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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.)
Ever launch a tennis ball up and, like, notice how it makes that perfect arc?
Speaker 2:Or, like, those awesome fountains where the water jets, they make these smooth curves, you know.
Speaker 1:Totally.
Speaker 2:Ugh.
Speaker 1:So turns out there's some really cool math happening behind those everyday things, and that's actually what we're gonna dive into today.
Speaker 2:Quadratic relationships.
Speaker 1:Quadratic relationships. We're using an actual lesson plan from Illustrative Math. It's for algebra 1 students.
Speaker 2:Yeah. And this lesson plan unpacks how teachers can introduce this super important
Speaker 1:concept. Okay. So full disclosure. When I hear quadratic relationship, I kinda go blank. So, like, what does that even mean?
Speaker 2:So, basically, it's when one quantity changes in proportion to the square of another quantity.
Speaker 1:Okay.
Speaker 2:Instead of just adding or multiplying at a steady rate, we're squaring things.
Speaker 1:So it's not as straightforward as, like, 2, 4, 6, 8 Right. Or 5, 10, 20, 40.
Speaker 2:Big bet.
Speaker 1:This is, like, next level stuff.
Speaker 2:It is.
Speaker 1:And this lesson plan, it seems like they use some really interesting ways to help students understand it. I noticed they mentioned visual patterns, like squares within a larger square. What's up with that?
Speaker 2:Yeah. It's really cool. Our brains are naturally drawn to patterns. Right? So this lesson uses that to its advantage.
Speaker 2:When students analyze squares within a larger square, they're kind of being eased into that concept of squaring a number.
Speaker 1:Okay. So instead of just throwing an equation at them, they're visually experiencing it.
Speaker 2:Exactly. They might start thinking about different ways to find the total area, maybe breaking it down row by row or even, like, column by column, which all helps lay the groundwork for how a squared term behaves in an equation.
Speaker 1:That's really clever making it visual. Now I also saw that they use dots Mhmm.
Speaker 2:To
Speaker 1:teach us. Do we get to, like, relive or connect the dots days from childhood?
Speaker 2:Not exactly connect the dots, but you're getting warmer.
Speaker 1:K.
Speaker 2:The lesson, it puts 2 sets of dots side by side. One makes a linear pattern, the other a quadratic.
Speaker 1:Hold on. Linear pattern. Can you remind me what that is again?
Speaker 2:Oh, for sure. Linear is like adding the same amount every time. Like climbing stairs.
Speaker 1:Okay.
Speaker 2:You're taking the same size step each time. So you go up 2 steps and 4 steps and 6. You're adding 2 each time. Yeah. Exponential, though, that's more like multiplying effect.
Speaker 2:Think of, you know, that whole penny doubling thing.
Speaker 1:Right. Right. You
Speaker 2:start with 1 cent, then 2, then 4, then 8. It gets big fast.
Speaker 1:Okay. So linear steady exponential blows up, but quadratic's different.
Speaker 2:Exactly. Quadratic relationships, they're not about adding the same amount or multiplying by the same amount each time.
Speaker 1:Okay.
Speaker 2:The dots in the quadratic pattern, they grow in a way that actually shows that squaring action.
Speaker 1:Oh, okay.
Speaker 2:And by looking at these visual patterns and then, like, making tables of values, students can start to see patterns and then, like, making tables of values, students can start to see how linear and quadratic growth are different. When they graph it, they literally see the quadratic points forming a curve instead of a straight line, which is a big moment.
Speaker 1:So they're making those connections. Yeah. What they see, what's in front of them in the table, and then eventually, that equation. Yeah. It's like you're building this bridge from their eyes to their brain.
Speaker 2:Exactly. That's where teachers have to be really on their toes because students might struggle to visualize the pattern as it gets bigger and bigger.
Speaker 1:Right.
Speaker 2:They might accidentally slip back into linear or exponential thinking. It's like those ideas are just so stuck in our heads.
