INTERPRETING THE UNIT RATE AS SLOPE

A rate is a comparison of two quantities that have different units, such as miles and hours. A unit rate is a rate in which the second quantity in the comparison is one unit.

Example 1 :

A storm is raging on Misty Mountain. The graph shows the constant rate of change of the snow level on the mountain.

A.  Find the slope of the graph using the points (1, 2) and (5, 10). Remember that the slope is the constant rate of change.

Change in y-value/Change in x-value :

=  (10 - 2)/(5 - 1)

=  8/4

=  2

B. Find the unit rate of snowfall in inches per hour. Explain your method.

2 inches per hour; Sample answer : The point (1, 2) is on the line, and represents 2 inches snowfall in 1 hour.

C.  Compare the slope of the graph and the unit rate of change in the snow level. What do you notice ?

They are the same.

D.  Which unique point on this graph can represent the slope of the graph and the unit rate of change in the snow level ? Explain how you found the point.

(1, 2) ; Sample answer : the unit rate is the amount of snow in 1 hour. So I found the point with an x-coordinate of 1. That point is (1, 2), which, along with another point on the graph, gives 2 as the slope.

Example 2 :

The equation y = 2.75x represents the rate, in barrels per hour, that oil is pumped from Well A. The graph represents the rate that oil is pumped from Well B. Which well pumped oil at a faster rate ?

Solution :

Step 1 :

Use the equation y = 2.75x to make a table for Well A’s pumping rate, in barrels per hour.

Step 2 :

Use the table to find the slope of the graph of Well A.

Slope  =  Unit rate 

=  (5.5 - 2.75)/(2 - 1) 

=  2.75/1

=  2.75 barrels/hour.

Step 3 :

Use the graph to find the slope of the graph of Well B.

Slope  =  Unit rate 

=  rise/run 

=  10/4

=  2.5 barrels/hour.

Step 4 :

Compare the unit rates.

2.75 > 2.5

So Well A’s rate, 2.75 barrels/hour, is faster.

Reflect :

Describe the relationships among the slope of the graph of Well A’s rate, the equation representing Well A’s rate, and the constant of proportionality

The slope and the constant of proportionality equal the value 2.75 in the equation y = 2.75x.

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