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