Transcript Ch 18 PowerPoint - Gordon State College

Chapter 18
Proportional Reasoning
Elementary and Middle School Mathematics, 8th Edition
© 2013 Pearson Education, Inc.
All rights reserved.
Big Ideas
1. 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.
Elementary and Middle School Mathematics, 8th
Edition
Van de Walle, Karp, Bay-Williams
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© 2013 Pearson Education, Inc.
All rights reserved.
Content Connections
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Algebra (Chapter 14)
Fractions (Chapter 15)
Percents (Chapter 17)
Geometry (Chapter 20)
Data Graphs (Chapter 21)
Probability (Chapter 22)
Elementary and Middle School Mathematics, 8th
Edition
Van de Walle, Karp, Bay-Williams
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© 2013 Pearson Education, Inc.
All rights reserved.
Types of Ratios
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Part-to-part ratios
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Represents a part of a
whole
4 oranges for $1.00
9 females and 7 males in a
group, 9/7 meaning a ratio of
nine to seven (not a fraction)
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Part-to-whole ratios
9 girls to 16 students in the
group, 9/16 meaning ninesixteenths of the class
(fraction)
Van de Walle, Karp, Bay-Williams

Ratios as rates
miles 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
Comparison of part to a
whole
Elementary and Middle School Mathematics, 8th
Edition
Ratios as quotients
4
© 2013 Pearson Education, Inc.
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Proportional Reasoning
Proportional Thinkers
 Have a sense of covariation
 Recognize proportional relationships as distinct
from nonproportional relationships in realworld 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
Elementary and Middle School Mathematics, 8th
Edition
Van de Walle, Karp, Bay-Williams
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© 2013 Pearson Education, Inc.
All rights reserved.
Two Ways to Think About Ratio

Multiplicative
Comparison
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Wand 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)
Composed Unit
Thinking 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
How many times greater is
one thing than another?
Elementary and Middle School Mathematics, 8th
Edition
Van de Walle, Karp, Bay-Williams
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© 2013 Pearson Education, Inc.
All rights reserved.
Proportional Reasoning
Reasoning 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
Elementary and Middle School Mathematics, 8th
Edition
Van de Walle, Karp, Bay-Williams
7
© 2013 Pearson Education, Inc.
All rights reserved.
Try This One
Identify the proportional reasoning you used.
Elementary and Middle School Mathematics, 8th
Edition
Van de Walle, Karp, Bay-Williams
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© 2013 Pearson Education, Inc.
All rights reserved.
Additive and Multiplicative Comparisons
How are these two
tasks alike and how
are they different?
Decide which has more
and share your reasoning.
Elementary and Middle School Mathematics, 8th
Edition
Van de Walle, Karp, Bay-Williams
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© 2013 Pearson Education, Inc.
All rights reserved.
Covariation
Within and Between
Ratios
Ratio 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
Elementary and Middle School Mathematics, 8th
Edition
Van de Walle, Karp, Bay-Williams
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© 2013 Pearson Education, Inc.
All rights reserved.
Covariation in Geometry, Measurement
Elementary and Middle School Mathematics, 8th
Edition
Van de Walle, Karp, Bay-Williams
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© 2013 Pearson Education, Inc.
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Covariation and Algebra

Graphs and tables provide a way of
thinking about proportions and connect
proportional thoughts to algebra.
Elementary and Middle School Mathematics, 8th
Edition
Van de Walle, Karp, Bay-Williams
12
© 2013 Pearson Education, Inc.
All rights reserved.
Try This One
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The squares
represent the
recipes for
lemonade used in
each pitcher.
Which pitcher will
have the stronger
lemonade flavor?
Justify your
answer.
Elementary and Middle School Mathematics, 8th
Edition
Van de Walle, Karp, Bay-Williams
13
© 2013 Pearson Education, Inc.
All rights reserved.
Cross-Products
Central to teaching students to reason
proportionally is teach ideas and restrain
the quick path to computation.
Visual of correct
proportional equations
Elementary and Middle School Mathematics, 8th
Edition
Van de Walle, Karp, Bay-Williams
Line segments can model
rate and distance
14
© 2013 Pearson Education, Inc.
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Teaching Proportional Reasoning
1.
2.
3.
4.
5.
Use composed unit and multiplicative comparison
ideas.
Help students distinguish between proportional and
nonproportional comparison.
Provide ratio and proportion tasks in a wide range of
contexts.
Engage students in a variety of strategies for solving
proportions.
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.
Elementary and Middle School Mathematics, 8th
Edition
Van de Walle, Karp, Bay-Williams
15
© 2013 Pearson Education, Inc.
All rights reserved.