UNDERSTANDING PERCENTSmathandteaching.org/uploads/SampleLesson_PDFs/Lesson_17.2.pdf · 17.2...
Transcript of UNDERSTANDING PERCENTSmathandteaching.org/uploads/SampleLesson_PDFs/Lesson_17.2.pdf · 17.2...
17.2 Understanding Percents
Unit 5: Ratio, Proportion, and Percent (Teacher Pages) Week 17 – TP7
UNDERSTANDING PERCENTS In this lesson, students use sense-making strategies to find percents of numbers. The use of visuals, familiar contexts, and mental math is stressed. This lesson falls near the beginning of a cluster that deals with the important proportional reasoning topics of ratio, rate, proportion, and percent. In the cluster, the routine procedures used for ratio, proportion, and percent computations and problem solving will be introduced after the sense-making strategies are explored. Students will learn to find percents both mentally and procedurally. The link between percents and proportions will be made, and the cross multiplication procedure will be justified. Math Goals
(Standards for
posting in bold)
• Convert fractions to decimals and percents. (Gr5 NS1.2; Gr7 NS1.3; Gr7 NS1.5) • Find percents of numbers. (Gr5 NS1.2; Gr6 NS1.4; Gr7 NS1.3)
Summative Assessment
Future Week • Week 22: Percent Applications (Gr5 NS1.2; Gr6 NS1.4; Gr7 NS1.3)
17.2 Understanding Percents
Unit 5: Ratio, Proportion, and Percent (Teacher Pages) Week 17 – TP8
PLANNING INFORMATION
Estimated Time: 45 – 60 Minutes
Student Pages * SP5: Ready, Set, Go * SP6: Percents of
Containers * SP7: Percents of Other
Items 1 * SP8: Percents of Other
Items 2 SP9: Money and Other
Items
Materials
Reproducibles
Homework SP9: Money and Other
Items
Prepare Ahead
Management Reminders
Assessment * SP17: Knowledge Check 17 R71: Knowledge Challenge
17 A81: Weekly Quiz 17
Strategies for English Learners Encourage students to make a meaningful connection with “cent” (hundred). percent = “per hundred” centimeter = “hundredth of a meter”
Strategies for Special Learners Using known concepts to build new concepts gives added meaning to the new concepts, practice on the old concepts, a foundation for learning rules, and tools for retrieving rules if forgotten.
* Recommended transparency: Blackline masters for overheads 193-196 and 199 can be found in the Teacher Resource Binder.
17.2 Understanding Percents
Unit 5: Ratio, Proportion, and Percent (Teacher Pages) Week 17 – TP9
THE WORD BANK
ratio
A ratio is a comparison of two numbers by division. The ratio of a to b is denoted by a:b (read “ a to b ”), or by a
b.
Example: The ratio of 3 to 2 may be denoted by 3:2, or by 3
2= 1.5 .
rate A rate is a ratio in which the numbers have units attached to them.
Example: 20 miles15 minutes
is a rate.
proportion A proportion is a statement that two ratios are equal.
Example: The equation 3 12
25 100 = is a proportion. It can also be
written 3 : 25 = 12 :100 .
unit rate A unit rate is a rate for one unit of measure.
Example: 80 miles per hour, which can be written:
80 miles1 hour
or miles80 hour
.
unit price A unit price is a price for one unit of measure.
Example: $1.25 per pound, which can be written:
1.25 dollars1 pound
or dollars1.25 pound
.
percent A percent is a number expressed as units per 100. To convert a positive
number to percent, multiply the number by 100.
Example: 15100
15% = = 0.15
4 = 4 100% = 400%×
17.2 Understanding Percents
Unit 5: Ratio, Proportion, and Percent (Teacher Pages) Week 17 – TP10
MATH BACKGROUND
Using Sense-Making Strategies to Find Percents
To help all students develop a better understanding of the meaning of percent, challenge students to find percents in multiple ways, and then share strategies with each other.
Example: What is 75% of 600?
Math Background 1 Explore
Strategy 1:
Use benchmark percents
50% of 600 = 300
25% of 600 = 150 (half of 50%)
50% + 25% = 75%,
therefore 75% of 600 =
300 + 150 = 450
Strategy 2:
Use fractions
75% = 34
3 600 = 4504 1•
Strategy 3:
Use decimals
75% = 0.75
600 0.75 3000 42000450.00
×
17.2 Understanding Percents
Unit 5: Ratio, Proportion, and Percent (Teacher Pages) Week 17 – TP11
PREVIEW / WARMUP
Whole Class
SP5* Ready, Set, Go
• Introduce the goals and standards for the lesson. Discuss important
vocabulary as relevant. • Students fill in the table for fraction, decimal, and percent equivalents.
