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.

https://slideplayer.com/slide/8986590/

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.

https://www.brightstorm.com/math/algebra/linear-equations-and-their-graphs/finding-the-slope-of-a-line-from-an-equation-problem-2/

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).

https://www.teacherspayteachers.com/Product/Proportional-Relationships-2493738

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).

https://www.texasgateway.org/resource/predicting-effects-changing-slope-problem-situations

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.

https://jay-harold.com/health-problems-caused-by-lead-15-jobs-with-lead-exposure/african-american-grandfather-father-and-son-on-porch-3/

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

https://www.commoncoresheets.com/SortedByGrade.php?Sorted=7rp2b

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.

https://www.pinterest.ca/amp/pin/260434790923519917/

Sequence of Algebra 1 Videos

I have produced a Beta version of a sequence of algebra 1 videos (up to 1 step equations as of Aug 15, 2021 with more on the way).

My approach is to unpack the concepts before showing the steps so the student understand how the math works. The videos are kept shorter, when possible, and they build on each other.

I will eventually revise many if not most videos based on feedback. Also, I will create a practice worksheet for each. For now I am simply trying to get something out there for the start of the school year. Solving equations is an incredibly important math topic to master and I hope these help.

Multiplication Problems as Intro to Unit Rates and Slope (follow up)

I previously related elementary school word problems with math topics in secondary schools. The photo below shows a method to help elementary school students unpack the multiplication problem, to help middle school students identify the unit rate, and to help algebra students identify slope (you can focus on simple problems like this as an entry point to the linear function type problems).

In advance of this method, a review is conducted on the representation of multiplication using the groups of items model (below). By drawing a picture for the two parts of the problem that have a number, the students are guided to break the problem into parts and then to unpack the parts. The “5 boxes of candies” is represented by squares (or circles if you prefer) with no items inside. The “each box holds 6 candies” is represented by a single square with 6 items (dots) inside.

In turn, the drawing of the group of items leads to the multiplication statement, “6 candies x number of boxes.” I prompt students to include the items with the number as sometimes they will write this statement as “6 x number of candies”. I point out that 6 and candies go together. As seen in the previous blog post, the next step in this problem would be to replace “number of boxes” with the quantity given and then compute.

Counting Money at the Store

The way a student counts money in school on a school desk or table (top photo) is the way he or she will attempt count at the register as seen in the 2nd photo in which the student pulled all bills from his wallet then counted, with some bills folded. (Bonus if you can identify the woman in the photo!!!)

In the top photo (below) I had the student pull bills out from his wallet, with the bills unfolded and in order in his wallet (you can see he pulled a $20 bill first). In the next photo you can see that he is counting out the bills from the wallet as he did in practice.

Reading Scatterplots with Ford Mustangs

One step in reading and analyzing scatterplots is simply identifying what the dots on the graph represent. Students who do not understand the meaning of the points, including the position, will struggle to interpret the graph. This post outlines a Jamboard activity to support interpretation of the points.

Overview

I present the scatterplot of used Ford Mustangs on a Jamboard (image above) with ads for two used Mustangs along with a cutout of each car. The cutouts are used to help the students understand the reasoning behind the position of each point. Here is a FB Reel and a YouTube video showing how the Jamboard can be used. To access the Jamboard, you must make a copy. See image at bottom of post.

Steps

First, I take the cutout of the first car and “drive it” along the x-axis (top 3 photos in gallery below). This helps them understand the horizontal axis placement. Then I move the car up to the appropriate price (bottom row left). Finally, I replace the car cutout with the bigger blue dot that was placed by the ad with the car. We then discuss that a dot can be used to represent that car and the location on the scatterplot is based on the two values in the ordered pair (which can be typed into the ( , ) in the Jamboard next to each car.

The same steps are used for the other Mustang (see it “driving” along the x-axis below).

The next step would be to identify additional points on the scatterplot. I then revisit driving the cars and show that driving the car more miles results in a lower price and driving the car less miles results in a higher price.

Finally, we discuss that this is a general trend but that it is not always true for each car. I highlight a couple points where one of the cars has more miles and a higher price (below). This leads into a discussion about additional factors influencing price.

Complementary Activity

An related I have used is having students create their own scatterplot for mileage and price of used cars. They shop on Carmax.com. This allows them experience the scatterplot from a data and context point of view.

Make a copy to access Jamboard.