Speaker 1:Right. It's like they have to unlearn those other patterns to get this more complex one. Yeah. And the lesson actually helps teachers with that. Right?
Speaker 2:It does. It suggests having students draw the patterns themselves, then carefully connect what they're drawing to the numbers they put in their table. Then compare those tables to the ones they made for the linear and exponential patterns.
Speaker 1:It's like giving the teachers x-ray vision into how their students are thinking.
Speaker 2:I like that.
Speaker 1:That's awesome.
Speaker 2:It's about giving them the tools so they can help their students.
Speaker 1:So we've got squares, dots, patterns. It's all very, like, visual and hands on
Speaker 2:It is.
Speaker 1:Which I really like.
Speaker 2:Yeah.
Speaker 1:But don't we have to, like, bring in equations at some point? How does this lesson plan do that?
Speaker 2:That's where it gets really cool. Remember those squares within a square patterns?
Speaker 1:Yeah.
Speaker 2:So the lesson has students come up with different expressions to show the total number of squares for each step.
Speaker 1:So, like, there's more than one way to, like, count the squares?
Speaker 2:Exactly. Some students might count row by row, while others might see it as a big square with smaller squares taken out.
Speaker 1:Right.
Speaker 2:But the cool part is they end up with different expressions that all represent the same quadratic relationship.
Speaker 1:So they start to see that equations aren't just these random symbols.
Speaker 2:Right.
Speaker 1:They're actually different ways of describing something real.
Speaker 2:Exactly. They represent something.
Speaker 1:That's so cool.
Speaker 2:And then to really make that connection strong, they switch over to those dot patterns we talked about.
Speaker 1:Right. The linear versus quadratic ones. Yeah. Yeah.
Speaker 2:And speaking of those, the lesson plan mentions that teachers should be ready for some confusion when students are comparing them. Something about the squaring action not always being super obvious.
Speaker 1:Okay. So even when they've got those two patterns side by side, they might not get how that visual difference turns into, like, a math difference.
Speaker 2:That's right.
Speaker 1:Okay. So how does the lesson help them make that connection?
Speaker 2:By connecting what they see to the actual math expression. They use an example showing that the number of dots in that quadratic pattern can be shown with n squared, where n is the step number.
Speaker 1:Okay. So if it's the 3rd step, you'd have 3 squared, which is 9. Uh-huh. You could literally see those 9 dots.
Speaker 2:Exactly. It makes it real. Yeah. You got it.
Speaker 1:Oh, that's cool.
Speaker 2:It connects the abstract to the concrete. And then to really challenge them, the lesson throws in another curve ball. It's a visual pattern, but this time, the squares are growing in a more complicated way. The relationship is n squared plus 2.
Speaker 1:So it's not just recognizing n squared anymore.
Speaker 2:Right.
Speaker 1:Now they've gotta figure out those extra parts of the expression.
Speaker 2:Exactly. And this is where those misunderstandings we talked about might pop up again. Students might get thrown off by that plus 2 and start thinking linearly again, assuming they just add 2 each time.
Speaker 1:Because our brains like those simpler patterns.
Speaker 2:Right. Exactly. But that's why the lesson points out these potential problems. It wants teachers to get students to really explain why they're doing what they're doing, to compare different kinds of growth, and to make sure they're not relying on those old linear or exponential ways of thinking.
Speaker 1:So they're learning to think like mathematicians.
Speaker 2:Yes.
Speaker 1:Questioning themselves and looking for those deeper patterns.
Speaker 2:Exactly. And it reminds me of one of the key points from the deep dive presenter notes. Remember how they emphasize not just teaching the what, but the why behind quadratic relationships?
Speaker 1:Oh, absolutely. It's not enough to just memorize formulas. Right. Students need to grasp the bigger picture.
Speaker 2:And that's why this lesson ends with a really interesting activity. It's like a final test.
Speaker 1:Okay. What is it?
Speaker 2:Students see 3 brand new dot patterns
Speaker 1:Okay.
