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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 feel like you're picking between a slow and steady tortoise and a hare that's, well, taking a bit of a nap at the starting line?
Speaker 2:I I can see that.
Speaker 1:That's kinda what it's like comparing linear and exponential growth, isn't it?
Speaker 2:It really is.
Speaker 1:So today, we're diving into a lesson plan from illustrative math called which one changes faster. Hoping to give you all some tools to guide students through this whole thing, this whole world of linear and exponential growth.
Speaker 2:It's a fascinating topic.
Speaker 1:It really is and can be kinda surprising. So let's get into it. What hidden gems, what golden nuggets can we pull out of this lesson plan to make this concept really stick with our students?
Speaker 2:Good question.
Speaker 1:What do you think?
Speaker 2:Well, right off the bat, the lesson throws a curve ball.
Speaker 1:Oh, tell me more.
Speaker 2:Yeah. It presents a graph, and it could be either linear or exponential.
Speaker 1:Really?
Speaker 2:Yeah. And it all comes down to, like, how you define the domain and range.
Speaker 1:Oh, that's interesting. So it's not immediately obvious which one it is.
Speaker 2:Exactly.
Speaker 1:And that actually kinda gets at the heart of what we're talking about. Right?
Speaker 2:Yeah. We're talking about 2 totally different ways things can change over time.
Speaker 1:Okay. So break that down for me.
Speaker 2:Alright. So linear growth. Things increase by a constant amount. Right? Think about, like, adding $5 to your piggy bank every week.
Speaker 2:Every week, same amount.
Speaker 1:Okay. Reliable, consistent.
Speaker 2:Exactly. Then there's exponential growth. That's all about increasing by a constant percentage over time. They think about compound interest.
Speaker 1:Ah, yes. The money in your savings account just magically growing.
Speaker 2:Right. Might seem slow at first, but give it time. Those gains really start to snowball.
Speaker 1:It's not just about the numbers themselves then. It's about understanding the why, the how, those patterns behind how things change.
Speaker 2:Exactly. And the lesson really drives that home in the first activity. Presents a pretty classic scenario. Gotta choose between bonds with simple interest.
Speaker 1:Okay. So that's our steady linear growth there.
Speaker 2:Right. And then the other option, a savings account with compound interest. That's our exponential growth.
Speaker 1:The classic investment dilemma. But wait a minute. There's a twist.
Speaker 2:Oh, it's a twist.
Speaker 1:This savings account, it comes with a fee.
Speaker 2:It does.
Speaker 1:You wouldn't think a little old fee would change things so much. But It's
Speaker 2:a big difference.
Speaker 1:Does it?
Speaker 2:Students have to think beyond just the simple calculations.
Speaker 1:Because they gotta factor that fee in.
Speaker 2:Yep. They have to look at how those investments grow over time, and that means looking at both the interest earned and the impact of that fee.
Speaker 1:Oh, interesting. So they can't just look at the interest rate alone?
Speaker 2:Nope. Gotta think long term. What I really like is that the lesson actually includes, like, tables that show how those investments grow year by year.
Speaker 1:Oh, that's helpful.
Speaker 2:Yeah. So students can actually see that at the beginning. Those bonds might look like the better deal.
Speaker 1:Yeah. Because they're nice and predictable.
Speaker 2:Exactly. But as time goes on, those years start piling up that compound interest. It starts to work its magic. It's like the power of compounding. Even small differences in those interest rates, they can lead to huge gains over time.
Speaker 2:It's like why financial experts are always saying, start investing early.
Speaker 1:It's so true. Like, that saying, the best time to plant a tree was 20 years ago. The second best time is now. We can't go back in time. But we can understand how that compound growth works.
Speaker 1:Right? It can help us make better decisions going forward.
Speaker 2:A 100%.
Speaker 1:So it's really about thinking long term Yeah. Especially with investments.
Speaker 2:Absolutely. And to really bring that home, the next activity, it ditches the whole real world context, goes totally abstract.
Speaker 1:Oh, interesting. How so?
Speaker 2:It just gives students 2 functions, 1 linear, 1 exponential. Ask them to compare.
Speaker 1:Okay. And what's the catch?
Speaker 2:The exponential function looks totally pathetic at first.
Speaker 1:Really? Give me an example.
Speaker 2:Okay. So let's say you've got, a linear function, and it's, say, y equals 2x+5.
Speaker 1:Right. So it's increasing by 2 for every increase in x.
Speaker 2:You got it. And then let's say our exponential function is y equals 1.1x. So that's a 10% increase for every increase in x.
Speaker 1:Okay. I'm following along.
Speaker 2:Now at first, the linear function is winning. Right?
Speaker 1:It looks way better.
Speaker 2:Plug in x 1. The linear function gives you 7. The exponential only gives you 1.1.
Speaker 1:Yeah. Exactly.
Speaker 2:But then you start plugging in bigger values for x, that 10% growth.