Encourage students to use benchmark fractions because these often lead to simpler calculations that can be done mentally.
• Encourage the sharing of methods, including mental math and
procedures.
INTRODUCE Whole Class
SP6* Percents of Containers
• Students estimate how full each container is as a percent
What percent represents a full container? 100% represents a whole container. How full is container B? About half full, which is 50%. Container C? One-fourth full, or half of 50%, which is 25%. Container D? Some students may estimate that this is one-eighth full, or half of 25%, which is 12.5%, or about 12%. Others may estimate that this is one-tenth full, or 10%. Accept all answers that are reasonable as long as they are justified.
• Students use logical reasoning to determine the number of ounces in
each container if the container has an 800-ounce capacity.
How many ounces of liquid are in container A? 800 oz. How do you know? The entire container is full. 100% of 800 = 800. Encourage students to write this equation. About how many ounces are in container B? 400 oz. How do you know? Half of the 800; 50% of 800 = 400. About how many ounces are in container C? 200 oz. How do you know? Half of 400, or one-fourth of 800; 25% of 800 = 200. About how many ounces are in container D? 100 oz. How do you know? If the estimate is 1
8 full, then possible a explanation is: half of 200, or one-
eighth of 800; 12.5% of 800 = 100.
17.2 Understanding Percents
Unit 5: Ratio, Proportion, and Percent (Teacher Pages) Week 17 – TP12
EXPLORE
Pairs/ Individuals
SP6* Percents of Containers
SP7-8* Percents of Other Items 1, 2
Math Background 1
• Monitor student progress by walking around the room. Encourage
mental, sense-making strategies in place of paper and pencil algorithms. Though students may recall some procedures for finding percents, remind them that it is often faster and easier to use benchmark percents and do simpler calculations mentally.
• An alternative way to complete problem #5 on SP7 is to provide a strip
of adding machine tape measuring one meter long to each group. Students locate each given percent A-J on the adding machine tape. When groups finish, tape all the strips vertically aligned on the wall to compare results. The class then discusses strategies on how the percents were located.
SUMMARIZE
Whole Class
SP6* Percents of Containers SP7-8* Percents of Other Items 1, 2
• Students share their answers with the class. During this process require
students to thoroughly explain their reasoning. Write an explanation on the board, such as, “I know that the answer to ____ is ____ because ____.” Invite other students to restate these answers in their own words to ensure understanding.
EXTEND
Whole Class Math Background 1
• Students find 17% of $84. Some possible mental strategies include
adding 10%, 5%, and 1% twice, or subtracting 1% three times from 20%.
• Teach students the procedural method of multiplying to find the percent
of a number. For example, when finding 17% of $84, simply multiply 0.17 x 84 = 14.28.
Which method do you prefer for this problem? Why?
PRACTICE Individuals
SP9 Money and Other Items
• Have students find the percents of the given amounts. Emphasize
mental strategies and encourage using procedures as a check.
CLOSURE Whole Class
SP5* Ready, Set
• Review the goals and standards for the lesson.
17.2 Understanding Percents
Unit 5: Ratio, Proportion, and Percent (Teacher Pages) Week 17 – TP13
SELECTED SOLUTIONS
Fraction Decimal Percent
1. 12
0.5 50%
2. 710
0.7 70%
3. 6100
or 350
0.06 6%
4. 13
0.333333333… 33 13
%
5. 25100
or 14
0.25 25%
SP5 Warmup
6. 40100
or 25
0.4 40%
SP6 Percents of Containers
1. 100%, 50%, 25%, and 12.5%. 2. 800 oz, 400 oz, 200 oz, and 100 oz. 3. 900 oz, 450 oz, 225 oz, and 112.5 oz. 4. 300 oz
5. 150 oz 6. 450 oz 7. 60 oz 8. 120 oz
SP7 Percents of Other Items 1
1. 5 2. 34 3. 10 4. 76 5. A meter is 100 centimeters 6. Since there are 100 cm in one whole meter and 100% in a whole unit, x% will be
represented by x cm. SP8 Percents of Other Items 2
1. all, $40 2. half, 2, $20 3. one-fourth, 4, $10
4. one tenth, 10, $4 5. 8, double 10% 6. 2, take half of 10%
Money Find 10% Find 5% Find 15% Find 20% 1 $30 $3 $1.50 $4.50 $6 2 $42 $4.20 $2.10 $6.30 $8.40 3 $16 $1.60 $0.80 $2.40 $3.20 4 $150 $15 $7.50 $22.50 $30 5 $500 $50 $25 $75 $100 Items Find 10% Find 5% Find 50% Find 25% 6 80 games 8 games 4 games 40 games 20 games 7 60 apples 6 apples 3 apples 30 apples 15 apples 8 40 balloons 4 balloons 2 balloons 20 balloons 10 balloons 9 120 students 12 students 6 students 60 students 30 students
10 12 cars 1.2 cars 0.6 cars 6 cars 3 cars More Money Find 1% Find 0.5% Find 1.5% Find 150%
11 $1,000 $10 $5 $15 $1,500 12 $150 $1.50 $0.75 $2.25 $225
SP9 Money and Other Items
13 $7,500 $75 $37.50 $112.50 $11,250
0 10 20 30 40 50 60 70 80 90 100
ABC D E F GH I J
STUDENT PAGES
17.2 Understanding Percents
Unit 5: Ratio, Proportion, and Percent (Student Packet) Week 17 – SP5
UNDERSTANDING PERCENTS
Ready (Summary)
We will learn sense-making strategies for finding percents of numbers.
