Ratios, Proportions, and Percents. Cups Blue 246 Total Cups 369 Equivalent Ratios vs. Equivalent Fractions.

Ratios, Proportions, and Percents

Cups Blue 2 4 6

Total Cups 3 6 9

€

2

3

€

4

6

€

6

9

Equivalent Ratios vs. Equivalent Fractions

Equivalent Fractions

€

2

3

€

4

6

€

6

9

More parts; smaller partsSame whole amountSame portion

Equivalent Ratios

Cups Blue 2 4 6

Total Cups 3 6 9

More parts; same size partsMore total paintMore blue pigment

Ratios

If you know that 2:3 is a part-to-part relationship, when else can you deduce from that ratio?

Tape Diagrams

• Best used when the two quantities have the same units.

• Highlight the multiplicative relationship between quantities.

yellow

blue

Tape Diagrams

1. If you will use 10 quarts of blue paint, how many quarts of yellow paint will you need?

yellow

blue

2. If you will use 18 quarts of yellow paint, how many quarts of blue paint will you need?

3. If you want to make 25 quarts of green paint, how many quarts of yellow and blue will you need?

Double Number Lines

• Best used when the two quantities have different units.

• Help make visible that there are infinitely many pairs in the same ratio, including those with rational numbers

• Same ratios are the same distance from zero

Driving at a constant speed, you drove 14 miles in 20 minutes. On a “double number line”, show different distances and times that would give you the same speed. Identify equivalent rates below.

Double Number Lines

Distance0 miles 14 miles

0 minutesTime

20 minutes

28 miles

40 minutes10 minutes

7 miles

Laundry Detergent Comparison

A box of Brand A laundry detergent washes 20 loads of laundry and costs $6. A box of Brand B laundry detergent washes 15 loads of laundry and costs $5. What are some equivalent loads?

Brand A

Loads washed 20

Cost $6

Brand B

Loads washed 15

Cost $5

Unit RatesExplain how to fill in the next tables with unit rates. Then

use the tables to make statements comparing the two brands of laundry detergent.

Brand A

Loads washed 20

Cost $6 $1

Brand B

Loads washed 15

Cost $5 $1

Brand B

Loads washed 15 1

Cost $5

Brand A

Loads washed 20 1

Cost $6

3.33 3

$0.30 $0.33

Designing the Super Sandwich

Ratio Tables

It takes Paul 2 hours to bike 8 miles. How long will it take him to bike 12 miles?

Time(hours)

Distance(miles)

2 8

? 12

cc: Microsoft.com

Ratio Tables

It takes Paul 2 hours to bike 8 miles. How long will it take him to bike 12 miles?

Time(hours)

Distance(miles)

1 4

2 8

? 12

cc: Microsoft.com

Ratio Tables

It takes Paul 2 hours to bike 8 miles. How long will it take him to bike 12 miles?

Time(hours)

Distance(miles)

1 4

2 8

3 12

cc: Microsoft.com

x3 x3

Susan and Tim save at constant rates. On a certain day, Susan had $6 and Tim had $14. How much money did Susan have when Tim had $35?

1 2 3 4 5 6 7 8 9 10

2 4 6 8 10 12 14 16 18 20

3 6 9 12 15 18 21 24 27 30

4 8 12 16 20 24 28 32 36 40

5 10 15 20 25 30 35 40 45 50

6 12 18 24 30 36 42 48 54 60

7 14 21 28 35 42 49 56 63 70

8 16 24 32 40 48 56 64 72 80

9 18 27 36 45 54 63 72 81 90

10 20 30 40 50 60 70 80 90 100

1 2 3 4 5 6 7 8 9 10

2 4 6 8 10 12 14 16 18 20

3 6 9 12 15 18 21 24 27 30

4 8 12 16 20 24 28 32 36 40

5 10 15 20 25 30 35 40 45 50

6 12 18 24 30 36 42 48 54 60

7 14 21 28 35 42 49 56 63 70

8 16 24 32 40 48 56 64 72 80

9 18 27 36 45 54 63 72 81 90

10 20 30 40 50 60 70 80 90 100

1 2 3 4 5 6 7 8 9 10

2 4 6 8 10 12 14 16 18 20

3 6 9 12 15 18 21 24 27 30

4 8 12 16 20 24 28 32 36 40

5 10 15 20 25 30 35 40 45 50

6 12 18 24 30 36 42 48 54 60

7 14 21 28 35 42 49 56 63 70

8 16 24 32 40 48 56 64 72 80

9 18 27 36 45 54 63 72 81 90

10 20 30 40 50 60 70 80 90 100

1 2 3 4 5 6 7 8 9 10

2 4 6 8 10 12 14 16 18 20

3 6 9 12 15 18 21 24 27 30

4 8 12 16 20 24 28 32 36 40

5 10 15 20 25 30 35 40 45 50

6 12 18 24 30 36 42 48 54 60

7 14 21 28 35 42 49 56 63 70

8 16 24 32 40 48 56 64 72 80

9 18 27 36 45 54 63 72 81 90

10 20 30 40 50 60 70 80 90 100

1 2 3 4 5 6 7 8 9 10

2 4 6 8 10 12 14 16 18 20

3 6 9 12 15 18 21 24 27 30

4 8 12 16 20 24 28 32 36 40

5 10 15 20 25 30 35 40 45 50

6 12 18 24 30 36 42 48 54 60

7 14 21 28 35 42 49 56 63 70

8 16 24 32 40 48 56 64 72 80

9 18 27 36 45 54 63 72 81 90

10 20 30 40 50 60 70 80 90 100

3 7

2 6 14

5 35

6 14

35

Factor Puzzles

6 14

35

3 7

2

5

Factor Puzzles

15

Ratio Tables

Three sweaters cost $18. What is the cost of 7 sweaters?

