Ratios and Proportions
Grade 6th and 7th
CCSSM National Professional Development
5/5/12
Sara Anaya
Ayşe Şahin
DePaul
University
Calmeca
Academy,
CPS
Plan:
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Overview of the content progression from
6th to 7th
Tools emphasized in the Standards to help
students understand ratio and
proportional relationships.
Examples of applications of these tools to
problem solving.
Extensions of these to 7th and 8th grades.
The Standards for Mathematical Practice.
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6th Grade
6.RP.1 - Definition of
ratio
6.RP.2 - Unit rate(s)
associated to ratio
Whole numbers
(6.RP.3) Given a proportional relationship
represent it a variety of ways:
. A table
. Plotting points on a coordinate graph.
. Tape diagrams
7th Grade
7.RP.1 – Unit Rates
associated with ratios (with
fractions and decimals)
7.RP.2.
Using the representations from
6th grade
a. Decide whether two
quantities are in a proportional
relationship.
b. Identify the constant of
proportionality
. Double number line diagrams.
6.RP.3 –Use ratios and rate reasoning to
solve (single step)real world mathematical
problems using the representations above
and:
7.RP.3 Use proportional
relationships to solve
(multi-step) ratios and
percent problems.
. Using the unit rate
. Seeing % as a rate
. Using ratio reasoning and manipulating
units by multiplying and dividing
The Standards requires that the
strategies listed here are not replaced
by solving equations.
7.RP.2
c. Represent proportional
relationships by equations.
d. Understanding the
proportional relationship on a
graph most importantly (0,0)
and (1.r)
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The Ratio and Proportion strand starts in
6th and ends in 7th. It leads directly into:
Understand
the connections between proportional
relationships, lines, and linear equations.
8.EE.5.
Graph proportional relationships, interpreting the unit rate
as the slope of the graph. Compare two different proportional
relationships represented in different ways. For example, compare a
distance-time graph to a distance-time equation to determine which
of two moving objects has greater speed.
8.EE.6. Use similar triangles to explain why the slope m is the
same between any two distinct points on a non-vertical line in the
coordinate plane; derive the equation y = mx for a line through the
origin and the equation y = mx + b for a line intercepting the
vertical axis at b.
What happens in 8th grade?
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An example of how to introduce the notion
of a ratio:
Illustrative Mathematics 6.RP.1 Task
example:
Games at recess
6.RP.1: The language of ratio and
proportion.
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This recipe has a ratio of 3 cups of flour to
4 tablespoons of butter.
How many cups of flour would you use for
1 tablespoon of butter?
This question is asking for one of the unit
rates associated with this ratio. Find the
unit rate using at least two different
methods.
6.RP.2 and 7.RP.1
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Make one poster for each method used in
your group and post.
TASK
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# cups of
flour
# tbsp of
butter
3
4
x2
x3
x4
x½
x¼
x2
6
8
9
12
x4
12
16
x½
3/2
2
3/4
1
x3
x¼
Table
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Tape diagram representation
(6.RP.3a)
1 cup of flour
1 tbsp of butter
¼ cup of flour
1 cup of flour
1 cup of flour
¾ cups of flour
1 tbsp of butter
requires ¾ cup
of flour
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0c
?c
3c
Cups of flour
Tablespoons of
Butter
0 tbsp 1 tbsp
4 tbsp
Double Number Line
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¸4
0c
?c
3c
Cups of flour
Tablespoons of
Butter
4 tbsp
0 tbsp 1 tbsp
¸4
Using double number lines we directly
translate the problem into asking: what is
3 ¸4?
(Standard 5.NF.3) ¾ cup of flour for every
tablespoon of butter.
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If you change the problem slightly, the
double number line is no longer a good
tool:
This recipe calls for 3 cups of flour for every
4 cups of sugar. How many cups of flour
would you use for 1 cup of sugar?
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1
2
3
Cups
Cups
1
2
1
3
2
4
3
Cups of flour
Cups of sugar
1
2
3
4
Double number line representations don’t work as well when the units are the
same.
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Now let’s find the other unit ratio:
This recipe has a ratio of 3 cups of flour to 4 tablespoons of butter.
How many tablespoons of butter would you use for 1 cup of flour?
3
÷3
x2
x3
x4
x½
x¼
# of
cups of
flour
# tbsp
Of butter
1
4/3
3
4
6
8
1 tbsp of butter
÷3
12
12
16
3/2
3/4
2
1
3c
1c
1c flour
x2
x3
9
0c
1c flour
x4
1c flour
x½
1/3 tbsp of butter
x¼
0 tbsp ? tbsp
4 tbsp
3
1 cup of four requires
4/3 cup of butter
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The standards for 6th grade ask that
students learn these strategies for
understanding ratio and proportion to solve
real world mathematical problems.
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Suppose Abby’s orange paint is made by
mixing 1 cup red paint for every 3 cups
yellow paint and Zack’s orange paint is
made by mixing 3 cups red for every 5
cups yellow. Whose paint is yellower?1
1Progressions
for the CCSS in Math (draft)
Solve this problem using at least three of
the tools we have discussed so far.
6.RP.3 and 7.RP.3
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This recipe calls for ¾ cup of flour for every
½ tablespoon of butter. How many cups of
flour would you use for every 1 tablespoon
of butter?
Now unit rates involve fractions or
decimals.
A 7th grade unit ratio problem
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This recipe calls for ¾ cup of flour for every ½
tablespoon of butter. How many cups of flour
would you use for every 1/4 tablespoon of butter?
2
0 c flour
0 tbsp butter
? c flour
¾ c flour
¼ tbsp
butter
½ tbsp
butter
3/8 cup of
flour for
every ¼
tbsp of
butter
2
A tape diagram is an excellent tool to remediate
for students who can’t do this computation.
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½ tbsp of butter
¼ of a cup of flour
1cup of flour
3/8 of a cup of
flour for every
¼ of a tbsp of
butter
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
Standard error: if you don’t identify the
whole correctly you get the wrong answer
of 3/6 vs. 3/9.
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How well did our work today align with the
Standards of Mathematical Practice?
Use your handout to identify which
standard(s) for practice was (were) used in
our tasks.
The Standards for Mathematical
Practice
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1. Make sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning.
Standards for Mathematical
Practice
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