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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.)
Okay. So have you ever, like, started a rumor? I'm kidding. I'm kidding. But Yeah.
Speaker 1:You know how it goes. Someone tells 1 person, they tell 2 more. And before you know it
Speaker 2:Right. Right.
Speaker 1:Everyone's in the loop. Yeah. It's exponential growth. Yeah. And it's not just how gossip spreads.
Speaker 1:It's a big deal in algebra. So today, we're diving deep into teaching exponential functions, effectively using lesson 9 from Illustrative Mathematics Algebra 1 Curriculum as our guide.
Speaker 2:Sounds good. Yeah. Understanding exponential functions opens up so many possibilities from finance to science, and this lesson does a great job of, making those connections, real for students.
Speaker 1:That's awesome. So before we get into the thick of it, can you give us a quick overview? What's the core message? What are students walking away with?
Speaker 2:Imagine, like, you see a graph, an equation, or even a real world situation Yeah. And you can instantly see exponential growth or decay. Right? Yeah. That's what this lesson's all about, giving students the tools to do just that.
Speaker 2:They'll connect the dots between different ways of showing these functions
Speaker 1:Yeah.
Speaker 2:And see them in action in all sorts of contexts.
Speaker 1:I love it. So we're jumping into activity 1, equivalent or not, which tackles a common misconception that I'm pretty sure I've fallen for at some point.
Speaker 2:Oh, it's a classic pitfall, really. Mhmm. The activity zones in on the difference between, let's say, x squared, you know, x to the power of 2 versus 2 to the power of x.
Speaker 1:Okay.
Speaker 2:They look similar on the surface, but they act completely different.
Speaker 1:Yeah.
Speaker 2:And this activity uses a clever substitution to drive that point home. Students just plug in different values for x and see that while they might be equal sometimes
Speaker 1:Yeah.
Speaker 2:They're definitely not the same overall.
Speaker 1:Yes. It's like a myth busting activity where they can test it out and say, oh, these two things are not the same.
Speaker 2:Exactly. And the lesson even provides a handy table to illustrate this point.
Speaker 1:Oh, cool.
Speaker 2:Visualizing those results side by side makes all the difference, but it doesn't stop there. There are other ways to explore this too, like, what happens when x is odd versus even or even using graphs to see those differences.
Speaker 1:I love that there's flexibility built in. Each student learns in their own way, so having a variety of approaches makes a big difference.
Speaker 2:It does. It does.
Speaker 1:Now activity 2 shifts gears from the theoretical to the real world with cost of solar cells. What's the big idea here?
Speaker 2:Well, this activity taps into renewable energy, the fact that solar cells have been getting cheaper
Speaker 1:Right.
Speaker 2:And students get a graph showing this decrease, which, you guessed it, is exponential decay
Speaker 1:Oh, okay.
Speaker 2:In action.
Speaker 1:So we're not looking at, like, just graphs on a board. We're talking about a real trend that they hear about all the time.
Speaker 2:Absolutely. And it's not passive. They analyze the graph, turn it into math language function notation.
Speaker 1:Okay.
Speaker 2:And answer questions about what they see. They're putting it all together.
Speaker 1:That's cool. That's really cool. It's not just this abstract mathematical idea. It's something that's, like, shaking the world around
Speaker 2:them. Exactly.
Speaker 1:Okay. On to activity 3. This one sounds like origami, paper folding. What's the story here?
Speaker 2:This activity is great for giving students a real feel for how fast things can grow exponentially. Imagine, you have your students take a piece of paper and just fold it in half Mhmm. Again and again
Speaker 1:Over and over.
Speaker 2:And they see how thick it gets even though they're just doubling it each time.
Speaker 1:It's like that visual is so helpful for them to see how quickly exponential growth takes off just like the paper does.
Speaker 2:Exactly. And that's the point the activity drives home. The experience sets them up for learning about graphs of exponential equations. But here's the thing.
Speaker 1:Okay.
Speaker 2:It makes them think about domain and range, but specifically the number of folds. You can't have half a fold. Right?
Speaker 1:Can't fold a paper point 6 times. Mhmm.
Speaker 2:It has
Speaker 1:to be a whole number. Yeah.
Speaker 2:Exactly. And that's a perfect example of what we call discrete data.
Speaker 1:Uh-huh.