Set (Goals) • Convert fractions to decimals and
percents. • Find percents of numbers.
Go (Warmup)
Complete the table. Use mental math when possible. If another method is used, show the work clearly. Be prepared to explain your methods. Fraction Decimal Percent
1. 12
2. 7
10
3. 0.06
4. 0.3333333…
5. 25%
6. 40%
17.2 Understanding Percents
Unit 5: Ratio, Proportion, and Percent (Student Packet) Week 17 – SP6
PERCENTS OF CONTAINERS
Complete the problems below. 1. About how full is each container? Give estimates using percents. Container A Container B Container C Container D
2. You are now given the information that, when full, each container holds 800 ounces
of liquid. How many ounces do you estimate to be in each container?
3. A correction is made to the above information. These containers actually hold 900
ounces of liquid. Now how many ounces do you estimate to be in each container?
The container below holds 600 ounces of liquid when full. Determine how much liquid would make it: 4. 50% full ___________ 5. 25% full ___________ 6. 75% full ___________ 7. 10% full ___________ 8. 20% full ___________
17.2 Understanding Percents
Unit 5: Ratio, Proportion, and Percent (Student Packet) Week 17 – SP7
PERCENTS OF OTHER ITEMS 1 Complete each sentence. A dollar is _____ cents. Find the following percents of a dollar. 1. 5% of a dollar is _____ cents. 2. 34% of a dollar is _____ cents.
3. 10% of a dollar is _____ cents . 4. 76% of a dollar is _____ cents. 5. A meter is _____ centimeters. The rectangle below represents a meter stick. Write
some numbers on the meter stick to represent centimeters. The left edge of the meter stick represents 0 centimeters.
Going from left to right, write the letter on the meter stick above that represents each
of the following:
A. 100% of the way B. 50% of the way
C. 25% of the way
D. 75% of the way
E. 5% of the way
F. 10% of the way G. 20% of the way H. 0% of the way I. 80% of the way J. 55% of the way
6. How are finding percents and finding centimeters on a meter stick related? ___________________________________________________________________ ___________________________________________________________________
17.2 Understanding Percents
Unit 5: Ratio, Proportion, and Percent (Student Packet) Week 17 – SP8
PERCENTS OF OTHER ITEMS 2
Complete each sentence. 1. Finding 100% of something is the same as finding __________ of it.
So 100% of $40 is __________.
2. Finding 50% of something is the same as finding __________ of it.
This is the same as dividing by__________. So 50% of $40 is __________.
3. Finding 25% of something is the same as finding __________ of it.
This is the same as dividing by __________. So 25% of $40 is __________.
4. Finding 10% of something is the same as finding __________ of it.
This is the same as dividing by__________. So 10% of $40 is __________.
5. Finding 10% can help with finding other percents as well. Use your answer for 10% of $40 to find 20% of $40. Explain your strategy.
___________________________________________________________________ ___________________________________________________________________
6. Find 5% of $40. Explain your strategy. ___________________________________________________________________ ___________________________________________________________________
17.2 Understanding Percents
Unit 5: Ratio, Proportion, and Percent (Student Packet) Week 17 – SP9
MONEY AND OTHER ITEMS Complete the charts below.
Money Find 10% Find 5% Find 15% Find 20%
1. $30
2. $42
3. $16
4. $150
5. $500
Items Find 10% Find 5% Find 50% Find 25%
6. 80 games
7. 60 apples
8. 40 balloons
9. 120 students
10. 12 cars
More Money Find 1% Find 0.5% Find 1.5% Find 150%
11. $1,000
12. $150
13. $7,500