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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 notice how, like, the second you drive a brand new car off the lot, it's like it's already lost, like, a ton of its value? Yeah. It's like that new car smell is also the smell of depreciation Right. And hitting you right in the wallet.
Speaker 2:Wallet. Absolutely.
Speaker 1:So that's what we're diving into today. Yeah. Exponential decay. Yeah. And we're looking at this lesson plan that you shared with me, about exponential decay and how it tackles that concept Yeah.
Speaker 1:With a really clever approach, I think.
Speaker 2:What's so fascinating about this lesson is that it doesn't just throw formulas at students. Right. It really empowers them to understand how quantities
Speaker 1:Yeah.
Speaker 2:Like, the value of a car can shrink by a certain amount over time
Speaker 1:Right.
Speaker 2:Or by a consistent percentage over time. Yeah. It's really laying the groundwork for them to understand how things change in the real world.
Speaker 1:Yeah. Like, giving them the tools to decode the hidden math Exactly. That's happening all around them. Them.
Speaker 2:Yeah.
Speaker 1:So by the end of this, your students will be fluent in using equations and expressions to represent exponential decay. But it sounds like it goes even deeper than just memorizing the formulas.
Speaker 2:Absolutely.
Speaker 1:You know what I mean?
Speaker 2:The goal is to get them to be able to apply this concept Mhmm. To real life situation.
Speaker 1:Right.
Speaker 2:And I think the way this lesson does that through an activity called notice and wonder, 2 tables k. Is genius. Okay.
Speaker 1:How does that work?
Speaker 2:So it kinda throws students a curveball right off the bat by introducing exponential patterns using fractions.
Speaker 1:Okay.
Speaker 2:And so it's kinda like a sneak peek into these more complex scenarios that they're gonna encounter later.
Speaker 1:So it's easing them into the world of exponential relationships without overwhelming them with, like, big whole numbers right away.
Speaker 2:Exactly.
Speaker 1:Yeah.
Speaker 2:And it compares these 2 linear growth and these two tables.
Speaker 1:Okay.
Speaker 2:And so it's really beautifully highlighting those differences in a very subtle way. Yeah. And it's getting them to think beyond just simple addition and subtraction.
Speaker 1:I can see how that would really open up their minds to the different ways that things can change over time.
Speaker 2:Right.
Speaker 1:What comes next?
Speaker 2:So the next activity, what's left, is where things get really, really interesting.
Speaker 1:Okay.
Speaker 2:This is where the lesson shifts from that familiar subtraction to using multiplication when dealing with decreasing quantities.
Speaker 1:Now that sounds like a really big moment for students
Speaker 2:It is.
Speaker 1:To kinda wrap my head around that what's so powerful about approaching it through multiplication.
Speaker 2:So think about it this way.
Speaker 1:Okay.
Speaker 2:If you have something that's worth a $100 and it loses a fourth of its value
Speaker 1:Okay.
Speaker 2:You could say, okay. Well, that's just a $100 minus 14 times a $100.
Speaker 1:Right. Which is $75.
Speaker 2:Exactly. Basic subtraction.
Speaker 1:Yeah.
Speaker 2:But this lesson encourages a different approach
Speaker 1:Okay.
Speaker 2:Which is 34 multiplied by a $100.
Speaker 1:Okay. I see where you're going with this because it's still $75. Right. But by framing it as multiplication Yeah. You're emphasizing that it's the decrease is, like, proportional Aced to the original value.
Speaker 1:Exactly. Instead of just
Speaker 2:A fixed amount.
Speaker 1:Exactly.
Speaker 2:And that's really the key because then that leads into this idea of exponential decay. Yeah. It's not about subtracting a certain amount each time. It's about this percentage decrease.
Speaker 1:Right. This percentage of what remains. This is really clicking for me now. It's about understanding that the value keeps shrinking relative to what's left, not by some fixed amount each time.
Speaker 2:You got it. And the beauty of this lesson is it doesn't stop there with this abstract concept. It goes on to an activity called value of a vehicle.
Speaker 1:Okay. And this activity brings that idea to life in a way that students can really connect with. I bet they can all relate to that. Oh, yeah. So how's that activity work?
Speaker 2:So imagine this. Students are presented with a scenario, maybe a car worth $18,000 that loses a third of its value each year.
Speaker 1:Okay.
Speaker 2:And they had to figure out how much it's worth over time.
Speaker 1:Okay.
Speaker 2:And this is where those moments happen.
Speaker 1:Yeah. Because they start to see that it's not depreciating a consistent $6,000 every year.
Speaker 2:Right. Exactly.
Speaker 1:It's shrinking
Speaker 2:Yeah.
Speaker 1:As the value drops.
Speaker 2:It keeps getting smaller and smaller. And this activity also introduces 2 important terms, growth factor and decay factor. Okay.
Speaker 1:Those are two terms that my students always trip up on.
Speaker 2:Yeah.
Speaker 1:So how does this lesson differentiate between those
Speaker 2:2? So when we think about a growth factor, it could be any number that you multiply by repeatedly to see how a quantity changes.
Speaker 1:Okay.
Speaker 2:And a decay factor is simply a growth factor that's less than 1.
Speaker 1:Okay.
Speaker 2:Like that 34 we were talking about earlier.
Speaker 1:Oh, okay. So even when something is decreasing, we can still talk about a growth factor. It's just a fraction that represents that shrinking value.
Speaker 2:Exactly. And I love how this lesson really highlights that because it paves the way for students to really understand exponential functions. Mhmm.
Speaker 1:Down the line when they start learning about those Yeah.
