I’ve always needed a way to motivate the study of quadratics. In the past, I’ve used materials from some of Dan Meyer’s 3-Acts: Super Mario to get students to realize that linear predictions are sometimes wrong, and Will It Hit the Hoop? to specifically focus students on quadratic graphs. But even to my teacher ears, the jump to actual quadratics skills sounded cheap: “Now that we all agree quadratic functions are important, let me teach you to multiply things like (x+1)(x+2), because it’s really important for understanding parabolas, and I’ll explain why later.” Groan.

I’d like to share a new lesson that I really liked because it:

- Naturally focuses students on area models of quadratic expressions;
- Shows that quadratics are the way to model something that’s speeding up or slowing down;
- Has a really low barrier to entry.

A low barrier to entry means students can dabble their toes in this concept pretty easily at the start, without encountering hard math until they’ve played around a bit. Before we go on, let’s check that this blog post is worth your time. Here is the whole lesson I’m about to describe, fast-forwarded to be just 2 min long:

Still interested? Cool.

The most direct way to (a) introduce area models of quadratic expressions, and (b) make it seem like quadratic expressions are useful is to pose a question that’s directly related to area. Something like: Farmer Joe has 100 feet of fence and wants to make the largest sheep pen he can. What length and width should he use for the pen? [The answer is to model area as *A* = (*L*)(*W*) = (*L*)(100-2*L*) = 100*L* – 2*L ^{2} *, graph the quadratic function, and find its vertex].

In my experience, the Farmer Joe question doesn’t arouse much natural curiosity from students, and I think I know why: even students who naturally enjoy math puzzles have no inkling at the outset of their inquiry that their solution method will also help them understand the many faces of quadratics: projectiles, cars speeding up or slowing down, the famous handshake problem, etc. It’s not until you’re well into the problem, and you see that the graph of area vs length looks like the flight path of a projectile, that you have a chance of recognizing how significant quadratics might be. And by that point, you’ve already done enough hard math that you might be a bit tired or grumpy. Learning quadratics should be like hiking to a beautiful vista: look at all the things I can see from up here! The *ahhhh* experience of arriving at that vista needs to come sooner in the introduction or students end up feeling the way I felt on my last hiking trip in Montana: that’s a great view, but OMG I hate mosquitos–let’s get the #@%! out of here.

If you’re learning quadratics after learning linear functions, then the best way to notice you’re at a pretty awesome vista is to see that you’re looking at a pattern that’s *accelerating*. A pattern that’s accelerating is noticeably very different than all the patterns we’ve done so far. My class starts linear function by looking at dot patterns like Fawn’s–specifically, we focus on ones that visually distinguish the y-intercept from the slope. For example, looking at the pattern below, how many dots would be in Stage 10? Stage x?

Students get really used to asking, “How fast is the pattern growing?” or “How many dots does it add each stage?” We also do modified versions of Stacking Cups and Barbie Bungee to keep emphasizing that finding the rate is crucial for making a prediction.

In addition, the narrative in my room is that algebra is a way to predict the future by finding and expressing patterns. For example, when we study direct varation early in the year, students actually make short videos of a prediction experiment in their own lives.

Okay, against that backdrop, I present students with the following lesson to try a prediction that *finally breaks* the constraint of using constant-rate patterns and motivates area models for polynomial multiplication. Here’s the full, narrated video overview of the lesson.

**Update 6/21/16: **Here’s a Desmos activity to go with the visual “dot pattern” section at the end.

**Room for improvement: **As I was transitioning students to (x+2)(x+3) and drill problems, I felt that even though I’d gotten students to the vista, I need to do a better job of showing them everything they can see. What if they think these area patterns only work when the first difference in the pattern goes like +1, +3, +5, etc? I should show that if the first differences go +2, +6, +10, etc, then you can use 2x* ^{2}*…visually, just draw two of the x

*patterns. If you wanted +1, + 2, +3, you could use (1/2)x*

^{2 }*by drawing the x*

^{2 }*dot pattern and then cutting it in half. I should also make the connection to accelerating cars, psychology’s inverted U-shaped graph of stress vs performance graph, Farmer Joe, and everything else that’s quadratic. However, I think that’s best saved for the next lesson. We teach roughly 90-minute blocks, and I like each block to have some conceptual development and some practice. When you see kids every other day as it is, you need to squeeze in some practice to each lesson. So in the future, we’ll transition to (x+2)(x+3) and do drill just as shown above, but the following lesson I’ll take time point out all the landmarks you can see from this vista.*

^{2}**CCSS thought: **I’m not sure how this lesson would play in a Common Core state. Do you do arithmetic series in Algebra 1, and if so, do you do them before quadratics? That would probably make this whole shtick might seem kind of lame. We don’t do CCSS here. Our state test doesn’t really assess comprehension much, so I’m not sure how much this lesson will even improve my students’ standardized test scores. My students have always been able to multiply binomials without experiencing an intellectual need for doing so. But this lesson just felt so *satisfying.** *I hope it’s been worth your time to read about it.

**Sharing the file:** I’m happy to share a copy of the powerpoint file to anyone who’d like it. Just ask in the comments.