Proportional Relationship (with k) vs Linear Function (with m)

There have been interesting discussions on various Facebook Teacher pages about proportional relationships vs linear functions. This mirrors discussion about the constant of proportionality vs slope vs unit rates.

The difference between the proportional relationships and slope is context and the ratios. The ratio for the former addresses the variables themselves. The ratio for the slope addresses the change in the variables. This arises from context. To flesh this out let’s use the pay as a function of hours, with $15 an hour for the hourly rate.

If we focus on the fact that to compute the total pay, we multiply by the hours worked by 15 we have a proportional relationship and the $15 an hour is a constant of proportionality.

If we focus on the fact that every increase of 1 hour results in an increase of $15 in our total pay, we have a linear function and the $15 an hour is a rate of change (or slope of the line). Because of the context, we have different constants (k vs m).