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

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

6th Grade 6.RP.1 - Definition of ratio 6.RP.2 - Unit rate(s) associated to ratio Whole numbers 6.RP.3 –Use ratios and rate reasoning to solve (single step)real world mathematical problems using the representations above and: . Using the unit rate . Seeing % as a rate . Using ratio reasoning and manipulating units by multiplying and dividing (6.RP.3) Given a proportional relationship represent it a variety of ways: . A table . Plotting points on a coordinate graph. . Tape diagrams . Double number line diagrams. The Standards requires that the strategies listed here are not replaced by solving equations. 7th Grade 7.RP.1 – Unit Rates associated with ratios (with fractions and decimals) 7.RP.3 Use proportional relationships to solve (multi-step) ratios and percent problems. 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 7.RP.2 c. Represent proportional relationships by equations. d. Understanding the proportional relationship on a graph most importantly (0,0) and (1.r) Tape diagrams are also known as strip diagrams or fraction bars. Make sure you mention this so teachers can recognize the tool from their own contexts. Point out that the different size boxes are not meant to imply more or less important, just how the text fits. Anaya and Şahin IM&E CCSSM National PD

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. What happens in 8th grade? Anaya and Şahin IM&E CCSSM National PD

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 6.RP.1: The language of ratio and proportion. Anaya and Şahin IM&E CCSSM National PD

This recipe has a ratio of 3 cups of flour to 4 tablespoons of butter 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

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

Table # cups of flour # tbsp of butter 3 4 6 8 9 12 16 3/2 2 3/4 1 x 2 Anaya and Şahin IM&E CCSSM National PD

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 Anaya and Şahin IM&E CCSSM National PD

Double Number Line 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

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

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

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

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? # of cups of flour # tbsp Of butter 1 4/3 3 4 6 8 9 12 16 3/2 2 3/4 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 Facilitator notes: “unit rate is ¾” is never correct, must state ¾ of what per 1 of what.: connect to mathematical practices standard. (precision, contextualizing). Use this as a segue to ask what other mp standards they used as they solved and in discussion? 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

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

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. Change the content of the problem based on student’s, Milk and chocolate – ELL’s or color blind students. Suppose Abby makes hot chocolate by mixing 1 cup of milk for every 3 tablespoons of chocolate syrup and Zack makes his hot chocolate by mixing 3 cups of milk red for every 5 tablespoons of chocolate syrup. Whose drink is more chocolaty? 6.RP.3 and 7.RP.3 Anaya and Şahin IM&E CCSSM National PD

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. Facilitator notes: Should discuss what the differences are in the two problems, how has this become more complicated? For Sara and Ayse: when we make the solution slides for this we should make sure and include the reference for the standard which requires facility with complex fractions (for some states it’s coming early, check Illinois standards) A 7th grade unit ratio problem Anaya and Şahin IM&E CCSSM National PD

3/8 cup of flour for every ¼ tbsp of 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

3/8 of a cup of flour for every ¼ of a tbsp of butter Anaya and Şahin IM&E CCSSM National PD

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

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. The Standards for Mathematical Practice Anaya and Şahin IM&E CCSSM National PD

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. Standards for Mathematical Practice Anaya and Şahin IM&E CCSSM National PD