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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 get that feeling like, you know, maybe numbers are playing tricks on you? Yeah. Like, when you see a sale and you're like, oh, awesome. 20% off.
Speaker 2:Yeah.
Speaker 1:And then you're like, oh, I've got a coupon for another 15% off.
Speaker 2:Yeah.
Speaker 1:And you, like, stack those discounts
Speaker 2:Right.
Speaker 1:And it's still not as cheap as you hoped it would be.
Speaker 2:Right. Yeah. Happens all the time.
Speaker 1:So today's deep dive is gonna, like, equip you with the knowledge to see through those, numerical illusions, I guess you could call them
Speaker 2:Okay.
Speaker 1:Especially when it comes to your hard earned cash rate.
Speaker 2:Yes.
Speaker 1:We're gonna be taking a look at a lesson plan all about teaching compound interest.
Speaker 2:Okay.
Speaker 1:But don't worry. This isn't about, like, dusting off your old textbooks or anything like that.
Speaker 2:Yeah.
Speaker 1:This is about understanding a concept that, honestly, it affects everyone, whether you're dealing with loans, investments, or even just trying to resize a photo for, like, I don't know, your social media feed or something. So, like, I think what's really fascinating about compound interest is that it often contradicts our, like, intuitive sense of how percentages work. Absolutely. And the lesson plan we're diving into today, it really highlights this perfectly.
Speaker 2:Yeah.
Speaker 1:And it really emphasizes, like, the difference between just, like, simple increases Mhmm. And then, like, the snowball effect of compounded growth.
Speaker 2:Yeah. What's really interesting about teaching compound interest or learning about it is that, you know, it is it is one of those things that kinda goes against what your brain is telling you is right. Yeah. So the lesson plan that we're looking at today, one of their mathematical goals that they really highlight is understanding the difference between a simple percent increase versus a compounded percent increase, and then also why if you apply that percent increase multiple times, it ends up being different than if you were to just apply, one larger increase.
Speaker 1:Okay. So let's, like, break that down a little bit. So the lesson plan really wants students to understand that applying a percentage increase multiple times, it leads to a different result than, like, just using one bigger increase.
Speaker 2:Right.
Speaker 1:Can you give us an example of what they mean by that?
Speaker 2:Sure. So imagine you have, like, a vintage comic book collection.
Speaker 1:Okay.
Speaker 2:And it's worth, say, $1,000 today.
Speaker 1:Okay.
Speaker 2:And let's say that the value of that collection increases by 10% each year for 2 years.
Speaker 1:Okay.
Speaker 2:Now what our brains wanna do is just kinda simplify that. They wanna say, okay. 10% plus 10%, that's 20%.
Speaker 1:Right. Yep.
Speaker 2:So after 2 years, it'll be worth $1200.
Speaker 1:Yeah.
Speaker 2:But that's not actually how compounding works.
Speaker 1:Oh, okay.
Speaker 2:It's a little sneakier than that.
Speaker 1:I'm already sensing, like, what those moments going on. What's the what's the catch?
Speaker 2:So the catch is that, yes, after that 1st year, the value increases to $1100. Right? You've added 10%.
Speaker 1:Right.
Speaker 2:But in year 2, that 10% increase isn't calculated based on the original $1,000. It's calculated based on that new value of $1100.
Speaker 1:So instead of just gaining an extra $100, you're gaining a little bit more because of that initial growth.
Speaker 2:That's exactly it. So that second year, you're actually gaining a $110, not a $100. Wow. And this is the fundamental idea behind compounding. Right?
Speaker 2:It's like the interest you earned in year 1 starts earning its own interest in year 2.
Speaker 1:That makes way more sense than how my brain usually wants to approach percentages.
Speaker 2:And that's the thing. Our minds you know, we wanna linearize things. Right? We wanna think in a straight line. 10 plus 10 equals 20.
Speaker 1:Yeah.
Speaker 2:But compounding, it's not linear at all. It's exponential. It's a whole different bowl game.
Speaker 1:Okay.
Speaker 2:And the lesson uses a really clever example, to visually demonstrate this, and it involves resizing an image.
Speaker 1:Oh, yeah. I remember this. They had, like, 2 students, Andre and May, and they're both trying to enlarge an image for a project. Right. Sounds simple enough.
Speaker 1:Right?
Speaker 2:Yeah. Deceptively simple.
Speaker 1:Yeah. But their different approaches really bring this whole compounding idea to life.
Speaker 2:Exactly. So we have Andre and Mai. They're both starting with a one 100 pixel image.
Speaker 1:Yeah.
Speaker 2:They're both starting with a 100 pixel image.
Speaker 1:Okay.
Speaker 2:Andre decides to play it safe.
Speaker 1:Right.
Speaker 2:He's gonna increase the size by 10% twice.
Speaker 1:Okay.
Speaker 2:Mai, feeling a little more confident, decides to just go all in with a single 20% increase.
