I’m kicking off a series of posts on a quadratics unit designed to have a simple narrative arc — one that’s clear enough to allow our weakest students to follow the conceptual plot. The previous post contained the outline of the whole unit. Now we’ll dive into the first lesson.

Dan Meyer has suggested his popular *Will it Hit the Hoop* as a good launch for quadratics study. And it *is* good, but the launcher is aimed in the wrong direction. Instead of aiming kids’ attention at vertex form, I think it should set up the transition to factored form. Here’s how the official Desmos version ends by making connections to vertex form:

But if we anchor our unit on factored form instead, we can expose the value of factoring early (and often). That’s great, since factoring is the skill I’m most worried kids will resist. In addition, factored form provides the clearest link between the two halves of a quadratics unit: graphing parabolas, and solving quadratic equations. Without that link, our weakest students might feel the two halves are basically unrelated to each other.

I revised the lesson to refocus it on factored form. Here it is on Desmos, and here’s how it works now:

The goal is to prompt this question: what kind of equation makes a parabola? Here are the last 2 screens of my Desmos lesson:

Hopefully, students are ready to care about zeros of a function, so it’s time for some instruction: we transition to the handout below (pdf version here). Students should **NOT** have access to calculators for this handout. If they have calculators, they won’t think the tedious problems are tedious, and they won’t start looking for patterns and shortcuts. I want them muttering to themselves. Here’s the handout:

You might have noticed the “Prep” questions at the top of the first page: 8 + (9) and 8(9), and (3) + (2). What’s up with those? They’re designed to remind kids that not all problem with parentheses are multiplication problems. That’s the kind of misconception that can throw our weakest students off track for the whole rest of the lesson. After that little reminder, the key connection between parabolas and zeros happens in #1, part c. That’s where the class substitutes the zeros into the equation and sees why they’re zeros in the first place, and it’s worth some class discussion to flesh it out.

At this point, we could probably start teaching kids how to take a parabola’s zeros at 2 and 5, and create the factors of (*x*-2) and (*x*-5). But personally, I don’t think they’re ready for that. Not until they feel in their bones that

(*x *– 2)(*x – *5) is like a having a light saber

(*x –* 2) + (*x* – 5) is like wearing stormtrooper armor

(In case you never noticed, stormtrooper armor doesn’t seem to be too helpful to the stormtroopers). The rest of the handout asks a series of simple yes/no questions that gradually reveals the glory of factored form. Hopefully, this understanding emerges organically. Dan is fond of saying that if factoring is the aspirin, we first need students to experience the headache. The tedious addition problems are my headache here, and I actually think they’re a bit more effective than his headache because they’re more tightly connected to the actual concept of factoring.

**Even if you disagree with everything I’ve written so far, I don’t recommend the new & improved version of Will it Hit the Hoop.**

Folks who read Dan’s work regularly will find the irony here, but I think when Desmos revised this lesson, they added some new fancy tech features that detract from its power. The features help you gather individual student opinions on whether the basketball shots will go in or not (students vote “in” or “out” by clicking on their screens). But this way of voting sucks energy from something that should be a lively and social. In the class where I used them, students silently clicked through their opinions. In the classes where I didn’t, they agonized aloud — together — about where the shots would go. So I recommend not using them. Instead, have kids vote with their feet: move to one side of the room for “In” and another side for “Out”. My version of the Desmos lesson doesn’t have clickable voting.

**Next blog post — Lesson 2: “You don’t play with Legos one block at a time” **** **

**Update:** The official Desmos version does have one great feature my version doesn’t: the videos of the shots going in or out. When I teach this lesson, I just click over to the Desmos version to show the answers for each shot.

Hi Kevin, thanks as always for your thoughtful commentary. I’m resisting the factored form here because the zeros don’t actually have much meaning in the context, while the vertex

does. What am I missing?Hi Dan, I wonder which is your larger point: (a) that my lesson doesn’t have enough mathematical continuity from the basketball part to the zeros part; or (b) that students might *feel* like the zeros part isn’t well-connected to what came before. I think I’m quite comfortable proceeding with a lesson that’s a bait-and-switch as long as students don’t feel like it is. Not sure I’d ever admitted that publicly before. So how will students feel about focusing on zeros vs the vertex after doing the activity?

Hmm, this makes me think I should have them create the equations of 1-2 parabolas in my handout phase, to make the handout feel more connected (FYI, that will be the main focus of the next lesson). Thanks.

All this makes me curious — what are the students are wondering when the basketball part ends? Since this is their first exposure to non-linear stuff, I think it’s probably some version of “Why would anybody use parabolas to predict something in sports?” or “That was fun, but I’m not convinced it’s worth learning the math behind flying basketballs.”

My goal was to quickly impress them with a new power they can develop — something to show that this nonlinear stuff will be worth their effort. The rules about zeros are pretty easy to discover, and hopefully create that sense of empowerment. But it is kind of a bait-and-switch, and do students notice that and resent it a bit? I’m not sure.

I’m more sure that students who have just been introduced to parabolas for the first time are not wondering, “How are the coordinates of the vertex reflected in the equation of the parabola?”

Mostly, though, I’m just not sure how I would introduce quadratic functions with vertex form being the starting point. It’s the larger unit plan that’s driving my thinking. I wonder if other folks start quadratics with vertex form.

Anyways, thank you very much for your thoughts. Your critique wasn’t what I expected, and provided good food for thought. Speaking of food, my pie is done baking. Gotta go.

I also start with factored form before vertex form.

I like the click to vote (although I use Pear Deck instead of Desmos usually) but then I have them turn & talk to their group members (sit in 3s randomly assigned daily) to justify / argue over their choice.

Great post!

Hi! Thanks for commenting. I like Pear Deck, too, though Desmos’ classroom conversations tools are gradually making Desmos more Pear-Decky.

FYI a few weeks ago, I tried visibly random groups with vertical whiteboards, toward the end of this school year. I’m pretty sure I first learned of both of those from you.

Awesome … How did you and your students like it?