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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.)
Alright, math fans. Buckle up. Because today, we're launching into the world of quadratic equations. We're doing a deep dive into a lesson plan designed for algebra 1. It's all about getting students to really understand the when and the why behind using these tools.
Speaker 2:Not just the how. Exactly. Right. Because it's not enough to just know how to solve them. This lesson plan really tries to get at the why why are these equations actually useful?
Speaker 2:We're looking at when and why do we write quadratic equations. Got the teacher's guide right here. It's got some interesting ideas in it.
Speaker 1:Yep. I see that right here in the notes. Algebra 172, lesson teacher guide 1 dot pdf. Love having the source material handy. Okay.
Speaker 1:So it sounds like this lesson plan is going straight for the heart of the matter. Like, why should students actually care about these equations?
Speaker 2:Totally. That's what it's all about, building that intrinsic motivation, showing students that these aren't just some random abstract concept. They're actually tools for understanding stuff in the real world. No. Abstract concept.
Speaker 2:They're actually tools for understanding stuff in the real world. Now that's what gets me excited. And I see here they
Speaker 1:use the example of a, wait for it, a flying potato to bring it to life. Yeah. Oh, yeah. Classic flying potato. Who doesn't
Speaker 2:love a good potato projectile? Right.
Speaker 1:It's relatable. It's engaging. So they use this to illustrate how a quadratic equation can actually map out the path of something like, well, a potato flying through the air.
Speaker 2:Exactly. They even give the equation h equals negative 16 squared plus 80 t plus 96, where h is the potato's height and t is time.
Speaker 1:Okay. Let's unpack that a bit. I know from my own experience teaching this, sometimes students have a hard time connecting that abstract equation to the actual, you know, potato flying through the air. How does this lesson plan bridge that gap?
Speaker 2:It's all about building understanding in stages. Yeah. So they start with an activity called how many tickets. This one actually doesn't even involve quadratics yet. It uses a simple linear equation to ease students into the idea of equations representing real stuff.
Speaker 1:So starting with something familiar, building a foundation.
Speaker 2:Exactly. It lets them see how even a basic equation can help solve a real problem. Yeah. They're calculating the cost of theater tickets, something they can wrap their heads around.
Speaker 1:Getting those mathematical gears turning, so they're warmed up, they're thinking about equations, and then, bam, the flying potato comes back. What happens in this activity?
Speaker 2:Here's where it gets interesting. In the flying potato again, students are tasked with figuring out when does the potato hit the ground. But there's a catch. They're told they can't just graph it to find out.
Speaker 1:Oh, I can see the wheels turning in their heads now, like, wait a minute. I can't just punch this into my calculator and be done with it.
Speaker 2:Right. And that's the whole point. The lesson plan anticipates that students might try to solve it just by isolating the variable the way you would with a linear equation. But quadratic equations, they're a different beast.
Speaker 1:Oh, I remember those moments as a student scribbling away, trying to get that tier by itself, and then realizing I'm going in circles.
Speaker 2:But that's a valuable experience too. They hit that roadblock, and it makes them more ready to learn the new strategies they actually need.
Speaker 1:So they hit a wall, and that forces them to find a detour. But before we get to that detour, I'm curious, what are some of the ways students might approach this flying potato problem if they're left to their own devices?
Speaker 2:Well, some might fall back on guess and check plugging in different numbers for t and seeing how close they get to a height of 0.
Speaker 1:Makes sense, but not the most efficient method. Right?
Speaker 2:It can take forever, and you still might not get the exact answer. Right. The lesson plan actually points this out. Sometimes students need a nudge towards those more elegant solutions.
Speaker 1:Which brings us to the next activity where things get even more interesting. What's up next for our mathematical explorers in this lesson? What's revenue from ticket sales all about?
Speaker 2:So revenue from ticket sales, this is where they shake things up a bit. They give students 2 different equations for calculating revenue. One's in factored form, and the other is in standard form.
Speaker 1:Okay. I'm sensing a theme here. Setting up a little comparison maybe to show off how useful that factored form can be.
Speaker 2:Exactly. They ask students to figure out what ticket price would actually give you zero revenue. With the factored form, it's super easy to see. You just have to find the ticket price that makes one of the factors 0.
Speaker 1:Right. Because if one factor is 0, the whole thing's gotta be 0. Boom. There's your answer. Very elegant.
