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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. So imagine this. A student shares a hilarious meme. Right? And suddenly, boom.
Speaker 1:It's everywhere spreading like wildfire.
Speaker 2:Talk about exponential growth.
Speaker 1:Exactly. And that's what we're diving into today, how teachers can help students, well, grasp this whole idea of exponential rates of change.
Speaker 2:And, you know, it's funny you use that example because it's not just memes. Right? It's it's everything. Think about the spread of, like, a virus or even the stock market.
Speaker 1:It's everywhere.
Speaker 2:It really is. Understanding this stuff, it's not just about, like, acing algebra. It's about making sense of the world.
Speaker 1:Absolutely. And to guide our deep dive today, we are focusing on this really cool algebra 1 lesson plan. It's called looking at rates of change.
Speaker 2:Oh, I like it.
Speaker 1:So teachers, get ready because we're gonna break down the core ideas here, see how the activities make it all come alive, and, you know, let's be real. We'll tackle those head scratching moments your students might have along the way.
Speaker 2:And believe me, there are always a few of those.
Speaker 1:Right.
Speaker 2:But that's part of the fun. Right? What I find fascinating about this lesson is how it tackles a concept that often kinda well, it trips students up. You know?
Speaker 1:Oh, tell me more.
Speaker 2:It's the difference between linear and exponential functions, specifically their rates of change.
Speaker 1:Yeah. That can be tricky. It's like linear functions. Those are like a nice leisurely walk in the park. Right?
Speaker 2:Exactly.
Speaker 1:Predictable, steady.
Speaker 2:You know what to expect.
Speaker 1:Exactly. But exponential functions, oh, boy. They're like, you're hopping on a rocket ship.
Speaker 2:Hold on tight.
Speaker 1:Because that steepness, that growth, or even decay, it's constantly changing. And for students who are used to those nice straight lines
Speaker 2:It can feel wild.
Speaker 1:A little bit of a roller coaster ride.
Speaker 2:And this is where it gets really interesting because this difference, it's not just some abstract math concept.
Speaker 1:Right.
Speaker 2:It's crucial in countless fields.
Speaker 1:Oh, absolutely. Give me an example.
Speaker 2:Like, let's say you're trying to maul the spread of a virus. Yep. Right? Linear thinking. Not gonna cut it.
Speaker 2:Nope. You need exponential functions to really capture that that rapid, sometimes alarming rate of change.
Speaker 1:Which brings us to the heart of this lesson, the activities. And it's broken down into 3 main parts, each one building on the last.
Speaker 2:And as we go through them, keep an eye out for those moments because that's where abstract math, it becomes tangible.
Speaker 1:Love it. Alright. So first up, we have falling prices.
Speaker 2:Okay.
Speaker 1:Students get to analyze a table. It shows how the cost of solar energy has gone down over time.
Speaker 2:Interesting.
Speaker 1:It's a really cool way to ease them into this whole calculating the average rate of change thing, but using, you know, real data.
Speaker 2:I like it. Though it is worth mentioning, sometimes average cost and average rate of change, those can get jumbled up in student's mind. You know?
Speaker 1:Oh, 100%. It's like thinking about the price tag on something versus how quickly that price tag is changing.
Speaker 2:Yeah. Totally. Like, they're looking at a solar panel, but are they thinking about the price, or are they thinking about how much that price is dropping each year?
Speaker 1:It's 2 totally different
Speaker 2:things. Exactly. So teachers might wanna spend some extra time really unpacking that distinction, make it super clear.
Speaker 1:Great point. Okay. So next activity, we're getting a little more complex now, coffee shops.
Speaker 2:I'm listening. I'm listening. Coffee shops, everybody loves them.
Speaker 1:Right. So students are exploring, you guessed it, the growth of a coffee shop chain.
Speaker 2:Makes sense.
Speaker 1:And surprise, surprise, it follows that exponential pattern. Right?
Speaker 2:Naturally.
Speaker 1:But here's where it gets really interesting. This activity, it drives home the point that the average rate of change. It's not one size fits all.
Speaker 2:Oh, I see where you're going with this.
Speaker 1:It changes depending on what time frame you're looking at.
Speaker 2:And the graph is key here. Seeing that visual, that curve getting steeper, that's when it clicks.
Speaker 1:Yeah.
Speaker 2:You know, a single rate of change, that can't tell a whole story.
Speaker 1:Exactly. And speaking of seeing the bigger picture, our third activity, revisiting the cost of solar cells, it brings everything full circle.
Speaker 2:Okay. Back to the solar cells.
Speaker 1:Remember that table from the first activity, the one with the falling solar energy costs? Well, now students get to see that data in a whole new light.
Speaker 2:Literally.
Speaker 1:Literally. Because now they're analyzing a graph. They get to calculate and interpret those rates of change, but over different chunks of time.
Speaker 2:This is where it gets really into the analysis. Right? Yes. It's not just plugging in numbers anymore. It's what does this actually mean?
Speaker 1:They're basically becoming financial analysts using math to make predictions about the future of solar energy.
Speaker 2:And that's that's exciting. That's when they start to get it. You know? Yeah. Hey.
Speaker 2:I can use this for something real.
Speaker 1:Now wouldn't it be nice if learning always followed such a smooth upward trajectory?
