Alright. So we're diving into exponential decay today. Yeah. Something that, I think we all kinda remember from high school math class. Right?

Yeah. But but it's one of those things. It sounds scarier than it actually is.

Yeah. Definitely. It's one of those things that sounds very intimidating, but it really is pretty straightforward. And it's all over the place. Right?

Yeah.

It's kinda remarkable how often you see examples of exponential decay.

Oh, absolutely. Yeah. And I think what we're gonna be looking at today in this deep dive is really gonna highlight that. Like, this is a lesson plan for high school algebra, and it actually does a great job of of of illustrating just how much we encounter exponential decay in everyday life.

Yeah. It's a really interesting approach. The way they've set up this lesson plan, it really does take something that could feel very abstract and makes it super relevant to students. Mhmm. It's not just like, here's the formula.

Memorize it. It's like, no. Here's how this actually works in a way that you can connect with.

Yeah. And I think that's so crucial because once you see those connections to your own life

Mhmm.

It's like, oh, okay. This actually this matters.

Yeah.

This isn't just some random thing. Right. So so how does this lesson actually get started?

Well, they kick things off with a warm up activity. Pretty simple at first glance. You know, students are asked to kind of look at what happens when you keep dividing a number in half, starting with a half then going to a 4th then an 8th.

Oh, interesting. Yeah. I see where you go with this, like patterns.

Right? Exactly. Exactly. But but here's what's kinda cool about it. By working with those fractions, right, they're kinda they're subtly introducing this idea of a decay factor, like, even before they formally give it a name.

Mhmm. And I think that really helps build that intuition.

Right. It's like they're easing them in without them even realizing it.

Yeah. Exactly. And then later on, when you get to those more more complex ideas, they're like, oh, wait a minute. I've seen this before.

Right. Yeah. It's not so intimidating anymore. It's like sneaking the veggies into the meal before they even know.

Exactly.

And did I see that they even have them, like, look for patterns

Yeah.

In the in the decimal form of those fractions as well?

Oh, yeah. Absolutely. It's it's a really nice way to bring in that that extra number sense work, without without sort of taking away from the the main goal of the lesson.

Right. Right. So they're already starting to think about how those numbers are changing, how they're shrinking. Yeah. Okay.

I like that. But then, it looks like we jump from that into something that's that's pretty relatable, depreciating cell phones. Because, I mean, who hasn't experienced that? You buy a brand new phone and the second you walk out of the store, it's like it's already lost half its value.

It's it's so true. It's like the the minute you open the box, it's like, oh, it's depreciating.

Yep.

And this is where they have this activity called falling and falling.

I love it.

Which, you know, immediately, you're like, okay. I I get that.

Yeah.

And and in this activity, students get to compare the depreciation of 2 different phones, but they do it using graphs.

Okay.

And the goal is, can they come up with an equation that kind of represents this decline in value?

So it's it's actually a really practical skill that they're developing here.

Absolutely. Yeah. And, you know, this actually highlights a really common misconception. Even among experienced teachers, sometimes we fall into this trap, and that's focusing on the absolute change in price rather than the proportional change.

Oh, interesting.

Right? It's easy to look at, you know, a phone and be like, oh, wow. This one lost, you know, a $100 in value in a year. Like, that's that's a lot.

That's a terrible deal.

Right. But the real key is to look at, okay, what percentage of its value did it lose each year?

Right.

Because that's where the decay factor comes in.

Which makes sense. Because if you have a phone that's initially more expensive, of course, those yearly price drops are gonna look more dramatic on the surface even if it's actually depreciating at a slower rate than a cheaper phone.

Exactly. Exactly. And that's why having them compare 2 different phones side by side is so so effective.

Mhmm.

It forces them to think more critically

Yeah.

About, okay, what is this depreciation rate actually telling me?

So they're not just plugging and chugging. They're really having to think about, like, the why behind it.

Right. Right. And it's that deeper understanding that allows them to then take that and apply it in different situations.

Absolutely. I love that. So we've gone from, you know, these these shrinking fractions, which seems very abstract, to something that's, like, very concrete cell phones. Everybody's got one. Everybody gets it.

Yep. But then it looks like this lesson plan takes us to a matching game of sorts.

Kind of. It's it's more of a card sort, really, where students get a set of cards, and each card describes a different scenario. And some are exponential growth and some are exponential decay.

Okay.

And they have to figure out which graph goes with which card. Gotcha. So it sounds simple, but I'm guessing there's a there's gotta be a twist.

There's gotta be some, yeah, some challenges built in there.

Oh, there are. Absolutely. Yeah. It's designed to kind of, you know, mess with those assumptions a little bit, you know, about how how exponential functions work because you might have, you know, some scenarios where the starting point is pretty similar, but the rate at which they grow or decay is

very different.

Right. So you gotta really look closely.

Okay. So how do students even begin to tackle this? Where do they start with something like that?

