Ratios and Proportions Grade 6 th and 7 th

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

2. 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. Plan: Anaya and Şahin IM&E CCSSM National PD

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4. 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? Anaya and Şahin IM&E CCSSM National PD

5. 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. Anaya and Şahin IM&E CCSSM National PD

6. 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 Anaya and Şahin IM&E CCSSM National PD

7. Make one poster for each method used in your group and post. TASK Anaya and Şahin IM&E CCSSM National PD

8. x 2 x 2 x 3 x 3 x 4 x 4 x ½ x ½ x¼ x¼ Table Anaya and Şahin IM&E CCSSM National PD

9. 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 Tape diagram representation(6.RP.3a) Anaya and Şahin IM&E CCSSM National PD

10. 0 c ? c 3 c Cups of flour Tablespoons of Butter 4 tbsp 0 tbsp 1 tbsp Double Number Line Anaya and Şahin IM&E CCSSM National PD

11. 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 Anaya and Şahin IM&E CCSSM National PD

12. 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? Anaya and Şahin IM&E CCSSM National PD

13. 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 don’t work as well when the units are the same. Anaya and Şahin IM&E CCSSM National PD

14. 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? 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 ? tbsp 4 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 Anaya and Şahin IM&E CCSSM National PD

15. 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

16. 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 Anaya and Şahin IM&E CCSSM National PD

17. 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 Anaya and Şahin IM&E CCSSM National PD

18. 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. 3/8 cup of flour for every ¼ tbsp of butter 0 c flour ? c flour ¾ c flour 0 tbsp butter ¼ tbsp butter ½ tbsp butter Anaya and Şahin IM&E CCSSM National PD

19. 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 Anaya and Şahin IM&E CCSSM National PD

20. 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

21. 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 Anaya and Şahin IM&E CCSSM National PD

22. 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 Anaya and Şahin IM&E CCSSM National PD