proportional reasoning and strip diagrams jessica cohen western washington university presented at...
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Proportional Reasoning and Strip Diagrams
Jessica CohenWestern Washington UniversityPresented at NWMC, Oct 2014
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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.
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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
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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.
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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.
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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
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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
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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
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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
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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?
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• 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.
100
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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?
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FishChicken
• The four parts represent 160 guests, so each part must represent 40 guests. 40 guests chose fish and 120 guests chose chicken.
160
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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?
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Gus Ike
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Gus Ike
15
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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?
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250
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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?
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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
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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.
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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.
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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
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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