Well, I think a good starting point, and and this is a strategy that they suggest, is just to kind of, you know, do that initial sort of growth decay. Get them in the right piles.

Okay.

But but here's here's where things get interesting. Just knowing it's growth or decay, it's not enough.

Right.

You gotta then look at, okay, but what's the rate of change?

Right. Right. Because 2 things can be decaying, but at very different speeds.

Exactly. And so that's where that real deep dive into the graphs comes into play.

Got it. Got it. So it's kind of like this gradual process of elimination Yeah. But but one that requires them to really look at the graphs carefully.

Absolutely. Yeah. You can't just glance at it and be like, oh, yeah. That one that one.

That one looks right.

You gotta you gotta really analyze it. And, of course, you know, there's there's some things built in to make it a little trippier.

Sure.

You know, for instance, the scales on the graphs might be different.

Right. Right. Yeah.

So if you're not paying attention, that'll trip you up.

It's easy to get thrown off by that.

Exactly. Or you might misinterpret, you know, how fast or slow something is growing or decaying, especially when you're comparing things that have similar starting points.

Right. Those are all really good points. And I imagine as a teacher, you know, you're you're watching students go through this. You're probably seeing some of those common mistakes pop up. Yeah.

So so what can teachers do to kinda help guide them through that process? What are some, like, key questions or prompts that they can use?

Well, one thing that the lesson plan actually does really well is it provides teachers with some really insightful questions to be asking during this activity.

Right.

And one of the things they emphasize is talking about that yi intercept. Right? Like, what does that tell us about the scenario? But here's the catch. The way they've designed these descriptions, the way intercept alone often isn't enough to make the match.

Oh, interesting. Okay. So it's another layer of of challenge built in there.

Exactly. It forces you to think a little bit harder. And then another thing they suggest is to really focus on what that growth factor can tell us.

Okay.

And that's where I think things get really interesting because when you start analyzing that growth factor, you can start to differentiate between, you know, scenarios that might look similar on the surface, but they're gonna behave very differently over time.

Right.

It's like if you think about, you know, like investing money. Right? You could have 2 different investments. They both start at the same amount, but, you know, one of them has a slightly higher growth factor. It's gonna, you know, earn way more in the long run.

Well, that's a great analogy.

Yeah.

Yeah. It really brings it home. And and it highlights how, you know, understanding these concepts isn't just about passing a math test. It's about, like, you know, making smart decisions in life.

Exactly. Exactly. Yeah.

But, you know, I imagine even with all of these, you know, guiding questions and and strategies, there are still gonna be some students who who just struggle to to really grasp these concepts.

Oh, for sure. Yeah. And and you know what? That's fine. That's part of the process.

Right? I mean, that's what makes this lesson plan so great is that it actually acknowledges that there are gonna be these bumps in the road, and it gives teachers ways to to deal with that.

Yeah. Okay. So after after the card sort, what what comes next?

So they wrap things up with this really, really nice cool down activity that kinda brings everything together.

Oh, I love a good cool down.

Yeah. It's like that moment, you know, after after all the hard work, you get to kind of step back and and solidify things.

Exactly. And

in this cooldown, students are given this graph, and it shows the value of a camera going down over time. You know, depreciating. Another real world connection there.

Right.

But here's the twist. Instead of them having to come up with the equation themselves, they're given a few different, written descriptions.

Okay.

And they have to choose the one that actually matches the graph.

Oh, so they're kinda working backward. They've got the visual, and now they have to match it to the word.

Exactly. Exactly. And and to do that, they have to use everything they've learned about, like, how to analyze those graphs.

Right. You

know, where is that initial value? Is this thing is it is that growth factor bigger than 1 or smaller than 1? And then what is that specific value?

Okay. So it's a really clever way to assess their understanding without just saying, okay. Now go write the equation.

Exactly. Yeah. It's not just about getting the right answer. It's can you show me that you understand how all these pieces fit together? Yeah.

The graph, the equation and and that real world story.

Love it. So we've seen how this lesson, you know, unfolds from from that kind of, you know, sneaky warm up all the way to this this final, challenge. But if we were to, like, distill it down to its essence, what are those big takeaways for teachers? What should they really be hammering home when they're teaching this concept?

Yeah. Well, I think, you know, the lesson plan itself does a fantastic job of summarizing those key points. And one of the things they really stress is the importance of having students use that graph to find that information. Right? Like, what's that starting amount?

Is the is the growth factor greater than 1 or less than 1? Right. And then what is that actual value? And and they really encourage teachers to, like, slow down. Let the kids really wrestle with those ideas even if it means, you know, letting them struggle a little bit.

Yeah. Because that's how you get that deep understanding.

It's about giving them the tools to kinda construct their own understanding, not just, you know, handing them a formula and saying, okay. Now go memorize this.

