Ratios and Proportions / Connecting Ratios to Equations

Activity 10 of 15
Students generate slope triangles from the graph of a collection of equivalent ratios, thus leading to the notion of a proportional relationship. They use parallel grids to compare the tables, graphs, and equations for different unit rates.

Planning and Resources

Objectives
Students should understand and identify the constant of proportionality, k, as the unit rate or the slope of a line through the origin. They can connect ratios to equations of a line through the origin: a:b → y= $\frac{b}{a}$ x

Vocabulary
proportional relationship
constant of proportionality
slope triangle

Standard:

Lesson Snapshot

Understanding

A proportional relationship is formally associated with an equation, and the slope of the line representing the proportional relationship is associated with the notion of unit rate.

What to look for

This lesson focuses on the fact that a proportional relationship can be described by an equation of the form y = kx, where k is a positive constant, often called a constant of proportionality. The rate of change in y for a given unit in x will be constant for the points on the line.

Sample Assessment

Order the equations below in terms of steepest slope to least steep.
a. y = $\frac{2}{3}$ x
b. y = 2x
c. y = $\frac{1}{3}$ x
d. y = x

Answer: b, d, a, c

The Big Idea

A proportional relationship can be described by an equation of the form y = kx, where k is a positive constant, often called a constant of proportionality.

What are the students doing?

Students compare the slopes of two lines and make conjectures about the relationship between the steepness of the slopes and the value of value of the slope.

What is the teacher doing?

Have students explain how they can use the equation to predict points on the line before graphing them. Encourage them to use the TNS lesson to demonstrate their reasoning.