Subscribe
Share
Share
Embed
Lesson by lesson podcasts for teachers of Illustrative Mathematics®.
(Based on IM 9-12 Math™ by Illustrative Mathematics®, available at www.illustrativemathematics.org.)
Ever feel like your students could use a little more, real world in their math lessons?
Speaker 2:Definitely. It's all about finding those relatable hooks to grab their attention. Right?
Speaker 1:Exactly. And that's what I love about the lesson plan we're diving into today. It all starts with something everyone understands, discounts.
Speaker 2:Discounts. Now you're speaking my language. Who doesn't love a good sale?
Speaker 1:Seriously? But this goes beyond just snagging a deal. We're talking about a lesson plan that uses discounts as a springboard to understanding a powerful mathematical concept, repeated percent increase.
Speaker 2:Now it's getting interesting. So we're not just talking about calculating a single discount, but seeing how those percentages play out over time.
Speaker 1:You got it. It might sound simple on the surface, but this idea underpins some pretty significant math concepts, like the exponential growth we see in compound interest.
Speaker 2:Okay. Now you're talking about something that makes a real difference in people's lives and something I wish I'd understood better back in my school days.
Speaker 1:Me too. Yeah. But that's why we're here, right, to empower teachers to break down these sometimes intimidating concepts and make them relevant and engaging for their students.
Speaker 2:Absolutely. So how does this magical discount themed lesson plan actually work?
Speaker 1:Well, the first activity is called dandy discounts. Yep. And it starts with a scenario I think a lot of us can relate to. Imagine your students walking into their favorite bookstore.
Speaker 2:Hold on. Hold on. Are you talking about a bookstore that actually exists or, like, the ideal bookstore they dream of?
Speaker 1:Let's go with the dream bookstore.
Speaker 2:Now we're talking.
Speaker 1:So they walk into this awesome bookstore, and boom, every single book is 25% off.
Speaker 2:Okay. I'm already cooked.
Speaker 1:They find this amazing book. Let's say it's originally priced at $32.
Speaker 2:$32? That's a steal already. What kind of amazing book are we talking about here?
Speaker 1:Let's just say it's a math textbook that magically makes algebra fun.
Speaker 2:Okay. I'm listening.
Speaker 1:So they're about to grab this book, but wait, there's more. They also have a coupon for an additional 25% off.
Speaker 2:Wow. 25% off on top of 25% off. That's like I can't even calculate that in my head.
Speaker 1:That's kind of the point. The real learning comes from understanding how those discounts are applied. See, that second 25% off, it's not taken off the original $32 price tag. It's calculated on the already discounted price.
Speaker 2:Oh, so it's like a discount on top a discount. Sneaking.
Speaker 1:Exactly. That's the core idea of repeated percent change.
Speaker 2:Yeah.
Speaker 1:Each change is calculated based on the new amount, not that initial amount. And this, my friend, is the foundation of how compound interest works.
Speaker 2:Okay. I'm starting to see how this simple bookstore scenario connects to the bigger picture of, like, finances and stuff. But how does the lesson plan actually get students to make that connection?
Speaker 1:Instead of having students calculate each discount step by step, it encourages them to think about it using, wait for it, multiplication.
Speaker 2:Multiplication. Really? That's their secret weapon.
Speaker 1:It's more powerful than you might think. So instead of taking 25% off of $32 then another 25% off that new amount, they're encouraged to think of it like this. Mhmm. Multiply that $32 by 25%, which is the same as taking off 25%.
Speaker 2:Right. Because 0.75 is, like, what's left after you take away that 25%. Right?
Speaker 1:Exactly. And then for that second discount, they just multiply that result by 0.75 again.
Speaker 2:So instead of a bunch of subtraction and percentages flying around, they're streamlining it all into a couple of multiplication problems. I like it. It's elegant in a mathematical kind of way.
Speaker 1:And more importantly, it subtly introduces the idea of exponents. Remember, they're multiplying by 0.75 twice, which sets the stage for understanding exponential expressions later on.
Speaker 2:Okay. So this seemingly simple discount at the bookstore is actually their first step into dot done done done dot the world of exponents. I like where this is going.
Speaker 1:Okay. So we've tackled discounted books, uncovered the magic of multiplication, and now it's time to see how this all connects to something a little closer to home loans.
Speaker 2:Loans, Now that's a topic that's sure to get those students' attention. It's getting real now.
Speaker 1:Exactly. And that's where the owing interest activity comes in. It's all about understanding how interest charges on loans can really add up over time.
Speaker 2:Yeah. I think we've all been there, maybe not in the classroom, but definitely in real life. Right? Whether it's student loans, car payments, mortgages, those interest charges can be a real eye opener.
Speaker 1:Absolutely. And this activity doesn't shy away from that. It dives right into the concept of annual interest calculated on a loan.
Speaker 2:Okay. So walk me through it. How does it build on that idea of repeated percent change we were exploring with the discounts?
