1. Chapter 18 Proportional Reasoning

2. Big Ideas
A ratio is a multiplicative comparison of two quantities or measures.
Ratios and proportions involve multiplicative rather than additive comparisons.
Rate is a way to represent a ratio.
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.

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

4. Types of Ratios Part-to-part ratios Part-to-whole ratios
Represents a part of a whole
9 females and 7 males in a group, 9/7 meaning a ratio of nine to seven (not a fraction)
Part-to-whole ratios
Comparison of part to a whole
9 girls to 16 students in the group, 9/16 meaning nine-sixteenths of the class (fraction)
Other examples of ratios
Ratios as quotients
4 oranges for $1.00
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

5. Proportional Reasoning
Proportional 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

6. Two Ways to Think About Ratio
Multiplicative Comparison
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)
How many times greater is one thing than another?
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

7. 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

8. Try This One Identify the proportional reasoning you used.

9. Additive and Multiplicative Comparisons
How are these two tasks alike and how are they different?
Decide which has more and share your reasoning.

10. 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

11. Covariation in Geometry, Measurement

12. Covariation and Algebra
Graphs and tables provide a way of thinking about proportions and connect proportional thoughts to algebra.

13. 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.

14. 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

15. Teaching Proportional Reasoning
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.