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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.)
Hey, everyone. Welcome back. Are you ready for another deep dive? Today, we're tackling exponential functions, and let me tell you this is one lesson plan you'll wanna, you know, really get right because we're not just talking about helping your algebra students solve equations. We want them to actually see how these functions work, especially when it comes to understanding those tricky concepts of growth and decay.
Speaker 2:It's about making it visual. Right?
Speaker 1:Absolutely. So to help us unpack this, we're turning to some excerpts from, drum roll, please, an algebra curriculum guide. And let me tell you, they have a really interesting approach to introducing exponential functions, spending gift money.
Speaker 2:Relatable. Students get that. Right?
Speaker 1:Exactly. Everyone's had that experience or at least wished they had. So picture this. Jada gets a nice little sum of gift money, a $180 to be exact. Now every week, Jada spends a third of what she has left, and here's where the learning comes in.
Speaker 1:Students have to figure out which equation represents the money she has remaining after a certain number of weeks.
Speaker 2:That's clever. Using a scenario like that, it eases them into this idea of exponential decay without, you know, overwhelming them with the abstract stuff right away.
Speaker 1:It's like sneaking in a little math lesson while they think they're just figuring out Jada's spending habits.
Speaker 2:Right? Exactly. And it suddenly introduces that concept of b, the base of the exponential function, as, like, a factor of change over time.
Speaker 1:Oh, I see where you're going with this. So in this case, b represents the 2 thirds of her money that's, like, remaining each week.
Speaker 2:Precisely. Not the 1 third she's spending, which is where some students might, you know, get a little tripped up.
Speaker 1:It's almost like a little trap in the problem itself. You're making them really think about what the equation is actually representing.
Speaker 2:And that's the key. Right. Because the way the problem's worded, it would be easy to just grab that 1 third and go. Yeah. But the lesson anticipates that.
Speaker 2:It emphasizes the remaining fraction that 2 thirds to really drive home the point that b represents what's left, not what's gone.
Speaker 1:Oh, those sneaky curriculum writers. I like it. It's all about those subtle nuances. But that's what sets a strong foundation. Right?
Speaker 1:Getting those little details right from the get go.
Speaker 2:Absolutely. It makes all the difference.
Speaker 1:So we've got Jaida. We've got her shrinking pile of cash. We're starting to understand how this whole b thing works. I'm guessing it's time to dive into the, the actual equation now. Right?
Speaker 2:Time to unravel why is NEPX, my friend.
Speaker 1:The real fun begins. And by fun, I mean, the slightly terrifying world of graphs and transformations.
Speaker 2:Right. You know it. And this is where things get really interesting for students because they actually get to, like, play around with a and b in the equation and see how those changes play out on a graph.
Speaker 1:Like, they're in the driver's seat of this mathematical machine. Speaking of which, I'm seeing that the lesson encourages using graphing technology, things like Desmos.
Speaker 2:Crucial. It lets them see those changes happening in real time, which is so powerful for understanding.
Speaker 1:I bet. Instead of just, like, imagining in their heads, they're seeing those lines move and groove based on how they tweak the equation?
Speaker 2:Exactly. That dynamic exploration, it's gold. Yeah. It takes the abstract and makes it tangible, you know.
Speaker 1:I love it. But I'm also wondering as students start to, like, experiment and play around with those equations and graphs, are there any common misconceptions or roadblocks that teachers should, like, be prepared
Speaker 2:for? Oh, there are always a few tricky spots. One I see a lot is, students struggling to choose a good graphing window. Like, they zoom in too much or not enough, and they miss the big picture of how the function is behaving. Yeah.
Speaker 1:It's like trying to see a whole, I don't know, a whole elephant through a straw or something. Right? You only get a tiny piece of what's going on.
Speaker 2:Exactly. So helping them adjust that window, finding that right zoom level, that's key. They need to be able to see it all, the e intercept, how steep the curve is, the whole shape of the thing.
Speaker 1:It's about giving them those tools to, like, frame the problem visually. Right?
Speaker 2:Love it. And speaking of seeing the whole picture, we've been talking a lot about exponential growth, but this lesson doesn't just leave it there, does it? I mean, we gotta talk about decay too. Right? The the flip side of the coin.
Speaker 2:You can't have one without the other. And that's where things get really interesting, I think, because it challenges some assumptions.