Speaker 1:Okay. So at first glance, it seems like they'd both end up with the same size image. Right?
Speaker 2:Right.
Speaker 1:I mean, 10% plus 10% equals 20%. Right?
Speaker 2:That's what our intuition wants us to think.
Speaker 1:Right. Yeah.
Speaker 2:That's the trap.
Speaker 1:Yeah.
Speaker 2:So let's break down Andre's approach.
Speaker 1:Okay.
Speaker 2:First 10% increase takes that image from a 100 pixels to a 110 pixels. Okay. Pretty straightforward.
Speaker 1:Yeah. So far, so good.
Speaker 2:But here's where the magic happens.
Speaker 1:Okay.
Speaker 2:That second 10% increase isn't calculated on the original 100 pixels.
Speaker 1:Right.
Speaker 2:It's calculated on that new size of a 100 10 pixels.
Speaker 1:Oh, I see where this is going. Which means Instead of adding another 10 pixels, he's adding a little bit more because the image is already bigger.
Speaker 2:Exactly. So that second 10% increase actually adds 11 pixels this time What? Bringing that final image size to a 121 pixels. Okay. Meanwhile, May, over there, with her single 20% increase, she's sitting at a 120 pixels.
Speaker 1:Wow. So even though they both technically, like, you know, they increased the image size by what seems like 20%.
Speaker 2:Right.
Speaker 1:The way they did it, like, bit by bit versus all at once.
Speaker 2:Mhmm.
Speaker 1:It leads to a different outcome.
Speaker 2:Different outcome.
Speaker 1:That's so interesting.
Speaker 2:That's compounding in action.
Speaker 1:And this scales up to, like, much bigger things than just image pixels.
Speaker 2:Right? Absolutely.
Speaker 1:Like, think about population growth.
Speaker 2:Yeah. Population growth.
Speaker 1:Like, a small difference in, you know, the growth rate each year Yeah. Can lead to huge differences in population size over decades or even centuries.
Speaker 2:Right? Exactly.
Speaker 1:That reminds me of that section in the lesson plan where they ask, are you ready for more?
Speaker 2:Okay.
Speaker 1:And they talk about how if you multiply something by 1.01 10 times, you get a much bigger result than if you just added 0.01 to it 10 times. Yeah. Right?
Speaker 2:It's the it's the, it's the idea between simple interest and compound interest. So simple interest is where you would just add that point 0110 times. Right. Compound interest is where you're multiplying it by that 1.01, and it makes a huge difference in in the long
Speaker 1:run. Yeah. It does. It's like the difference between adding pennies to a jar Yeah. And, like, having your money grow in a savings
Speaker 2:account Exact.
Speaker 1:Which, speaking of, is exactly where the lesson plan takes us
Speaker 2:next. Okay.
Speaker 1:It dives into the world of bank accounts and interest rates.
Speaker 2:Right. Yeah. And and this is where things can get
Speaker 1:Oh, yeah. I've definitely This
Speaker 2:is where I think, yeah, people can get a little
Speaker 1:have my fair share of confusion
Speaker 2:Concern.
Speaker 1:Trying to decipher
Speaker 2:Yeah.
Speaker 1:Those interest rate disclosures.
Speaker 2:For sure.
Speaker 1:They always seem to throw around terms like, you know, nominal rate and effective rate. Uh-huh. And I'm never quite sure, like, what the difference is.
Speaker 2:Yeah. It's a really common point of confusion. Yeah. But think of it this way. The nominal interest rate, that's like the sticker price.
Speaker 1:Okay.
Speaker 2:It's what the bank advertises.
Speaker 1:Okay.
Speaker 2:It's the initial rate they tell you you're getting. But the effective interest rate, that's the real MVP.
Speaker 1:Right.
Speaker 2:That's the one that factors in how often that interest is being compounded.
Speaker 1:Okay.
Speaker 2:So it gives you a much clearer picture of how much your money is actually going to grow.
Speaker 1:Okay. So the effective rate takes into account that, like, snowball effect that we were talking about earlier where, like, the interest you earn starts earning its own interest. Exactly. Okay.
Speaker 2:It's like, you know, when you're ordering a coffee Okay. You can order a plain coffee Yeah. Or you can get all the bells and whistles.
Speaker 1:Yeah.
Speaker 2:Right?
Speaker 1:Right.
Speaker 2:You can get the, you know, the whipped cream and the caramel and all that. The effective rate's giving you the full picture
Speaker 1:Right.
Speaker 2:Add ons and all.
Speaker 1:Okay. I like that I like that analogy.
Speaker 2:Yeah.
Speaker 1:So in the case of a savings account, is the effective interest rate always higher than the nominal rate?
Speaker 2:It is if the interest is compounding, and it pretty much always is.
Speaker 1:Okay.
