Systems of Equations: Graphing

It was time to start systems of equations which I like, but starting with the graphing method, I don't. My instructional coach suggested I use graphing calculators since it is practically impossible to do by hand. Light bulb! It's investigation time. :)

I wanted to make something up where students would graph two lines and then find the intersection point. Thanks go to @bbrennan- he suggested that I have the students calculate the intersection instead of just estimating. Which worked out wonderfully by the way!

I typed up step-by-step directions so that students could work individually for once.

I had students read the first three sentences and stop. I had hoped from a previous investigation on parallel and perpendicular lines that students would understand the concept of solution and no solution. Unfortunately, they did not. We kinda scrapped that investigation anyway. So once I explained solution, the students figured out quickly what no solution meant. From there, I had them flip the paper over and go down through each system to decide if the lines were parallel or perpendicular based on their previous knowledge of slope.

Then we flipped back to the front page and I let students work on their own through the example. I circulated the room making sure that everyone got the correct answer before allowing them to move on.

Once everyone had the process under control, I set them loose on the back page. Fingers were flying, buttons were clicking, math was being done! When students came upon systems of equations that were parallel, they still typed it in the calculator and then got an error. When they asked for help I guided them back to their graph and asked what kind of lines they saw. As soon as they noticed the parallelism (te-he), they realized their mistake. Maybe next time I should put more emphasis on actually looking at the graphs before calculating.

The solutions for number 7 and 8 were ugly decimals that students whined about writing down. But once they cried me a river, built a bridge, and got over it, they realized it was only 8 digits of writing.  I think this activity went well. Students thought it was cool to have the calculators doing the work for them and the process became almost a mantra: "Second, trace, 5, enter, enter, enter".

Compared to graphing by hand, I LOVED this method. I think it was a great visual, an actually effective way to incorporate technology, and is a good foundation for understanding the point of systems. Also, I planned for it to take whole period, and only one period, and it did! By golly, I might be getting the hang of this.


  1. Nice lesson. I think you may have been on to something in having the students get an error when they used trace on parallel equations. What I mean is that your students got some practice over using the trace feature first, and then once that was mastered, they got the second instruction to view the graphs for parallelism; otherwise, it may have been too many steps to learn at once if you taught enter the equations, graph, look at the graph for parallelism, enter the steps for trace, and record the point of intersection.

    One suggestion, which may be irrelevant if graphing by hand won't be on the test. To get some practice in manual graphing on paper, you could have the students enter the system of equations in the Y= editor and then use the TABLE function to get a couple points for each line and then graph them on paper. For your first set of equations, you could even have them scan through the table looking for the point of intersection--they could see how creating a table is helpful when the point of intersection is two integers, but not too useful when x and y are not.

    Paul Hawking
    The Challenge of Teaching Math
    Latest post:
    Teaching factoring quadratics when a>1

  2. Paul,
    Thanks for your perspective, I hadn't thought of it that way.

    I probably should show the students how to use the table function as well.