As I prepared to teach students about working with systems of equations, I reflected on last year’s students and came to a painful realization- there were several students who just never really got what they were doing. Sure, most students got pretty good at using the substitution method or the elimination method to solve for two systems of equations, but even for those who could do well with test questions, did they really understand at a conceptual level what they were doing? It was humbling to realize that for most of them, the answer was probably no, they didn’t. So for the past week I took a different approach to the topic, applying the Discovery Lab approach I’ve seen work so well for other topics. And it was awesome to see how they responded to the Systems of Equations Discovery Lab.

### Getting Students Hands-On with Systems of Equations

In order to break it down and show students visually exactly what we’re looking for when solving systems of equations (where the two lines will intersect) and the three possible outcomes (one solution, no solutions, or infinite solutions) here’s what I did:

Step #1- Break it down and illustrate it. Starting with just graphs of various systems of equations, I asked students to make observations about the lines in each graph. They saw graphs with one solution or intersection, no solutions (parallel lines), and infinite solutions (one line shown).

Step #2- Have students share their observation about the graphs they’ve examined. Monitor if students are noticing the differences between the graphs. Encourage students to start drawing conclusions about what rules exist that determine how many solutions are present for a system of equations.

Step #3- Give students a short overview of the three types of solutions. For each type of solution, give students three graphs of systems of equations and the equations for the two lines represented in the graphs. Ask students to make observations about the system of equations:

When I first gave students a graph and two equations and asked them to talk about their observations, after about 30 seconds of blank stares I realized that a little modeling was needed. So, we talked about the first set together, describing what we noticed in the equations. We talked about the y-intercepts and slopes, and how those looked on the graph. Then, I let them loose and heard really great math talk. I heard appropriate uses of math vocabulary terms: “Oh, those lines are parallel,” “These lines intersect here,” and “they have the same slope different y-intercepts.”

Step #4- Have students reflect on what they’ve learned about working with systems of equations.

### Increasing Student Engagement

I saw this great quote on Instagram today from Jen Jones at Hello Literacy:

That’s exactly what I saw happening with this approach to systems of equations. I really felt like the students were sitting in the driver’s seat. They were actively making connections and drawing conclusions, rather than waiting for me to just tell them the next thing to do. To be honest, there’s always a little fear involved with releasing control to students- a moment of panicked thoughts like *what if they don’t get it *or *what if this whole class period just spirals out of control. *But every time, at the end of a lesson where I’ve consciously stepped back and let the students take the lead on their learning, I’ve been so pleased with the results. And honestly, if things do spiral a bit out of control, or if the students are just super lost, it’s not a big deal. I can just step in and do a bit of modeling, or a quick reminder, and then set them on their way again. Oh, those darn irrational fears- they can be downright annoying.

Did I mention this was the day that my principal was in the room for my formal observation? My principal was so intrigued with the lesson that she went into the classrooms of my two team mates to see the same lesson in action again (yes, I’m trying to convert them to Discovery Labs as well! And they’re coming over to the dark side, he he!!) She commented to me in the post conference that she wished that she had learned math through more activities like what she saw my students doing, rather than the stand and deliver focus on steps and procedures approach that all her math classes followed over the years. Don’t we all?!

At the end of the day, I feel like this year’s students are in a much better place to not only do pretty well on answering test questions, but also actually remembering in the long run what they’re even being asked to do when solving systems of equations. I’m putting this lesson in the win column. How do you help students to really conceptually understand systems of equations? What guided inquiry approaches have worked for you? If you haven’t really dipped your toe into the guided inquiry approach, give it a shot!

If you want to check out the whole lesson, the download is available in my TPT store.

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