63

8.6 Time and TemperatureYour book states that 1 year = 365 ¼ days, so we use this conversion. More precisely, 1 year = 365.24 days, which is why we have a leap year every 4 years, but we skip the leap year at the turn of the century (years ending in 00).

Ex a 4 minutes = ______________ seconds

Ex b A movie has a run time of 160 minutes. How many hours is this?

Ex c Convert 7200 seconds to hours

Ex d Convert 68o F to Celsius

Ex e A patient has a fever of 40o C. What is this in Fahrenheit degrees?

Practice Problems

1. How many hours are in a week? 2. 900 seconds = ____________ hours.

3. Convert 68o F to Celsius 4. 100 o C = ______________ o F

64

9.1 Perimeter and ApplicationsPerimeter – the sum of the lengths of the sides:Ex a Find the perimeter of each object:

Rectangle: has 4 angles which are 90o (right angles) Question:

Formula for Perimeter of a rectangle: Formula for Perimeter of a square: P = P =

Ex b A living room is 18 ft X 12 ft. The doorway into the living room is 6 ft wide. a) If baseboard costs $2.25/foot, what is the cost of installing baseboard?

b) If baseboard is only sold in 8-ft segments for $12 each, what is the cost?

65

Practice Problems 1. Find the perimeter of a square picture frame with 9.5 inches on each side.

2. A 9’ X 10’ room is decorated with border paper. If each roll is 12 ft, how many rolls are needed?

3. A yard is enclosed with chicken wire fencing. If each roll of 50 ft. costs $26, how much does it cost to enclose a 60 ft X 30 ft back yard?

66

9.2 AreaFormula for Area of a rectangle: Formula for Area of a square: A = A =

Ex a Find the area of a rectangle that is 4 yards X 5 yards.

Ex b Convert 4 yards and 5 yards to feet, and find the area using these units.

Triangles

Formula for Area of a Triangle:

Ex c Find the areas:

67

Formula for Area of a Parallelogram:

Ex d Find the area of a parallelogram whose base is 4½ inches and height is 5¾ inches.

Formula for Area of a Trapezoid:

Practice Problems - Identify the shapes, then find the areas:

68

9.3 Circles Formulas:

Ex a Find the radius and the diameter of each circle

Circumference – the distance around the _____________________________

Formula for Circumference: C =

Also: C =

Estimates for

Ex b Find the circumference of a circle whose radius is 10 inches. Use 3.14 as an estimate for .

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Formula for Area of a CircleA =

Ex d Find the area of a circle with radius = 3 ft. Use 22/7 as an estimate for

Practice Problems:1. Which has more pizza: a 12-inch square pizza, or a 14-inch round pizza?

2. Find the perimeter of this shape.

3. Find the area of the shape above

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9.4 Volume

Rectangular Solids (fill with cubes)

Formula for Volume: V =

Ex a Find the volume of a box whose dimensions are 8” X 10” X 3”

Cylinders

Formula for Volume: Volume = Area X height V =

Ex b Find the volume of a can whose diameter is 14 cm and whose height is 10 cm.

Practice Problems:1. How many cubic feet is a refrigerator that is 2½ ft wide, 1½ ft deep, and 5 ft tall?

2. A tower in Germany has a height of 110 m and radius of 21 m. Find the volume, using 3.14 for .

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10.1 The Real Numbers

Integers – positive and negative whole numbers.

Ex a May’s bank balance is $20 overdrawn. Write the balance as an integer

Ex b Represent the following numbers on the number line: 3, -1, 0, -4

Rational Numbers - Include integers, fractions, and decimals that can be written as fractions.

Ex c Graph the numbers:

Decimal Notation for Rational Numbers (any fraction can be converted to decimal)

Ex d Write − 3

5 as a decimal

Ex e Write − 5

11 as a decimal

Alternate formats:03 = –

31100 - 1¾ –

32

Real NumbersIrrational numbers - the decimals do not stop or repeat

72

Ex f Use < or > to write a true statement for each pair3 - 8 0 - 3 1.5 - 2.7

- 12 - 5−2

3 - 5 - 4 ½

Absolute Value – always positiveIf number is positive, the absolute value is the same as the numberIf the number is negative, the abs. value has the opposite sign

| 5 | = | - 7 | |-3 47| |0| |2.58|

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10.2 Additing Real Numbers (esp. negative numbers)

Ex a 4 + ( - 7) Ex b (-3) + 2

Ex c - 3 + (- 6) Ex d - 3.5 + 0

Adding without number lines:1. Add positive numbers - add as before2. Add negative numbers - add absolute values; the final sign is negative3. Add a positive and a negative

a) Subtract the amountsb) Take the sign of the “dominant” number (greater absolute value)

Ex e 3.2 + (-7.8)

- 30 + (-14)

−23 +

16

- 11.2 + 11.2

Opposites (Additive Inverses)

Ex f Find the opposite of: 17 - 4.1

16 0

The sum of a number and its opposite Is

The opposite of the opposite:

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10.3 Subtracting Real NumbersSubtraction – same as adding the opposite of the number

Ex a 2 – 6

Ex b 3 – (-1)

Ex c - 2 – 3

Ex d - 4 – (-3)

Practice Problems1. 6 – 11 2. -8 – 5

3. - 7.2 – (- 3.1) 4. 1.8 - 4

5. 0 – 22.9 6. −1

6+ 5

6

7.

35−(−2

3 )8.

− 511

+ 511

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10.4 Multiplication of Real Numbers

Product of a positive and a negative number – the result is __________________Tip: Determine the sign and set it aside, then multiply the absolute values separately.

Ex a 7(-6)= (-3.61)(5)=( 3

7 )⋅(−116 )

=

Product of 2 negative numbers – the result is __________________

Ex b (-5)(-3) = -2(-7.95) =(−25

7 )⋅(−1415 )

=

Multiplying More than 2 NumbersEvery pair of multipled negative numbers produces a positive number.For more than 2 negatives: ____________________________________________ produces a positive number.

_____________________________________________ produces a negative number.

Ex c (- 4)(- 3)(2)(-1) =

Ex d (- 5)(4)(- 3)(- 2)(- 1) =

Ex e (−1

6 )⋅(− 23 )(−3

5 )=

76

10.5 Division of Real NumbersThe rules for signs in division are the same as for multiplication.

Quotient of a positive and a negative number – the result is __________________

Ex a

-305 =

27-9

Quotient of 2 negative numbers – the result is __________________

Ex b

-42-6 = −4 .8÷(−1 .2 )=

Division and Zero

Related equations:

-243

=− 8

-240

=x

0-24

=x

Ex c

0-19

-520

Reciprocals - the product of 2 reciprocals is ____________

Ex d Find the reciprocal for:

number:

23 4

−12 - 5

reciprocal:

The signs of a number and its reciprocal are:

77

Dividing Fractions – Recall “Keep, Change, Flip”

Ex e −2

3÷(−4

9 )

Ex f Solve:

54x=−3

2

Dividing Decimals

Ex g – 6.48 ¿ (– 4)

Ex h 3.51 ¿ (–0.3)

Sign Placement – equivalent forms:

−47=-4

7= 4

-7

The sign cannot be “moved” for a mixed number: −2 1

3

Practice Problems:

1. −4⋅( 7

12 )2.

-3 . 4-20 3.

−4 35⋅(−10 )

4. 5 ÷(−2 1

2 )5.

(-7+73 )

6. (2.1)(-0.4)