Question

Four students were discussing how to find the unit rate for a proportional relationship. Which method is valid?
“Look at the graph of the relationship. Find the y-value of the point that corresponds to x = 1. That value is the unit rate.”
“Look at the graph of the relationship. Count the number of units up and the number of units to the right one must move to arrive at the next point on the graph. Write these two numbers as a fraction.”
“Look at the graph of the relationship. Find the x-value of the point that corresponds to y = 2. That value is the unit rate.”
“Look at the graph of the relationship. Find two points which have y-values that are one unit apart. The unit rate is the difference in the corresponding x-values.”

Answer 1

Valid methods: the first and second only.

Reasoning:
- Unit rate for a proportional relationship y = kx is k = y/x.
- Method 1: Find y when x = 1. That gives y(1) = k, so this is valid.
- Method 2: Count the units up (rise) and to the right (run) between two points and take rise/run. That slope = Δy/Δx = k, so this is valid.
- Method 3: Finding x when y = 2 gives x = 2/k, not k, so it’s incorrect.
- Method 4: If two points have y-values one unit apart, their x-values differ by 1/k. The difference in x is the reciprocal of the unit rate, not the unit rate itself, so this is incorrect.