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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 scrolled through social media and seen a post just blow up, go viral?
Speaker 2:Yeah. Yeah. Totally.
Speaker 1:Or maybe watched your savings grow, you know, thanks to compound interest.
Speaker 2:Mhmm.
Speaker 1:What seemed like, I don't know, totally different things, they actually have a really powerful principle working behind the scenes. It's exponential growth.
Speaker 2:Right. Right.
Speaker 1:And today, we're gonna take, I think, a fascinating approach to understanding it.
Speaker 2:Okay.
Speaker 1:We're actually gonna peek into a high school eligible lesson plan.
Speaker 2:Oh, interesting.
Speaker 1:Yeah. Now before you think we're, like, going back to school here, stick with us.
Speaker 2:I see. Yeah.
Speaker 1:This isn't about memorizing formulas or anything like that. Right. It's about seeing how experts take a really complex idea
Speaker 2:Yeah.
Speaker 1:And break it down into those clear, understandable steps Uh-huh. Which honestly is something I think we can all use. Right?
Speaker 2:Absolutely. Yeah.
Speaker 1:So the lesson plan that we are diving into today is from illustrative mathematics.
Speaker 2:Okay.
Speaker 1:And it uses a pretty, I think, engaging example to illustrate this whole idea of exponential growth, and it's bacteria.
Speaker 2:Oh, okay. Interesting. Yeah.
Speaker 1:So picture this. You know, you got your petri dish
Speaker 2:Yeah.
Speaker 1:And it's teeming with all these bacteria. Right? So small you can barely even see them individually.
Speaker 2:Right.
Speaker 1:But here's where it gets really interesting. Let's say this particular bacteria, it doubles every hour.
Speaker 2:Okay. So that doubling is key there.
Speaker 1:Yeah.
Speaker 2:That highlights what we call the growth factor, like, the defining characteristic of exponential growth.
Speaker 1:Right. And I think this is where, you know, people get tripped up sometimes. Right? They think, oh, it just keeps growing by the same amount over time.
Speaker 2:Right. Exactly. And that is what we call linear growth, right, where you're just adding the same amount consistently.
Speaker 1:Yeah.
Speaker 2:Exponential growth, we're talking about multiplying.
Speaker 1:Right.
Speaker 2:So in this case, that bacteria scenario, our growth factor is 2. Right? Okay. Every hour, that population's being multiplied by 2.
Speaker 1:And the lesson does a really smart thing here. Yeah. They have students actually write out what's happening each hour
Speaker 2:Okay.
Speaker 1:As an expression. Right? Yep. So let's say we're starting with, I don't know, 500 bacteria.
Speaker 2:500 bacteria. Okay.
Speaker 1:An hour later, how many do we have?
Speaker 2:Well, we've got 500 times 2.
Speaker 1:Exactly. 500 times 2. Right. That Yeah. It's 500 times 2 times 2 and so on and so forth.
Speaker 2:Yeah. It's getting big fast.
Speaker 1:It's like that game we used to play as kids, telephone, where you whisper something
Speaker 2:Oh, yeah.
Speaker 1:And it changes. By the end, it's totally different. Yeah. Except here, instead of changing the phrase, we're multiplying those bacteria.
Speaker 2:Right. Exactly. And just like that childhood game can have some pretty hilarious outcomes. This mathematical game of, like, repeated multiplication leads to a really important moment that the lesson plan actually very cleverly guides students towards.
Speaker 1:I like that. I like that.
Speaker 2:Yeah. They start to recognize the pattern. Right? 500, then 500 times 2, 500 times 2 times 2. They realize there's gotta be, you know, a better, more elegant way to express this.
Speaker 2:Right.
Speaker 1:There's gotta be a shortcut here.
Speaker 2:Exactly. Exactly. And that's where this comes in. Y f 4 to 502 x.
Speaker 1:Boom. There it is. Our exponential equation.
Speaker 2:Exactly.
Speaker 1:But, you know, we don't just wanna throw the equation at you and move on. Right. We gotta unpack what each part of this actually means.
Speaker 2:Yeah. Absolutely.
Speaker 1:Because that's where the true understanding, I think, comes in, don't you think? A
Speaker 2:100%. Yeah. In this equation, we've got y, which represents the total number of bacteria after a specific time.
Speaker 1:Okay. So that's, like, our total count.
Speaker 2:Exactly. Our total count. Yeah. And then that 500, that's just our starting point, our initial amount.
Speaker 1:Right.
Speaker 2:Yeah. Like we talked about, that's our growth factor, that consistent multiplier.
Speaker 1:Makes sense.
Speaker 2:And lastly, we've got our x, which is our time variable, which in this case, we're measuring in hours.
Speaker 1:Now that equation, y 502x Yeah. Might look a little, I don't know, abstract.
Speaker 2:Right. It's
Speaker 1:just kinda sitting there on its own. Right. But this is where the lesson plan takes us next.
Speaker 2:Okay.
Speaker 1:It's from equations to graphs.