Speaker 2:So if the interest compounds, the effective rate will be higher than the nominal rate. To break this down a little bit further, the lesson plan gives us an example of
Speaker 1:a
Speaker 2:$1,000 account. Okay. And it has a 1%
Speaker 1:monthly
Speaker 2:interest rate. Right. And they walk through how with compounding, that account grows much faster than than if it were just earning a simple 12% annual interest.
Speaker 1:Wait. So a 1% monthly interest rate, like, that doesn't just translate to a simple 12% interest rate at the end of the year.
Speaker 2:It does not, which is, you know, kind of again, it's one of those things where you think, okay. Well, 1% a month, 12 months in a year, that must be 12%.
Speaker 1:Right. Yeah.
Speaker 2:But it's not because of the compounding.
Speaker 1:Right.
Speaker 2:So in that case, that 1% monthly interest compounded monthly actually results in an effective annual rate of about 12.68%.
Speaker 1:Wow. Okay.
Speaker 2:Yeah. Now that extra, you know, 0.68% might not seem like a lot. Right. But over time, believe me, it adds up.
Speaker 1:Okay. This is definitely making me rethink how I look at interest rate. So when I'm, like, comparing savings accounts or, like, trying to decode the fine print on a loan
Speaker 2:Right.
Speaker 1:How can I, like, actually spot the nominal versus the effective rates? Yeah. They can't, like, make it easy for us. Right?
Speaker 2:No. They definitely they definitely like to make it make it a little bit confusing sometimes. Yeah. But but here's the thing.
Speaker 1:Okay.
Speaker 2:Look for keywords. So for savings accounts, keep an eye out for APY.
Speaker 1:Okay.
Speaker 2:That stands for annual percentage yield.
Speaker 1:Okay.
Speaker 2:And, usually, that means they're talking about the effective rate. Okay. The one that shows you the real deal after all that compounding magic.
Speaker 1:Okay. So APY is my friend when I'm trying to, like, choose a good savings account. Got it.
Speaker 2:Exactly. Yeah. That's the one you wanna focus on Okay. For savings.
Speaker 1:What about loans? Uh-huh. What, like, what terms should I be watching out for there?
Speaker 2:So loans can be a little trickier. Uh-huh. They'll often use APR Okay. Which stands for annual percentage rate.
Speaker 1:Right.
Speaker 2:And that does factor in some fees.
Speaker 1:Okay.
Speaker 2:But oftentimes, it doesn't fully capture the impact of compounding Okay. Especially if the loan has, like, a really long term or kind of a weird payment schedule.
Speaker 1:Okay. So it's, like, the main takeaway is don't just blindly trust that first number you see.
Speaker 2:Right. Then do that.
Speaker 1:Like, I need to channel my inner detective and, like, hunt down those keywords. Read the fine print.
Speaker 2:Absolutely.
Speaker 1:And maybe, yeah, like, pull out a calculator too just to be extra sure.
Speaker 2:Yeah. It never hurts. Okay. Especially with something as important as your money. Right?
Speaker 1:Right. Right. For sure.
Speaker 2:It's worth taking those extra couple of minutes.
Speaker 1:Yeah. I think this deep dive has really driven home the point that those, like, seemingly small percentages can have a huge impact over time.
Speaker 2:Absolutely. They're wild. Compounding is a powerful, powerful force.
Speaker 1:Yeah. We've covered a lot of ground today from resizing images. Right. To, like, understanding how interest rates really work.
Speaker 2:Yeah. It's amazing how this one concept
Speaker 1:It's amazing how this one concept compounding. It, like, pops up in so many different areas of life.
Speaker 2:Oh, it's everywhere.
Speaker 1:It really is.
Speaker 2:Yeah. And the more aware you are of it, the better you're gonna be at, you know, making those decisions
Speaker 1:Gotcha.
Speaker 2:Whether it's with your money or even just other things in your life.
Speaker 1:Right.
Speaker 2:So, you know, remember, even small consistent actions, like, they can lead to really remarkable results over time
Speaker 1:Yeah.
Speaker 2:Whether it's saving a little extra each month or, you know, just being more mindful of those, like you said, those sneaky interest charges.
Speaker 1:Yeah.
Speaker 2:Understanding the power of compounding can truly, truly pay off.
Speaker 1:That's a fantastic takeaway. Yeah. So to recap, I guess Yeah. Pay attention to the details, especially when numbers are involved.
Speaker 2:Yes.
Speaker 1:And, like, never underestimate the power of small, consistent actions over time.
Speaker 2:Love it.
Speaker 1:It's like that old saying, you know, slow and steady wins the race Right. Especially when compounding is on your side.
Speaker 2:Couldn't have said it better myself.
Speaker 1:Well, a huge thank you to, the authors of Illustrative Math for the insightful lesson material
Speaker 2:Yes.
Speaker 1:And to our listeners for joining us on this deep dive into the fascinating world of compound interest.
Speaker 2:Yes. Thank you, Guy.
Speaker 1:Remember, knowledge is power. So keep those brains engaged, and stay curious.
Speaker 2:Always stay curious.