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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.)
Ready to unpack some serious algebra. Today, we're doing a deep dive into how students learn about patterns of growth.
Speaker 2:It's more than just memorizing formulas, though. We're giving teachers the tools to help students understand how things in the real world change.
Speaker 1:So we're talking about seeing patterns and then using math to, like, describe them Mhmm. To predict what'll happen next. What are we looking at specifically?
Speaker 2:The heart of this lesson is linear and exponential growth.
Speaker 1:Okay.
Speaker 2:Think of it this way. Linear growth is like you're driving a car and you set the cruise control. You're going the same speed, right, same distance, same amount of time. It's predictable.
Speaker 1:Makes sense.
Speaker 2:But exponential growth, now that's like slamming on the accelerator. Things change faster and faster as you go.
Speaker 1:I like that. That. So if my students are looking at a table of values by the end of this lesson, what should they be able to tell me?
Speaker 2:They should be able to instantly tell you just from looking at the numbers if the pattern is linear or exponential.
Speaker 1:Oh, interesting.
Speaker 2:They should be able to see if it's growing by the same amount each time, that's linear, or if there's a common factor, which would be exponential growth.
Speaker 1:And then they can use that to, like, make predictions.
Speaker 2:Exactly. They can use that information to write an equation and even predict what the values will be in the future. It's like having a crystal ball.
Speaker 1:I love it. How does this lesson actually teach them to do that, though? What are some of the activities they'll do?
Speaker 2:Well, it starts with something called which one doesn't belong. They'll get 4 tables of values. Right. But one of them won't fit the linear or exponential pattern.
Speaker 1:Oh, so they have to really think about those relationships. Sometimes as teachers, we know just seeing that contrast, it just makes it click.
Speaker 2:Exactly. They'll have to decide. Are these numbers going up by a certain amount, or are they being multiplied by something?
Speaker 1:So they're really understanding the difference between linear and exponential, not just memorizing a definition, and they're doing that right from the start. What's next? Then they
Speaker 2:get into activities like growing stores.
Speaker 1:Mhmm.
Speaker 2:In this one, there are two scenarios. One company is expanding its stores at a steady rate. That's linear growth.
Speaker 1:Yeah.
Speaker 2:The other one's doubling its stores every year. That's exponential.
Speaker 1:Wow. Big difference. So they get to see how those patterns actually play out in different situations? Do they get to see it visually too?
Speaker 2:Absolutely. They'll build tables, calculate the differences and factors. This will help them understand how those numbers show the growth pattern. Then there's flow and followers. That one helps them make connections between the different ways we can represent growth.
Speaker 1:So, like, word problems and equations. What else?
Speaker 2:Exactly. Word problems, tables, and expressions. They have to figure out which ones go together like a puzzle.
Speaker 1:That's a fun way to do it. I bet the students who like a challenge will love that one. And what's this one? Meow Island. That one sounds kinda fun.
Speaker 2:Oh, yeah. In that one, we've got 2 islands with cats on them.
Speaker 1:Okay.
Speaker 2:One island's cat population is growing at a steady rate linear growth, But on the other island, the cats are multiplying really fast, and that's exponential.
Speaker 1:So they have to look at the tables, figure out the pattern, and then what? Predict how many cats there will be.
Speaker 2:You got it.
Speaker 1:I love that. It takes something that could be kind of abstract, like these math concepts, and makes it real. I mean, who doesn't love cats?
Speaker 2:Exactly. It's all about finding those examples that make sense to them.
Speaker 1:And speaking of making sense of things, teachers are probably already thinking about how their students will react to this lesson. What are some of the things they might find tricky about linear and exponential growth?
Speaker 2:Yeah. I mean, no matter how cool the activities are, there are always gonna be some things that trip students up. What are some of the things they have trouble with when it comes to linear and exponential growth?
Speaker 1:Well, sometimes they just get the 2 types of growth confused.
Speaker 2:Makes sense.
Speaker 1:Like, they get that you add the same amount each time with linear growth. But then when it comes to multiplying by the same factor for exponential, it's like
Speaker 2:It's like they forget about multiplication. They wanna keep adding. Exactly. That's why it's so important to show them lots of different examples and have them really think about what's happening. Like, are we going up by a certain amount, or are we multiplying by something?
Speaker 2:Having them use their own words to explain the difference is really helpful.
Speaker 1:It helps them make those connections. Yeah. What other misconceptions should teachers be ready for?
Speaker 2:This one might seem kinda obvious, but it trips them up more than you'd think.
Speaker 1:Okay.
Speaker 2:It's when they have to take words like triples or doubles and turn them into, you know, actually multiplying.
Speaker 1:Oh, I see. It seems easy, but I guess when they're thinking about everything else, it can be tricky.
Speaker 2:Yeah. To help with that, we can make sure to emphasize that, like, triples always means times 3, doubles means times 2, and so on. I think visuals help a lot with this too. Diagrams, pictures, anything to really show them what it means.
Speaker 1:I love that. Make it stick. So we've got these concepts. We've got some cool activities. What's something teachers can leave their students thinking about even after the lesson is over?
Speaker 2:Here's a good one. What if the growth isn't perfectly linear or exponential?
Speaker 1:Yeah. Right. Because the real world doesn't always work that way.
Speaker 2:Exactly. We can ask them to think about things like the stock market or how populations grow when there are other factors involved. It's not always a simple pattern.
Speaker 1:So they see that these types of growth are important, but they don't tell the whole story.
Speaker 2:Exactly. And if we encourage them to think beyond just the simple cases, it helps them apply what they've learned in a more, I don't know, a more sophisticated way.
Speaker 1:For sure. And who knows? Maybe it'll even get them interested in data and modeling.
Speaker 2:Oh, that would be amazing.
Speaker 1:It would. This deep dive has been so helpful. We talked about the basics of linear and exponential growth. We talked about what might be tough for students and even how to make them think even deeper.
Speaker 2:Absolutely. It's been great breaking it all down with you.
Speaker 1:And a huge thank you to the authors of Illustrative Math for giving us the material for this deep dive. If you're listening, teachers, you can find a link to the lesson plan in our show notes. And until next time, keep those minds growing. I think that's so cute. You know, helping them see that math is everywhere.
Speaker 2:Absolutely. And this lesson does a really nice job of, like, giving them a starting point. They learn about linear and exponential growth, and then they can start to see it all around them.
Speaker 1:Yeah. Those moments. Right? When they're like, oh, that's exponential growth.
Speaker 2:Exactly. Those are the best, aren't they? When they realize math isn't just something you do in school, it's a way to understand how things work.
Speaker 1:It's like giving them a new set of glasses to see the world through.
Speaker 2:I like that. And this lesson with all its cool activities and connections to, you know, real stuff, it's a great way to start them on that path.
Speaker 1:It really is. This deep dive has been so helpful even for me. We covered what linear and exponential growth are, how students might struggle with them, and even how to push their thinking further.
Speaker 2:It's been great. I hope teachers feel ready to take this lesson and make it their own.
Speaker 1:And a big thank you to Illustrative Math for the materials for this deep dive. Teachers, you can find a link to the lesson in our show notes. And until next time, keep those minds growing.