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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 how experts predict how populations will change, you know, like, way into the future?
Speaker 2:Yeah. It's like seeing into the future of cities and countries.
Speaker 1:Today, we're diving deep into population modeling.
Speaker 2:We'll be breaking down a lesson plan.
Speaker 1:A A lesson plan that helps students, well
Speaker 2:Helps students predict those populations.
Speaker 1:And they do it like pros.
Speaker 2:Using math.
Speaker 1:Okay. So think about this. We're talking about using math equations.
Speaker 2:To figure out how cities grow. Or even shrink. It's pretty cool when you think about it.
Speaker 1:It's like those moments you get in math class.
Speaker 2:Where it all suddenly clicks.
Speaker 1:Exactly. So the big picture here is getting students comfy using these, what are they called?
Speaker 2:Linear and exponential functions.
Speaker 1:Right. Right. To really grasp how populations change over time.
Speaker 2:And it's not just about memorizing the formulas. You
Speaker 1:No. No. Definitely not just plugging numbers into equations.
Speaker 2:It's about really understanding how to analyze the data.
Speaker 1:Data from the real world.
Speaker 2:Exactly. Like, thinking critically about how to represent that data.
Speaker 1:Choosing the right tools for the job.
Speaker 2:Because different populations have different stories. You know?
Speaker 1:Totally. And that's where choosing between linear and exponential comes in.
Speaker 2:Right. Linear functions. That's like steady, consistent change.
Speaker 1:Think about it like, like clockwork.
Speaker 2:Yeah. Precisely. But exponential.
Speaker 1:Exponential's a whole other ballgame.
Speaker 2:That's where things can really take off.
Speaker 1:Because of the constant percentage change. Right?
Speaker 2:Exactly. Things can explode fast with that percentage constantly changing. It's like, think about how quickly a video can go viral online.
Speaker 1:That's a perfect example. Or, like, the growth of well, almost anything that spreads really rapidly.
Speaker 2:And this lesson takes these kinda abstract mathematical ideas
Speaker 1:and grounds them in something students can really connect with.
Speaker 2:Their world, populations, the future. So how does this lesson plan actually get students using these functions?
Speaker 1:Well, it throws them right into the deep end.
Speaker 2:They get their hands dirty with some real case studies.
Speaker 1:3 to be exact.
Speaker 2:Three cities.
Speaker 1:Paris, Austin, and Chicago.
Speaker 2:And each city.
Speaker 1:Shows how populations can change over time. So we're talking, like, students get actual population data
Speaker 2:Real number.
Speaker 1:From these cities over decades.
Speaker 2:And they're pretty different, these cities.
Speaker 1:Yeah. You've got Paris, kinda steady growth.
Speaker 2:And Austin booming.
Speaker 1:And Chicago well, Chicago's a bit more,
Speaker 2:A bit more of a roller coaster.
Speaker 1:Exactly. Ups and downs. You know?
Speaker 2:And the lesson plan wants teachers to point those differences out.
Speaker 1:It's not about here's the right answer. It's more.
Speaker 2:It's about seeing how different each city's story is.
Speaker 1:And then the students become these, like, mathematical detectives.
Speaker 2:They have to figure out what's driving those changes.
Speaker 1:It linear, exponential
Speaker 2:Right.
Speaker 1:Or something else completely.
Speaker 2:They've gotta look at the data, calculate differences year to year.
Speaker 1:And they even use something called quotients.
Speaker 2:Yeah. Quotients.
Speaker 1:Now, I'll be honest. I had to, refresh my memory on that one.
Speaker 2:Oh, yeah.
Speaker 1:What are quotients, and why are they important here?
Speaker 2:A quotient is just what you get when you divide one number by another.
Speaker 1:Okay.
Speaker 2:So, like, a city's population 1 year divided by the population the year before.
Speaker 1:Ah, I see.
Speaker 2:It helps see if the growth is steady, like linear growth.
Speaker 1:Insistent.
Speaker 2:Or if it's changing, maybe hinting at exponential growth.
Speaker 1:So it's like, are we growing at a steady pace, or is it taking off like
Speaker 2:Like that viral video.
Speaker 1:Exactly.
Speaker 2:And the cool thing is the lesson plan hints. One city is perfect for a linear model.
