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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. So let's jump in. We're doing a deep dive today into a lesson that helps students really get those quadratic functions. And it uses something I think we all can relate to, visual patterns.
Speaker 2:Yeah. It's really smart. It's from the Illustrative Mathematics Algebra 1 Curriculum. And, basically, this lesson shows students that you can express the same quadratic function in a bunch of ways. You've got equations, visual representations.
Speaker 2:You can even bring in the concept of area.
Speaker 1:So it's more than just, like, memorizing formulas. They're really understanding what's behind a quadratic function.
Speaker 2:Exactly. And it all starts with a warm up activity. That's a little sneakier than it looks at first.
Speaker 1:Oh, I like it. Sneaky warm up. Okay. What's going on?
Speaker 2:So they have students figuring out the area of squares, but there's a twist. These squares have smaller squares that have been added or removed from them. So it seems kinda simple. Right? But it actually, like, secretly introduces quadratic expressions.
Speaker 2:But it's all within the context of area, which is something they've already learned about.
Speaker 1:So it gets them thinking about how when you change the dimensions of something, it's area changes. And it's more complicated than just those, like, straightforward linear relationships.
Speaker 2:Exactly. They're getting those foundational pieces for understanding quadratics maybe without even realizing it yet.
Speaker 1:I see. And that's probably helpful for students who get a little freaked out maybe jumping straight into the abstract equations.
Speaker 2:Totally. Plus, it gets them thinking about area in different ways, which is really important for where this lesson goes next.
Speaker 1:Alright. So tell me more about that. What's the next activity?
Speaker 2:So activity 3.2, expanding squares. This one's where you see that moment. Imagine you have a square. You add 4 smaller squares to the corners. Then with each step, you add another layer of those corner squares.
Speaker 2:So the shape just keeps getting bigger, but those little corner squares, they stay the same.
Speaker 1:Okay. I can picture it. Like, almost those nesting dolls, but squares. So what are the students actually doing with this pattern?
Speaker 2:They have to figure out what's the mathematical relationship between the step number and how many total squares are in the pattern. It's kinda magical because they're taking something visual and turning it into an algebraic expression.
Speaker 1:Interesting. So is there one right answer there, or could they approach it in different ways?
Speaker 2:And that's what's so called there's not. There isn't just one right answer. Right? Like, students, they might see different patterns, and they might come up with different expressions, but they end up being equivalent.
Speaker 1:So they're all solving the same puzzle, but maybe taking different paths to get there.
Speaker 2:Yeah. Exactly. And this is so good because it gets them talking about how expressions can be equivalent in algebra even if they look totally different.
Speaker 1:I could see how that would be a big moment for some students. Okay. So what are some of the ways students might approach this expanding squares pattern?
Speaker 2:Okay. So some students, they might zero in on that square in the middle of each step. Mhmm. Right? So they see it as, like, an n by n square, where n is the step number.
Speaker 2:Then you just add those 4 extra squares that are always on the corners, and that gives them the expression n plus 4.
Speaker 1:Makes sense. It's really visual.
Speaker 2:Totally. But then some students, they might look at it a totally different way. They might think of it as this bigger square. That includes those corner squares. Right?
Speaker 2:So now that bigger square has sides that are n plus 2 long, but then they have to subtract those 4 little rectangles that are in the corners, and each of those has an area of n.
Speaker 1:So that's where you get the n plus 2, 4 n.
Speaker 2:Bingo. It seems kinda complicated maybe when you first look at it, but it's really the same pattern as n plus 4. And the best part is the lesson actually, like, walks them through figuring that out, that these two things are the same, which really shows them that, hey. A quadratic function can be written in different ways.
Speaker 1:Love that. More than one way to write a quadratic function.
Speaker 2:Yes. And there's more. It's also kinda sneaking in this idea of a function. So the step number is like the input. Right?
Speaker 2:And then all the total squares, that's, like, your output. It's getting them ready for more advanced algebra stuff.
