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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 wonder about the math behind how things grow? Not like, plants and stuff. More like, how about when information spreads online or, say, a stack of cash multiplying? Yeah. Buckle up because today's deep dive is all about spotting those hidden patterns of, you know, linear and exponential growth.
Speaker 2:And you might be surprised how often these patterns, pop up. Right. Not just in, like, a textbook, but in everyday life.
Speaker 1:We're diving into an algebra one lesson plan. It's called growing and growing.
Speaker 2:I like it.
Speaker 1:And don't worry. You don't need to remember what a quadratic equation, is Right. Let alone how to solve one.
Speaker 2:Exactly.
Speaker 1:We're after those moments, the ones that help you kinda see the world a little differently.
Speaker 2:Yeah. It's about understanding the sort of the power behind those patterns, which can be, surprisingly useful Yeah. For for, like, making decisions and things like that.
Speaker 1:Exactly. This lesson starts off with, well, bacteria.
Speaker 2:Okay.
Speaker 1:Picture a single bacteria cell. Every hour, it splits into 3. 1 hour in, no big deal. You've got 3 cells.
Speaker 2:Yeah.
Speaker 1:But then hour 2 hits, Those 3 split again and so on and so on. Mhmm. You can probably see where this is going. Yeah.
Speaker 2:This this gives you a feel for just how rapidly exponential growth, works. It's not just, you know, that increases.
Speaker 1:Right.
Speaker 2:It it it snowballs.
Speaker 1:Yeah.
Speaker 2:Just keeps getting bigger and bigger.
Speaker 1:It's like that old penny doubled every day for a month riddle.
Speaker 2:Right?
Speaker 1:Seems small at first, but the sheer speed of that growth can be incredible.
Speaker 2:Yeah.
Speaker 1:And and a little bit scary when you think about things like, say, the spread of a virus.
Speaker 2:Yeah. Highlights why understanding these different kinds of growth is is so important Mhmm. Because it's not always obvious at first glance which is which Yeah. Or what the what the long term implications might be.
Speaker 1:And that's where this lessons genie example comes in. And, who doesn't love a good genie story?
Speaker 2:Right.
Speaker 1:It's a classic for a reason. It makes those abstract math concepts suddenly feel very, very real.
Speaker 2:I I think so too. It's a good way to,
Speaker 1:To illustrate it. So imagine a genie pops out and, offers you one of 2 purses. Purse a starts you off with a cool $1,000, and every day, it adds another $200. Not bad.
Speaker 2:Not bad at all.
Speaker 1:But then there's purse b, Starts with a single penny, but that penny doubles every single day. Day 1, 2¢. Day 2, 4¢. Day 3, 8¢.
Speaker 2:Mhmm.
Speaker 1:You see where this is going. Yeah. Which purse are you picking?
Speaker 2:It's like choosing between, a steady paycheck or, like, an investment
Speaker 1:Yeah.
Speaker 2:With a with a slow start but the potential for, explosive growth.
Speaker 1:Yeah. Yeah. Yeah.
Speaker 2:Like, way beyond the future.
Speaker 1:Most people, understandably, would go for the instant gratification of per se. Right?
Speaker 2:Oh, yeah.
Speaker 1:That $200 every day feels like a sure thing.
Speaker 2:Exactly. But this lesson, like, encourages us to think past that, you know Yeah. Initial feeling. Right. Like, what happens when those pennies start to double and double again?
Speaker 1:Yeah. And the lesson plan actually has students calculate this out. A week, 2 weeks, a month, and that's where it really hits home. That seemingly insignificant penny, because of the power of exponential growth, starts to leave the linear growth of purse a in the dust.
Speaker 2:This is, this is the, the magic of compound interest in action.
Speaker 1:Right.
Speaker 2:It's not just that you earn more, but your earnings start earning their own earnings.
Speaker 1:Right. It
Speaker 2:it's like a snowball effect but with money.