Speaker 1:What happened?
Speaker 2:It explodes. By the time x is 10, the linear function is at 25, but the exponential, it's already zooming past it. It's, like, 2.6 or something.
Speaker 1:Wow. That's a huge difference.
Speaker 2:Yeah.
Speaker 1:Talk about deceiving.
Speaker 2:Right. If you're only looking at the short term, you're missing the whole picture.
Speaker 1:Totally. It's like those time lapse videos where you see a plant just shoot up out of nowhere.
Speaker 2:Yeah.
Speaker 1:But in reality, it was growing underground this whole time.
Speaker 2:Exactly. And just like those videos, if we're only looking at one part, one slice of time, it's easy to misunderstand what's going on.
Speaker 1:That makes a lot of sense. So we need to see the whole picture.
Speaker 2:Yes. And that's why it's so important for students to be able to, like, interpret and switch between those different ways of showing those functions, you know, the tables, the graphs, the equations.
Speaker 1:To really get the full picture.
Speaker 2:Exactly.
Speaker 1:But I can already hear my students now. Yeah. Why bother? The linear function is crushing it at the beginning. Like, how do we answer that?
Speaker 1:Well,
Speaker 2:that's a really common misconception, and the lesson addresses that directly.
Speaker 1:That was it.
Speaker 2:Yeah. It's natural to think that what we see first is what we get. But this activity, it forces you to challenge that assumption.
Speaker 1:By showing how even small differences can make huge changes over time.
Speaker 2:You got it.
Speaker 1:It really makes you think about those small changes we make. You know? Mhmm. If we stick with them over time
Speaker 2:It can really add
Speaker 1:up. Exactly. It's like building good habits or learning something new consistently.
Speaker 2:It's all connected. And speaking of connecting, the lesson wraps up by making sure students really get that big takeaway.
Speaker 1:That exponential functions always win in the end.
Speaker 2:Exactly. Even if it's slow at first, that exponential function, it will always eventually overtake the linear one. It's a mathematical fact.
Speaker 1:So how can we, teachers, help drive that point home? How can we make sure our students really internalize that?
Speaker 2:I think it's all about getting them to use their own words. You know? Yeah. Have them explain those key features of each type of growth.
Speaker 1:So they really own those definitions.
Speaker 2:Exactly. What does linear growth look like? How is exponential growth different? The more they can articulate that, the more it'll stick.
Speaker 1:I like that. Get them talking, get them thinking. Any other activities? Anything else that can really drive that understanding home?
Speaker 2:Well, the lesson actually suggests having students create their own scenarios, like come up with their own examples of linear versus exponential growth.
Speaker 1:Oh, get creative with it. I like it.
Speaker 2:Yeah. And have them apply that concept to stuff they can relate to. You know? It makes it so much more real.
Speaker 1:That's such a good idea. I hadn't even thought about using it with salaries before, but that's a perfect example.
Speaker 2:It is. One salary with a fixed raise each year.
Speaker 1:And another with, like, a percentage base raise.
Speaker 2:There you go. A real world lesson in exponential growth.
Speaker 1:Even adults could learn from that.
Speaker 2:Right. It's all about bridging that gap, you know, between those abstract math concepts and our students' actual lives.
Speaker 1:Make it real. Make it relevant.
Speaker 2:Exactly. It's more meaningful that way, more memorable.
Speaker 1:And who knows? Maybe we'll even inspire some savvy investors in the process.
Speaker 2:You never know.
Speaker 1:Speaking of which, this all reminds me of that genie activity we talked about. Remember?
Speaker 2:Throwing with the $1,000,000?
Speaker 1:Yes. Students have to choose between getting a $1,000,000 right now.
Speaker 2:Or a penny that doubles every day for a month.
Speaker 1:Exactly. And even with that huge head start, that $1,000,000, it can't compete with the doubling penny. Exponential growth for the win.
Speaker 2:It's a powerful image for sure.
Speaker 1:It really sticks with you. And it highlights that sometimes the biggest, most important changes. They're not always obvious at first glance.
Speaker 2:It's about looking for those patterns, you know, understanding what's happening beneath the surface.
Speaker 1:So true. Well, as we wrap up our deep dive today, have you had a final thought? A little something to chew on.
Speaker 2:I'm all ears.
Speaker 1:What else besides what we're talking about today, what else in the real world operates on an exponential curve?
Speaker 2:Oh, that's a good one.
Speaker 1:Right. Think about it. Population growth, how fast information spreads online.
Speaker 2:And how quickly technology is advancing these days.
Speaker 1:Exactly. Exponential growth is everywhere. And when we understand it, it helps us understand the world around us, and the world is changing faster than ever. Huge thank you to the authors of illustrative math for giving us so much to think about. This deep dive has been really inspiring.
Speaker 1:I'm feeling energized ready to go back to my classroom and spark some moments of my own. Thanks for joining us on this deep dive. Until next time.