Hey, everyone. Welcome back for another deep dive. And today, we're tackling something I know can make some people nervous. Oh, yeah. Quadratic equations. But don't worry.
Not gonna lie. A little intimidating.
We're gonna break it down with a little help from our friends at Illustrative Math.
Always helpful.
They actually have a really interesting approach to teaching this in their algebra 1 curriculum, and luckily for us
Yeah.
They've got some detailed notes and lesson plans that give us a peek behind the curtain.
I like how they don't just jump right into formulas and equations. Right? They actually take the time to build up to it.
Yeah. And speaking of building up to it, they actually mention that students should already have some familiarity with quadratic functions, you know, from, like, a previous unit.
It's all about that prior knowledge. Right?
Yeah.
So students have probably already analyzed graphs, looked at those outputs of quadratic function.
Like, maybe projectile motion.
Exactly. That's the classic example. They've seen how changing the input affects the output. But here's the twist. This lesson flips the script. Now it's like, okay. We want a specific output. How do we actually figure out the input that gets us there?
That's so interesting. It's, like, they're anticipating that roadblock ahead of time and kinda smoothing it out for students before they even hit it.
Exactly.
So how do they introduce this whole new way of thinking?
Well, with potatoes, of course.
Okay. Now you've got my attention. Potato?
You heard me right. There's this really fun warm up activity they call the potato problem. Picture this. You've got a potato being launched straight up in the air.
Okay. I'm already more engaged than I ever was with parabolas.
Right. And they give you all the specifics too. Launched from 20 feet up, initial velocity of 92 feet per second, the whole 9 yards. And then students get a graph that shows the potato's height over time.
Alright. I can see where this is going. So they've got this graph. Now what?
Now the real fun begins. Students have to answer these questions about the potato's flight path.
Like what? Give me an example.
For example, will this potato ever actually reach a height of, let's say, a 120 feet?
Okay.
Or even something like, when will it hit the ground?
Those are great questions, and I can definitely see how a graph, while helpful, might not give you the complete picture.
That's exactly the point. It gets students thinking about the limitations of just relying on a graph alone.
You might be able to get a rough idea.
An estimate, for sure.
Yeah. But it's not precise.
Not at all. And that's what's so clever about this activity. Mhmm. It highlights that need for something more, something more accurate, something more well
Like a quadratic equation.
Boom. Go.
So we've launched potatoes into the air with math.
Hopefully, not real potatoes.
Definitely not. And speaking of not real potatoes
We've got those picture frames.
Yes. The framing problem. I have to say I love how illustrative math makes these problems so relatable.
Right. It's like, who hasn't struggled to frame a picture on a budget?
Seriously. And you can really see how these real world examples tie back to those big mathematical practices.
Totally. In the lesson plan, they specifically call out m p 2 and m p 4.
Reasoning abstractly and quantitatively and modeling with mathematics for those who don't have those memorized.
Which, let's be honest, is most of us.
True. But the point is those practices are woven into these activities, so students are already flexing those muscles before they even get to the equation part.
It's all about building those connections.
Absolutely. So we've got our potatoes. We've got our picture frames. Students are starting to feel the need for a more precise solution.
Okay. But hold on. We're not dropping the equation just yet.
Oh, right. There's a buildup.
It's all about the buildup. Yeah. 1st, they have translate that framing problem, you know, with all the dimensions and materials Yes.
All those details.
Into a proper mathematical equation.
So it's more about the process of representing the problem with math symbols, not actually solving it at this point.
You got it. And what's cool is there are a few different ways students might approach it.
The lesson plan actually mentioned that, didn't it? Like, some students might focus on the length and width of the frame.
Right. While others might think about it in terms of area.
So they're encouraging different ways of seeing the problem. I
that. Anna, here's the thing. They're not just anticipating different approaches.
They're thinking about the things that could trick students up too. Right?
Exactly. They specifically call out potential misconceptions.
Like what? Can you give an example?
For 1, they mentioned how students might accidentally combine numbers and variables incorrectly.
Oh, I can definitely see that happening.
Or even mixing up area and perimeter. That's another big one.
Those are so common. It's great that the lesson plan addresses those head on.
And even better, they offer strategies for tackling them.
Scaffolding questions, visual aids, all those good teacher tools.
So teachers can feel prepared to guide students through those tricky spots.
It's almost like they've got a crystal ball and can see into our future classrooms.
Right. And finally, after all that prep work.
The big reveal, the quadratic equation in all its glory.
Drumroll, please. They introduced the general form, axhacksplusbxpluscplusc where at can't be 0.
It's like they've set the stage so well that the equation doesn't seem so intimidating anymore.
Because now it's not just some random bunch of letters and numbers.
It's got meaning. It's got context.
And most importantly, they emphasize that the solutions actually represent something real.
So in our framing problem, a solution isn't just some abstract number.
It tells you the actual thickness the frame needs to be.
To use up all your framing material without wasting any.
It's like that satisfying moment.
I love it. This has been so helpful, but before we get completely lost in the world of picture frames and quadratic equations
We gotta wrap it up.
Do we actually see how to solve these equations in this particular lesson, or is that for another time?
They leave us hanging just a bit, which honestly is kind of brilliant.
A little math cliffhanger to keep us coming back for more.
Exactly.
So they leave us in suspense, but they don't leave us high and dry.
They give us a little something to think about. Right?
A little homework assignment for next time.
Exactly. It's like they wrap up by asking, how could you take this these ideas and make them work for your own students?
Okay. So not just how to solve the equations, but how to actually teach it.
It's about taking those big ideas
Yeah.
You know, connecting to things they already know, making it fun and engaging
Right. Building up that need for the equation.
And then letting teachers run with it.
It's almost like they're saying, we've given you the tools. Now go build something amazing.
Because at the end of the day, who knows their students poorer than the teachers themselves?
That's so true. I feel like we could talk about this all day, but sadly, all good things must come to an end.
They really do.
And this has been such an interesting deep dive. I feel like I learned a lot, not just about quadratic equations themselves, but about how to approach teaching them in a way that actually sticks.
I think that's what I love about illustrative math. They really put a lot of thought into the why behind the what.
Couldn't have said it better myself, so to Illustrative Math, thank you. And to everyone listening, thank you for joining us on another deep dive.
Until next time.
We'll see you then.