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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, everyone. Get ready because we're about to dive into a topic that might sound a little intimidating at first, exponential expressions.
Speaker 2:Don't worry.
Speaker 1:We promise it's way more interesting than it sounds.
Speaker 2:That's right.
Speaker 1:We're doing a deep dive into a lesson plan called expressed in Different Ways.
Speaker 2:And what we love about this lesson is that it takes these expressions, you know, these kind of abstract mathematical ideas Right. And shows how they play out in the real world
Speaker 1:Exactly.
Speaker 2:Specifically with US population growth Mhmm. From 1790 to 1860.
Speaker 1:So think about it. By the end of this deep dive
Speaker 2:you'll not only be able to wrangle those exponential expressions
Speaker 1:You'll also get a glimpse into how they were used to track.
Speaker 2:To really make sense of.
Speaker 1:Yeah. Exactly. Something as complex as population change over time, which is incredible.
Speaker 2:Yeah. And the lesson starts with, like, the nuts and bolts Okay. Of interpreting and evaluating these expressions, you know, really getting a handle on the basics. But where it gets really cool is when they start showing you how to,
Speaker 1:Manipulate them.
Speaker 2:Yeah. Manipulate them to reveal all these different facets of repeated percent increase.
Speaker 1:Which is basically how populations grow. Right?
Speaker 2:Exactly. Like compound interest, but instead of your savings account.
Speaker 1:It's a whole nation multiplying.
Speaker 2:Exactly. And speaking of multiplying, the lesson actually uses this equation. P equals 4000 times 1.031 to the t power.
Speaker 1:Okay. Breaking it down.
Speaker 2:Where t represents the number of years since 17/90.
Speaker 1:Okay.
Speaker 2:And p gives you the population, get this, in 1,000. In 1,000. Wow. And the fascinating thing here is how well that equation, simple as it seems Right. Actually models the US population during that period.
Speaker 2:It really highlights the power of these functions.
Speaker 1:Feel like it makes sense of these really big trends.
Speaker 2:Exactly. These real world trends that we see around us.
Speaker 1:Absolutely. And that's just the beginning. Right?
Speaker 2:Yeah. We're just getting started.
Speaker 1:So it's not like they just kinda throw this equation at you and say, alright. Go figure it out. They actually give you, like, specific things to do with it.
Speaker 2:Oh, absolutely. They have you really dig into the numbers. Like, one of the first things they ask you to do is find the population for specific years.
Speaker 1:Oh.
Speaker 2:So they say, alright. Using this equation, what was the population in 17/90? What was it in 18/60?
Speaker 1:So you're really kind of plugging it in.
Speaker 2:Exactly. You're plugging in the numbers for
Speaker 1:And seeing how it works.
Speaker 2:Yeah. And then they take it a step further and say, okay. Now let's figure out the annual percent increase.
Speaker 1:Oh, okay. So now we're talking about how much it's growing each year.
Speaker 2:Exactly. And, you know, that's kind of the first layer.
Speaker 1:Okay.
Speaker 2:But then here's where it gets really interesting. We're like, hold on a second. We don't just wanna look at it year by year. Let's zoom out. Let's see what happens over a decade.
Speaker 2:What's the percent increase per decade?
Speaker 1:Wow.
Speaker 2:And then, get this, they're like, let's zoom out even further. Let's look at a whole century. What's that percent increase look
Speaker 1:like? Oh, wow. So you're really getting a sense of how that growth
Speaker 2:It just it mushrooms. You know? It's not just this steady, like, oh, it's going up by this much every year. It's compounding on itself.
Speaker 1:It's really driving home that idea of exponential growth.
Speaker 2:Exactly. And then you know what I love about how they do it? They ask you to think about it.
Speaker 1:Okay.
Speaker 2:They say, which one of these timescales, whether it's years, decades, or centuries, gives you the most useful information about what was going on with the population during that time?
Speaker 1:That's such a good question.
Speaker 2:Right. Because it's not always about looking at the tiniest slice. Sometimes you need to zoom out to really understand the big picture trends.
Speaker 1:Absolutely.
Speaker 2:And so they're really encouraging you to think critically about, like, how are we looking at this data and what story is it telling us?
Speaker 1:I love that. Okay. So we've talked about population growth. And now let's jump into a totally different scenario that the lesson covers the printing business.
Speaker 2:Oh, yeah. This is a fun one. So imagine you have the small printing company. Right? And they're trying to modernize Get With the Times.
Speaker 2:Okay. So they launch an online ordering system, and suddenly, boom, their customer base starts growing like crazy.
Speaker 1:It's like that moment where everything changes for them.
Speaker 2:Exactly. Oh, that's awesome. And to model this growth Okay. They introduce a new equation. C equals 10 times 1.12 to the m power Okay.
Speaker 2:Where c represents the number of customers. Got it. And m is the number of months since they launched their online system.
Speaker 1:So this is where we're shifting from years to months.
Speaker 2:Exactly. And now they hit you with these really interesting questions.
Speaker 1:Like what?
Speaker 2:Well, they start by asking, what's the growth factor for a single month? But then they're like, okay. Now tell me, what's the growth factor for a whole year?
