Activities to Help Kids Understand Meanings of Slope and Y-intercept

I've been introducing linearity to my 9th-graders. I have some introductory linear activities that have always worked very well for me that I'd like to share with you. They work particularly well for regular 8th- and 9th- graders, but you can also adapt them to your struggling students in higher grades.

This first one is a one-day activity, spanning about 60 minutes. Kids go right into it without any teaching, and I only preface it by stating that there are multiple ways of representing changing quantities, and that they would have to reconcile the different representations against each other in order to compare the changing savings amounts throughout the year.





What I like about this first activity is that by the end of it, kids have a pretty intuitive understanding of what slope and y-intercept mean in an equation. They also get a preview of graphical systems of equations and "break-even points", if that is what your curriculum beckons next.

Then, I follow it up with this activity the next day, which again the kids try to do on their own without assistance from me.

For the most part, I observe that kids naturally will want to use either a proportional reasoning or a unit rate to solve the rate comparison problems, which is great! We go over why the unit rate is a direct connection with the idea of slope (once again), and why their process of making the predictions (while taking into account both the rate and the starting value) is very similar to using a linear equation. We also review visual connections with the graphs they made.

At the end, as a quick check in, we do a quick equations Mad Libbs, which takes 5 minutes and reinforces the idea of the meanings of slope and y-intercept. I explain while going over it that slope is "something per something else", and that this is illustrated through some of those Mad Libbs problems.

Again, my pet peeve is that kids often arrive in Algebra 2 without knowing what slope and y-intercept mean in the context of a problem, even if they know how to graph equations and how to write equations. --What is the point of having a bunch of algebraic skills if you don't know what they mean??

As a culmination for my higher-level kids (ie. Algebra 2 or Precalc), I think they should always be able to tell me given a regression equation, what x stands for, what y stands for, what the slope represents, and what the y-intercept represents. It's much harder for kids than you'd think, but I think it's super important for tying everything together.