A couple of weeks ago, I had a thought: Do all linear functions pass through a point where the x and y coordinates are the same? I had a fun time thinking through that question, so I decided to turn it into a lesson to see what my Algebra 1 students did with it.

For context, we had just spent the past week solving systems of equations by graphing and substitution. We had not yet discussed systems with no solution.

So at the start of class, I gave my students this prompt:

I gave my students ten minutes to work on the problem silently. Most students seemed to go about proving that y=1/2x+4 did indeed pass through (8,8) and then moved on to making up other functions and seeing where they had matching coordinates. Interestingly, almost nobody graphed the line y=x. I think that if I had said "My hypothesis is that every line crosses y=x" then students would have very quickly answered my question. By framing the question slightly differently, I got them to think a lot harder.

Once the 10 minutes elapsed, I grouped them randomly at my vertical white boards and asked them to work together to prove me right or prove me wrong. Kids had all sorts of approaches. Many groups tried to prove me right by exhaustion, trying function after function and finding the point where x and y matched. Other groups searched for a counter-example and found one.

Once everyone had gotten a chance to work and think through the problem, we would gather around one board and listen to that group explain their thought process. I loved this group's work because they were able to prove that x and y could never be equal, since y would always be 3 more than x regardless of the x value you chose.

Then I would ask for another example of a line that never has matching coordinates. Kids would say stuff like y=x+1 or y=x-2. One kid said "y=x is the opposite. It like *super* works for your hypothesis"

At that point, I pulled Desmos up and graphed y=1/2x+4. Then I graphed the point (8,8). I graphed a few more points and their matching points, then hid all the lines on Desmos so that only the points were left. I said "What do you notice?" Right away, kids were shouting that it made a line, and that the line was y=x. So then I asked students to give me equations for lines that *disprove* my hypothesis. We graphed y=x+3, y=x+1, y=x+1000000 and talked about what they all had in common. As the kids rightly noted, all those lines are parallel. So parallel lines never intersect, so they have no solution in a system of equations. I made a quick analogy: If I give Ashley $100 and give Elaine a nickel, then every day I give them $1 each, when will Elaine have as much money as Ashley? Never, because they have different starting amounts (y-intercepts) and the same rate of change (slope).

We ended the lesson with a quick talk about the difficulty of proving vs. disproving a hypothesis. You can't prove a hypothesis like this by trying 10 functions. After all, most functions do cross y=x somewhere, and there are always more functions to try. It's much more effective to try to *disprove* a hypothesis. Then, if you find a single counter-example, your job is done. And sometimes you learn something about the problem that makes it easier to actually prove the hypothesis. For once, I felt like I was getting my kids to do math in the way that I've always heard about university math (which I never took past multivariable calculus).

I think this is the most impactful way I've taught about systems that have no solution. I'll definitely be using this next year, and I really recommend it for other Algebra 1 teachers.

But that wasn't my favorite part of the lesson.

My favorite part of the lesson was this one kid, T, who came up with a stunning visual proof of the problem. When he first explained it to me, I honestly did not understand what he meant, but I could tell there was something interesting going on, so I switched on my camera.

I did a bad teacher move at the end, where I explained his proof to him rather than letting him explain it, but honestly I was so excited by his idea that my gleeful math brain took over.

I love this moment of class so much because T is seeing the problem in a way that I never would have come up with if I had spent a year thinking about this problem. And by the way, he's (almost) right! Every linear function would eventually make a square, if you include squares with an area of 0 that occur right at the origin.

The only problem is, his amazing insight doesn't prove the thing that T thinks it does. Every linear function forms a square in the way he describes, but some of those squares occur when the coordinates are opposites, such as the point (-3, 3) which would be on the line y=x+6.

But I was so excited by T's idea that I ran to my desk and made a quick Desmos graph to show the class how his theory worked. If you pay careful attention, you'll notice that this function, y=1/2x+4, actually makes two squares at different points, (8,8) and (-2,2). The lines like y=x+3 that disprove my hypothesis only make squares once, where the coordinates are opposites. By the way, you can play around with my Desmos graph here.

Anyway, after class T came up to me and said "Why did you film me if I was wrong?" I said "Because you had an amazing idea!" T said "Yeah but it was wrong so it wasn't useful."

Because we were heading to lunch, I did a poor job of explaining to him why I loved his idea so much. I'm going to try to do a better job here:

The reason I filmed that moment is because T's idea broadened my conception of the problem. It made me see a visual structure to linear functions that I had never seen before, and it made me wonder what other ideas I could prove or disprove with this theory. It made me wonder what types of rectangles you'd get from visualizing quadratic functions, exponential functions, logarithmic functions this way. It made me realize that if you graph all the rectangles that are made by y=20/x, you get an infinite amount of rectangles that all have an area of 20. And it made me want to share his video with as many other math teachers as possible so I can hear what they wonder.

In other words, T's idea made me feel like a mathematician, and I hope it made him feel the same way too. He wasn't immediately right, but neither was Andrew Wiles when he proved Fermat's Last Theorem. And his idea opened up a whole world of other opportunities for insight that I've barely been able to think about yet.

So T, if you're reading this (and you better be since I am emailing it to you as soon as I'm done writing), I hope you take this moment as a victory. You invented a new math idea, which most kids very rarely get the chance to do in math class. You should be proud of yourself. I'm sure proud of you.