More about Constant of Proportionality

Student Summary

When something is traveling at a constant speed, there is a proportional relationship between the time it takes and the distance traveled.

The table shows the distance traveled and elapsed time for a bug crawling on a sidewalk.

We can multiply any number in the first column by 23\frac23 to get the corresponding number in the second column. We can say that the elapsed time is proportional to the distance traveled, and the constant of proportionality is 23\frac23. This means that the bug’s pace is 23\frac23 seconds per centimeter.

Table with 2 columns and 4 rows of data. distance traveled (cm) and elapsed time (sec).
Table with 2 columns and 4 rows of data. The columns are: distance traveled (cm) and elapsed time (sec). The table has the ordered pairs (the fraction 3 over 2 comma 1), (1 comma the fraction 2 over 3), (3 comma 2) and (10 comma the fraction 20 over 3). Each pair has an arrow pointing from the value in the first column to the value in the second column. The arrow represents multiplying the first value by the fraction 2 over 3 to calculate the second value.

This table represents the same situation, except the columns are switched.

We can multiply any number in the first column by 32\frac32 to get the corresponding number in the second column. We can say that the distance traveled is proportional to the elapsed time, and the constant of proportionality is 32\frac32. This means that the bug’s speed is 32\frac32 centimeters per second. 

Table with 2 columns and 4 rows of data. elapsed time (sec) and distance traveled (cm). 
Table with 2 columns and 4 rows of data. The columns are: elapsed time (sec) and distance traveled (cm). The table has the ordered pairs (1 comma the fraction 3 over 2), (the fraction 2 over 3 comma 1), (2 comma 3) and (the fraction 20 over 3 comma 10). Each pair has an arrow pointing from the value in the first column to the value in the second column. Times the fraction three over 2 is below the table.

Notice that 32\frac32 is the reciprocal of 23\frac23. When two quantities are in a proportional relationship, there are two constants of proportionality, and they are always reciprocals of each other. When we represent a proportional relationship with a table, we say the quantity in the second column is proportional to the quantity in the first column, and the corresponding constant of proportionality is the number we multiply values in the first column by to get the values in the second.

Visual / Anchor Chart

Standards

Building On
5.MD.1

5.MD.A.1

Addressing
7.RP.2.b

7.RP.A.2.b

7.RP.2.b

7.RP.A.2.b