Speaker 2:Well, in a perfect world. Right. But we know it doesn't always work out that way.
Speaker 1:So true. Even with the best laid plans, there are always gonna be a few bumps in
Speaker 2:the road. Exactly.
Speaker 1:Speaking of bumps, what are some misconceptions students might have when it comes to this whole exponential rate of change thing? What should teachers be prepared for?
Speaker 2:Okay. So a big one is that students, they often default to linear thinking.
Speaker 1:Makes sense.
Speaker 2:You know, they spend so much time with those nice, predictable, linear equations. It's only natural for them to think, hey, rate of change, same deal. Right.
Speaker 1:It's like trying to fit a square peg in a round hole.
Speaker 2:Yeah.
Speaker 1:They've got the square peg of linear thinking, and we're throwing them this round hole of exponential growth.
Speaker 2:Perfect analogy. And that's where teachers need to step in. Right? Gently guide them out of that linear comfort zone. And this lesson actually gives them a great chance to do that.
Speaker 1:How so?
Speaker 2:By having students, you know, compare and contrast those 2 side by side.
Speaker 1:Oh, I like where you're going with this.
Speaker 2:Remember that coffee shop activity? Yeah.
Speaker 1:The one where they're looking at the growth of the coffee shop chain?
Speaker 2:Exactly.
Speaker 1:Yeah.
Speaker 2:That's perfect for this. Teachers could even have students, like, try to fit a straight line to that graph.
Speaker 1:Oh, that's good.
Speaker 2:They'll see really quickly. It just it doesn't work. Especially not long term. Right. They might be able to, like, make it fit for a tiny little section.
Speaker 2:But over time, no way. That straight line just cannot capture that that beautiful dynamic curve of exponential growth.
Speaker 1:It's like trying to contain a wildfire with a garden hose.
Speaker 2:Exactly.
Speaker 1:Okay. So that's one misconception. What about the other one you mentioned?
Speaker 2:Right. So the other big one is, well, it's connecting those average rate of change calculations, you know, to what the data actually means.
Speaker 1:So it's not enough to just crunch the numbers.
Speaker 2:Right. Students can get so caught up in the how, they miss the why. The bigger picture. Exactly. They're so focused on the individual trees, they miss the whole forest.
Speaker 1:Love that visual.
Speaker 2:And that's where those real world examples, they're so crucial. Instead of abstract numbers, have students think about the units, you know, the falling prices activity. Instead of just saying negative 7.55, they could say, the cost of solar energy, it dropped by an average of $7.55 each year.
Speaker 1:Oh, that's so much more powerful. It's not just a number anymore. It's a story.
Speaker 2:Exactly. When students connect those calculations to a story, suddenly, it clicks. It's meaningful. It's memorable.
Speaker 1:It's real.
Speaker 2:Exactly.
Speaker 1:So we've talked about the core math ideas. We've explored these engaging activities, and we've even, you know, mapped out some of those common misconceptions. I feel like we've covered a lot of ground here.
Speaker 2:We have.
Speaker 1:But before we wrap up this deep dive, I always love to leave our listeners with a little something to ponder, you know.
Speaker 2:I like it.
Speaker 1:A spark to ignite further exploration. What do you have for us today? This lesson does a fantastic job of exploring exponential growth and decay. But, you know, it makes me wonder, are there other real world scenarios that teachers could use to, like, really bring this concept home for students?
Speaker 2:Oh, tons. Think about how information spreads online.
Speaker 1:Oh, yeah.
Speaker 2:A single tweet going viral. Totally. Exponential. Or how about when a small investment just, like, snowballs over time because of compound interest?
Speaker 1:There it is again.
Speaker 2:Exponential again. Even something like, the growth of bacteria under the right conditions.
Speaker 1:It's all connected.
Speaker 2:It all comes back to this idea of change that just accelerates so quickly.
Speaker 1:Like that saying, it always seems impossible until it's done. Right?
Speaker 2:Yeah.
Speaker 1:Exponential growth can feel almost invisible at the beginning.
Speaker 2:Right.
Speaker 1:And then, boom, suddenly, huge changes are happening everywhere.
Speaker 2:Exactly. And, you know, when students start to see those connections for themselves, that's when math goes from being something they have to learn to something they're driven to understand.
Speaker 1:100%. So teachers, if you're listening, think about how you can bring these real world examples into your classroom. What would happen if your students became, like, viral marketing detectives? Oh, I love that. Right.
Speaker 1:Analyzing the growth of your students became, like, viral
Speaker 2:marketing detectives. Oh, I love that.
Speaker 1:Right. Analyzing the growth of online trends.
Speaker 2:So cool.
Speaker 1:Or maybe they could be like financial advisors for a day and explore how different investment strategies play out over time using, you know, what they're learning about exponential functions. The possibilities are truly endless.
Speaker 2:And that's what's so great about a deep dive like this, isn't it? Yeah. We're not just giving you the information. We're trying to spark those new ideas, get you excited to make these concepts come alive for your students.
Speaker 1:Absolutely. And on that note, a huge thank you to the creators of Illustrative Math for this amazing lesson plan and to you, our listeners, for joining us on this deep dive. Until next time. Keep those minds curious and those classrooms engaged.