Speaker 2:Exactly. But then they look at the same problem with the standard form equation, and it's not so simple. This can even get some students thinking, maybe I can rearrange this back into factored form, which is a great preview of what they'll be learning later on.
Speaker 1:I love that. They start to see that factored form is like having a shortcut. So it's not just about solving for the ticket price. It's about understanding that different forms of the equation can be more or less usable depending on what you're trying to do.
Speaker 2:Exactly. And they keep reinforcing that in the cool down activity, the movie theater. Students go back to finding ticket prices for different revenue goals. Again, they see the problem in both standard and factored form to really drive the point home.
Speaker 1:It's like they're building up a toolkit of different ways to approach these equations. But even with the best lesson plans, there are always gonna be some common misconceptions that trip students up. When it comes to quadratic equations, what are some of those things that teachers should watch out for?
Speaker 2:One big one is misinterpreting that constant term in a factored expression. Like, if you have x plus 3 by 4 plus 7.6, they might think that 7.6 represents a specific height, going back to that flying potato example.
Speaker 1:Instead of seeing it as part of the whole expression that defines the curve, they fixate on one number.
Speaker 2:Exactly. So making sure they understand that constant term in context is key.
Speaker 1:It's about the overall shape of that parabola, not just one point on it. What other misconceptions tend to come up?
Speaker 2:Another really common one is trying to solve quadratic equations the same way you would a linear equation, like trying to isolate the variable.
Speaker 1:Oh, yeah. I remember that. It's like trying to put a square block through a round hole.
Speaker 2:Exactly. Those squared terms mean you need a whole different set of tools. And then even after seeing it in action a bunch of times, some students still struggle to connect that factored form to the zeros of the function.
Speaker 1:So they might not make that leap that, hey, if back 2 is a factor, then 2 is gonna be a 0. It's about making those connections between factors, zeros, solutions, what it all actually looks like on a graph.
Speaker 2:You got it. Addressing those misconceptions is super important, and this lesson actually gives teachers some really concrete ways to do that.
Speaker 1:Give me the good stuff. What are some practical tips teachers can use to really tackle these misconceptions head on?
Speaker 2:Well, for me, visuals are always key.
Speaker 1:Definitely. Visuals, visuals, visuals. Teachers should be using graphs, diagrams, real world examples, anything to help students actually see these concepts in action.
Speaker 2:It's one thing to look at an equation on paper, but it's a whole other thing to see how it maps onto, like, a parabola on a graph or, you know, the path of a projectile potato. Gotta love a good visual with flying food. Exactly. And don't be afraid to show those quadratic equations in different ways. Have students write them out in standard form, then factored form.
Speaker 2:Make those connections clear.
Speaker 1:It's like learning a new language. Right? The more ways you see it used, the better you understand it.
Speaker 2:Totally. And you know what else really helps solidify understanding? Getting students talking to each other about math.
Speaker 1:Oh, for sure.
Speaker 2:Think pair share, group work, class discussions, all that good stuff. When they have to explain something to someone else, it just clicks.
Speaker 1:It's true. It's like you're teaching yourself at that point.
Speaker 2:Right. And even those misconceptions we talked about, those can be turned into learning opportunities.
Speaker 1:Oh, yeah. Those moments when you finally figure out what you were getting wrong the best.
Speaker 2:Exactly. So don't shy away from addressing those misconceptions directly. Help students figure out where they went off track.
Speaker 1:It's all part of the learning process. Right? Making mistakes, figuring it out, putting the pieces together.
Speaker 2:Absolutely. And I I think this lesson does a really nice job of embracing that whole process. Yeah. It's not just about finding the right answer. It's about exploring, asking questions, making those connections.
Speaker 1:It's about giving students the tools and the confidence to see themselves as mathematicians. You know? Like, they can totally handle this. This deep dive has been amazing. So many good takeaways.
Speaker 1:And I just have to say huge thanks to the authors of Illustrative Math for putting this lesson plan together.
Speaker 2:It's a good one for sure.
Speaker 1:Absolutely. Well, that's about all the time we have for our deep dive today. But before we go, I wanna leave you all with one final thought. How else could you use that flying potato scenario or a similar real world example to really get those quadratics to stick? Get creative, have some fun with it.
Speaker 1:Until next time. Happy teaching.