Introduction to Linear Functions – Buying a Used Car

When our 3rd child was born, we decided to buy a used Honda Odyssey as 3 young kids were not fitting into a sedan. Being the stats geek I am (master’s in statistics at the University of South Carolina – total geek) I collected mileage and price data for all the used Odysseys for sale on dealer sites throughout South Carolina. I then created a the scatterplot shown below. I went to a dealer, showed an agent my graph, and he immediately exclaimed “Where did you get that? We create graphs like that every week!”

It was this experience that led me to the idea of using used car data to introduce linear functions. Shopping for a used car has proven to be a relevant, real life activity the students enjoy.

Here is a link to a comprehensive activity that walks students through various components I use for introducing students to linear function topics.

  • Used car shopping to collect data on 10 used cars of a single make and model.
  • Creating a scatterplot for price vs mileage of the used car of choice.
  • Creating a line of best fit (regression line) to model the data.
  • Creating a linear bi-variate equation (regression equation) to model the data.

The activity is presented on a WORD document (feel free to revise). It shows screenshots to walk student through the Carmax website (subject to Carmax revising their website). The screenshots make it easy for the student to navigate, which increases independence. (NOTE: there is an ample number of Youtube videos on using Google Sheets for this activity.)

The end product looks like this. Note the importance of using 1000s of miles as the slope is more meaningful, -$140.64 per thousand miles, as opposed to 14 cents per mile. I would start with the scatterplot alone to unpack the variables, the relationship between the variables, and the ordered pairs. Then the line and equation can be introduced to show a meaningful use of the line and the equation. The y-intercept has meaning with “0 miles” equating to a new car (I do not explain that new cars have miles already accumulated until we unpack the math).

Monthly Budget Project

The images shown are excerpts from the latest iteration of a budget project I have used for years. The content addressed in this project can be used as stand alone activities and are relevant real life examples for our students. Even the younger students could benefit, e.g., learning addition by shopping for items online and recording the prices (for older students throw in computing tax). These topics are especially useful for multiplication word problems, rate, single variable equations, and linear functions (slope being rate of change such as car payment per month).

Here is an overview. You graduate from high school and are living on your own. You have a job, but your car is getting old. You need to figure out how to save for a down payment in your budget and for when you must pay a car payment and insurance. (You will have to get your OWN insurance.)

The image below shows the table for all monthly expenses.

The students have imbedded activities such as

  • estimating monthly food costs by estimating cost for meals for a single day
  • shopping for disposable household items
  • shopping for car insurance based on the car they shop for (more on that at the end) NOTE: they do not share personal information other than a school email address (or my email address) to receive the quote
  • searching for a job with a hourly pay and estimate after tax income

They shop for a car last as the idea is they need to save up for a down payment. The amount they can save is based on how much money is left over after paying all other bills. How much they save will be converted to how much they can spend on a car payment and monthly insurance payment.

Absolute Value is Absolutely not the Opposite

I have frequently encountered the presentation of absolute value as a positive value or opposite. This is part of the repertoire of memory devices we (certainly I have in the past) use as a short cut to learning how to do the steps for a problem. The meaning of the absolute value of a number is it’s distance from 0 (below).

Below is an image of a Do Now or Initiation handout I use to introduce absolute value. From the start I focus like a laser on the meaning of distance for absolute value. I start with a situation that may be prior knowledge for them. Then take a step towards the mathy part with the numbers and slowly make my way to the symbol.

Graphing Calculator and Miami Vice

The TI-83 appeared only 6 years after Miami Vice but it and the upgrade versions are still suggested or even required in SOME* US colleges (see gallery of math syllabi below). This has implications for math classes in high schools, as seen in many teacher Facebook posts.

*In a previous iteration of this post I wrote “many” and wanted to clarify.

Teachers are faced with a dilemma, do they use Miami vice era technology because the higher institutes of learning may require it or do they avail themselves and their students of user-friendly and effective technology like Desmos, which is FREE!

I suggest using Desmos (or similar technology) to unpack topics and then assigning practice with the TI model of choice, with it used on the tests as well. This will mirror what students will likely see in college.

To make this situation even more disjointed, a commonly used math placement test for colleges does not allow either Desmos or a TI calculator.

Clockwise from top left: syllabi from CCSU (Connecticut), Gordon State, Texas A&M Commerce, THE Ohio State University, University of Kentucky, and University of Oregon.