Ratios and Proportions Grade 6th and 7th

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Ratios and Proportions Grade 6th and 7th

• CCSSM National Professional Development
• 5/5/12
• Sara Anaya
• Ayse Sahin

Calmeca Academy, CPS
DePaul University
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Plan

• 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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What happens in 8th grade?

• 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.

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6.RP.1 The language of ratio and proportion.

• An example of how to introduce the notion of a
  ratio
• Illustrative Mathematics 6.RP.1 Task example
• Games at recess

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6.RP.2 and 7.RP.1

• 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.

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TASK

• Make one poster for each method used in your
  group and post.

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Table
cups of flour tbsp of butter
3 4
6 8
9 12
12 16
3/2 2
3/4 1
x 2
x 2
x 3
x 3
x 4
x 4
x ½
x ½
x¼
x¼
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Tape diagram representation(6.RP.3a)
1 tbsp of butter
¼ cup of flour
1 cup of flour

1 cup of flour

1 cup of flour

1 tbsp of butter requires ¾ cup of flour
¾ cups of flour

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Double Number Line
0 c
? c
3 c
Cups of flour
Tablespoons of Butter
4 tbsp
0 tbsp
1 tbsp
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• 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.

0 c
? c
3 c
Cups of flour
Tablespoons of Butter
4 tbsp
0 tbsp
1 tbsp
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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
3
4
1
2
3
Cups of flour
Cups of sugar
1
2
3
4
Double number line representations dont work as
well when the units are the same.
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• Now lets 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?

of cups of flour tbsp Of butter
1 4/3
3 4
6 8
9 12
12 16
3/2 2
3/4 1
1 tbsp of butter
3
3
3 c
0 c
1 c

1c flour
x 2
x 2

x 3
1c flour
x 3
0 tbsp
4 tbsp
? tbsp
x 4

x 4
1c flour
x ½
x ½
1/3 tbsp of butter
1 cup of four requires 4/3 cup of butter
x¼
x¼
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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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6.RP.3 and 7.RP.3

• Suppose Abbys orange paint is made by mixing 1
  cup red paint for every 3 cups yellow paint and
  Zacks 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.

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A 7th grade unit ratio problem

• 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.

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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?
• A tape diagram is an excellent tool to remediate
  for students who cant do this computation.

3/8 cup of flour for every ¼ tbsp of butter
? c flour
¾ c flour
0 c flour
0 tbsp butter
¼ tbsp butter
½ tbsp butter
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¼ of a cup of flour
½ tbsp of butter
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 dont identify the whole
  correctly you get the wrong answer of 3/6 vs. 3/9.

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The Standards for Mathematical Practice

• 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.

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Standards for Mathematical Practice

• 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.