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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 dive into this lesson plan on exponential functions, specifically focusing on how to interpret graphs. It's one thing to plot points. This is about really understanding what those curves tell us, you know?
Speaker 2:Absolutely. It's like we're giving your students x-ray vision for exponential graphs. They'll be able to connect a graph to its equation without breaking a sweat, really grasp the nuances of growth and decay, and even make predictions about, like, real world scenarios.
Speaker 1:Sounds impressive. And the lesson plan we're looking at today starts with this intriguing warm up activity called which one doesn't belong. It gives students 4 functions, y 2 to the x power, y 8 times 0.25 to the x power, y 8 times 2 to the x power, and y into x plus 8, their task, to figure out which one doesn't fit with the rest.
Speaker 2:I love this approach because it really gets those wheels turning. Instead of jumping straight into graphing, they have to stop and analyze each equation. They're asking themselves, okay, is this exponential growth decay? What's the initial value here?
Speaker 1:It's like a logic puzzle for math. And I think this sets the stage for a bigger idea in the lesson. Right? That constant interplay between equations, graphs, and tables.
Speaker 2:Exactly. We don't want our students to just be robots plotting points. They need to see how each of those representations tells the same story just in a different language, and that's where those light bulb moments happen.
Speaker 1:Speaking of light bulb moments, the lesson plan that moves into this really cool scenario called the value of a computer activity. We're talking about depreciating assets, super relevant to real life, but there's a twist. Right?
Speaker 2:Oh, yeah. There's definitely a twist. This activity presents students with a graph that shows how a computer's value depreciates over time, but here's the catch. The graph doesn't use those nice neat 1, 2, 3 values on the x axis.
Speaker 1:So they can't just zip over to year 2 on the graph and instantly see what the value is. That's where the real thinking kicks in.
Speaker 2:Exactly. And that's what makes this activity so great. It encourages a more, what I call strategic exploration. They might try guess and check, testing out different decay factors to see which one best matches the graph, or maybe they'll use their estimation skills.
Speaker 1:It's great that it forces them to think outside the box instead of just relying on rote memorization.
Speaker 2:You got it. And when I did this activity with my students, some of them were definitely thrown off at first by those uneven time jumps on the x axis. But once I encourage them to create a table and test out different decay factors, that's when they started to have those moments.
Speaker 1:And those moments are priceless. Yeah. Okay. So we got those brains warmed up, and we hit them with the value of a computer. What challenge awaits next in this grand exponential adventure?
Speaker 2:Well, next up is the moldy wall activity, and it really kicks things up a notch. It's like imagine this. You got these 2 types of mold. Right? And they're both growing on this wall, but they're growing at different rates.
Speaker 2:One doubles in size every day, and the other one, it triples. Now the students are given 2 graphs, but there are no numbers on the axis. They only have the single point where the graphs intersect that's labeled for them, and they have to figure out which graph represents each type of mold.
Speaker 1:Wow. Okay. So they really have to use their understanding of how those growth factors, that doubling versus tripling, affect the shape and the steepness of the graph. Right?
Speaker 2:Exactly. It's all about those visual cues. So the graph that represents the mold that's tripling, that one's gonna have this really steep, almost vertical climb compared to the one that's just doubling. And this activity really drives home the point that the steeper that curve, the faster that growth rate.
Speaker 1:I love it. It's like a visual puzzle. But I imagine that single intersection point where the graphs meet, that's gotta lead to some interesting discoveries.
Speaker 2:Oh, absolutely. That point represents the moment when both types of mold cover the same area on the wall. It's like this powerful visual representation of how those different growth rates actually play out over time.
Speaker 1:It's amazing how a single point on a graph can hold so much meaning. I bet this activity sparks some really interesting discussions in the classroom.
Speaker 2:It definitely does, and it helps students understand that interpreting these graphs isn't just about plugging numbers into formulas. It's about seeing patterns, making connections, extracting meaning from those visual skills that go way beyond just math class.
Speaker 1:It sounds like this activity really pushes students to think critically and creatively about how we can use graphs to make sense of, like, real world situations. But before we, get too far ahead of ourselves, let's talk about some potential misconceptions students might have along the way because even the brightest minds can stumble sometimes. Right?
Speaker 2:Absolutely. It's all part of the process. Right? And with exponential functions, there are definitely some common sticking points.
Speaker 1:Like that whole thing with a graph steepness Mhmm. And what it actually tells us about the growth rate, it's really easy to look at a steep graph and just assume it automatically means faster growth, even if it starts lower down on the axis.
