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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. Check this out. We're diving deep into algebra today, but with a fun twist.
Speaker 2:Oh, a twist. I like twists. Algebra could be a bit dry sometimes.
Speaker 1:Tell me about it. But no worries. This time, we're exploring a lesson plan on, get this, bouncing balls.
Speaker 2:Bouncing balls in algebra. That sounds intriguing.
Speaker 1:It is. It uses the concept of bouncing balls to teach students about exponential functions.
Speaker 2:Okay. I'm intrigued. Exponential functions, those can be a bit tricky to grasp. Right?
Speaker 1:Right. But this lesson plan makes it way more engaging. We've got the actual lesson plan to look at.
Speaker 2:So tell me, how do bouncing balls and exponential functions even relate to each other?
Speaker 1:Well, think about it. Each time a ball bounces, it doesn't go up as high as the previous bounce. Right? Yeah.
Speaker 2:That's true. The height of each bounce keeps decreasing.
Speaker 1:Exactly. And that decrease can be modeled using an exponential function.
Speaker 2:I see. So students get to analyze the relationship between the bounce number and the height.
Speaker 1:Yes. They get to see how this mathematical concept plays out in a real world scenario.
Speaker 2:I like that. It's so much more interesting than just looking at numbers on a page. Yeah. So walk me through it. How does this lesson plan actually work?
Speaker 1:Well, it starts by emphasizing how important it is to choose the right kind of model for the data, figuring out if it's linear, exponential, or something else entirely.
Speaker 2:Right. Because you can't just assume everything follows a straight line. Sometimes things grow or decay much faster. Right?
Speaker 1:Exactly. And that's where the exponential functions come in. But then the lesson takes a step further and dives into graphs.
Speaker 2:Graphs? Are we talking about plotting those bounces on a graph?
Speaker 1:You got it. But it's not just about plotting the points. The lesson also highlights how even a tiny change in how you set up that graph can completely change how the
Speaker 2:data looks. Oh, I see. You mean, like, changing the scale of the axis can make a big difference. Exactly. And this is where things get really interesting because it forces
Speaker 1:students to think critically about the tools they're using and how those tools can actually influence their interpretation of the data.
Speaker 2:So it's like teaching them to be savvy consumers of information. They can't just blindly trust a graph without understanding the context and how it might be skewed. That's a valuable skill in any field. So are we talking about students collecting their own data on bouncing balls?
Speaker 1:They can. The lesson plan includes an optional hands on activity where students experiment with different types of balls and record their bounce heights.
Speaker 2:That's awesome. So they get to experience firsthand how even small variations in the type of ball can affect its bounciness. It's like a mini science experiment combined with math.
Speaker 1:Exactly. And this hands on experience can really help solidify their understanding of the concepts.
Speaker 2:So they're not just passively learning about exponential functions. They're actually seeing them in action. But real world data can be messy, can it? I mean, I doubt those bouncing balls will create perfectly smooth curves on a graph.
Speaker 1:You hit on a really important point. Real world data rarely fits perfectly into those neat little boxes we find in textbooks, and you are right. Bouncing balls are no exception.
Speaker 2:So how does this lesson plan deal with that? How do you teach exponential functions when the data itself is all over the place?
Speaker 1:That's where this lesson plan truly shines. Instead of focusing on finding the perfect exponential function, it encourages a more exploratory approach, prompting students to try different strategies for analyzing that data.
Speaker 2:Oh, I like that. So it's less about rote memorization and more about problem solving and critical thinking. What kind of strategies are we talking about here? Give me an example.
Speaker 1:Well, they might look at the differences between each bounce height. You know, like, does the ball consistently lose half its height with every bounce? Or is there a different pattern?
Speaker 2:I see. So they're looking for patterns and relationships within the data itself, which can help them understand if an exponential model is even a good fit.
Speaker 1:Exactly. And the lesson plan encourages them to visualize the data in different ways too, maybe even by plotting it on a graph. They can experiment with different curves to see what fits best.
Speaker 2:So they're like data detectives trying to uncover the hidden mathematical structure beneath the surface. I love that.
Speaker 1:Me too. And it really highlights how even when something seems random, there might be underlying patterns and rules governing its behavior. But let's face it, not every kid is a natural born data detective. I'm guessing there are common pitfalls students might fall into when trying to wrangle all this bouncy data.
Speaker 2:Oh, absolutely. One of the biggest things they struggle with is that they expect that real world data is going to behave like the textbook problems.
Speaker 1:Right. They want that perfectly smooth curve every single time.
Speaker 2:Precisely. And when they get data that's a little messier, it throws them off.
Speaker 1:It's like they've never dropped an actual ball outside of a physics simulation.
Speaker 2:Exactly. So this lesson plan does a really smart thing by talking about finding the best fitting model.
Speaker 1:Okay. So acknowledging that it might not be a perfect match. Right? Like, the the real world is messy.
