# Lesson 4

Comparing Proportional Relationships

### Lesson Narrative

In this fourth lesson on proportional relationships, students expand on the work of the previous lesson by comparing two situations that are represented in different ways. For example, one situation might specify a rate of change, while the other is represented by a table of values, a graph, or an equation. Students move flexibly between representations and consider how to find the information they need from each type. They respond to context-related questions that compare the two situations and solve problems with the information they’ve garnered from each representation.

### Learning Goals

Teacher Facing

• Compare the rates of change for two proportional relationships, given multiple representations.
• Interpret multiple representations of a proportional relationship in order to answer questions (in writing), and explain the solution method.
• Present a comparison of two proportional relationships (using words and multiple other representations).

### Student Facing

Let’s compare proportional relationships.

### Student Facing

• I can compare proportional relationships represented in different ways.

### Glossary Entries

• constant of proportionality

In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. This number is called the constant of proportionality.

In this example, the constant of proportionality is 3, because $$2 \boldcdot 3 = 6$$, $$3 \boldcdot 3 = 9$$, and $$5 \boldcdot 3 = 15$$. This means that there are 3 apples for every 1 orange in the fruit salad.

number of oranges number of apples
2 6
3 9
5 15
• rate of change

The rate of change in a linear relationship is the amount $$y$$ changes when $$x$$ increases by 1. The rate of change in a linear relationship is also the slope of its graph.

In this graph, $$y$$ increases by 15 dollars when $$x$$ increases by 1 hour. The rate of change is 15 dollars per hour.