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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 heard that story? You know, the tortoise and the hare, slow and steady, they say, wins the race.
Speaker 2:Yeah. Classic.
Speaker 1:Well, get ready because our deep dive today, it's about a math showdown where the tortoise well, it's got a secret weapon. We're talking exponential growth going head to head with quadratic growth.
Speaker 2:Oh, this is gonna be good.
Speaker 1:Buckle up because we're diving into some algebra. Don't worry. Not your boring old lecture. We've got these excerpts from a real teacher's guide all about comparing these two types of functions.
Speaker 2:My fun.
Speaker 1:And we're here to pull out the good stuff, the little things that'll make your students go, woah. I get it now.
Speaker 2:I love those moments.
Speaker 1:Right. So this lesson plan, it jumps right in, sets the stage for this epic math battle, even lays out the goal. Wanna know what it is
Speaker 2:Hit me with it.
Speaker 1:To show everyone, and I mean everyone, that even though both types of growth seem impressive, exponential functions, they play the long game. They end up way ahead, leaving those quadratic functions in the dust.
Speaker 2:It's true. They may start slow, but once they get going
Speaker 1:No stopping them.
Speaker 2:Exactly. And what I find fascinating is this lesson plan doesn't just throw formulas at you. You know? It builds on what students should already know. Like, remember linear versus exponential growth.
Speaker 1:Oh, yeah. That was a fun one.
Speaker 2:Right. So this lesson, it creates this, like, connected understanding of how things change over time.
Speaker 1:So it's all about building those connections. Makes sense.
Speaker 2:Exactly. And, honestly, understanding how different things grow, it's like a superpower. Suddenly, you can look at investments, population, even, like, how a virus spreads and kinda start to see where things might go.
Speaker 1:No way. Predict the future.
Speaker 2:Okay. Maybe not predict, but definitely make smarter guesses.
Speaker 1:That's still super useful. Oh. Okay. I'm hooked. How does this lesson actually break down this math showdown?
Speaker 2:Well, let me tell you.
Speaker 1:Because I was looking, and it seems like it's all built around activities, hands on stuff, which I'm always a fan of.
Speaker 2:Oh, absolutely. Way more engaging than just listening to someone drone on and on about equations.
Speaker 1:Right.
Speaker 2:And this lesson, it nails it.
Speaker 1:There's this super cool warm up where students have to use their brains, put expressions in order, least to greatest. But here's the catch. No calculators?
Speaker 2:Oh, sneaky.
Speaker 1:They gotta use what they know about exponents strategically.
Speaker 2:I like it. Gets them thinking.
Speaker 1:I know. Right? Yeah. I can already picture my students getting fired up for that channel.
Speaker 2:Twinkle friendly competition.
Speaker 1:Exactly. And then it gets even better. Activity 4.2. Which one grows faster? It's where it gets really good.
Speaker 1:Students have to actually predict which function will win the whole growth race. Yeah. Quadratic, exponential, place your bets.
Speaker 2:I'm on the edge of my seat.
Speaker 1:Right. But I love that they have to think about it, you know, intuitively before even looking at the actual equations.
Speaker 2:It's like setting the stage for a magic trick. You think you know how it'll unfold and then bam.
Speaker 1:Mind blown.
Speaker 2:Exactly. And to represent the functions, they use these visual patterns, pattern a and pattern b, super clear, which is so important, especially for those visual learners who might not be as comfy with just the abstract equation.
Speaker 1:Oh, that makes sense. Seeing the growth visually can be huge.
Speaker 2:It's a game changer.
Speaker 1:Although, I can see where some students might get tripped up.
Speaker 2:Oh, yeah. Like what?
Speaker 1:Like, looking at pattern b, seeing all that doubling and just assuming, oh, that's gotta be linear. Right? Yeah. Easy to fall into that trap.
Speaker 2:True. True. It does seem that way at first glance. Right. But you gotta look closer.
Speaker 1:Gotta dig deeper. So they've got their visual patterns. Yeah. Then what? How do they actually see this growth unfold?
Speaker 2:Well, that's where our good friend, the table, comes in. The lesson has students make tables to track each function as it grows.
Speaker 1:Nice and organized.
Speaker 2:Right. And this is where the magic really happens, where the differences between quadratic and exponential growth, they become crystal clear.
Speaker 1:So we're talking those changing growth factors in action.
Speaker 2:You got it. The tables really drive home how those growth factors in quadratic functions, they're all over the place, always changing.
Speaker 1:Well, exponential functions have that steady constant growth factor.
Speaker 2:Right.
Speaker 1:Their secret weapon.
Speaker 2:Exactly. It's what lets them pull ahead in the end.
Speaker 1:Okay. That's making more sense now. But it's not just about numbers in a table, is it? The lesson encourages bringing in visuals too. Right?
