Ratios and Proportions / Proportional Relationships

Understanding

Ordered pairs that satisfy equations of the form y = ax + b with b ≠ 0 will not lie on a line through the origin and so will not belong to a collection of equivalent ratios.

What to look for

To find a line representing a particular proportional relationship, students might first identify two points in the associated ratio and move the points on the screen to these points using the horizontal and vertical arrows.

Sample Assessment

Cost in dollars, c, and amount of fruit in pounds, p, are related by the equation c = 2.5p.

Which of the following is true about the proportional relationship?
a. The cost increases by $1 for every increase of 2.5 pounds.
b. The cost increases $2 for every half-pound increase in weight.
c. The cost increases by $2.50 for every 1-pound increase in weight.
d. There is not enough information to determine how the cost will increase.

Answer: c) The cost increases by $2.50 for every 1-pound increase in weight.

The Big Idea

A proportional relationship (y = kx with k > 0) is often referred to as direct variation; the variable y varies directly with the variable x. Or as x increases, so does y at a constant rate.

What are the students doing?

Students write an equation to describe a proportional relationship and use unit rate - or constant of proportionality - in the equation to generate ordered pairs along the line.

What is the teacher doing?

As students compare graphs, remind them to think about the role of the independent and dependent variable for a given context.