Speaker 2:I see.
Speaker 1:And I get it. Graphs can sometimes feel a little intimidating. Like, you need some kind of secret decoder ring to figure them out.
Speaker 2:Right.
Speaker 1:But trust me, they're really just visual stories.
Speaker 2:Yeah. Yeah.
Speaker 1:And in this case, this story is all about the incredible speed
Speaker 2:Okay.
Speaker 1:Of exponential growth.
Speaker 2:I like that.
Speaker 1:You know?
Speaker 2:Yeah.
Speaker 1:It's a good way to put it, I think.
Speaker 2:Graphing this equation isn't just about connecting the dots just for the sake of it.
Speaker 1:Right.
Speaker 2:It's about seeing that bigger picture.
Speaker 1:Yeah.
Speaker 2:That cumulative effect of that repeated multiplication.
Speaker 1:Exactly. Exactly. Yeah. And, thankfully, the lesson plan actually provides the graphs for the students here
Speaker 2:Okay. Good.
Speaker 1:Which is really helpful.
Speaker 2:Yeah. We
Speaker 1:could see that initial 500 bacteria.
Speaker 2:Right.
Speaker 1:That's our starting point Right.
Speaker 2:Yeah.
Speaker 1:Marked clearly on the graph.
Speaker 2:Uh-huh.
Speaker 1:But then watch out. Because as we move along that timeline
Speaker 2:Yeah.
Speaker 1:Each hour, the line starts to climb.
Speaker 2:Okay.
Speaker 1:And then it's like
Speaker 2:It takes off.
Speaker 1:Yeah. It shoots upward Yes. Like a rocket ship or something.
Speaker 2:That's a great visual. Yeah. That's what that steep curve really drives home. It's like that's the visual representation of that growth factor of Tutor in action.
Speaker 1:Yeah.
Speaker 2:And, you know, I think this is why understanding exponential growth is so important Yep. Whether we're talking about bacteria multiplying, investments compounding, even going back to, like, your social media example. Right? Like Totally. Like, a post catching fire spreading like wildfire.
Speaker 1:Yeah. Yeah.
Speaker 2:Things don't always just increase at a nice steady pace. They can take off.
Speaker 1:Yeah. They can take off. That image of a rocket really fits, I think.
Speaker 2:Totally.
Speaker 1:Speaking of things taking off, I kinda wanna shift gears from the, bacteria here
Speaker 2:Okay.
Speaker 1:And look at another example that this lesson plan brings up.
Speaker 2:Uh-huh.
Speaker 1:And it's a mouse population growing in a forest.
Speaker 2:Okay. Yeah. So, I mean, what's really interesting here
Speaker 1:I know. Right?
Speaker 2:Is we can apply that same exponential equation
Speaker 1:Okay.
Speaker 2:That y was a b x Yeah. To this totally different scenario.
Speaker 1:Yeah. Yeah.
Speaker 2:So let's say we've got, I don't know, 10 mice to start.
Speaker 1:Ten mice.
Speaker 2:And their population doubles every 6 months. So our s, our initial amount would be 10.
Speaker 1:Got it.
Speaker 2:Our b, the growth factor, would be 2. Right? Because they're doubling.
Speaker 1:Right. Uh-huh.
Speaker 2:But here's the thing. Our x, that time variable Yeah. It's not in hours anymore. Important to keep track of. Yeah.
Speaker 2:Absolutely. Yeah.
Speaker 1:So if we wanted to see, like, how many mice there would be
Speaker 2:after, I don't know, 18
Speaker 1:months Okay. We'd plug in 3 for x. Exactly. Because that's 3 of those 6 month periods. Exactly.
Speaker 1:Yeah.
Speaker 2:Exactly. So it would be 10 times 2 to the power of 3
Speaker 1:Oh, good.
Speaker 2:Which gives us 80 mice.
Speaker 1:Eighty mice. Okay.
Speaker 2:Now 80 mice might not sound like a whole lot.
Speaker 1:Right.
Speaker 2:But remember, this kind of growth, it accelerates real quick.
Speaker 1:Okay.
Speaker 2:So let's say we look at 5 years down the line.
Speaker 1:Okay.
Speaker 2:That's 10 6 month period.
Speaker 1:Oh, I got it.
Speaker 2:Plug in 10 for that x.
Speaker 1:Right.
Speaker 2:And suddenly, we're dealing with over 10,000 mice.
Speaker 1:Oh, wow.
Speaker 2:10,240 to be exact. Yeah.
Speaker 1:See, that's a lot of mice.
Speaker 2:That's the power of exponential growth.
Speaker 1:It is. It is.
Speaker 2:And it shows why even a small starting point Yeah.
Speaker 1:Can lead to really significant outcomes.
Speaker 2:It's a good reminder that these seemingly, you know, simple equations we work with in math class
Speaker 1:Right.
Speaker 2:Can reveal some pretty, I think, profound things about the world around us. Right?
Speaker 1:Absolutely. Yeah. And I think this lesson does a really nice job of highlighting that. Yeah. It really does.
