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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 wonder how those, like, graceful parabolic curves, you know, the ones, like, in bridges or the path of basketball, well, how are they shaped by math?
Speaker 2:It's more than just recognizing the equation. We're gonna you'll actually grasp how changing those equations, even slightly, can, like, shift and reshape those those iconic parabolas.
Speaker 1:So today's deep dive takes us into the heart of quadratic functions. We're drawing our insights from, let me see, a high school algebra lesson plan.
Speaker 2:Yeah. About changing the vertex of a parabola.
Speaker 1:Exactly. Think of this deep dive as like a a crash course or maybe a refresher on how those visual representations of math actually work.
Speaker 2:Yeah. It's all about making those parabolas dance to your tune just by understanding, the magic, I guess, of vertex form.
Speaker 1:Okay. So vertex form of a quadratic equation, you know, the one that looks like yxh2plusk. Right. I know. I know.
Speaker 1:Sounds like we're speaking another language.
Speaker 2:But but this form is actually incredibly useful. Unlike the standard quadratic equation, vertex form gives you, you know, a clear snapshot of the parabola shape and position just from the equation itself. And the lesson plan we're exploring today uses a well, I think it's brilliant.
Speaker 1:It's called, get this, peanut jumping over a wall to really drive this point home.
Speaker 2:This activity, it it cleverly addresses a common hurdle for learners, connecting that abstract equation to the visual representation of the graph, you know. So
Speaker 1:imagine you're trying to animate a little peanut.
Speaker 2:Yeah.
Speaker 1:Jumping over a wall using, you guessed it, a quadratic function.
Speaker 2:Right.
Speaker 1:What are some common struggles this activity is designed to overcome?
Speaker 2:Well, students often struggle to visualize how changes to the equation, you know, translate to the Peanuts trajectory. They might change a number and expect the Peanuts to, like, teleport across the screen, not realizing it needs to follow a smooth curved path dictated by the equation.
Speaker 1:I can see that happening. So how does the activity help them bridge that gap?
Speaker 2:Peanut jumping over a wall encourages, kind of a trial and error approach. Students can tinker with the equation, changing the starting point of the jump, adjusting the peak of the parabola, and see how those tweaks play out visually.
Speaker 1:It's like giving them a virtual sandbox
Speaker 2:Yeah.
Speaker 1:To experiment and really get a feel for how those numbers influence the graph.
Speaker 2:Precisely. And the beauty of it is that there isn't always just one right answer.
Speaker 1:Oh, interesting.
Speaker 2:Just like in real life, there can be multiple ways to make that peanut clear the wall.
Speaker 1:I like that.
Speaker 2:This helps dismantle the misconception that math is all about finding the, quote, one true solution.
Speaker 1:And it highlights the importance of understanding the underlying concepts rather than just memorizing formulas. But let's not forget this isn't just about animated peanuts. Right?
Speaker 2:Exactly. This type of thinking is applicable in a wide range of scenarios. Imagine you're designing a bridge or a skate ramp or even mapping the trajectory of a rocket launch.
Speaker 1:Speaking of real world applications, this lesson plan seems to be full of them. We've covered the peanut jumping, but I hear there's also something about a smiley face graph. Okay. Smiley face graphs. Tell me more.
Speaker 2:So the, smiley face activity, it kinda builds upon the concept we were talking about with, peanut jumping over a wall.
Speaker 1:Okay.
Speaker 2:This time, students are challenged to construct the different parts of a smiley face by manipulating quadratic equations.
Speaker 1:Interesting.
Speaker 2:Yeah.
Speaker 1:So it's like taking those basic parabolas and and turning them into building blocks for more complex shapes.
Speaker 2:Precisely. They're given a basic parabola, and they have to figure out how to transform it to create, you know, an eye, the curve of a smile Mhmm. And so on.
Speaker 1:I see.
Speaker 2:This involves combining different transformations, like shifting the parabola up or down, stretching or or compressing it, even flipping it upside down.
Speaker 1:So they're really flexing those equation manipulation muscles. What else makes this activity stand out?