Speaker 2:It's all connected.
Speaker 1:It's so clever how this lesson kind of introduces that concept without explicitly calling it out. It's like planting that seed for their future math exploration.
Speaker 2:Exactly. And the lesson wraps up with this really nice summary of exponential decay. It emphasizes that key point that the growth factor is gonna be less than 1, and it provides this nice little formula that they can use.
Speaker 1:It's almost like they've been on this journey of discovery, and now they have the tools and the language to express what they've learned.
Speaker 2:Yeah.
Speaker 1:But, you know, even with the most engaging lessons, I feel like there's always those little things that students can misinterpret. Right? Oh, absolutely. Did the source material highlight any potential stumbling blocks that teachers should watch out for?
Speaker 2:Yeah. So one common misconception is that students really struggle to let go of subtraction.
Speaker 1:Okay.
Speaker 2:You know, they want it they see that value decreasing, and they just wanna subtract.
Speaker 1:Right.
Speaker 2:So they might get stuck on trying to subtract a fixed amount instead of using that multiplication approach.
Speaker 1:How does a lesson plan recommend we tackle that as teachers?
Speaker 2:Slow down.
Speaker 1:Okay.
Speaker 2:And really break down those calculations step by step. Yeah. Especially in that what's left activity.
Speaker 1:Okay.
Speaker 2:When you're looking at each part of that expression Yeah. Talk about what it represents.
Speaker 1:Yeah. Get them to see past just the arithmetic and really understand, like, the why behind the concept. What other misconceptions came up?
Speaker 2:Another one is this idea of constant rate.
Speaker 1:Okay.
Speaker 2:You know, they might think that that car is losing $6,000 every year instead of thinking about it as a percentage.
Speaker 1:Oh, yeah. That classic linear thinking trap. It feels much more intuitive Right? To think about things changing at a steady rate.
Speaker 2:It does. And so what I love about this lesson is that it encourages teachers to use visual aids. Okay. Especially with that value of a vehicle activity.
Speaker 1:Yeah.
Speaker 2:You can use graphs.
Speaker 1:Right.
Speaker 2:You can use tables.
Speaker 1:Yeah. Seeing it visually really helps to differentiate those two types of decay. Absolutely. Okay.
Speaker 2:And what I think is so cool about this lesson is that it doesn't just end there. Yeah. It leaves the students with a question.
Speaker 1:Oh, I like that. A little cliffhanger.
Speaker 2:A little cliffhanger.
Speaker 1:To get them thinking?
Speaker 2:Yes.
Speaker 1:What's the question?
Speaker 2:So remember those two tables from the beginning?
Speaker 1:Yeah.
Speaker 2:That compared the linear and the exponential growth with the fractions?
Speaker 1:I do I do a
Speaker 2:It poses this question. Can a quantity growing linearly ever catch up to a quantity that's decaying exponentially even if they start at the same value? Yeah. And I think that's such a cool question to leave them with.
Speaker 1:I do too. Because if I'm remembering those tables right at the beginning, they start out pretty close together. They do. And so it really does make you wonder, like
Speaker 2:It does.
Speaker 1:Is that linear growth ever going to catch up to that exponential decay?
Speaker 2:Yeah. And I love that it doesn't just say, here's the answer.
Speaker 1:Right.
Speaker 2:It lets them explore that idea.
Speaker 1:It encourages that deeper thinking
Speaker 2:Yeah.
Speaker 1:Which is really what it's all about. Right?
Speaker 2:Exactly.
Speaker 1:It's like we're low key setting them up to think like they're, like like, in, economists or mathematicians
Speaker 2:Right.
Speaker 1:Or even just savvy car buyers. Exactly. And that's what I love about this lesson.
Speaker 2:Yeah. It doesn't just teach exponential decay, like, here's the formula. Go apply it.
Speaker 1:Right.
Speaker 2:It connects it to these real world situations that students are really interested in.
Speaker 1:It's giving them that why Yes. That they can really connect with.
Speaker 2:And hopefully sparking that interest. Yeah. That makes them wanna keep learning.
Speaker 1:And honestly, as teachers, isn't that
Speaker 2:Yes.
Speaker 1:The ultimate goal
Speaker 2:It is.
Speaker 1:To get them excited about learning.
Speaker 2:Yeah.
Speaker 1:This has been so great.
Speaker 2:I've loved talking about this with you.
Speaker 1:It's a good reminder for me too that even these concepts that seem really complex, like, when you hear exponential decay, you're like, oh.
Speaker 2:Right.
Speaker 1:That sounds really hard.
Speaker 2:It does.
Speaker 1:But when you connect it to something that students understand and experience
Speaker 2:Yeah.
Speaker 1:It makes it so much more accessible.
Speaker 2:It does. And even exciting.
Speaker 1:It's all about meeting them where they are
Speaker 2:Yeah.
Speaker 1:And giving them those tools to make sense of the world Yeah. And to unlock these mathematical wonders I
Speaker 2:love that.
Speaker 1:Even in something like card appreciation.
Speaker 2:So to all of you listening out there who are about to teach exponential decay
Speaker 1:Mhmm.
Speaker 2:Remember, you're not just teaching formulas. You're opening the door to this whole world
Speaker 1:Yes.
Speaker 2:Of really cool patterns and connections.
Speaker 1:And you might even inspire the next generation of mathematicians or economists or or savvy car buyers.
Speaker 2:Yeah.
Speaker 1:That's right. I love it. Well, this has been awesome.
Speaker 2:It has.
Speaker 1:Such a good deep dive. Until next time.