Proportional Reasoning and Strip Diagrams Jessica Cohen Western Washington University Presented at NWMC, Oct 2014 Jessica.Cohen@wwu.edu

What is proportional reasoning? Maybe a better question: what are some characteristics of proportional thinkers? Sense of covariation Recognize proportional relationships as distinct from nonproportional relationships Develop a variety of strategies for solving proportions, many of which are nonalgorithmic Understand ratios as representations of a relationship, separate from the quantities compared.

Why is proportional reasoning important? The cornerstone of higher mathematics Similarity Linear relationships Dilations Scaling Slope Rates Percent Trig ratios Probability Inverse and direct relationships

CCSS and proportional reasoning 2.OA.4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. 4.OA.2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. 5.MD.1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

CCSS Grade 6 6.RP.1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities 6.RP.2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship 6.RP.3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

CCSS Grade 7 7.RP.1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units 7.RP.2. Recognize and represent proportional relationships between quantities 7.RP.3. Use proportional relationships to solve multistep ratio and percent problems

What does research tell us? More than half of the adult population are not proportional thinkers (Lamon, 1999) Focusing on reasoning, instead of a formula, can improve student ability to reason proportionally (Lamon, 1999) There are some factors associated with helping students develop proportional thinking

Helping students reason proportionally Provide ratio and proportion problems in a variety of contexts Encourage discussion and experimentation in predicting and comparing ratios Help children relate proportional reasoning to existing processes Recognize that symbolic or mechanical methods for solving proportions do not foster reasoning

Strip Diagrams In a terrarium, the ratio of grasshoppers to crickets is 6:5. There are 48 grasshoppers. How many crickets are there? Grasshoppers: Crickets: We distribute the grasshoppers evenly among the squares on top, so we have 8 grasshoppers in each square (48 ÷ 6 = 8). Then each of the “cricket” squares also has to represent 8, so there are 5 x 8 = 40 crickets

Solve a problem Suppose you are mixing blue paint and yellow paint in a 2:3 ratio to make green paint. How many pails of each color would you need to make 100 pails of green paint? Use a strip diagram to solve Solve with cross-multiplication Compare your two solution methods. What does your use of the strip diagram tell you about the algorithm?

The 5 parts of paint together make 100 pails, so each part represents 20 pails. Then you need 40 pails of blue paint and 60 pails of yellow paint. 𝑥 2 = 100 5 →5x=200 →x=40 100

Try another one: At a wedding, guests had a choice between fish and chicken. Three times as many guests chose chicken as fish. If 160 guests attended the wedding, how many chose chicken and how many chose fish? Solve this problem in any way Solve using a strip diagram Compare your solution strategies. In what way does each strategy help build proportional thinking?

Fish Chicken The four parts represent 160 guests, so each part must represent 40 guests. 40 guests chose fish and 120 guests chose chicken. 160

And another one: Gus and Ike are playing cards. The ratio of Gus’s cards to Ike’s is 5 to 3. After Gus gives Ike 15 cards, they each have the same number of cards. How many do they have now? How many did each have to start? How would you solve with cross multiplication? Solve using a strip diagram. Why is this a proportion problem?

Gus Ike

Gus Ike 15

Connecting to Fractions: A small aquarium holds 2/3 as much water as a large aquarium. If the two aquariums hold 250 gallons together, how much does each aquarium hold? Solve using a strip diagram. Could you solve this with cross multiplication? Why is this a proportion problem? How is this different from the previous problems?

250

Connecting to Percents: Ike got 30 questions right on a test and scored 40%. Assuming each question is worth the same number of points, how many questions were on the test? Solve in any way Use a strip diagram to solve How are percents and proportions related?

4 parts represent 30 questions, so every 2 parts represents 15 questions. Then 10 parts represents 5*15 = 75 questions, meaning there were 75 total questions on the test. 30

Connecting to Cross Multiplication One of the strongest criticisms of the cross-multiplication algorithm is that it depends on students making equivalent fractions, yet there is often no sense-making instilled to help students ensure the fractions used are equivalent. How can strip diagrams be used to properly set up a cross multiplication? Choose one of the problems we have solved today and use it to show how cross-multiplication works.

Strip Diagrams and Fraction Division If it takes 4/5 of a cup of flour to make 2/3 of a batch of cookies, how much flour do you need for a full batch of cookies? 2/3 batch means we have two out of the three parts that we need for a full batch. Use this to solve this problem using a strip diagram Use your work with the strip diagram to explain why the division algorithm for fractions works.

2 parts represent 4/5 cup, so each part must represent 2/5 cup. The full recipe requires 3 parts, or 6/5 cup of flour. 4/5 cup

Sources Lamon, S. J. (1999) Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. Mahwah, NJ: Lawrence Erlbaum Van de Walle, J. A., Karp, K.S., and Bay-Williams, J. M. (2010) Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Allyn & Bacon

Questions/comments Jessica.cohen@wwu.edu