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).

In my experience, many students struggle with translating word problems modeled by a proportional relationship or a linear function into a algebraic expression or equation. (Image below.)

Here is a link to a Jamboard that can be used to introduce the algebraic expression to model the unit rate. (See image below on how to copy and edit.)

Here is how you can use this to introduce modeling the word problem.

Start with the unit rate concept. In this case there is $45 “in” every hour. This is modeled in slide 1 (top 2 photos).

The next 2 photos show slide 2 in which the student duplicates the $45 image and fills 2 hours, with $45 “in” each. They complete the multiplication expression by multiplying by 2.

This is followed by the same steps for 3 hours (photo bottom left) and sequentially to 6 hours.

In the last slide there are no hours shown because the # of hours is unknown. This leads to using “X” to represent the unknown NUMBER of hours (I don’t let students get away with x=hours) and finally the algebraic expression (bottom photo).

An effective instructional strategy is to make the new math topic meaningful. A fellow Facebook group member asked about teaching the topic constant of proportionality. My suggestion is to use hourly wages as an introduction.

I created a handout that starts with students finding a job with an hourly pay stated and then completing a time sheet.

This is followed by unpacking the relationship between hours and pay.

This establishes a context and a situation that many if not most students may find interesting and to connect to the math topic. This handout is intended as an introduction and not the formal unpacking of the term.

A question was posed recently that I find intriguing and important. The question was, “what is the difference between slope, constant of proportionality, unit rate, and rate of change?” I researched the answer to this to evaluate my own understanding of these terms. Here is what I found (and am open to feedback as my understanding evolves).

First, I found a credible website that provides a glossary of math terms, the Connected Math Program page on the Michigan State University website.

In this glossary I found the definitions of the terms in question, along with the term rate.

I then found examples from a Google search that provided more of a visual image of each term.

Note that when we identify slope in an equation, we are identifying the slope of a line from the equation of the line. Slope is a measure of steepness of a line so the number likely should be thought of in that context.

The constant of proportionality can be found in different representations, but it is a constant while the slope is a measure of steepness – a special type of constant. (See definitions from MSU).

A rate, as seen in the definition above, involves units. A rate of change is the change of input and output variables so this is a little different than a simple rate as units may not be cited but could be cited (see last two images below).

The confusion for many of us is that there is overlap between these terms, depending on context. This is similar to the man on the right who can be referred as father or son, depending on the context. Slope is a rate of change for a line. A constant of proportionality is the slope for a linear representation of a proportional relationship.

The constant of proportionality is a constant but can be interpreted in a given context.

Slope is a ratio that can be interpreted in a given context. In the example below, slope = 100/1 and is interpreted as a rate of change of $100 per month.