Speaker 1:It's the same principle just applied to a different scenario. So let's say someone borrows, oh, I don't know, $1,000 at an interest rate of 5% per year. Now after that 1st year, they don't just owe an extra $50 in interest.
Speaker 2:Right. Because it's not like a simple interest situation where you just calculate 5% of the original amount and you're done.
Speaker 1:Exactly. We're talking compound interest here, which means that 5% interest for year 2, it's calculated on the new principal, that housing $50 and so on and so on.
Speaker 2:Wait. So the interest is calculated on the interest from the previous year too? That just doesn't seem fair.
Speaker 1:Well, that's the nature of compound interest, and that's precisely why it's so important for students to grasp this concept early on.
Speaker 2:For sure. But seeing those numbers on paper is one thing. Do you students actually get to, like, visualize how quickly those interest charges can snowball?
Speaker 1:That's where the next activity, comparing loans, comes into play. It brings in the power of visuals, specifically graphing technology, to illustrate the impact of different interest rates.
Speaker 2:Graphing calculators. Now we're really taking it back to high school.
Speaker 1:Hey. Those graphing calculators are still around for a reason. Visualizing data is a powerful tool no matter what level of math you're doing. And in this activity, students get to compare what happens when 3 people take out loans for the same amount. Let's stick with $1,000, but at different interest rates.
Speaker 2:Okay. So they're seeing how those interest rates, even if they're just a few percentage points apart, can really affect how much they end up owing over time.
Speaker 1:Exactly. And the lesson plan emphasizes choosing appropriate graphing windows so students can really see those differences in loan growth without one loans graph, you know, exploding off the chart before the others even get going.
Speaker 2:Makes sense. You wanna make sure those comparisons are clear and, well, impactful.
Speaker 1:Right. Now for teachers who are feeling extra adventurous and wanna give their students a bit more of a challenge, there's an optional activity called comparing average rates of change.
Speaker 2:Oh, now we're getting fancy. This sounds a bit more advanced. What's the main takeaway for students with this one?
Speaker 1:It dives into the idea that exponential growth, like we see with compound interest, isn't linear. It accelerates over time.
Speaker 2:So it's not like a steady, predictable increase, more like hitting the gas pedal on a car.
Speaker 1:Perfect analogy. Students calculate and compare the average rate of change for each loan over different periods of time. What they'll see is that the longer that loan goes unpaid, the more dramatic that compounding effect becomes.
Speaker 2:Okay. So it's really driving home that snowball effect we talked about earlier. Only this time, it's not snow, it's money, and not the fun kind.
Speaker 1:I can only imagine. With all this talk about snowballing and accelerating, it seems like there's a lot of room for things to get, well, a little confusing. What are some of the common misconceptions students might have about repeated percent change even with these engaging activities?
Speaker 2:One of the biggest ones is that they sometimes forget we're dealing with a new amount each time that interest is calculated.
Speaker 1:So instead of calculating the interest on the new total, they might accidentally
Speaker 2:Base it on the original amount. Like, in our loan example, they might just keep calculating 5% of that initial $1,000 year after year instead of factoring in the interest that's already accumulated.
Speaker 1:I see. It's like they're stuck on that original number even though the amount is constantly growing.
Speaker 2:Exactly. And that can really throw off their calculations, especially as those time periods get longer.
Speaker 1:Right. Those little errors can snowball pretty quickly. Pun intended. Are there any other common stumbling blocks teachers should be aware of?
Speaker 2:Another tricky part is transitioning from that repeated multiplication idea, you know, multiplying by 1.05 over and over again to using exponential notation.
Speaker 1:So they might grasp that multiplying by 1.05 represents a 5% increase.
Speaker 2:Right. But when it comes to writing it in that more concise form, like 1.05 raised to a certain power, that can feel like a whole new ballgame.
Speaker 1:It's like they're learning a secret code. So how can teachers help their students decode this, exponential notation and really overcome these misconceptions.
Speaker 2:This is where visual aids become super helpful. Breaking down those calculations step by step, maybe even using color coding or creating tables to track the changes over time, those visual cues can make a world of difference in helping students see the why behind the what. You know?
Speaker 1:Absolutely. Sometimes all it takes is a clear visual to turn that light bulb moment on. So if we were to distill this whole deep dive from discounted books to snowballing loans into one key takeaway for our listeners, what would it be?
Speaker 2:For me, it's about emphasizing the power and the importance of understanding repeated percent change. It's not just an abstract math concept. It's a tool that students can use to make sense of so many things in their lives from understanding how interest works on loans and investments
Speaker 1:to even wrapping their heads around things like population growth or the spread of information online. It's exponential growth in action.
Speaker 2:Exactly. And when students can connect those dots between the classroom and the real world, that's when learning becomes truly empowering.
Speaker 1:Couldn't agree more. A huge thank you to Illustrative Math for creating such an engaging and practical lesson plan. And to our listeners, keep exploring those real world connections to exponential growth. You might be surprised at just how prevalent it really is.