Speaker 2:It's true. We're kinda conditioned to think that way, aren't we? Steep equals fast. But the thing about exponential growth is that initial value where things start, that plays a huge role too. You know, I always like to bring up the example of investments with my students.
Speaker 2:Imagine, you know, one investment, it starts off small, but it has this really high interest rate. Then there's another investment, starts off way bigger, but has a lower interest rate.
Speaker 1:So even though that second investment might look more impressive at first because it's starting higher, that little investment with its supercharged growth rate, that one could end up winning the race in the long run.
Speaker 2:Exactly. And when they see that play out on a graph, graph, when those two curves intersect, it really drives home the point that a steeper line doesn't always equal a bigger value over time.
Speaker 1:It's like that classic story of the tortoise and the hare, but for exponential functions.
Speaker 2:Okay. So speaking of potentially misinterpreting graphs, I feel like another big one is figuring out that growth or decay factor, especially when those input values on the x axis aren't nice and consecutive. We saw a bit of that with the value of a computer activity, but it seems like that could throw students off in a lot of different situations.
Speaker 1:Totally. When you don't have those nice even jumps on the x axis, it can be really easy to get stuck trying to calculate the factor for each little step. Mhmm. You know? But when I teach this, I always encourage my students to look for what I call those, sweet spots on the graph.
Speaker 2:Sweet spots.
Speaker 1:Like yeah. So I'm talking about points where you can clearly see that the output value has, say, doubled or tripled compared to a previous point. Like, let's say we're looking at a graph of bacterial growth, and we see that at hour 2, the population was 1,000, and then boom, at hour 4, it's jumped to 2,000.
Speaker 2:So even though we don't have data for hour 3, we can still tell that the doubling time is 2 hours because the population doubled over that time period. Exactly. And once they get that, figuring out that growth factor becomes a lot less daunting even with those gaps on the x axis. It's all about looking for those key relationships.
Speaker 1:This is so helpful. I'm already feeling more prepared to tackle exponential functions with my students. But before we wrap up, I really wanna hear more about that thought provoking question you mentioned earlier. The one that's supposed to, like, blow our students' minds and leave them thinking about exponential growth and decay long after the lesson is over.
Speaker 2:Oh, yeah. The grand finale. I think it's so important to connect this stuff to what's happening outside of school. Right? We've talked about mold, computers, even investments.
Speaker 2:But exponential relationships, they're everywhere. We just have to know how to look
Speaker 1:for them. That's so true. Once you start seeing the patterns, it's like you can't unsee them. But how do we actually get our students to have that moment of realizing how prevalent these functions are in their own lives?
Speaker 2:You know, I've actually found that one of the best ways is to kinda turn the tables and make them the explorers. Instead of just giving them examples, challenge them to find examples of exponential growth and decay themselves. Tell them to look for it in their own lives, in the news, even in history books.
Speaker 1:That's such a great idea. Mhmm. It takes it from being this abstract concept to something that's relevant and engaging.
Speaker 2:Right. Mhmm. It's
Speaker 1:like instead of saying, hey. Remember that compound interest formula? We can say, go find something in your world that shows exponential growth or decay in action and tell us about it.
Speaker 2:Exactly. Let them be the detectives, and you know what? They'll start noticing it everywhere. Like, oh, there's exponential growth happening in this rabbit population or that's exponential decay happening as this radioactive material breaks down. And, hey, even the way things can go viral on social media, that's exponential too.
Speaker 1:It really is amazing how it all connects. And I imagine the discussions you could have when they bring those real world examples back to class. I mean, a student who's, like, totally fascinated by a news story about, say, honey bee populations declining, they could actually graph that data and see that exponential decay happening right before their eyes.
Speaker 2:And that's when it becomes real for them. Right? It's not just some abstract formula anymore. It's like, wow, this math stuff, it's actually showing up in this thing I care about.
Speaker 1:Absolutely. This whole deep dive has been such a good reminder that teaching exponential functions, it's about so much more than just, you know, plotting points on a graph. It's about, like you said, making it real. Yeah. Getting them to think critically, to explore, and to really be, like, wowed by the math that's happening all around them.
Speaker 2:Totally. And when we can get our students to see the world through that exponential lens, it just opens up so many possibilities.
Speaker 1:So true. And on that note, I think we'll leave our listeners with that challenge to go out there and find those exponential relationships hiding in plain sight. Thanks for joining us on this deep dive into the fascinating world of exponential functions. See you next time.