Linear equations are a math concept that spans multiple grade levels. Students start seeing graphs and talking about proportional relationships in 7th grade math, according to Common Core State Standards. Then as they move on to 8th grade, students begin talking about slope, y-intercept, linear equations, and comparing linear functions. As they progress to high school and Algebra 1, it becomes about interchanging the forms of linear functions and systems of equations.
As an 8th grade math teacher, most of the year is spent looking at a graph. Whether it is about linear equations, functions, geometric transformations, or something else, graphs are vital.
One thing to always remember is to start small. As someone who teaches at a Title 1 school, I see a wide level of math abilities. I’m talking about students who come into 8th grade with as low as a 2nd grade math level all the way to on-grade level abilities. This proves a challenge in many ways, one being how to engage all of these students and help them learn 8th grade material. This is how I teach to reach all of these students.
Students learn about proportional relationships in 7th grade. However, don’t assume they remember or understood it. I start off the section working solely with proportional relationships. This allows for students who remember proportional relationships to start with something they know. Or for those who didn’t retain or didn’t understand the concept, it allows for an easier transition into full-blown 4-quadrant linear equations.
I only use 1-quadrant graphs for the first portion of teaching linear equations. It drives focus and isn’t as overwhelming as a 4-quadrant graph.
First thing I teach is lattice points. Lattice points are the crossing points on the graph. I use this term because it helps students visualize and have afor where the linear graph crosses through the grid lines of the graph. And it assists in the process for rise and run.
Ina classroom setting, it can be easier to point out lattice points. I had windows in my classroom that aided in my ability to describe lattice points. But due to online learning, I had to use Jamboard instead.
To create this, I create the graph I want in PowerPoint (Thanks to Lindsay Bowden!). Then I save it as a picture and insert it into Jamboard. To create the lattice points, I use the circle tool. I shrink it as small as possible. Then using the tool bar at the top, I will change the fill to “Transparent” and the outline to whatever color I need. (I color-code based on use.)
Using the transparent fill allows students to track along the graph to see where it perfectly crosses through the grid lines.
Now that students have found the lattice points on the graph, it is time to create the rise and the run.
In the classroom, I make students complete a “horrible” exercise. I take them to the bottom of the closest staircase. We have a chat about how when you walk up stairs, you have to pick up your foot before moving it over the stair, otherwise you can stub your toes. Ouch. I then introduce “Rise over Run”. It is what gets us from lattice point to lattice point. So we have to “rise” our foot before you “run” your foot when going up stairs. Then the annoying part for students comes next: I make them walk up the stairs saying “Rise, Run” every time they move their foot up and over! Great, right?
With online learning, we had to visualize it a little more. But Jamboard came in clutch to help us out.
This is where those lattice points come in handy. From here, we create our rise and our run. It’s what gets us from lattice point to lattice point.
To create the lines, I use the pen, click the little arrow, choose highlighter. The highlighter allows for a thicker line. To make it straight, hold down the SHIFT key and then draw the line. Again, I color-code to help in the process later.
Putting It All Together
Now for the equation, I find that students get totally lost at first with the y = mx. When I teach this part, I explain that “m” is for mountain and mountain’s have slopes. But I also rewrite the equation to look like this:
y = (slope)x OR sometimes y = (rise/run)x
to help with students struggling with putting slope in the right order.
What About the Y-Intercept?
When we finally reach the point where we can talk about the y-intercept, I show two graphs with the same slope, but one starts at the origin while the other starts higher up.
In person, I would have the students do a turn & talk with their table partner. In online learning, I used a quick Google Forms or a Desmos activity to gather data on what students think.
Then once we have come to the conclusion that they start at different locations along the y-axis, we move into how to write the equation. I use the similar writing as above:
y = (slope)x + (y-intercept) OR y = (rise/run)x + (y-intercept)
Once students understand the 1-quadrant linear equations, we will move onto the 4-quadrant graphs. Sometimes I will ease our way by doing lines that cross through the origin, but I find that most of the time a quick review is all students need.
From there we talk about the non-positive slopes: negative, zero, and undefined. I always have my students watch The Adventures of Slope Dude. It gives students an actionable visual to see what the different types of slope looks like. Plus it aides me when I try and prompt a student on the slope: “Is it puff puff positive or nice negative? What would Slope Dude say?”
Overall, you know your students best. Take baby steps and don’t be afraid to go back to the basics of linear equations. Think mastery over time. Students have multiple opportunities to learn linear equations in the form of graphing throughout their 8th grade and Algebra 1 years. The better the foundation, the better students can build on their knowledge for future years of math.
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