Speaker 2:Right. Because who doesn't love a good graph?
Speaker 1:Exactly. Give students that moment where they see that exponential curve just taking off like a rocket ship.
Speaker 2:Makes you wanna shout, look out.
Speaker 1:I know. Right? And they even include this little teacher tip about adjusting the graphing window to make the difference even clearer. It's all in the details.
Speaker 2:They really thought of everything. It really is. It's like, boom. There it goes.
Speaker 1:Taking off into the stratosphere.
Speaker 2:Yeah.
Speaker 1:But you mentioned earlier that the lesson plan, it anticipates some struggles students might have.
Speaker 2:It does. Yeah. It's like they've been in the classroom, you know.
Speaker 1:The best kind of resource. So besides that thing with pattern b, thinking it's linear because of the doubling Right. Right. What other hurdles might students hit when they're exploring these ideas?
Speaker 2:Well, one thing that trips them up sometimes is really getting why exponential growth, it ultimately beats quadratic growth.
Speaker 1:Even when the quadratic function starts out looking all strong and mighty.
Speaker 2:Exactly. Like, they can see it in the tables, the graphs, but wrapping their heads around the why, the math behind it all, that can be tough.
Speaker 1:Makes sense. So how does the lesson plan tackle that why question? Because, honestly, if we can't explain it clearly to our students
Speaker 2:Or are we even teaching it? That's right. Exactly. Well, this is where this lesson play it really shines. It says to focus on those growth factors, those changing growth factors we were talking about.
Speaker 1:Right. Right. Keep
Speaker 2:going. Remember how quadratic functions, their growth factors are all over the place, no consistency?
Speaker 1:Yeah. Yeah. And exponential functions, they've got that steady, unchanging growth factor.
Speaker 2:Their secret weapon. It's like, imagine the hare in that race. It keeps getting tired, slowing down.
Speaker 1:But the tortoise, slow and steady?
Speaker 2:With that steady pace, eventually, it catches up and zooms past that constant growth factor. It's the tortoise's superpower.
Speaker 1:Okay. That analogy really clicks. I could see that being super helpful for students.
Speaker 2:Especially those visual learners, right, who maybe aren't as into the abstract math.
Speaker 1:Exactly. Yeah. But the lesson doesn't just leave it at the abstract level either, does it?
Speaker 2:It doesn't. No way.
Speaker 1:It encourages teachers to help students connect these ideas to the real world. Right?
Speaker 2:Absolutely. Real life stuff.
Speaker 1:Find those examples of exponential and quadratic growth all around them.
Speaker 2:Because let's face it. If they don't see how the math applies to their world
Speaker 1:They'll probably just tune out. Yeah. So how does it suggest making those real world connections? Does it give specific examples?
Speaker 2:Oh, yeah. It does. It talks about stuff like investment returns.
Speaker 1:Oh, interesting.
Speaker 2:Where, like, simple interest, that might show you quadratic growth. But compound interest, that's where you see the exponential growth.
Speaker 1:Oh, that's a good one.
Speaker 2:What else? And then there's population growth. That can often be exponential, especially early on.
Speaker 1:Makes sense. And what about, like, how information spreads online?
Speaker 2:Oh, absolutely.
Speaker 1:That's gotta be exponential. Right?
Speaker 2:Think about a viral video, how fast it goes from a couple shares to, like everyone's seen it.
Speaker 1:It's like that exponential curve just taking off.
Speaker 2:Exactly. That's what this lesson is all about, helping students see that these aren't just random math things. They're patterns, real patterns shaping our world.
Speaker 1:Yeah. They're everywhere once you open your eyes to it.
Speaker 2:Isn't it cool? And what I love about this lesson plan is it doesn't just, like, throw these big ideas at teachers and say, figure it out. It gives you a real road map. You know?
Speaker 1:Totally. It's like having a coach in your corner helping you through each step.
Speaker 2:Exactly. It tells you those misconceptions students might have, gives you clear ways to address them, even suggest questions to ask to get those moments.
Speaker 1:It's like they've thought of everything. So as we're wrapping up this deep dive into exponential and quadratic growth Yeah. What's the big takeaway? What do you hope listeners walk away with?
Speaker 2:I think the biggest thing is that idea of the constant growth factor and exponential functions. It might not seem like a big deal at first, but it's what lets exponential growth, you know, eventually blow past even the most impressive quadratic growth.
Speaker 1:Like that saying, slow and steady wins the race, but with a math twist. The tortoise has a secret engine or something.
Speaker 2:Exactly. And that's such a powerful idea for students to grasp.
Speaker 1:It really is. Well, huge thanks to the authors of Illustrative Math. This lesson is fantastic. And to our listeners, thanks for joining us on this little math adventure. Keep exploring.
Speaker 1:Keep asking those questions, and we'll see you next time for another deep dive.