## Slope, Constant of Proportionality, Unit Rate, Rate, Rate of Change

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.

## Introduction to Slope

Slope is one of the the most important topics in algebra and is often understood by students at a superficial level. I suggest introducing slope first by drawing upon prior knowledge and making the concept relevant (see photo above). Â This includes presenting the topic using multiple representations: the original real life situation, rates (see photo above) and tables, visuals, Â and hands on cutouts (see photos below).

A key aspect of slope is that it represents a relationship between 2 variables. Color coding (red for hours, green for pay) can be used to highlight the 2 variables and how they interact – Â see photo above and below.

The photo below can be used either in initial instruction, especially for co-taught classes, or as an intervention for students who needs a more concrete representation of a rate (CRA). The clocks (representing hours) and bills can be left in the table for or cut out.

## Starbucks would benefit from some math intervention

Starbucks has changed their rewards program to “boost participation.” The chief strategy officer apparently could use some math intervention.

From USA Today (emphasis mine):

“Currently, customers with “gold” status have to visit 12 times to earn a free food or drink item. Under the new program, those customers will have to earn 125 stars to get a free reward.

Most customers spend around \$5 with each visit, said Matt Ryan, global chief strategy officer. At that rate, a customer would still need to visit about 12.5 times, or spend a little more than \$60, before earning a free reward.”

In other words, “most customers” will need to visit more often to get the same reward benefit.