Speaker 2:Whole numbers, not a continuous curve. And the source material even points out that many students might need guidance to actually put this on a graph correctly.
Speaker 1:It's easy to forget that you can't just connect all the dots with a smooth line. Right?
Speaker 2:Right. Right.
Speaker 1:Especially in this case.
Speaker 2:Exactly.
Speaker 1:Okay. Cool. Is there more to this paper folding extravaganza?
Speaker 2:Oh, there's more. The are you ready for more section kicks things up a notch. Students calculate how many folds to get to 1 meter thick and then a whole kilometer.
Speaker 1:Oh, 1 kilometer. Talk about putting exponential growth into perspective.
Speaker 2:Right. It really opens their eyes to how fast these functions can grow, and it even encourages them to look up the world record for paper folding.
Speaker 1:I love that. That's really cool. That's a really cool activity. What does activity 4 have in store?
Speaker 2:This one's called InfoGap smartphone sales, and it connects exponential functions to deck trends, something today's students know. Now this one's optional, but it's packed with potential. It throws students into a scenario with real world data on smartphone sales.
Speaker 1:Okay. So we're stepping out of those picture perfect textbook examples.
Speaker 2:Precisely. Students pair up. 1 gets a problem about the sales. The other gets the data, but no context. They have to work together, ask the right questions, and figure it out.
Speaker 1:I like this information gap idea. Sounds like it really gets them communicating and problem solving.
Speaker 2:Absolutely. And it challenges them to estimate, strategize, and find the best way to model the data even if it doesn't fit perfectly into those need equations.
Speaker 1:Because let's face it, the real world is messy.
Speaker 2:And that's an important lesson too. Plus, it pushes students to think about the limits of exponential growth. We're talking about things like market saturation where everyone who wants a smartphone already has one, and that growth just has to slow down.
Speaker 1:So it's about recognizing that even though math models are powerful, they're still just representations of a bigger, more complex reality.
Speaker 2:Exactly.
Speaker 1:I love how this lesson takes them on a journey from abstract equations to real life scenarios like solar cells and smartphones.
Speaker 2:And we're just scratching the surface. The lesson synthesis ties it all together beautifully. I
Speaker 1:was just thinking that, like, how does this all come together? What's the big takeaway?
Speaker 2:Well, the lesson summary highlights how all these activities from folding paper to smartphone sales contribute to a deeper understanding of exponential functions.
Speaker 1:Oh, that's cool. So it's about showing that thread that connects all these seemingly random activities.
Speaker 2:Precisely. Speaking of connecting the dots, the lesson wraps up with a relatable example, caffeine breakdown.
Speaker 1:Oh, interesting.
Speaker 2:It demonstrates how you can use graphs, function notation, and context to understand how caffeine levels decrease over time.
Speaker 1:Oh, that's a good one. We've all been there trying to shake that caffeine buzz.
Speaker 2:Yeah.
Speaker 1:It shows how math is literally happening in our bodies.
Speaker 2:Precisely. And this lesson gives students the tools to model and understand those processes.
Speaker 1:This has been an awesome look at lesson 9, but I have a feeling there's still more to unearth.
Speaker 2:You're absolutely right. We've covered a lot, but there are more insights to uncover, particularly around the challenges teachers might encounter when they actually teach this.
Speaker 1:Okay.
Speaker 2:But we'll have to save that for our next segment.
Speaker 1:So, you
Speaker 2:know, when you were asking earlier about other real world examples of where understanding exponential functions is really important
Speaker 1:Yeah.
Speaker 2:Well, a few things come to mind. Like, think about how fast a virus can spread.
Speaker 1:Right.
Speaker 2:That's exponential growth in action. And it's not a happy example, but it's real. Epidemiologists, they use these same mathematical tools we've been talking about to model those outbreaks
Speaker 1:Oh, wow.
Speaker 2:And understand how bad they could get.
Speaker 1:Yeah. It's not just about predicting the spread though. We're talking about understanding how things like getting vaccinated or social distancing can change the whole game.
Speaker 2:Absolutely. Understanding how those actions can actually flatten the curve Right. And slow down that scary growth, that's math having a real impact on public health, maybe even saving lives.
Speaker 1:That's a powerful connection to make for students, especially for those thinking about careers where they can, you know, make a real difference. What else comes to mind?
Speaker 2:Well, since we are talking about solar power before, we should probably mention climate change too.
