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Ratios and Proportions Ratios and Proportions Grade 6 Grade 6 th th and 7 and 7 th th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul Universit y
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Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

Mar 26, 2015

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Page 1: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

Ratios and Proportions Ratios and Proportions Grade 6Grade 6thth and 7 and 7thth

CCSSM National Professional Development

5/5/12Sara AnayaAyşe Şahin

Calmeca Academy, CPS

DePaul Universit

y

Page 2: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

Plan: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.

Anaya and Şahin IM&E CCSSM National PD

Page 3: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

Anaya and Şahin IM&E CCSSM National PD

Page 4: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

What happens in 8What happens in 8thth grade? 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.

Anaya and Şahin IM&E CCSSM National PD

Page 5: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

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

An example of how to introduce the notion of a ratio:

Illustrative Mathematics 6.RP.1 Task example:Games at recess

Anaya and Şahin IM&E CCSSM National PD

Page 6: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

6.RP.2 and 7.RP.16.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.

Anaya and Şahin IM&E CCSSM National PD

Page 7: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

TASKTASK

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

Anaya and Şahin IM&E CCSSM National PD

Page 8: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

TableTable

# cups of flour

# tbsp of butter

3 4

6 8

9 12

12 16

3/2 2

3/4 1

Anaya and Şahin IM&E CCSSM National PD

x 2

x 3

x 4

x ½

x 2

x 3

x 4

x ½

Page 9: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

Tape diagram representationTape diagram representation(6.RP.3a)(6.RP.3a)

Anaya and Şahin IM&E CCSSM National PD

1 cup of flour

1 cup of flour

1 cup of flour

¾ cups of flour

1 tbsp of butter

1 tbsp of butterrequires ¾ cup of flour

¼ cup of flour

Page 10: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

Double Number LineDouble Number LineAnaya and Şahin IM&E CCSSM National PD

3 cCups of flour

Tablespoons ofButter

0 tbsp 4 tbsp

0 c

1 tbsp

? c

Page 11: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

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.

Anaya and Şahin IM&E CCSSM National PD

3 cCups of flour

Tablespoons ofButter

0 tbsp 4 tbsp

0 c

1 tbsp

? c

Page 12: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

Anaya and Şahin IM&E CCSSM National PD

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?

Page 13: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

Anaya and Şahin IM&E CCSSM National PD

1 2 3Cups

Cups

1 2 3 4

1 2 3Cups of flour

Cups of sugar

1 2 3 4

Double number line representations don’t work as well when the units are the same.

Page 14: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

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?

Anaya and Şahin IM&E CCSSM National PD

# of cups of flour

# tbspOf

butter

1 4/3

3 4

6 8

9 12

12 16

3/2 2

3/4 1

x 2

x 3

x 4

x ½

÷ 3 ÷ 3

x 3

x 4

x ½

x 21c flour

1c flour

1c flour

1 tbsp of butter

1 cup of four requires4/3 cup of butter

1/3 tbsp of butter

3 c

0 tbsp 4 tbsp

0 c

? tbsp

1 c

3

3

Page 15: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

The standards for 6th grade ask that students learn these strategies for understanding ratio and proportion to solve real world mathematical problems.

Anaya and Şahin IM&E CCSSM National PD

Page 16: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

6.RP.3 and 7.RP.36.RP.3 and 7.RP.3

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?11Progressions for the CCSS in Math (draft)

Solve this problem using at least three of the tools we have discussed so far.

Anaya and Şahin IM&E CCSSM National PD

Page 17: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

A 7A 7thth grade unit ratio grade unit ratio problemproblem

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.

Anaya and Şahin IM&E CCSSM National PD

Page 18: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

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 can’t do this computation. Anaya and Şahin IM&E CCSSM

National PD

? c flour

0 tbsp butter

3/8 cup of flour for every ¼ tbsp ofbutter

¼ tbsp butter

¾ c flour

2

½ tbsp butter

2

0 c flour

Page 19: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

Anaya and Şahin IM&E CCSSM National PD

1cup of flour

½ tbsp of butter

3/8 of a cup of flour for every ¼ of a tbsp of butter

¼ of a cup of flour

Page 20: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

Standard error: if you don’t identify the whole correctly you get the wrong answer of 3/6 vs. 3/9.

Anaya and Şahin IM&E CCSSM National PD

Page 21: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

The Standards for Mathematical The Standards for Mathematical PracticePractice

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.

Anaya and Şahin IM&E CCSSM National PD

Page 22: Ratios and Proportions Grade 6 th and 7 th CCSSM National Professional Development 5/5/12 Sara Anaya Ayşe Şahin Calmeca Academy, CPS DePaul University.

Standards for Mathematical Standards for Mathematical Practice 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.

Anaya and Şahin IM&E CCSSM National PD