Number Cost

3 18

Ratio Tables

Three sweaters cost $18. What is the cost of 7 sweaters?

Number Cost

3 18

61

Ratio Tables

Three sweaters cost $18. What is the cost of 7 sweaters?

Number Cost

3 18

61

7 42

Your Turn

The ratio of Kate's stickers to Jenna's stickers is 7:4. Kate has 21 stickers. How many stickers does Jenna have?

Kate’s Stickers

Jenna’s Stickers

7 4

21 ???

Solution StrategiesStrategy Description

Build-up strategy Students use the ratio to build up to the unknown quantity.

Unit-rate strategy Students identify the unit rate and then use it to solve the problem.

Factor-of-change strategy Students use a “times as many strategy.

Fraction strategy Students use the concept of equivalent fractions to find the missing part.

Ratio Tables Students set up a table to compare the quantities.

Cross multiplication algorithm

Students set up a proportion (equivalence of two ratios), find the cross products, and solve by using division.

Cross Multiplication AlgorithmHow does this work?

Step 1: Start with two equal fractions =

€

2

6

€

3

9

Source: IES Practice Guide:Developing Effective Fraction Instruction for Kindergarten Through 8th Grade

Step 2: Find a common denominator using each of the two denominators.

Multiply by , which is multiplying by 1

Multiply by , which is multiplying by 1

€

9

9

€

3

9

€

6

6€

2

6

Cross Multiplication Algorithm

Step 3: Calculate the result: (2 x 9) (3 x 6) (6 x 9) (9 x 6)

=

Step 4: Note that the denominators are equal. If two equal fractions have equal denominators, then the numerators are also equal. So, (2 x 9) (3 x 6)=

Source: IES Practice Guide:Developing Effective Fraction Instruction for Kindergarten Through 8th Grade

Comparing MixturesThere are two containers, each containing a mixture of 1 cup red punch and 3 cups lemon lime soda. The first container is left as it is, but somebody adds 2 cups red punch and 2 cups lemon lime soda to the second container.

•Will the two punch mixtures taste the same? Why or why not?

Mixture 1 Mixture 2

PERCENTS

Percents

0% 100%50%

80400

75%25%

6020

x 3

x 3

Percents

0% 100%50%

800

70%20% 30%10% 40% 90%80%60%

8 16 5648403224 64 72

5%

4

÷10

÷10

Percents

0% 100%50%

800

70%20% 30%10% 40% 90%80%60%

8 16 5648403224 64 72

5%

4

Problem Strings

• Cathy Fosnot

• Problem string for a particular strategy are meant to be done more than once

• Not intended to be used all at once, handed out as worksheets or used as independent work for the students

• Helps secondary students construct mental numerical relationships

Percents – Start Unknown

• _____ is 100% of 40

• _____ is 200% of 40

• _____ is 50% of 40

• _____ is 25% of 40

• _____ is 10% of 40

• _____ is 5% of 40

• _____ is 1% of 40

• _____ is 6% of 40

• _____ is 0.5% of 40

• _____ is 13.5% of 40

Percents – Percent Unknown

• 10 is what percent of 20

• 5 is what percent of 20

• 15 is what percent of 20

• 2 is what percent of 20

• 3 is what percent of 20

• 5 is what percent of 50

• 15 is what percent of 50

• 2 is what percent of 50

• 17 is what percent of 50

• 39 is what percent of 40

Percents – Result Unknown

• 3 is 100% of _____

• 3 is 50% of _____

• 3 is 25% of _____

• 3 is 10% of _____

• 3 is 1% of _____

• 3 is 12% of _____

• 6 is 50%of _____

• 12 is 50% of _____

• 12 is 25% of _____

• 6 is 25% of _____

Percents

Jean has 60 text messages. Thirty-five percent of them are from Susan. How many text messages does she have from Susan?

Percents

Your parents took your family out to dinner. They wanted to give the waiter a 15% tip. If the total amount of the dinner was $42.00, what should be paid to the waiter as a tip?

Percents

If 60 is 100% then 6 is 10% and 3 is 5%. Multiply 5% by 7 to get to 35% and 3 by 7 to get 21.

0% 100%

600

35%

x

5% 10%

63

x 7

x 7

Percents

I know 10% is 6 and 5% is 3, so

10% 610% 610% 6 5% 335% 21

0% 100%

600

35%

x

5% 10%

63

Percent of Decrease

• A coat selling for $120 is discounted 25%. What is the sale price?

0% 100%

0

Percent of Decrease

• A coat selling for $120 is discounted 25%. What is the sale price?

0% 100%

0 120

Percent of Decrease

• A coat selling for $120 is discounted 25%. What is the sale price?

0% 100%

0 120

75%

x

Percent of Increase

• In a retail store the prices were increased 60% What would be the price of an item if the original price was $20?

0% 100%

0

Percent of Increase

• In a retail store the prices were increased 60% What would be the price of an item if the original price was $20?

0% 100%

0 20

160%

x

Percent of Increase

• In a retail store the prices were increased 60% What would be the price of an item if the original price was $20?

0% 100%

0 20

160%

x

Percent of Increase

• A price of a pair of shoes is increased from $24 to $80. What is the percent of increase?

0% 100%

0

Percent of Increase

• A price of a pair of shoes is increased from $24 to $80. What is the percent of increase?

0% 100%

0 80

x

24

Resources

04/10/23 • page 54

Developing Effective Fractions Instruction for Kindergarten Though 8th Grade IES What Works Clearinghouse

www.commoncoretools.wordpress.com

It’s All Connected: The Power of Proportional Reasoning to Understand Mathematics Concepts Carmen Whitman (Math Solutions)