## A Positive Diversion with Desmos

We’ve just come back from a two-week vacation. I was getting really burned out before the break, so it was much needed. But some of my classes I only see once a week this term, so I was concerned about how much they’d have retained.

Today was to be the last day of quadratics for my TTO 2 class (Dutch bilingual class; American grade 8), with the day’s topic being solving systems of equations with quadratics. The textbook introduces it so matter-of-fact, like just set the two equations equal to each other and solve, but I wanted my students to come up with that themselves, which of course means: Desmos time.

Which means we ended up falling down the rabbit hole, but I couldn’t be happier because the whole class ended up stumped, and at the end the correct answer was given by a very quiet girl who struggles a bit and has literally not raised her hand to give an answer the entire year. Her shy smile when she was the only one who knew just completely made my day.

It reminded me: Going off track in your lessons is not a bad thing when it gives a chance for the whole class to be puzzled and for someone new to bring their insight.

Her insight below the jump:

I started by asking the students for an example of a quadratic expression, a quadratic equation, and a quadratic function. The expression they gave me was $x^2 + 3x$. This was a particularly fortuitous choice because it gave us a different kind of quadratic function than they’d ever seen. In TTO2 they only work with graphing and applying quadratic functions in the form $y = ax^2 + c$, so they had no idea what the $3x$ would do.

One thing I love about Desmos is that it starts graphing immediately as you type. So as soon as I typed in $y = x$ we got to have a mini-review about the gradient and y-intercept of a line; add the exponent, review the features of the basic parabola. Before I added the $+ 3x$, I asked the students what they thought would happen; one thought the y-intercept would be 3 and another thought it would move to the right (my mistake in not asking for reasoning in either student nor asking the class to vote on what they thought would happen; missed opportunity!).

They were surprised when it moved to the left and down. We talked about the connection between the x-intercepts and the solutions to the quadratic equation we’d done and why that worked. They wanted to see one with a “real” y-intercept, so I changed it to $y = x^2 + 3x - 4$. More surprise…and then the rabbit hole:

J.: “Hey, how can you get a squiggle?” Oh man, I live for this, when the kids are really legitimately curious about the math the see.

Me: “What do you think? Let’s try something.”

C: “Make a cup and cap parabola line up next to each other” — actually a cool idea but super difficult to execute well and clearly not going to give a squiggle without extra unwanted pieces.

A: “Divide the entire thing by something.”

Me: “Like what?”

The first suggestion was 6, which of course wasn’t interesting, but that inspired me to suggest they try to divide by something more exciting. Which is how we ended up with:

$y = (x^2 + 3x - 4)/6x$

Complete amazement ensues!

It was great to have a discussion about what was different about this one and why. Boy were they stumped for a really long time (to the point where I started to worry that maybe this diversion wasn’t such a good idea after all), with some good ideas but mostly bad ones about what was happening and why. One student offered that x could be positive or negative, to which I added, “but?” but no takers…

I nudged a bit: “This one looks really weird, but some things are the same: look, the x-intercepts are the same as they were before we divided by $6x$.”

J.: “Hey, what’s the y-intercept?” — we zoomed out and out and it looked like maybe it was lying along the y-axis, but someone said, “No, I think it doesn’t actually touch it at all” and I confirmed that was true and said, “But why? How can it be there is no y-intercept?”

The words were barely out of my mouth when someone suddenly gasped, “x can’t be zero because you can’t divide by zero!” So much fun! But my favorite moment came two minutes later:

Me: “Ok, but we still didn’t get the squiggle! Who has an idea?”

…silence. Long protracted silence. And then a hand that has never gone up before.

M.: “What if you did x to the power of 3?”

And her little smile. MAN, that is why you go off plan.