# Proportional Reasoning

## Look at direct variation and proportional reasoning. This investigation will help you to differentiate between relative and absolute meanings of "more" and to compare ratios without using common denominator algorithms. Topics include differentiating between additive and multiplicative processes and their effects on scale and proportionality, and interpreting graphs that represent proportional relationships or direct variation.

### In This Session

Part A: Two Different Meanings of “More”
Part B: The Mixture Blues
Part D: Speeds, Rates, Steepness, and Lines
Homework

In Session 3, we looked at functions and found them to be a relationship between inputs and outputs where there is exactly one output for each input. In this session we’ll look at a special kind of functional relationship: the proportional relationship. We will develop proportional reasoning skills by comparing quantities, looking at the relative ways numbers change, and thinking about proportional relationships in linear functions. Note 1

### Learning Objectives

In this session, we’ll explore direct variation and proportional reasoning. We will:

• Differentiate between relative and absolute meanings of “more” and determine which of these is a proportional relationship
• Compare ratios without using common denominator algorithms
• Differentiate between additive and multiplicative processes and their effects on scale and proportionality
• Interpret graphs that represent proportional relationships or direct variation

### Key Terms

New in This Session:

Absolute Comparison: An absolute comparison is an additive comparison between quantities. In an absolute comparison, 7 out of 10 is considered to be larger than 4 out of 5, since 7 is larger than 4.

Relative Comparison: A relative comparison is a multiplicative, or proportional, comparison between quantities. In a relative comparison, 4 out of 5 is considered to be larger than 7 out of 10, since 4/5 is larger than 7/10.

Proportional Relationship: A proportional relationship is a relationship between inputs to outputs in which the ratio of inputs and outputs is always the same.

Direct Variation: A direct variation is a relationship between inputs and outputs in which the ratio of inputs and outputs is always the same.

Origin: The origin of a coordinate system is the point (0, 0).

### Notes

Note 1

Many people have trouble reasoning proportionally. Typically, when people first begin to think about proportions, they think in absolute, rather than in relative, terms. These different ways of thinking correspond to using additive (how much more is 12 than 10?) rather than multiplicative (what is the ratio of 10 to 20 compared to 12 to 25?) reasoning.

This session introduces the idea that there are different meanings of “more” and distinguishes between relative and absolute comparisons. To familiarize ourselves with the idea of equivalent ratios, we will use both additive and multiplicative methods to explore different ways of making similar figures. We will look at mixture problems and explore ratios without using algorithms to convert them to common denominators. Finally, we will examine characteristics of equations and graphs that represent direct variation.

Materials Needed: Graph paper, rulers, handouts of Quadperson, blank overheads

Review
Groups: Discuss any questions about the homework. If time allows, take a few minutes to try out the number games with a partner. Pairs should show their networks to one another. One partner can choose a (secret) input, run it through the network, and reveal only the output. Then the other partner can use the “undoing” network to find the original number.

Groups: Take a minute and discuss iteration, along with Problems H4 and H5 from Session 3. It is likely to be a new idea.