Speaker 2:Yes. It's about understanding that those models, those equations, they're representations.
Speaker 1:Not necessarily a one to one with reality.
Speaker 2:Exactly. And this ties into another potential pitfall for students.
Speaker 1:Oh, do tell. I live for these gotchas in learning.
Speaker 2:This one has to do with the difference between discrete data and continuous data.
Speaker 1:Okay. I'll admit, sometimes those terms trip me up. Remind me, what's the distinction again?
Speaker 2:Think of it this way. Discrete data, it's like counting. You can have 1 bounce, 2 bounces, but you can't have 1 and a half bounces.
Speaker 1:Right. You can't have half a bounce just like you can't have half a student.
Speaker 2:Exactly. But with continuous data, you can divide it up however you want.
Speaker 1:Like on a graph, you could draw a line connecting the bounces even though in reality, there's no such thing as halfway between bounces.
Speaker 2:Exactly. And this lesson really wants teachers to hit that idea home because it matters for how we analyze things.
Speaker 1:Because it's not just about the math. It's about understanding the nature of what you're measuring. Right?
Speaker 2:Right. It's about helping students become better thinkers about data, not just calculators.
Speaker 1:And that idea of real world application, it seemed like the lesson hinted at other areas where exponential functions pop up.
Speaker 2:It did. It gave a little shout out to things like, oh, how about sound decay? That's an exponential function at work.
Speaker 1:Woah. Really? I never thought about that, but it makes sense the way sound fades away.
Speaker 2:Exactly. Or how about population growth under ideal conditions? Exponential.
Speaker 1:So this lesson, it's like giving students a peek into a bigger picture.
Speaker 2:It is. And for teachers, that opens up so many possibilities.
Speaker 1:Okay. So if you're a teacher about to use this lesson plan, what are your top takeaways? Give me the short version.
Speaker 2:1st, don't be afraid of real world data. Let the students wrestle with it.
Speaker 1:Let them see that math isn't always neat and tidy.
Speaker 2:Exactly. 2nd, don't shut down different approaches. Let them explore different ways to analyze that bouncy data.
Speaker 1:So more discussion, less lecturing.
Speaker 2:You got it. And finally, don't shy away from those misconceptions we were talking about. Use those as learning moments.
Speaker 1:Turn those moments into teaching gold.
Speaker 2:Precisely. Guide them to question, to think critically.
Speaker 1:It's about giving them the tools to be mathematical thinkers.
Speaker 2:Yes. And, you know, it's funny. We keep coming back to those real world applications.
Speaker 1:Well, those are always fun.
Speaker 2:Remember that rebound factor we talked about with the balls? Yeah. How high they bounce back up?
Speaker 1:Right. Like, how much energy they keep after hitting the ground.
Speaker 2:That idea, it applies to more than just balls.
Speaker 1:Oh, give me an exam. What else can rebound?
Speaker 2:Sound. Think about how sound fades away in a room. Okay.
Speaker 1:Yes, ma'am.
Speaker 2:Basically exponential decay happening.
Speaker 1:Wait. Really? So the sound waves are losing energy with each bounce just like the ball?
Speaker 2:Precisely. Each time the sound hits a wall, some energy is absorbed.
Speaker 1:So hold on. Does that mean the rebound factor for sound, it's different depending on the room?
Speaker 2:Absolutely. Think about a big empty room, hard surfaces everywhere.
Speaker 1:Lots of echo. Right.
Speaker 2:Exactly. High rebound factor because the sound waves bounce around longer.
Speaker 1:Versus, like, a recording studio, all the soft panels.
Speaker 2:Tons of absorption going on there, so the sound dies out quickly, Low rebound factor.
Speaker 1:So that echo I get in my apartment, I can blame it on a high rebound factor.
Speaker 2:There you go. You're getting it. But it goes even further than that.
Speaker 1:Oh, no. You're gonna blow my mind again, aren't
Speaker 2:you? Maybe just a little. This rebound idea, it even applies to medicine.
Speaker 1:Yeah. That's just crazy talk.
Speaker 2:Not at all. Think about when you take a pill.
Speaker 1:Okay.
Speaker 2:The amount of medicine in your blood, it decreases over time.
Speaker 1:Yeah. That's why you gotta take more doses.
Speaker 2:And that decrease, often, it's exponential decay.
Speaker 1:So the rebound factor there is, like, how much of the medicine is left after a while?
Speaker 2:Exactly. And that affects how often you need to take it to keep it working.
Speaker 1:Wow. We started with bouncing balls, and now we're talking about life saving drugs.
Speaker 2:It all comes back to math. It's everywhere.
Speaker 1:It really is amazing. Big thanks to the illustrative math folks for this lesson plan.
Speaker 2:Such a clever way to teach this stuff.
Speaker 1:And thanks to you for breaking it all down. Until next time, everyone.