Speaker 1:So you're really starting to understand the nuances of how that exponent works.
Speaker 2:Exactly. You have to think about it.
Speaker 1:Yeah.
Speaker 2:M is months. Right? So if we're talking about a year, well, how many months are in a year?
Speaker 1:Right. 12.
Speaker 2:12.
Speaker 1:Yeah.
Speaker 2:So you're basically looking at what happens to that expression when it's 12, but it doesn't stop there.
Speaker 1:It keeps going.
Speaker 2:Oh, yeah. They're like, alright. We've got this equation in terms of months. Can you rewrite it in terms of years?
Speaker 1:Oh, wow. So it's like flipping the script.
Speaker 2:Exactly. It's like taking that same idea, that same growth, but expressing it in a different way.
Speaker 1:So you're really understanding the relationship between all the pieces.
Speaker 2:Exactly. And then just to hammer home that long term impact
Speaker 1:Okay.
Speaker 2:They ask you to calculate the growth factor over a whole decade.
Speaker 1:Wow. So you can really see.
Speaker 2:It's like, look how much this company could grow in just 10 years.
Speaker 1:And I think it's really cool how they're connecting it back to that initial population growth example.
Speaker 2:Totally. They're showing that this whole concept of exponential change, it's not just some abstract math thing. It's happening all around us.
Speaker 1:In so many different contexts.
Speaker 2:Yeah. We see it with populations, with businesses, even with things like compound interest. Yeah. It's everywhere.
Speaker 1:It's like they found a way to make exponential expressions, I don't know, actually exciting.
Speaker 2:Right. And that's the thing about this lesson. It's not just about, like, here's the formula, plug in the numbers, and you're done.
Speaker 1:Right. Exactly.
Speaker 2:They're really pushing you to understand the why behind it all.
Speaker 1:Yeah. To really grasp, like, what does this actually mean?
Speaker 2:Exactly. And they do such a good job of connecting it back to the real world. Like, throughout the lesson, they're really careful about defining all the key terms.
Speaker 1:Oh.
Speaker 2:So it's not just, like, throwing jargon around. They really wanna make sure that you understand.
Speaker 1:Okay. What does growth rate actually mean? What does growth factor actually mean? And how are those two things related?
Speaker 2:Right. Because once you have that solid foundation
Speaker 1:Right.
Speaker 2:You can apply it to all sorts of situations, whether you talk about populations, businesses, anything that involve this kind of repeated percent increase.
Speaker 1:Yeah. It makes it so much more applicable.
Speaker 2:Right. And I love that, you know, for those visual learners out there
Speaker 1:Oh, yeah.
Speaker 2:They even suggest sketching a graph of, like, that printing company's growth.
Speaker 1:Oh, that's cool.
Speaker 2:Because when you see that exponential curve, it's like, woah.
Speaker 1:You really get a sense of how quickly things can take
Speaker 2:off. Exactly.
Speaker 1:I'm a very visual learner, so I appreciate that.
Speaker 2:Me too.
Speaker 1:So we've talked about, you know, the different activities, the places where people might get tripped up. But one thing I really wanna emphasize is the lessons focus on critical thinking.
Speaker 2:Oh, absolutely. They don't want you to just be robots, you know, just plugging in numbers and spitting out answers.
Speaker 1:Right.
Speaker 2:They want you to think about the limitations of using a mathematical model to represent something as messy and complex as, like, population growth.
Speaker 1:Like, math is a great tool.
Speaker 2:It's a powerful tool.
Speaker 1:But it's not the be all, end all.
Speaker 2:Exactly. It's not crystal ball.
Speaker 1:There are other factors at play.
Speaker 2:And so they're encouraging you to, like, question assumptions, analyze different representations of the data.
Speaker 1:Right. Look at it from different angles.
Speaker 2:Exactly. And think about the bigger picture. You know, what are the limitations of this model? What other factors might be influencing this trend?
Speaker 1:That's so important.
Speaker 2:Right. Because at the end of the day, they're not just trying to teach you about exponential expressions. They're trying to teach you how to think Yeah. Like a mathematician.
Speaker 1:To really, like, use your brain.
Speaker 2:Exactly. To be critical, to be discerning. And those are skills that are gonna serve you well.
Speaker 1:In any field.
Speaker 2:In any field. No matter what you do.
Speaker 1:Exactly.
Speaker 2:Whether you're a mathematician or, you know, a musician.
Speaker 1:Right. Or anything in between.
Speaker 2:Yes, ma'am.
Speaker 1:So if you're looking for a way to spice up your understanding of exponential expressions, whether you're a student, a teacher, or just someone who's curious about the world
Speaker 2:This lesson is a fantastic resource.
Speaker 1:It's called expressed in different ways, and, of course, we'll link in the show notes.
Speaker 2:Absolutely.
Speaker 1:And a huge thank you to Illustrative Math for creating such an awesome engaging lesson.
Speaker 2:They did a great job.
Speaker 1:Until next time, keep exploring, keep asking questions, and we'll catch you on the next deep dive.
Speaker 2:See you then.