Speaker 1:Yeah. And suddenly, those math equations from school, you know, they're not just numbers on a page anymore. They're explaining how wealth can build over time or how trends explode in popularity.
Speaker 2:Exactly. But this lesson goes beyond just, like, crunching numbers. It wants students to really see the difference between these kinds of growth, which is where graphs come out.
Speaker 1:Because who doesn't love a good graph?
Speaker 2:Right? Well, they they can be surprisingly insightful. Yeah. Remember how we talked about linear growth being like a steady line? Yeah.
Speaker 2:On a graph, that's exactly what it looks like. Mhmm. But exponential growth, that's a curve. Starts slow and then just, like, takes off like a rocket.
Speaker 1:Yeah. Straight up. And when you see those two lines side by side, the difference is is stark.
Speaker 2:Yeah.
Speaker 1:It really brings home the power of that compounding effect in a way that just talking about it can't quite capture.
Speaker 2:Exactly. Exactly. It helps us visualize how those, seemingly insignificant changes early on Right. Can lead to, like, massive differences down the line.
Speaker 1:Okay. So we've got bacteria splitting, genies handing out magical purses, but this lesson plan has one more trick up its sleeve. This time, there are no calculations, no tables of numbers to work with, just a graph.
Speaker 2:And this is where you really, you really get to test your understanding. Yeah. Can you look at the shape of a line and tell what kind of growth it represents?
Speaker 1:Right.
Speaker 2:Can you spot the better deal just by looking?
Speaker 1:It's like the lesson is saying, alright. You've learned the basics. Now let's see if you can apply it.
Speaker 2:Exactly.
Speaker 1:Right.
Speaker 2:And the skill, being able to interpret graphs, it's it's huge. Yeah. It's not just about math class.
Speaker 1:Right.
Speaker 2:It's about, understanding data in the real world. Yep. Whether you're looking at stock market trends, population growth, or even the spread of information online, being able to decipher those, visual cues, it's incredibly valuable.
Speaker 1:Yeah. It's like learning a whole new language. Right? But it's visual. And once you start to see it, you can't unsee it.
Speaker 1:Suddenly graphs and news articles or presentations at work, they take on a whole new meaning.
Speaker 2:Yeah. It's like you've unlocked, a secret code. But instead of, you know, hidden messages, you're uncovering trends and patterns that were always there Yeah. Just, waiting to be understood.
Speaker 1:Exactly. And what's so cool about this lesson plan is that it doesn't just, like, throw a bunch of formulas at you. It uses these really clever scenarios, like, you know, like those splitting bacteria or the genie's purses
Speaker 2:Right.
Speaker 1:To help you actually get the difference between linear and exponential growth.
Speaker 2:It's about making those abstract concepts, click. Yeah. Not just through, you know, calculations, but through, like, visualization and storytelling.
Speaker 1:It's like, you might not remember every formula from algebra class, but you'll definitely remember that image of the doubling every day. Right?
Speaker 2:Right. It creates an moment that that sticks with you.
Speaker 1:Exactly.
Speaker 2:And that's what makes this lesson, you know, so effective.
Speaker 1:And that's what I love about these deep dives. It's like we started with a simple algebra lesson, but we end up talking about everything from the spread of viruses to interpreting graphs in the real world.
Speaker 2:Yeah. It shows how math isn't just this, like, isolated subject you learn in school. It's it's the language of patterns.
Speaker 1:Right.
Speaker 2:And those patterns, they're everywhere.
Speaker 1:So the next time you see a graph showing, say, the rising cost of something or the growth of a social media platform, take a second look. What kind of growth are you seeing? Linear, exponential. It might just change your whole perspective.
Speaker 2:And that's the beauty of it, isn't it? The more we understand these patterns, the better equipped we are to, you know, make sense of the world around us.
Speaker 1:Couldn't have said it better myself. A huge thanks to the minds behind for creating such an insightful and engaging lesson. And to our listeners, keep those moments coming. Until next time, happy exploring.