Speaker 2:It does. It does. Yeah. Now I know what you're thinking.
Speaker 1:What's that?
Speaker 2:Enough about the mice already.
Speaker 1:Okay. Yeah.
Speaker 2:Let's talk about something a little closer to home.
Speaker 1:Okay.
Speaker 2:Something a little bit closer to maybe our wallets. That's right. We're going from mice to money.
Speaker 1:Okay.
Speaker 2:Specifically, financial bonds.
Speaker 1:Right.
Speaker 2:And this is where that equation, that yoquel a a e x, it really shows that it's not just some abstract, like, mathematical concept.
Speaker 1:Right.
Speaker 2:It's a really powerful tool for understanding how, you know, investments can grow over time.
Speaker 1:Absolutely. Yeah. It's the same principle as the bacteria in the mice.
Speaker 2:Right? Right. Exactly.
Speaker 1:Just applied in a different context.
Speaker 2:Yeah.
Speaker 1:So let's say you buy a bond. Right?
Speaker 2:For $500 Right. And it promises to double in value every 10 years.
Speaker 1:Okay.
Speaker 2:That doubling. That's our growth factor. Right?
Speaker 1:There it is again. Exactly.
Speaker 2:Exactly.
Speaker 1:So our a in the equation, our initial amount would be $1,000.
Speaker 2:Yep.
Speaker 1:Our b, our growth factor is 2.
Speaker 2:Right.
Speaker 1:And our x is the number of decades.
Speaker 2:Exactly. Precisely. So now if you wanna know what that bond might be worth after, say, 30 years, what would you do?
Speaker 1:Well, you plug in 3 for
Speaker 2:x. Exactly. Because that's 3 decades. Right?
Speaker 1:3 decades.
Speaker 2:So you'd have 500 times 2 to the power of 3, which equals
Speaker 1:$8,000. Talk about seeing your money grow.
Speaker 2:Right.
Speaker 1:I mean but before we all go rushing out to buy bonds, it's important to remember, you know, this is kind of a simplified example.
Speaker 2:Absolutely. Yeah. Real world investing, much more complex.
Speaker 1:Oh, yeah.
Speaker 2:Lots of other factors that can influence how those investments perform.
Speaker 1:Of course. Of course. It's a good reminder that while, you know, math gives us these powerful tools to, like, model and understand this kind of growth, there are always other things going on.
Speaker 2:Or ways.
Speaker 1:Speaking of things that can, you know, trip us up, I like how this lesson plan actually goes into some common misconceptions that students often have about exponential growth.
Speaker 2:Oh, yeah. Yeah. One of the biggest ones is, like, mistaking that repeated multiplication for just adding the growth factor, you know, multiple times.
Speaker 1:Oh, I see. Yeah.
Speaker 2:Yeah. It's easy to think if it doubles every hour, then after 3 hours, I just multiply the starting amount by 2, then 2 again, then 2 again.
Speaker 1:Right. Like, it's 2 plus 2 plus 2.
Speaker 2:Exactly. Instead of 222.
Speaker 1:Right. Right.
Speaker 2:And that's where having that clear equation that y is obx can really help, I think.
Speaker 1:Totally. Yeah.
Speaker 2:It's a good visual reminder of how that exponent really changes things.
Speaker 1:Yeah. It really emphasizes the difference, right, between that linear growth where we're consistently adding the same amount
Speaker 2:Right.
Speaker 1:Versus that exponential growth where we're multiplying.
Speaker 2:Big difference. Huge.
Speaker 1:Like, it's the difference between walking up a staircase and, you know, hopping on a rocket ship.
Speaker 2:That's a great analogy. And it highlights, I think, a key takeaway from this whole lesson. Being able to look at a situation and go, okay, is this exponential growth?
Speaker 1:Right. 1st identify it.
Speaker 2:Exactly. And then if it is, you can use those equations and those graphs to really represent and analyze what's going on.
Speaker 1:And to think, we started with this seemingly simple lesson about bacteria.
Speaker 2:It's amazing. Right? A single mathematical concept can connect things like the growth of bacteria to the potential of investments to even, like we said, that social media post going viral.
Speaker 1:It really makes you wonder, like, what other hidden connections are out there, you know, just waiting to be discovered.
Speaker 2:Totally.
Speaker 1:Well, big thanks to the illustrative mathematics team for this really thought provoking lesson plan.
Speaker 2:Yeah. Great stuff.
Speaker 1:And to all of you listening, we hope this deep dive has given you a fresh perspective on exponential growth and just all the ways, honestly, that it shapes our world.
Speaker 2:Absolutely. And next time you see something rapidly multiplying out there in the world, whether it's, you know, good news spreading or maybe those leftovers in the back of your fridge
Speaker 1:Oh, no.
Speaker 2:You'll know the math behind it.
Speaker 1:That's right. You'll be able to impress your friends.
Speaker 2:Exactly.
Speaker 1:Alright. Until next time.