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Ratios, Proportions, and Percents

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Cups Blue 246 Total Cups 369 Equivalent Ratios vs. Equivalent Fractions

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

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Equivalent Ratios Cups Blue 246 Total Cups 369 More parts; same size parts More total paint More blue pigment

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

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Tape Diagrams Best used when the two quantities have the same units. Highlight the multiplicative relationship between quantities. yellow blue

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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?

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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

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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 Distance 0 miles 14 miles 0 minutes Time 20 minutes 28 miles 40 minutes 10 minutes 7 miles

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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 washed20 Cost$6 Brand B Loads washed15 Cost$5

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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 washed20 Cost$6$1 Brand B Loads washed15 Cost$5$1 Brand B Loads washed151 Cost$5 Brand A Loads washed201 Cost$6 3.33 3 $0.30$0.33

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Designing the Super Sandwich

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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) 28 ?12 cc: Microsoft.com

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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) 14 28 ?12 cc: Microsoft.com

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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) 14 28 312 cc: Microsoft.com x3

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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?

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12345678910 2468 1214161820 36912151821242730 481216202428323640 5101520253035404550 6121824303642485460 7142128354249566370 8162432404856647280 9182736455463728190 102030405060708090100

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12345678910 2468 1214161820 36912151821242730 481216202428323640 5101520253035404550 6121824303642485460 7142128354249566370 8162432404856647280 9182736455463728190 102030405060708090100

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12345678910 2468 1214161820 36912151821242730 481216202428323640 5101520253035404550 6121824303642485460 7142128354249566370 8162432404856647280 9182736455463728190 102030405060708090100

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12345678910 2468 1214161820 36912151821242730 481216202428323640 5101520253035404550 6121824303642485460 7142128354249566370 8162432404856647280 9182736455463728190 102030405060708090100

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12345678910 2468 1214161820 36912151821242730 481216202428323640 5101520253035404550 6121824303642485460 7142128354249566370 8162432404856647280 9182736455463728190 102030405060708090100

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37 2614 535

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614 35 Factor Puzzles

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614 35 3 7 2 5 Factor Puzzles 15

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Ratio Tables Three sweaters cost $18. What is the cost of 7 sweaters? NumberCost 318

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Ratio Tables Three sweaters cost $18. What is the cost of 7 sweaters? NumberCost 318 6 1

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Ratio Tables Three sweaters cost $18. What is the cost of 7 sweaters? NumberCost 318 6 1 7 42

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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? Kates Stickers Jennas Stickers 74 21???

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Solution Strategies StrategyDescription Build-up strategyStudents use the ratio to build up to the unknown quantity. Unit-rate strategyStudents identify the unit rate and then use it to solve the problem. Factor-of-change strategy Students use a times as many strategy. Fraction strategyStudents use the concept of equivalent fractions to find the missing part. Ratio TablesStudents 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.

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Cross Multiplication Algorithm How does this work? Step 1: Start with two equal fractions = Source: IES Practice Guide: Developing Effective Fraction Instruction for Kindergarten Through 8 th Grade Step 2: Find a common denominator using each of the two denominators. Multiply by, which is multiplying by 1

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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 8 th Grade

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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: (a)using 3 cups of ice cream (b) to make 3 cups of milkshake.

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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 1Mixture 2

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PERCENTS

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Percents 0% 100%50% 80 400 75%25% 6020 x 3

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Percents 0%100%50% 80 0 70%20% 30% 10% 40%90% 80% 60% 81656484032246472 5% 4 ÷10

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Percents 0%100%50% 80 0 70%20% 30% 10% 40%90% 80% 60% 81656484032246472 5% 4

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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

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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

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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

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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 _____

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

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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?

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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% 60 0 35% x 5% 10% 6 3 x 7

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

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

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Percent of Decrease A coat selling for $120 is discounted 25%. What is the sale price? 0%100% 0120

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Percent of Decrease A coat selling for $120 is discounted 25%. What is the sale price? 0%100% 0120 75% x

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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

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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

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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

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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

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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

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Resources 6/11/2014 page 54 Developing Effective Fractions Instruction for Kindergarten Though 8 th Grade IES What Works Clearinghouse www.commoncoretools.wordpress.com Its All Connected: The Power of Proportional Reasoning to Understand Mathematics Concepts Carmen Whitman (Math Solutions)

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