Exactly. Exactly. And I think, you know, what's so brilliant about this lesson is that it does all of that while still being so engaging. Yeah. Like, the kids are into it.

They're not just learning about some abstract math concept.

Right.

They're seeing how this actually plays out in in their world. Right? Whether it's the value of a phone or or a camera or or any number of things.

Like, you almost have to, like, unlearn some of those those initial instincts. You know? Right. Because you look at graph and you see it going down, you think, okay. It's it's decreasing.

But you have to really look closely to see, is it decreasing at a steady rate? Or is it is it that rate itself changing, which is, you know, what we're talking about with exponential decay?

Yeah. And that's why I think it's so helpful, like you were saying, to to actually have those visuals. Mhmm. To put that linear graph next to an exponential decay graph and really let students see the difference.

Yeah. And encourage them to really, like, zoom in.

Right? Yes.

Not just look at the big picture, but look at those individual changes from one step to the next.

Exactly. Exactly. And and ask those questions like, okay. Is it going down by the same amount each time, or is that amount getting smaller and smaller?

Right.

Sometimes just, like, reframing the question in that way can make a big difference.

It's all about helping them develop that that lens, that analytical lens

Absolutely.

To really dissect what they're seeing. Okay. So confusing it with linear decrease, that's one potential pitfall. What else what else do you see tripping students up when they're first learning about exponential decay?

Well, that growth factor can definitely be a tricky one. Okay. Because, you know, we're so used to thinking about growth factors in terms of things getting bigger. And then suddenly, we're talking about decay, and it's like, wait a minute. It's between 0 and 1 now.

Right. Because a growth factor greater than 1 means things are actually growing.

Exactly. So you gotta kinda shift your thinking there. And and one thing that I found really helpful, and this might sound kinda basic, but I think it's really important, is just being really intentional about the language that we use when we're talking about those growth factors.

Oh, interesting. Okay.

So it's not just saying, like, okay. The growth factor is 0.75. It's like, okay. That means each year, the value is only 75% of what it was the year before.

So you're you're almost, like, translating the math into into, like, everyday language

Yeah.

Making it less abstract.

Exactly. Because that's how you make it stick. Right? It's not just about memorizing a formula. It's understanding the concept behind it.

Absolutely. Now before we before we wrap up this deep dive, I do wanna touch on one more thing. Mhmm. And and it's not, you know, specific to exponential decay. Mhmm.

I think it's something that that a lot of students struggle with, especially when they get those those dreaded word problems.

Word problems. Yeah. Every student's like, worst nightmare.

I know. Right? Yeah. It's like it's like their kryptonite. But but, honestly, I think for a lot of students, the hardest part is just, like, knowing where to start.

Right.

Right? They've got this this big chunk of text, maybe a diagram, some numbers thrown in there.

It's overwhelming.

It is. It is. It's like where do I even where do I even begin?

Mhmm.

So so what are some strategies that that teachers can use to kinda help students break down those word problems? Mhmm. How do you how do you make it less intimidating?

You know, one thing I always tell my students is read the problem multiple times.

Okay.

And each time you read it, focus on something different.

Oh, interesting.

So that first read through, don't even worry about the numbers.

Okay.

Just what's the story here? What's happening in this problem? What are they asking me to find?

So it's about, like, setting the scene Exactly. Yeah. Yeah. Creating that mental model.

Right. Right. And then the second time through, now you can start zeroing in on those details. What are the actual quantities? What are we measuring in?

Are there any keywords that kind of hint at what I'm supposed to be doing? What operation I need to use.

It's like they're they're gathering clues

Exactly. Exactly.

Before they even try to, like, solve the case.

Right.

And sometimes and I know this sounds maybe a little out there, but sometimes it can even be helpful to have them, like, rephrase the problem in their own words.

Yes. Love that. Or even

just draw a quick sketch. You know? Yeah. Anything to make it feel more more real to them.

Absolutely. Yeah. Because those visuals can be so powerful.

So powerful. Okay. Those are those are some seriously great tips. And and, you know, that strikes me that those strategies, while they're, you know, incredibly helpful in math, they're not just math strategies. Right.

They're they're life strategies.

Absolutely. Yeah. I mean, we're talking about problem solving

Right.

Critical thinking.

Attention to detail.

Exactly. Yeah. Those are skills that you need no matter what you're doing.

Couldn't agree more. Well, it seems like we have reached the end of our deep dive. Any final any final thoughts before we resurface?

You know, I think just to kind of bring it back to this lesson plan, I think it does such a nice job of showing us how we can make even, you know, tricky math concepts like exponential decay, how we can make those accessible and engaging. It's not just about teaching kids to solve equations. It's about helping them understand how to use math to make sense of their world.

Beautifully said. And a huge thank you to the authors of illustrative math for providing such, you know, rich and thought provoking material. And to all of you amazing listeners out there, thanks for joining us on this deep dive. Until next time, stay curious.