CCSSM National Professional Development Ratios and Proportions Sara Anaya Ayşe Şahin Diana White Grades 6/7

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 c. See % as a rate d. 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. 6 th graders are not asked to express proportional relationships algebraically 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)

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: Illustrative Mathematics Games at recess Anaya, Şahin, White 3

6.RP.2 and 7.RP.1 This recipe has a ratio of 3 cups of flour to 4 cups of sugar. How many cups of flour would you use for 1 cup of sugar? – Answer this question using as many different methods as possible. Anaya, Şahin, White 4

Tape Diagram Representation Author's Name IM&E CCSSM National PD 1 cup of flour ¾ cups of sugar

Table # cups of flour# cups of sugar /22 3/41 Author's Name IM&E CCSSM National PD x 2 x 3 x 4 x ½ x¼ x 2 x 3 x 4 x ½ x¼

Author's Name IM&E CCSSM National PD 123 Cups Cups of flour Cups of sugar 1234 Double number line representations don’t work as well when the units are the same.

Changing the problem slightly, a double number line becomes an appropriate representation: This recipe calls for 3 tablespoons of butter for every 4 cups of sugar. How many tablespoons of butter would you use for 1 cup of sugar? Author's Name IM&E CCSSM National PD 3 tbs Tablespoons Cups 0 c 4 c 0 tbs

Using double number lines we directly translate the problem into asking: what is 3 /4? (Standard 5.NF.3) ¾ tbs of butter for every cup of sugar. Author's Name IM&E CCSSM National PD 3 tbs 4 c Tablespoons Cups 0 c 0 tbs 1 c ? tbs

This recipe has a ratio of 3 cups of flour to 4 cups of sugar. – How many cups of sugar would you use for 1 cup of flour? Solve this problem using a method you did not use for the previous question. Anaya, Şahin, White 10

7th Grade Unit Rate Problem This recipe calls for ¾ cup of flour for every ½ cup of sugar. How many cups of flour would you use for every 1 cup of sugar? Answer this question using as many different methods as possible. Anaya, Şahin, White 11

This recipe calls for ¾ cup of flour for every ½ cup of sugar. How many cups of flour would you use for every 1/4 cup of sugar? Answer this question using as many different methods as possible. Anaya, Şahin, White 12

6.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? Anaya, Şahin, White 13

7 th Grade Extension Suppose Abby makes orange paint by mixing 1 cup red paint for every 3 cups yellow paint. She can only find 100 cups of the red paint, and as much of the yellow paint as she likes. How many cups of orange paint can she make? Answer this question using as many different methods as possible. Anaya, Şahin, White 14