ratios and proportions in the common core - nc...
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Ratios and Proportions in the
Common Core
NCCTM State Mathematics Conference
Robin Barbour [email protected]
www.ncdpi.wikispaces.net
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Understand ratio concepts and use ratio reasoning to solve problems. 1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”1
Ratios and Proportions 6.RP
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Understand ratio concepts and use ratio reasoning to solve problems. 3. Use ratio and rate reasoning to solve real-world and
mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. a. Make tables of equivalent ratios relating quantities with
whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
b. Solve unit rate problems including those involving unit pricing and constant speed.
Ratios and Proportions 6.RP
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Understand ratio concepts and use ratio reasoning to solve problems. 3. Use ratio and rate reasoning to solve real-world and
mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. c. Find a percent of a quantity as a rate per 100 (e.g.,
30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Ratios and Proportions 6.RP
Cups Blue 2 4 6 Total Cups 3 6 9
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23
€
46
€
69
Equivalent Ratios vs. Equivalent Fractions
Equivalent Fractions
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23
€
46
€
69
More parts; smaller parts Same whole amount Same portion
Equivalent Ratios
Cups Blue 2 4 6 Total Cups 3 6 9
More parts; same size parts More total paint More 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 40 quarts of blue paint, how many quarts of yellow paint will you need?
2. If you will use 48 quarts of yellow paint, how many quarts of blue paint will you need?
3. If you want to make 100 quarts of green paint, how many quarts of yellow and blue will you need?
yellow
blue
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
Distance 0 miles 14 miles
0 minutes Time
20 minutes
28 miles
40 minutes 10 minutes
7 miles
PERCENTS
Percents
0% 100% 50%
80 40 0
75% 25%
60 20
x 3
x 3
Percents
0% 100% 50%
80 0
70% 20% 30% 10% 40% 90% 80% 60%
8 16 56 48 40 32 24 64 72
5%
4
÷10
÷10
Percents
0% 100% 50%
80 0
70% 20% 30% 10% 40% 90% 80% 60%
8 16 56 48 40 32 24 64 72
5%
4
Percents
Jean has 60 text messages. Thirty-five percent of them are from Susan. How many text messages does she have from Susan?
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
x 7
Percents
I know 10% is 6 and 5% is 3, so
10% 6 10% 6 10% 6 5% 3 35% 21
0% 100%
60 0
35%
x
5% 10%
6 3
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 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
Brand B Loads washed 15 1 Cost $5
Brand A Loads washed 20 1 Cost $6
3.33 3
$0.30 $0.33
Ratio Tables
It takes Paul 2 hours to bike 8 miles. How long will it take him to bike 12 miles?
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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?
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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?
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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
3 7
2 6 14
5 35
6 14
35
Factor Puzzles
6 14
35
3 7
2
5
Factor Puzzles
15
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Analyze proportional relationships and use them to solve real-world and mathematical problems. 1. Compute unit rates associated with ratios of fractions, including ratios of
lengths, areas and other quantities measured in like or different units. 2. Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
c. Represent proportional relationships by equations d. Explain what a point (x, y) on the graph of a proportional relationship
means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
Ratios and Proportions 7.RP
Solution Strategies
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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.
Cross Multiplication Algorithm How does this work?
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Step 1: Start with two equal fractions =
€
26
€
39
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
€
99
€
39
€
66
€
26
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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 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: (a) using 3 cups of ice cream
(b) to make 3 cups of milkshake.
€
12
€
34
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
Turn and Talk 1. How can you make sure you are posing
problems that will allow all children to be able to access the content – yet provide challenges for all students?
2. What would you describe as an example you can use in instruction to compare additive and multiplicative thinking?
3. How would you get students to describe the different meanings of ratio?
4. How would you help students to understand the difference between proportional vs.
non-proportional relationships?
Perplexing Puzzle
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cc: Microsoft.com
Make a rectangle out of the pieces in the envelope.
Perplexing Puzzle
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A
B
D
E
C F
G
H
Perplexing Puzzle Directions
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We are going to make a larger puzzle the same shape as the smaller puzzle.
Enlarge your piece so that if the edge of a piece measures 4 cm in the old puzzle, it will measure 6 cm in the new puzzle.
Enlarge your piece of the puzzle so that it will fit into the new larger puzzle.
Graph of Perplexing Puzzle
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0
2
4
6
8
10
12
14
16
18
0 2 4 6 8 10 12
New
Len
gth
(cm
)
Original Length (cm)
Perplexing Puzzle
Wrap Up
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1. Describe the graph (shape, starting point, etc.) 2. What does the ordered pair (8, 12) mean in this
problem? 3. Compare the ratios of for each ordered pair
graphed. What is significant about all these ratios? 4. Write a rule to describe the relationship between the
new length and original length. 5. What does the coefficient tell you about the
relationship?
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7th Grade Geometry: Draw construct, and describe geometrical figures and describe the relationships between them. 1. Solve problems involving scale drawings of
geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Geometry 7.G
Resources
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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)
QUESTIONS
COMMENTS
