# Ratios, Proportions, and Percents

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Ratios, Proportions, and Percents

Equivalent Ratios vs. Equivalent Fractions
Do these two terms mean the same thing? Turn and talk. Cups Blue 2 4 6 Total Cups 3 9

Equivalent Fractions More parts; smaller parts Same whole amount
Same portion

Equivalent Ratios More parts; same size parts More total paint
Equivalent ratios have the same unit rate More parts; same size parts More total paint More blue pigment Cups Blue 2 4 6 Total Cups 3 9

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

Double Number Lines 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. Distance 0 miles 7 miles 14 miles 28 miles 0 minutes Time 10 minutes 20 minutes 40 minutes

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 In the ratio tables that follow, fill in equivalent rates of loads washed per dollar. Include some examples where the number of loads washed is less than 15 and the cost is less than \$5. Explain your reasoning. Brand B Loads washed 15 Cost \$5

Unit Rates Explain 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 3.33 3 Brand A Loads washed 20 1 Cost \$6 Brand B Loads washed 15 1 Cost \$5 \$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 Solve the following problem using one of the discussed strategies. We chose to use a table. 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 Dividing by 2 gives a unit rate of 4 miles for every hour. 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 x3 Multiply both quantities by 3 to get a time of 3 hours. Notice the Between and within relationship between equivalent ratios. Each time the distance can be found by multiplying by 4 (the between relationship) – foundational to writing the equations. Also note the relationships between the different rows which is the within relationship. x3 cc: Microsoft.com

Susan and Tim save at constant rates
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 12 14 16 18 20 15 21 24 27 30 28 32 36 40 25 35 45 50 42 48 54 60 49 56 63 70 64 72 80 81 90 100

1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 15 21 24 27 30 28 32 36 40 25 35 45 50 42 48 54 60 49 56 63 70 64 72 80 81 90 100

1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 15 21 24 27 30 28 32 36 40 25 35 45 50 42 48 54 60 49 56 63 70 64 72 80 81 90 100

1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 15 21 24 27 30 28 32 36 40 25 35 45 50 42 48 54 60 49 56 63 70 64 72 80 81 90 100

1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 15 21 24 27 30 28 32 36 40 25 35 45 50 42 48 54 60 49 56 63 70 64 72 80 81 90 100

3 7 2 6 14 5 35

Factor Puzzles 6 14 35

Factor Puzzles 3 7 6 14 35 2 15 5

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 6 1

Ratio Tables Three sweaters cost \$18. What is the cost of 7 sweaters?
Number Cost 3 18 6 1 42 7

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 Strategies Strategy 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. While the cross-product algorithm is efficient, it has little meaning. In fact, it is impossible to explain why one would want to find the product of contrasting elements from two different rate pairs. This algorithm should not be introduced until 7th grade. According to the IES What Works Clearinghouse: Developing Effective Fractions Instruction for Kindergarten Though 8th Grade, a need for using the cross-product algorithm should be used based on the numbers in the problem. Either way, students should understand why it works.

Cross Multiplication Algorithm
How does this work? Step 1: Start with two equal fractions = 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 Source: IES Practice Guide: Developing Effective Fraction Instruction for Kindergarten Through 8th Grade

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

Solving Ratios with Rational Numbers
Chandra made a milkshake by mixing cup of ice cream with cups of milk. How many cups of ice cream and milk Chandra should use if she wants to make the same milkshake for the following amounts: using 3 cups of ice cream (b) to make 3 cups of milkshake.

Comparing Mixtures There 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 x 3 20 40 60 80 25% 50% 75% 100% 0% x 3

Percents ÷10 4 8 16 24 32 40 48 56 64 72 80 0% 5% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% ÷10

Percents 4 8 16 24 32 40 48 56 64 72 80 0% 5% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

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 5% of 40 _____ is 200% of 40 _____ is 1% of 40 _____ is 50% of 40 _____ is 6% of 40 _____ is 25% of 40 _____ is 0.5% of 40 _____ is 10% of 40 _____ is 13.5% of 40

Percents – Percent Unknown
10 is what percent of 20 5 is what percent of 50 5 is what percent of 20 15 is what percent of 50 15 is what percent of 20 2 is what percent of 50 2 is what percent of 20 17 is what percent of 50 3 is what percent of 20 39 is what percent of 40

Percents – Result Unknown
3 is 100% of _____ 3 is 12% of _____ 3 is 50% of _____ 6 is 50%of _____ 3 is 25% of _____ 12 is 50% of _____ 3 is 10% of _____ 12 is 25% of _____ 3 is 1% 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 x 7 0% 100% 60 35% x 5% 10% 6 3 x 7 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.

Percents I know 10% is 6 and 5% is 3, so 10% 6 10% 6 5% 3 35% 21 0%
100% 60 35% x 5% 10% 6 3 I know 10% is 6 and 5% is 3, so 10% 6 10% 6 5% 3 35% 21

Percent of Decrease A coat selling for \$120 is discounted 25%. What is the sale price? 0% 100%

Percent of Decrease A coat selling for \$120 is discounted 25%. What is the sale price? 0% 100% 120

Percent of Decrease A coat selling for \$120 is discounted 25%. What is the sale price? x 0% 100% 120 75%

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%

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% 20 x 160%

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% 20 x 160%

Percent of Increase A price of a pair of shoes is increased from \$24 to \$80. What is the percent of increase? 0% 100%

Percent of Increase A price of a pair of shoes is increased from \$24 to \$80. What is the percent of increase? 0% 100% 24 80 x

Resources Developing Effective Fractions Instruction for Kindergarten Though 8th Grade IES What Works Clearinghouse It’s All Connected: The Power of Proportional Reasoning to Understand Mathematics Concepts Carmen Whitman (Math Solutions) Reference handout with resources.

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