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Lesson by lesson podcasts for teachers of Illustrative Mathematics®.
(Based on IM 9-12 Math™ by Illustrative Mathematics®, available at www.illustrativemathematics.org.)
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.
Speaker 1:But don't worry.
Speaker 2:Not gonna lie. A little intimidating.
Speaker 1:We're gonna break it down with a little help from our friends at Illustrative Math.
Speaker 2:Always helpful.
Speaker 1:They actually have a really interesting approach to teaching this in their algebra 1 curriculum, and luckily for us
Speaker 2:Yeah.
Speaker 1:They've got some detailed notes and lesson plans that give us a peek behind the curtain.
Speaker 2:I like how they don't just jump right into formulas and equations. Right? They actually take the time to build up to it.
Speaker 1: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.
Speaker 2:It's all about that prior knowledge. Right?
Speaker 1:Yeah.
Speaker 2:So students have probably already analyzed graphs, looked at those outputs of quadratic function.
Speaker 1:Like, maybe projectile motion.
Speaker 2: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.
Speaker 2:Now it's like, okay. We want a specific output. How do we actually figure out the input that gets us there?
Speaker 1: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.
Speaker 2:Exactly.
Speaker 1:So how do they introduce this whole new way of thinking?
Speaker 2:Well, with potatoes, of course.
Speaker 1:Okay. Now you've got my attention. Potato?
Speaker 2: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.
Speaker 1:Okay. I'm already more engaged than I ever was with parabolas.
Speaker 2: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.
Speaker 1:Alright. I can see where this is going. So they've got this graph. Now what?
Speaker 2:Now the real fun begins. Students have to answer these questions about the potato's flight path.
Speaker 1:Like what? Give me an example.
Speaker 2:For example, will this potato ever actually reach a height of, let's say, a 120 feet?
Speaker 1:Okay.
Speaker 2:Or even something like, when will it hit the ground?
Speaker 1:Those are great questions, and I can definitely see how a graph, while helpful, might not give you the complete picture.
Speaker 2:That's exactly the point. It gets students thinking about the limitations of just relying on a graph alone.
Speaker 1:You might be able to get a rough idea.
Speaker 2:An estimate, for sure.
Speaker 1:Yeah. But it's not precise.
Speaker 2: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
Speaker 1:Like a quadratic equation.
Speaker 2:Boom. Go.
Speaker 1:So we've launched potatoes into the air with math.
Speaker 2:Hopefully, not real potatoes.
Speaker 1:Definitely not. And speaking of not real potatoes
Speaker 2:We've got those picture frames.
Speaker 1:Yes. The framing problem. I have to say I love how illustrative math makes these problems so relatable.
Speaker 2:Right. It's like, who hasn't struggled to frame a picture on a budget?
Speaker 1:Seriously. And you can really see how these real world examples tie back to those big mathematical practices.
Speaker 2:Totally. In the lesson plan, they specifically call out m p 2 and m p 4.
Speaker 1:Reasoning abstractly and quantitatively and modeling with mathematics for those who don't have those memorized.
Speaker 2:Which, let's be honest, is most of us.
Speaker 1: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.
Speaker 2:It's all about building those connections.
Speaker 1: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.
Speaker 2:Okay. But hold on. We're not dropping the equation just yet.
Speaker 1:Oh, right. There's a buildup.
Speaker 2:It's all about the buildup. Yeah. 1st, they have translate that framing problem, you know, with all the dimensions and materials Yes.
Speaker 1:All those details.
Speaker 2:Into a proper mathematical equation.
Speaker 1:So it's more about the process of representing the problem with math symbols, not actually solving it at this point.
Speaker 2:You got it. And what's cool is there are a few different ways students might approach it.
Speaker 1:The lesson plan actually mentioned that, didn't it? Like, some students might focus on the length and width of the frame.
Speaker 2:Right. While others might think about it in terms of area.
Speaker 1:So they're encouraging different ways of seeing the problem. I
Speaker 2:that. Anna, here's the thing. They're not just anticipating different approaches.
Speaker 1:They're thinking about the things that could trick students up too. Right?
Speaker 2:Exactly. They specifically call out potential misconceptions.
Speaker 1:Like what? Can you give an example?
Speaker 2:For 1, they mentioned how students might accidentally combine numbers and variables incorrectly.
Speaker 1:Oh, I can definitely see that happening.
Speaker 2:Or even mixing up area and perimeter. That's another big one.
Speaker 1:Those are so common. It's great that the lesson plan addresses those head on.
Speaker 2:And even better, they offer strategies for tackling them.
Speaker 1:Scaffolding questions, visual aids, all those good teacher tools.
Speaker 2:So teachers can feel prepared to guide students through those tricky spots.
Speaker 1:It's almost like they've got a crystal ball and can see into our future classrooms.
Speaker 2:Right. And finally, after all that prep work.
Speaker 1:The big reveal, the quadratic equation in all its glory.
Speaker 2:Drumroll, please. They introduced the general form, axhacksplusbxpluscplusc where at can't be 0.
Speaker 1:It's like they've set the stage so well that the equation doesn't seem so intimidating anymore.
Speaker 2:Because now it's not just some random bunch of letters and numbers.
Speaker 1:It's got meaning. It's got context.
Speaker 2:And most importantly, they emphasize that the solutions actually represent something real.
Speaker 1:So in our framing problem, a solution isn't just some abstract number.
Speaker 2:It tells you the actual thickness the frame needs to be.
Speaker 1:To use up all your framing material without wasting any.
Speaker 2:It's like that satisfying moment.
Speaker 1:I love it. This has been so helpful, but before we get completely lost in the world of picture frames and quadratic equations
Speaker 2:We gotta wrap it up.
Speaker 1:Do we actually see how to solve these equations in this particular lesson, or is that for another time?
Speaker 2:They leave us hanging just a bit, which honestly is kind of brilliant.
Speaker 1:A little math cliffhanger to keep us coming back for more.
Speaker 2:Exactly.
Speaker 1:So they leave us in suspense, but they don't leave us high and dry.
Speaker 2:They give us a little something to think about. Right?
Speaker 1:A little homework assignment for next time.
Speaker 2:Exactly. It's like they wrap up by asking, how could you take this these ideas and make them work for your own students?
Speaker 1:Okay. So not just how to solve the equations, but how to actually teach it.
Speaker 2:It's about taking those big ideas
Speaker 1:Yeah.
Speaker 2:You know, connecting to things they already know, making it fun and engaging
Speaker 1:Right. Building up that need for the equation.
Speaker 2:And then letting teachers run with it.
Speaker 1:It's almost like they're saying, we've given you the tools. Now go build something amazing.
Speaker 2:Because at the end of the day, who knows their students poorer than the teachers themselves?
Speaker 1:That's so true. I feel like we could talk about this all day, but sadly, all good things must come to an end.
Speaker 2:They really do.
Speaker 1: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.
Speaker 2:I think that's what I love about illustrative math. They really put a lot of thought into the why behind the what.
Speaker 1: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.
Speaker 2:Until next time.
Speaker 1:We'll see you then.