2 Equivalent Ratios vs. Equivalent Fractions Do these two terms mean the same thing? Turn and talk.Cups Blue246Total Cups39
3 Equivalent Fractions More parts; smaller parts Same whole amount Same portion
4 Equivalent Ratios More parts; same size parts More total paint Equivalent ratios have the same unit rateMore parts; same size partsMore total paintMore blue pigmentCups Blue246Total Cups39
5 RatiosIf you know that 2:3 is a part-to-part relationship, when else can you deduce from that ratio?
6 Tape Diagrams Best used when the two quantities have the same units. Highlight the multiplicative relationship between quantities.yellowblue
7 Tape DiagramsyellowblueIf 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?
8 Double Number LinesBest 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 numbersSame ratios are the same distance from zero
9 Double Number LinesDriving 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.Distance0 miles7 miles14 miles28 miles0 minutesTime10 minutes20 minutes40 minutes
10 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 ALoads washed20Cost$6In 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 BLoads washed15Cost$5
11 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 ALoads washed20Cost$6$1Brand BLoads washed15Cost$5$13.333Brand ALoads washed201Cost$6Brand BLoads washed151Cost$5$0.30$0.33
13 Ratio TablesIt takes Paul 2 hours to bike 8 miles. How long will it take him to bike 12 miles?Time(hours)Distance(miles)28?12Solve the following problem using one of the discussed strategies.We chose to use a table.cc: Microsoft.com
14 Ratio TablesIt takes Paul 2 hours to bike 8 miles. How long will it take him to bike 12 miles?Time(hours)Distance(miles)1428?12Dividing by 2 gives a unit rate of 4 miles for every hour.cc: Microsoft.com
15 Ratio TablesIt takes Paul 2 hours to bike 8 miles. How long will it take him to bike 12 miles?Time(hours)Distance(miles)1428312x3Multiply 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.x3cc: Microsoft.com
16 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?
25 Ratio Tables Three sweaters cost $18. What is the cost of 7 sweaters? NumberCost318
26 Ratio Tables Three sweaters cost $18. What is the cost of 7 sweaters? NumberCost31861
27 Ratio Tables Three sweaters cost $18. What is the cost of 7 sweaters? NumberCost31861427
28 Your TurnThe ratio of Kate's stickers to Jenna's stickers is 7:4. Kate has 21 stickers. How many stickers does Jenna have?Kate’s StickersJenna’s Stickers7421???
29 Solution Strategies Strategy Description Build-up strategy Students 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 strategyStudents 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 algorithmStudents 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.
30 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 1Multiply by , which is multiplying by 1Source: IES Practice Guide:Developing Effective Fraction Instruction for Kindergarten Through 8th Grade
31 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
32 Solving Ratios with Rational Numbers Chandra made a milkshake by mixing cup of icecream with cups of milk. How many cups of icecream 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.
33 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 1Mixture 2
38 Problem Strings Cathy Fosnot Problem string for a particular strategy are meant to be done more than onceNot intended to be used all at once, handed out as worksheets or used as independent work for the studentsHelps secondary students construct mental numerical relationships
39 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
40 Percents – Percent Unknown 10 is what percent of 205 is what percent of 505 is what percent of 2015 is what percent of 5015 is what percent of 202 is what percent of 502 is what percent of 2017 is what percent of 503 is what percent of 2039 is what percent of 40
41 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 _____
42 PercentsJean has 60 text messages. Thirty-five percent of them are from Susan. How many text messages does she have from Susan?
43 PercentsYour 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?
44 Percentsx 70%100%6035%x5%10%63x 7If 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.
45 Percents I know 10% is 6 and 5% is 3, so 10% 6 10% 6 5% 3 35% 21 0% 100%6035%x5%10%63I know 10% is 6 and 5% is 3, so10% 610% 65% 335% 21
46 Percent of DecreaseA coat selling for $120 is discounted 25%. What is the sale price?0%100%
47 Percent of DecreaseA coat selling for $120 is discounted 25%. What is the sale price?0%100%120
48 Percent of DecreaseA coat selling for $120 is discounted 25%. What is the sale price?x0%100%12075%
49 Percent of IncreaseIn a retail store the prices were increased 60% What would be the price of an item if the original price was $20?0%100%
50 Percent of IncreaseIn a retail store the prices were increased 60% What would be the price of an item if the original price was $20?0%100%20x160%
51 Percent of IncreaseIn a retail store the prices were increased 60% What would be the price of an item if the original price was $20?0%100%20x160%
52 Percent of IncreaseA price of a pair of shoes is increased from $24 to $80. What is the percent of increase?0%100%
53 Percent of IncreaseA price of a pair of shoes is increased from $24 to $80. What is the percent of increase?0%100%2480x
54 ResourcesDeveloping Effective Fractions Instruction for Kindergarten Though 8th Grade IES What Works ClearinghouseIt’s All Connected: The Power of Proportional Reasoning to Understand Mathematics Concepts Carmen Whitman (Math Solutions)Reference handout with resources.