Speaker 2:And they have to figure out which one shows a quadratic relationship. But here's the catch, they don't get any equations.
Speaker 1:Woah. So they have to use everything they've learned about those visual patterns and tables of values Exactly. To solve it.
Speaker 2:Yes. It brings everything together and encourages a deeper understanding. It's really clever.
Speaker 1:I love it. I love how this lesson plan really lets students figure things out for themselves. Yes. But before we move on, we've been talking about quadratic expressions, but we haven't really defined it. What does that even mean?
Speaker 2:You're right. It's an important part of this whole thing. A quadratic expression is basically a math expression with a squared term. So that n squared, we keep talking about.
Speaker 1:Yeah.
Speaker 2:That's a perfect example of a quadratic expression.
Speaker 1:So even though they don't use that exact term in the lesson plan, students are working with quadratic expressions the whole time.
Speaker 2:Exactly. They're understanding how they work and how they connect to those visuals and tables without getting bogged down in definitions. It's pretty cool.
Speaker 1:It's like they're learning a new language without even realizing it.
Speaker 2:That's a great way to put it.
Speaker 1:Which is how we all learn our first language. Right? You just kinda you're immersed in it.
Speaker 2:Yes.
Speaker 1:You make mistakes, and you figure it out.
Speaker 2:Exactly.
Speaker 1:That's a
Speaker 2:good one. True.
Speaker 1:Yeah.
Speaker 2:It's very cool.
Speaker 1:That's a great point. Mhmm. And it shows how cool this lesson is. It throws students into the world of quadratics and gives them the tools to explore and discover.
Speaker 2:And make mistakes.
Speaker 1:And make mistakes. Exactly. Yeah. It's amazing how much goes into designing a lesson like this. They really thought of everything to make it interesting and easy to understand for students.
Speaker 2:I know. Right? They did a great job using visuals, hands on stuff, and just enough of the math language to get them ready for what's next.
Speaker 1:Speaking of what's next, we've been talking so much about this lesson plan that I almost forgot to ask. Mhmm. Where else do we see these quadratic relationships, like, out in the real world?
Speaker 2:Oh, they're everywhere. Once you start noticing them, you'll see them all over.
Speaker 1:Okay. Give me an example.
Speaker 2:Remember we were talking about throwing a tennis ball earlier?
Speaker 1:Yeah.
Speaker 2:The path it makes, that arc, that's a quadratic relationship in action.
Speaker 1:Oh, wow. I never would have thought of that. Yeah. So this math, it's not just some theory. It's actually how things work in the real world.
Speaker 2:Exactly. And it's not just sports. Think about bridges or arches and buildings. Oh. The way they're designed to hold weight and be strong, that's all thanks to quadratic relationships.
Speaker 1:That's really cool. It's like we're surrounded by these invisible math rules that control how things move and work.
Speaker 2:And once you understand those rules, you can actually use them to make things happen or even predict what's gonna happen.
Speaker 1:That's really powerful. Yeah. So for someone listening who's like, quadratic what? What's the big takeaway here?
Speaker 2:I hope they realize that something like a quadratic equation, which might seem kinda abstract, can actually have a huge impact on our daily lives.
Speaker 1:That's a great point. You might not see it, but it's there.
Speaker 2:Exactly. And maybe this deep dive will make them curious to spot quadratic relationships in their own lives.
Speaker 1:I love that. So to wrap things up, this episode really opened my eyes. Before today, quadratic relationships sounded about as interesting as, I don't know, watching grass grow. But now I get it. It's elegant, it's powerful, and it's all around us.
Speaker 1:A huge thank you to the creators of Illustrative Math for this lesson plan.
Speaker 2:Yes. They made something that could be really tough to grasp, easy to understand. That's pretty awesome.
Speaker 1:Absolutely. And everyone listening, keep your eyes peeled for those quadratic relationships. You might be surprised where you find them. Until next time.