Speaker 2:It introduces the concept of domain restriction.
Speaker 1:On domain. That's about defining the boundaries of where the parabola exists. Right?
Speaker 2:Yeah. Yeah.
Speaker 1:Like saying, you could only exist between these two points on the graph.
Speaker 2:Exactly. For example, to make a smile, the parabola can't just go on, you know, forever in both directions.
Speaker 1:Right. Right.
Speaker 2:Students need to learn how to contain it within a specific range of x values.
Speaker 1:I'm starting to see how these activities are carefully designed to, like, layer upon each other, each one building upon the previous concept.
Speaker 2:Right.
Speaker 1:1st, understanding the basic shifts, then applying that to a fun scenario like the the peanut jump, and now incorporating domain restrictions to create, you know, specific shapes.
Speaker 2:Absolutely. And and there's another layer of, well, I think brilliance woven throughout this lesson plan.
Speaker 1:Okay.
Speaker 2:The emphasis on explanation.
Speaker 1:Okay.
Speaker 2:Students aren't just aiming for the right equation.
Speaker 1:Yeah.
Speaker 2:They're encouraged to explain their reasoning, to articulate why their chosen transformations achieve the desired outcome.
Speaker 1:That's such a critical aspect of learning, isn't it? Being able to explain something clearly, I think, demonstrates true understanding.
Speaker 2:Precisely. It moves them beyond rote memorization
Speaker 1:Yeah.
Speaker 2:And into the realm of genuine comprehension. And that's what's so fantastic about this this deep dive. We're not just revisiting high school algebra.
Speaker 1:Right.
Speaker 2:We're rediscovering the, you know, elegant language of shapes and patterns that surround us every day.
Speaker 1:It's like we're peeling back the layers of the everyday world Yeah. To to reveal the mathematical framework underneath.
Speaker 2:Yes.
Speaker 1:So as we, you know, approach the end of this deep dive, what are some some key takeaways our listeners should, you know, hold on to?
Speaker 2:Well, we started with the power of vertex form.
Speaker 1:Right.
Speaker 2:This this alternative way of expressing quadratic equations that unlocks, you know, a deeper understanding of a parabola's shape and position.
Speaker 1:And we saw how those what seem like abstract equations can really be used to model, you know, real world situations. Yeah.
Speaker 2:Like that jumping peanut.
Speaker 1:The trajectory of a jumping peanut or even, like, the curve of a smile.
Speaker 2:Exactly. And it helps us see those connections between, like, you're saying the equation and that visual representation. And you know what? That brings us to another important takeaway.
Speaker 1:What's that?
Speaker 2:The importance of being able to explain your reasoning.
Speaker 1:Yeah. That
Speaker 2:was Being able to explain, like, why a certain equation results in that particular parabolic curve.
Speaker 1:Right. It's not just about getting the right answer.
Speaker 2:Yeah. It's about understanding why.
Speaker 1:It's about understanding those, like, underlying principles.
Speaker 2:Exactly.
Speaker 1:So the next time that you see a parabola, you know, whether it's, the arch of a bridge or the path of, like, a ball someone's thrown through the air, even just a curve in a design You're
Speaker 2:gonna start seeing those hidden equations everywhere, shaping the world around us.
Speaker 1:It's like you're gaining a a secret decoder ring for the universe.
Speaker 2:I like that.
Speaker 1:A huge thank you to the authors of Illustrative Math for, you know, giving us this really rich and thought provoking lesson plan to to explore and talk about.
Speaker 2:It really is beautifully done, the way they make math accessible and engaging.
Speaker 1:And relevant to our lives.
Speaker 2:Yeah. Absolutely.
Speaker 1:And to you, dear listener, we hope this deep dive has, left you with a new appreciation.
Speaker 2:A renewed appreciation.
Speaker 1:Yeah. A renewed appreciation for the power and the beauty of mathematics.
Speaker 2:Here. Here.
Speaker 1:Until next time, keep those parabolas in your sights, and we'll catch you on our next deep dive.