Speaker 1:Okay.
Speaker 2:It's a big one. One. The rate at which those greenhouse gases are building up in the atmosphere, a key part of climate change, that can be modeled using, you guessed it, exponential functions.
Speaker 1:So understanding these functions helps us see just how fast these gases accumulate, which is, like, essential for predicting those warming trends and trying to find ways to fix it.
Speaker 2:Exactly. It's really amazing how something we're discussing in the context of an algebra class can give us such valuable insights into something as complex as climate change.
Speaker 1:Seriously. Math is everywhere. And speaking of everyday stuff, what about compound interest?
Speaker 2:Oh, the classic.
Speaker 1:Right. It's like understanding how compound interest works is like having a secret weapon when it comes to your own money.
Speaker 2:For sure. It can work for you or against you depending on if you're saving or borrowing.
Speaker 1:Right. And it all comes back to those core principles we're talking about. It's honestly kind of mind blowing when you stop and think about it.
Speaker 2:It really is.
Speaker 1:So with all these real world examples in mind, let's talk about how this lesson actually approaches teaching this stuff. What stood out to you about this approach?
Speaker 2:You know, we've gone into the activities themselves, but the big picture thing here is this, mathematical modeling. The whole lesson emphasizes that.
Speaker 1:Oh, interesting. So it's not just about, like, plugging numbers into formulas. We're talking about really using math to make sense of real world situations.
Speaker 2:Exactly. And it encourages that right from the start, giving real scenarios
Speaker 1:Yeah.
Speaker 2:And walking students through how to create models that represent them mathematically.
Speaker 1:It's like giving them the tools to be math detectives in a way.
Speaker 2:Exactly. And once they have those models, they can answer questions, make predictions Yeah. Get a much deeper understanding of what's going on.
Speaker 1:It's like turning something passive, just observing a pattern into this active process.
Speaker 2:And that's what makes this curriculum so cool. It gets that math isn't just procedures. It's a way of thinking.
Speaker 1:Absolutely. Are there any activities in this lesson that really highlight that mathematical modeling piece?
Speaker 2:2 come to mind right away, the cost of solar cells and the info gap, smartphone sales.
Speaker 1:Yeah. We talked about them briefly earlier. Tell me more about how they bring this modeling idea to life.
Speaker 2:Okay. So with cost of solar cells, it doesn't just tell students, hey. This graph shows how solar energy prices have gone down. It makes them analyze that relationship shown in the graph
Speaker 1:Yep.
Speaker 2:And define it using math with that function notation.
Speaker 1:So they're not just looking at the graph. They're figuring out the math behind it and using that to understand the bigger picture.
Speaker 2:You got it. That's mathematical modeling in a nutshell.
Speaker 1:Awesome. And then what about InfoGap smartphone sales? Where does that fit in?
Speaker 2:That one takes it up a notch by using something you don't always see, messy data.
Speaker 1:Because real world data is rarely perfect.
Speaker 2:Exactly. This activity embraces that and shows students that it's okay. They get a dataset that doesn't fit those perfect equations.
Speaker 1:Right.
Speaker 2:So they have to make judgment calls, estimate, and really think about the best way to model it, just like data analysts in the real world.
Speaker 1:That's such an important skill because they're not always gonna have all the answers, and things are not always gonna be clear cut.
Speaker 2:Exactly. So to sum it up, this lesson isn't just about teaching the mechanics of these functions. It's about giving students the power to use those functions as tools to understand the world.
Speaker 1:That's amazing. Yeah. I'm so glad we took this deep dive. It's amazing how much is packed into just one lesson.
Speaker 2:Me too. And we're not done yet. There's more to uncover. There are also challenges teachers might come across when teaching this.
Speaker 1:Oh, k.
Speaker 2:But we'll have to save that for the next part.
Speaker 1:Alright. We'll be right back with more. Alright. So we're back. And now, we're diving into the challenges that teachers might bump into while teaching this stuff.
Speaker 1:Because let's be real, even with the coolest math lessons, there are always those moments.
Speaker 2:Right? Absolutely. Those moments when you finish explaining something and it's like you can hear crickets chirping.
Speaker 1:Right.
Speaker 2:Or you get those blank stares that make you wonder if you accidentally started speaking a different language.
Speaker 1:Yeah. Totally. So what advice would you give to teachers who might be, you know, a little nervous about tackling those hurdles?