Speaker 1:Love those little sneaky moves these lessons do. Okay. So what about activity 3.3? Growing steps. What's going on there?
Speaker 2:So this one mixes it up a bit. Instead of all those squares, now they have a rectangle. And this rectangle, it grows in length and width with every step.
Speaker 1:Okay. I'm picturing it. So how does it grow exactly?
Speaker 2:It's actually pretty simple, but it really gets them thinking. Step 1 is just a 1 by 2 rectangle. Okay? Then step 2, it becomes 2 by 3. Step 3, 3 by 4.
Speaker 2:You see how it's expanding in both directions.
Speaker 1:Yeah. Like one of those ladders firefighters use. Each part is longer than the one below it. But this doesn't seem quadratic right away, does it? Seems more like it would be linear.
Speaker 2:And that's the point. That's what the lesson wants them to think about. Is it linear? And then they're given 2 different equations, n plus 1, and then also m plus n. Both of those show the pattern just in different ways.
Speaker 1:So are they trying to decide which equation is the correct one?
Speaker 2:It's more about getting them to look deeper.
Speaker 1:Right.
Speaker 2:They might think, oh, yeah. It's linear just because the sides are increasing steadily. But the lesson is designed to make them really think about it.
Speaker 1:Look deeper seems to be the theme here. So what are the kinds of questions they should be asking themselves as they're looking at this rectangle pattern? Yeah. Like those optical illusions. Or you think you see something and then, wait a minute, There's more to it.
Speaker 2:That's it. It's like the lesson saying, okay. Ask yourself this. Does the area go up by the same amount each time? Because that's how you tell if it's linear.
Speaker 2:Right?
Speaker 1:Right. Linear is all about that constant rate of change. Right. But with this rectangle, it's increasing more and more each time.
Speaker 2:Exactly. That's where you see it's quadratic. And here's the cool part. So that expression n plus 1, you might not see the squared term right away, but it's actually the same as n+n. So again, it's showing them quadratics can look different, but be the same thing.
Speaker 1:Like they're learning to spot a quadratic function even if it's in disguise. So this lesson, it does a great job introducing these concepts, but what about those common mistakes students make with quadratics? The teacher's guide, it actually talks about those, doesn't it?
Speaker 2:It does, and it's got some good advice for teachers. So one thing is students might not see the quadratic relationship if it's not using squares visually. Like, we think square. We picture a square. Right?
Speaker 2:Makes sense.
Speaker 1:So how can teachers help with that?
Speaker 2:Visuals, definitely. Break it down step by step or even have them build the patterns themselves, like, with blocks or something. It's all about making that connection between the algebra and what they can actually see.
Speaker 1:Right. Makes sense. What other misconceptions come up?
Speaker 2:Oh, this one's common. They might think anytime a variable goes up by 1, it's automatically linear.
Speaker 1:Yeah. I can see that happening.
Speaker 2:Yeah. And the rectangle pattern from activity 3.3, that's perfect for showing them why that's not always true. Because, sure, the sides go up by 1 each time, but the area doesn't. That's where you see it's quadratic, not linear. That whole idea of a constant rate of change, that's key here.
Speaker 1:This lesson is great because it doesn't just tell them about quadratics. It helps them figure it out themselves.
Speaker 2:I agree. By the end of this, they're not just memorizing. They're understanding, you know, like, what makes a function quadratic. They're seeing it. They're writing it out.
Speaker 2:It's all connecting in their minds.
Speaker 1:And it sounds like a lot of that comes from really letting them be active learners, not just listening, but figuring things out, making mistakes even, and then, boom, that moment. Powerful stuff.
Speaker 2:Absolutely. And gotta give it up for the authors at Illustrative Math. Really well designed lesson. Really gets you thinking.
Speaker 1:For sure. This was awesome getting to really dig into quadratic functions and how to teach him in a way that really clicks. And for you listening, here's something to think about. Could you use this visual approach for other math concepts like, what about exponential growth or sequences? Lots of ways to make those moments happen.
Speaker 1:Thanks for joining us on the deep dive. See you next time.