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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

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Plan: Overview of the content progression from 6 th to 7 th 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 7 th and 8 th grades. The Standards for Mathematical Practice. Anaya and Şahin IM&E CCSSM National PD

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6 th 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. 7 th 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 6 th 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)

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What happens in 8 th grade? The Ratio and Proportion strand starts in 6 th and ends in 7 th. 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

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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 MathematicsIllustrative Mathematics 6.RP.1 Task example: Games at recess Anaya and Şahin IM&E CCSSM National PD

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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. Anaya and Şahin IM&E CCSSM National PD

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

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Table # cups of flour # tbsp of butter /22 3/41 Anaya and Şahin IM&E CCSSM National PD x 2 x 3 x 4 x ½ x¼ x 2 x 3 x 4 x ½ x¼

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Tape diagram representation (6.RP.3a) Anaya and Şahin IM&E CCSSM National PD 1 cup of flour ¾ cups of flour 1 tbsp of butter requires ¾ cup of flour ¼ cup of flour

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

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

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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?

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Anaya and Şahin IM&E CCSSM National PD 123 Cups Cups of flour Cups of sugar 1234 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? Anaya and Şahin IM&E CCSSM National PD # of cups of flour # tbsp Of butter 14/ /22 3/41 x 2 x 3 x 4 x ½ x¼ ÷ 3 x 3 x 4 x ½ x¼ x 2 1c flour 1 tbsp of butter 1 cup of four requires 4/3 cup of butter 1/3 tbsp of butter 3 c 0 tbsp4 tbsp 0 c ? tbsp 1 c

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The standards for 6 th 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

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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 1 Progressions for the CCSS in Math (draft) Progressions 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

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A 7 th 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. Anaya and Şahin IM&E CCSSM National PD

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

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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

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Standard error: if you dont identify the whole correctly you get the wrong answer of 3/6 vs. 3/9. Anaya and Şahin IM&E CCSSM National PD

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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. Anaya and Şahin IM&E CCSSM National PD

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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. Anaya and Şahin IM&E CCSSM National PD

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