To find the unit rate in a proportional relationship, we need to identify how much of the output (y-value) corresponds to one unit of the input (x-value). Let's evaluate the methods suggested by the four students:
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“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.”
- This method is valid because if you find the y-value when x = 1 in a proportional relationship, that y-value directly represents the unit rate.
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“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.”
- This method is also valid because counting the rise (change in y) over the run (change in x) between two points gives you the slope of the line, which represents the unit rate for the proportional relationship.
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“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.”
- This method is not valid. The unit rate is defined in terms of the output (y) per one unit of input (x), so finding the x-value when y = 2 does not give us the unit rate.
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“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.”
- This method is not valid. The unit rate should be measured as the output per unit of input, not by taking the difference in x-values when only the y-values are considered.
In conclusion, the first and second methods are valid for finding the unit rate in a proportional relationship.