Speaker 2:Well, first things first, just breathe. It's completely normal for students to struggle with these kinds of abstract concepts like exponential functions.
Speaker 1:I think every teacher needed to hear that.
Speaker 2:It's true. Don't get discouraged. It's all part of the process. Just anticipate those challenges and have some go to strategies ready. And one of the most effective, which this lesson does a great job with, is making it real.
Speaker 2:Ground the learning in things they can actually relate to.
Speaker 1:So less about formulas on the board, more about bringing those formulas to life.
Speaker 2:Exactly. When students see these functions playing out in situations, they understand. It just clicks so much faster.
Speaker 1:Like they say show, don't tell.
Speaker 2:Right? Right. Or in this case, show and tell. And that's where those hands on activities, like the paper folding one, are so powerful. They actually get to experience exponential growth firsthand.
Speaker 2:It's true.
Speaker 1:And sometimes you just gotta get those kids up and moving. Right? Breaks up the day a little.
Speaker 2:For sure. Movement, real world connections, showing the same idea in different ways, it's all helpful. Now let's talk about those pesky misconceptions.
Speaker 1:Oh, yes.
Speaker 2:Those can really trip students and teachers up.
Speaker 1:They're like invisible speed bumps.
Speaker 2:Right. A student might seem like they're getting it
Speaker 1:Yeah.
Speaker 2:And then, bam, misconception.
Speaker 1:It's so frustrating too because sometimes you feel like you've explained it.
Speaker 2:But Exactly. Clear explanations are key, but you've gotta back them up with different representations of the idea too.
Speaker 1:Okay. So don't just tell them the right way to think about it, but show them visually from all these different angles.
Speaker 2:Yes. Visuals are your best friend. Graphs, diagrams, anything. Add in real world examples. Have them explain things back to you in their own words.
Speaker 2:The more ways they connect with a concept, the stronger that understanding.
Speaker 1:It's about building that solid foundation from all sides. Okay. Another challenge. This one's a bit sneaky rote memorization.
Speaker 2:Yes. The old memorizing without understanding trick.
Speaker 1:It happens everywhere, but math seems to be a magnet for it.
Speaker 2:It really is. And, honestly, the best antidote is fostering a classroom where it's okay to be curious.
Speaker 1:So it's less about finding that one right answer
Speaker 2:Yeah.
Speaker 1:And more about encouraging them to actually ask the questions.
Speaker 2:Yes. Have them explain their thinking, debate ideas respectfully, and don't be afraid to say I don't know every now and then.
Speaker 1:Yeah.
Speaker 2:Turn it into a learning opportunity. I don't know, but let's find out together.
Speaker 1:I love that.
Speaker 2:It becomes journey you take together.
Speaker 1:Absolutely. And speaking of journeys, we have to talk more about mathematical modeling. It's such an important part of this lesson.
Speaker 2:Definitely.
Speaker 1:But it can also be really intimidating, both for teachers and students.
Speaker 2:For sure. It's like we're taking all those neat equations from the textbook and throwing them out into the messy real world.
Speaker 1:Where data is messy. Mhmm. And there isn't always a nice, neat answer.
Speaker 2:Exactly. It's about making choices, simplifying, and finding models that help make sense of things.
Speaker 1:So what helps teachers who are, like, trying to guide their students through that? Yeah. Where do they start?
Speaker 2:Start slow. Give them lots of support. And here's the big one. Embrace the mess.
Speaker 1:Love that. Embrace the mess.
Speaker 2:Because that's what real world data is. Messy, incomplete, rarely perfect. But that's where the real learning happens.
Speaker 1:Right. And activities like the info gap, smartphone sales, those are perfect for that. They're given data that isn't perfect. They have to wrestle with it.
Speaker 2:Exactly. And they come out with more than just math skills. They're learning to deal with information, make decisions when things aren't clear cut.
Speaker 1:Yeah. Those are such vital skills for life.
Speaker 2:Exactly.
Speaker 1:Well, that's a perfect place to wrap things up, I think. To all the teachers out there about to embark on this exponential adventure, embrace those challenges, celebrate those moments, and most importantly, have fun with it.
Speaker 2:I totally agree. Big thanks to Illustrative Mathematics for creating such a thought provoking lesson and to you all for joining us. Keep those questions coming.