Speaker 1:Yeah. Because we tend to think exponential. We think bigger, bigger, bigger.
Speaker 2:Right? But with decay, we're talking about things getting smaller over time.
Speaker 1:Mhmm.
Speaker 2:And that's where the base, that b value, takes on a whole new meaning.
Speaker 1:Okay. So walk us through that. What happens when b is, say, less than 1?
Speaker 2:So the next activity in this lesson, it has students compare functions where b is less than 1. This is where they really start to see that, hey, exponential functions, they don't always go up, up, up. Sometimes they go down, down, down.
Speaker 1:Okay. I remember this clicking for me as a student, and it was like, woah. Mind blown. Because it's it's kind of counterintuitive at first. Right?
Speaker 2:Totally. Because a smaller base, like closer to 0, that actually means faster decay. It's like the closer b is to 0, the quicker things shrink.
Speaker 1:It's like that saying going down in flames, but for math, the faster you burn, the quicker you're gone. Right?
Speaker 2:You got it. And, you know, a good real world example can really help solidify this concept. Think about radioactive decay, for example.
Speaker 1:Oh, that's a good one. Or even just, like, the value of, I don't know, the value of a car going down over time.
Speaker 2:Precisely. It's about connecting those mathematical concepts to something tangible that students can, you know, grasp onto.
Speaker 1:And probably relate to a little bit more than, you know, abstract numbers and graphs. Right?
Speaker 2:Yeah.
Speaker 1:So once they've played around with both growth and decay, I'm guessing the lesson then brings it all together. Time to synthesize. Right?
Speaker 2:Exactly. And they do it in a really engaging way too. This last activity, it's all about prediction. They're given different scenarios where the a and b values are changing, and they have to sketch what they think the graph will look like before actually graphing it.
Speaker 1:Oh, that's cool. So it's not just about plugging in numbers anymore. It's about really internalizing how those changes impact the overall shape and behavior of the graph.
Speaker 2:It's like they're becoming exponential detectives. They see the clues, those a and b values, and have to predict how the graphs can play out.
Speaker 1:I love that analogy. But I imagine this is probably where some students might need a little extra support. Right?
Speaker 2:Absolutely. And the lesson actually suggests starting with, like, simpler base values. You know, like, maybe start with 2 and then, kind of build from there.
Speaker 1:Yeah. Give them a little confidence boost before throwing in, like, fractions and decimals.
Speaker 2:Exactly. Baby steps. But, you know, what I love about this whole lesson, it's not just about memorizing formulas. Right? It's about empowering students to actually see the math, to understand the why behind the what.
Speaker 1:It's like giving them x-ray vision for exponential functions. They look at an equation, bam, they could picture the graph. Or they see a graph, and they're like, I know what you did there. Equation.
Speaker 2:Right. It's connecting that symbolic language to the visual language
Speaker 1:Mhmm.
Speaker 2:And understanding how those 2 work together to, you know, unlock these really cool dynamics.
Speaker 1:Totally. And I feel like this whole deep dive has been like that for me. You know. Just really eye opening to see how thoughtfully this lesson was designed. It's not just about throwing information at students.
Speaker 1:It's about anticipating those little hiccups and those common misconceptions and then guiding them through it.
Speaker 2:Absolutely. It's about giving them the tools and then letting them explore. Right?
Speaker 1:Exactly. So for all of our listeners out there, the ones who are, like, getting ready to bring this lesson to life for their own students, Any final thoughts? Any parting wisdom you wanna leave them with?
Speaker 2:You know, I think a really powerful question for educators to consider is this. How can we extend this even further? How can we help students connect exponential functions to real world applications that, you know, really resonate with their lives?
Speaker 1:Oh, I love that. Make it relevant.
Speaker 2:Exactly. Challenge them to find examples of exponential growth or decay in, like, current events or, I don't know, science, music, whatever they're into.
Speaker 1:Yes. Help them see that math isn't just this thing stuck in a textbook. Right? It's everywhere.
Speaker 2:It's the language of the universe.
Speaker 1:Oh, I like that. Well said. And on that note, a huge thank you to the authors of Illustrative Math for providing these incredible resources. And to all of you listening, keep exploring, keep questioning, and keep diving deep into the wonders of mathematics.