Elementary and Middle School Mathematics, 8 th Edition © 2013 Pearson Education, Inc. All rights reserved. Chapter 18 Proportional Reasoning

Elementary and Middle School Mathematics, 8th Edition

© 2013 Pearson Education, Inc.All rights reserved.

Chapter 18Proportional Reasoning

Elementary and Middle School Mathematics, 8th EditionVan de Walle, Karp, Bay-Williams

© 2013 Pearson Education, Inc.All rights reserved.2

Big Ideas1. A ratio is a multiplicative comparison of

two quantities or measures.2. Ratios and proportions involve

multiplicative rather than additive comparisons.

3. Rate is a way to represent a ratio.4. Proportional thinking is developed by

comparing and determining the equivalence of ratios and solving proportions in problem-based contexts and situations without recourse to rules or formulas.



Content Connections Algebra (Chapter 14) Fractions (Chapter 15) Percents (Chapter 17) Geometry (Chapter 20) Data Graphs (Chapter 21) Probability (Chapter 22)



Types of Ratios Part-to-part ratios

Represents a part of a whole9 females and 7 males in a group, 9/7 meaning a ratio of nine to seven (not a fraction)

Part-to-whole ratiosComparison of part to a whole9 girls to 16 students in the group, 9/16 meaning nine-sixteenths of the class (fraction)

Part-to-whole ratios Other examples of ratios

Ratios as quotients4 oranges for $1.00

Ratios as ratesmiles per gallon, square yards—different units and how they relate to each other

Ratios are closely related to fractions with the use of the fraction bar



Proportional ReasoningProportional Thinkers Have a sense of covariation Recognize proportional relationships as

distinct from nonproportional relationships in real-world contexts

Develop a wide variety of strategies for solving proportions or comparing ratios

Understand ratios as distinct entities representing a relationship different from the quantities they compare



Two Ways to Think About Ratio Multiplicative

ComparisonWand A is 8 inches and Wand B is 10 inches

Two ways to compare:A is eight-tenths as long as B (or four-fifths the length)B is ten-eighths as long as A (or five-fourths)

How many times greater is one thing than another?

Composed UnitThinking of the ratio as a unit:

4 oranges for $1.00 or 8 for $2.00, 16 for $4.00

2 oranges for $0.50 1 oranges for $0.25



Proportional ReasoningReasoning proportionally starts early with one-to-

one correspondence, place value, fraction concepts, and multiplicative reasoning before middle school.

Proportional Thinkers Understand ratios as distinct entities Recognize proportional relationships as distinct

from nonproportional relationships Sense of covariation Develop a wide variety of strategies



Try This One Identify the proportional reasoning you used.



Additive and Multiplicative Comparisons

Decide which has more and share your reasoning.

How are these two tasks alike and how are they different?



CovariationWithin and Between RatiosRatio of oranges to money is a within ratio

Ratio of the original number of oranges (4 to $1.00) to the number of oranges (16 to $4.00) in a second situation is a between ratio



Covariation in Geometry, Measurement



Covariation and Algebra Graphs and tables provide a way of

thinking about proportions and connect proportional thoughts to algebra.



Try This One The squares

represent the recipes for lemonade used in each pitcher.

Which pitcher will have the stronger lemonade flavor?

Justify your answer.



Cross-Products

Central to teaching students to reason proportionally is teach ideas and restrain the quick path to computation.

Visual of correct proportional equations Line segments can

model rate and distance



Teaching Proportional Reasoning1. Use composed unit and multiplicative

comparison ideas.2. Help students distinguish between proportional

and nonproportional comparison.3. Provide ratio and proportion tasks in a wide

range of contexts.4. Engage students in a variety of strategies for

solving proportions.5. Recognize that symbolic or mechanical

methods do not develop proportional reasoning and should not be introduced until students have had many experiences with intuitive and conceptual methods.

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Elementary and Middle School Mathematics, 8 th Edition © 2013 Pearson Education, Inc. All rights reserved. Chapter 18 Proportional Reasoning