1. Ratios, Proportions, and Percents

2. 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

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 rate
More parts; same size parts
More total paint
More blue pigment
Cups Blue 2 4 6
Total Cups 3 9

5. Ratios
If 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.
yellow
blue

7. 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?

8. 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

9. 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

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 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

11. 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

12. 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

13. 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

14. 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

15. 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?

16. Factor Puzzles 6 14 35

17. Factor Puzzles 3 7 6 14 35 2 15 5

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

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

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

21. 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
???

22. 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.

23. 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.

24. 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

25. PERCENTS

26. Percents
x 3 20 40 60 80
25%
50%
75%
100% 0%
x 3

27. 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

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

29. 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

30. 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

31. 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

32. 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 _____

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

34. 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?

35. 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.

36. 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

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

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

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

40. 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%

41. 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%

42. 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%

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

44. 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

45. 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.