Speaker 1:Another one just screams exponential.
Speaker 2:But here's the kicker.
Speaker 1:There's one that just doesn't fit.
Speaker 2:The rebel of the group.
Speaker 1:Messes up our nice, neat models.
Speaker 2:It's a great way to show that a perfect fit, sometimes you don't get one.
Speaker 1:It really depends on what the data is telling you.
Speaker 2:No one size fits all in the real world.
Speaker 1:And that in itself is a pretty important lesson.
Speaker 2:For sure. The world's complex, our models, Just approximations sometimes.
Speaker 1:But even more important than finding the perfect model
Speaker 2:is being able to explain why you chose that model.
Speaker 1:What's the reasoning?
Speaker 2:What's it telling us about how those populations are changing? That's the good stuff.
Speaker 1:Okay. So we've looked at Paris, Austin,
Speaker 2:Chicago. Individual cities.
Speaker 1:But what about the whole planet?
Speaker 2:The big picture.
Speaker 1:That's where the second activity takes us.
Speaker 2:Looking at world population.
Speaker 1:How it's changed and, And
Speaker 2:what might happen in the future.
Speaker 1:No pressure. Right.
Speaker 2:It's a big one.
Speaker 1:It's not just those hypothetical cities anymore.
Speaker 2:This is real global stuff.
Speaker 1:And it's a lot more, I don't know, open ended.
Speaker 2:Definitely. Students have to do some real digging.
Speaker 1:Yeah. Because they don't get the data handed to them this time.
Speaker 2:They've gotta find it.
Speaker 1:Reliable population data, and that's a whole other skill. Right?
Speaker 2:Absolutely.
Speaker 1:It's like real research.
Speaker 2:Figuring out where to go, who to trust.
Speaker 1:Who to trust with those numbers. Yeah.
Speaker 2:Exactly.
Speaker 1:And those numbers tell a pretty wild story. Right?
Speaker 2:Oh, yeah.
Speaker 1:Back in 1804, world population, get this.
Speaker 2:Hit a 1000000000.
Speaker 1:1,000,000,000. Wow. Fast forward to 2011
Speaker 2:7,000,000,000.
Speaker 1:7,000,000,000. That's a lot of people.
Speaker 2:Yes. Huge growth.
Speaker 1:It really shows you how powerful exponential growth can be.
Speaker 2:At least for a while.
Speaker 1:But it's not always predictable. Right?
Speaker 2:Not at all. No.
Speaker 1:You can't just draw one of those curves and say, that's it.
Speaker 2:Nope. Real life doesn't work like that.
Speaker 1:It's messy.
Speaker 2:Way messier.
Speaker 1:And the lesson really emphasizes that. You know?
Speaker 2:That our models are just tools.
Speaker 1:Linear, exponential, all helpful, but
Speaker 2:They have limits.
Speaker 1:The world is complex.
Speaker 2:Super complex.
Speaker 1:So those models, sometimes they just I don't know.
Speaker 2:They don't capture everything.
Speaker 1:Yeah.
Speaker 2:Exactly. There's always more going on than what's in the equation.
Speaker 1:Just like that rebellious city.
Speaker 2:Right. Sometimes the data just does its own thing.
Speaker 1:Throws you a curveball. So how does the lesson handle that when students are looking at, you know, world population, all that data?
Speaker 2:Well, it encourages them to think beyond just fitting the data to a model.
Speaker 1:Not just finding the perfect line.
Speaker 2:Exactly. It's about asking bigger questions. Like what? Like, what could slow down population growth in the future?
Speaker 1:Or even make it go down.
Speaker 2:Yeah. What factors play into that?
Speaker 1:The social stuff, economic stuff, the environment.
Speaker 2:All of it.
Speaker 1:It makes you think. Right?
Speaker 2:Definitely.
Speaker 1:We've seen these functions linear, exponential, how they help us understand populations changing.
Speaker 2:From single cities to the entire world.
Speaker 1:But the real world.
Speaker 2:It always keeps us guessing.
Speaker 1:It's never as simple as plugging in a formula.
Speaker 2:That's for sure.
Speaker 1:So listeners, until next time. Keep questioning those numbers.
Speaker 2:And keep exploring.
Speaker 1:There's always more to discover about the world and how it's changing.