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.

Slippery Slope – 3rd Grade Multiplication Word Problems to Slope in Algebra

Top left is a scaffolding I use to help students learn to solve math problems using multiplication (3rd grade). The situations are typically rate problems (e.g., 5 pumpkins per plant or $2 per slice of pizza) although the term “rate” is not used yet. The same concept of rate plays out in high school with slope of a line, applied to real life situations (top right).

These types of problems start in 3rd grade (below, top left), play out in 6th grade (below, top right), into 8th grade (below, middle), and into high school algebra and statistics (below, bottom). I referenced this connection previously regarding word problems and dominoes. This highlights how crucial it is that strategically selected gaps in a student’s math education be addressed in context of long range planning.

Introduction to Slope

which-jobSlope 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).  This includes presenting the topic using multiple representations: the original real life situation, rates (see photo above) and tables, visuals,  and hands on cutouts (see photos below).

10-dollars-per-hour-graphA 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 –  see photo above and below.5-dollars-per-hour-graph

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.cut-outs-hours-bills

Interpreting Slope Intercept

Slope is one of the most important topics covered in high school algebra yet it is one of the least understood concepts. I have two observations about this. First, slope is often introduced with the formula and not as a rate of change. Second, students intuitively understand slope as rate of change conceptually when presented in a relevant, real life context. The challenge is compounded when slope is presented with the y-intercept.slope intercept scaffolding

In the photo I present slope and y-intercept in a context students can understand (money is their most intuitive prior knowledge). The highlighting makes it easier for them to see the context, specifically the variables. I have the students work on this handout and I circulate and ask questions.

Here’s a typical exchange – working through problems 11, 12:

  • Me: “Look at the table, what’s changing?”
  • Student: “the cost”
  • Me: “How much is it changing?”
  • Student: “20”
  • Me: “20 what?”
  • Student: “20 cost”
  • Me: “What are you counting when you talk about cost?”
  • Student: “money…dollars”
  • Me: “So the price is going up 20 what?”
  • Student: “Dollars”
  • Me: Show me this on the yellow” (student knows from before  to write +$20)
  • Me: “What else is changing?”
  • Student: “People”
  • Me: “By how much”
  • Student: “1 people…person”
  • Me: “write that on the green”
  • Me: “Now do this same thing on the graph. Where do you start?” (they put their pencil on the y-intercept
  • Me: “What do you do next?” (they typically know to move over and up)
  • Me: “Use green to highlight the over” (they highlight)
  • Me: “How much did you go over?”
  • Student: “1…1 person”
  • Me: “Now what?” (Student goes up.)
  • Me: “Highlight that in yellow.” (They highlight.)
  • Me: “How much did it go up?”
  • Student: “2…20…20 dollars”
  • Me: “What is a rate?” (I make them look at their notes until they say something about divide or fraction or point to a rate)
  • Me: “So what is the rate of change?”
  • Student: “$20 and 1 person”
  • Me: “Look at the problem at the top. What is the 20?”
  • Student: “$20 per person.”

I point out that you can find this rate or slope in the equation, the table and in the graph.

Intro to Slope as Rate of Change

Slope may be the most challenging concept to teach in algebra yet it is one of the most important concepts. I use the following sequence to introduce slope: rate of change, rise over run, rise over run as a rate of change. The first photo is a map of Manhattan with directions on counting city blocks. This builds on prior knowledge to introduce rise and run.

Manhattan map for rise and run

The photo below builds on the map and transitions students into coordinate planes. They are introduced to rise over run and positive and negative as indicators of the direction of a line.

rise over run on grid

The photo below combines rise and run with rate of change. The hourly wage is prior knowledge they can much more easily comprehend. A major issue is getting students to include units and to understand what units are. This would have been addressed in the previous unit on rates and proportions.

rise over run as rate of change

This is the handout used with a train activity in which I use battery operated trains and time them as they travel 200″. I project a stop watch on the screen as the train moves. The kids pick up right away that Percy is “slow.” As Percy is traveling I ask them how they know it is slow and get answers like “it takes a long time.” This is a concrete representation which they can draw upon as they work with the graph and calculations.

train activity

The photo below shows a scaffolded version of a Smarter Balance (Common Core assessment) test question. The original question simply shows the graph and asks for average rate of change from 0 to 20 years. Even with the scaffolding many problem areas appear: units vs variable (student wrote “value” as opposed to $), including $ with the 1000, finding unit rate, and even identifying the part of the graph at 0 years.

rate of change Smarter Balance problem scaffolded

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