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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.)
Okay. So today, we are diving into Yeah. Percentage change.
Speaker 2:Right.
Speaker 1:And this is something that I think Yeah. We probably all have to deal with at
Speaker 2:some point
Speaker 1:in our lives.
Speaker 2:A lot of the time.
Speaker 1:Whether it's sales or taxes or tips or whatever. It's like these percentages are always floating around out there.
Speaker 2:Right.
Speaker 1:But, I think what's really interesting is that there's, like, a lot more really interesting math Absolutely. Behind the scenes of all those percentages.
Speaker 2:Absolutely.
Speaker 1:So today, we're gonna be looking at a lesson plan
Speaker 2:Okay.
Speaker 1:That's designed for teachers.
Speaker 2:Okay.
Speaker 1:But I think, honestly, this is something that, like, anybody can get something out of.
Speaker 2:Absolutely.
Speaker 1:And it's called recalling percent Change.
Speaker 2:Recalling Percent Change. Okay. Yeah.
Speaker 1:I like it. So what I think is so cool about this
Speaker 2:Okay.
Speaker 1:Is that it, like, really breaks down all these, like Totally. Core concepts in a way that, like, makes it make sense.
Speaker 2:That's the key. Yeah. You know? I think so often people get caught up in, like, just trying to get the answers, and they don't understand, like, why Right. Why they're doing what
Speaker 1:they're doing. Exactly.
Speaker 2:And so I think this lesson does a really nice job of breaking that down.
Speaker 1:Yeah. And it's not just about, like Right. How to calculate. Like, here's how you get a percentage.
Speaker 2:Yeah. Yeah.
Speaker 1:But it's, like, why that matters and, like
Speaker 2:Why does it matter? What are the implications?
Speaker 1:Right. Exactly. Like, where do we even see this in the real world?
Speaker 2:Totally. Totally.
Speaker 1:And they actually have in this lesson plan
Speaker 2:Okay.
Speaker 1:All these different activities that cover, like, everything from, like, the price of scooters Just to, like, the mysteries of exponential growth.
Speaker 2:Right. I'm in. I'm in.
Speaker 1:So it's like
Speaker 2:Yeah.
Speaker 1:It runs the gamut. So our mission today is to kind of Okay. I don't know. Crack the code Yeah. Of these percentage changes
Speaker 2:I love that.
Speaker 1:And hopefully give you guys listening. Moment. Some
Speaker 2:I like it.
Speaker 1:Yeah. Exactly.
Speaker 2:So good.
Speaker 1:So they start off with this activity called wheels.
Speaker 2:Wheels. Okay.
Speaker 1:And they're like, okay. Picture this. You're in the market for a new scooter.
Speaker 2:Right.
Speaker 1:And you find one for a $160.
Speaker 2:Okay. A $160.
Speaker 1:But then you see this bike that you like Mhmm. And it's 20% more expensive than the scooter.
Speaker 2:Gotcha.
Speaker 1:So the question is
Speaker 2:Yeah.
Speaker 1:How do we figure out the price of that bike?
Speaker 2:Okay. So, I mean, the intuitive way to do that would be to just take 20% of the 160 and add that on.
Speaker 1:Right? Right. Exactly.
Speaker 2:But I'm guessing this lesson is gonna give us some sort of, like
Speaker 1:Well, so they shortcut? Or
Speaker 2:They do that, but then they're like Yeah.
Speaker 1:But, also You could just multiply it by 1.2.
Speaker 2:Okay. There we go.
Speaker 1:And I'll be honest. If I were, like, on the fly trying to figure this out Yeah. Yeah. I would 100%.
Speaker 2:For sure. Add and subtract.
Speaker 1:Add and subtract.
Speaker 2:Totally.
Speaker 1:But I guess my question is, like Yeah. Why is this multiplication thing
Speaker 2:Why is that important? Why do we care?
Speaker 1:Why do we care?
Speaker 2:Besides just speed. Right?
Speaker 1:Besides just speed. Right? Like, why is this important at all?
Speaker 2:Right. Right. Right.
Speaker 1:And what they say is that, like, it all has to do with this distributive property.
Speaker 2:Okay. The distributive property. Right.
Speaker 1:So, basically, instead of adding that 20%, we can just multiply that scooter price by 1.20.
Speaker 2:By 1.20 and get that bike price directly.
Speaker 1:Right. Right.
Speaker 2:So I get that.
Speaker 1:Yeah. I'm also a little bit like Okay. Why does 1.20?
Speaker 2:Yeah. Why does that why does that work? Yeah. Help me understand. Okay.
Speaker 2:Okay. Because,
Speaker 1:like, where does that come from?
Speaker 2:1.20, essentially, what we're doing there is we're saying a 100%
Speaker 1:Okay.
Speaker 2:Plus 20%. Right. Because 1.20 is just, you know, that as a decimal.
Speaker 1:Right.
Speaker 2:So when we multiply by that, we're multiplying by the original 100%
Speaker 1:k.
Speaker 2:And the extra 20%.
Speaker 1:Gotcha.
Speaker 2:All at once.
Speaker 1:Okay.
Speaker 2:It's kind of a cool little shortcut.
Speaker 1:Okay. You
Speaker 2:know? It's more elegant, I think.
Speaker 1:I mean, it is it is pretty elegant.
Speaker 2:It's slick. I like it.
Speaker 1:Yeah. It's slick. So it's not just about speed then. It's about, like, this is like
Speaker 2:It's about the structure, man.
Speaker 1:The structure of it.
Speaker 2:It's about the underlying structure of what we're doing. Yeah. It's It's deep, man.
Speaker 1:So I'm starting to see why this is called recalling percent change
Speaker 2:Okay.
Speaker 1:Because it's like Yeah. It's not just about, like Great. Here's the formula, plug in the numbers.
Speaker 2:It's not a formula. It's like Effectively. Concept. Understanding. Yeah.
Speaker 1:It's a concept. How this works. That's right. How it
Speaker 2:works.
Speaker 1:Yeah. That's cool. Yeah. And then this multiplication mindset. Yeah.
Speaker 1:I love that's what they're calling it now.
Speaker 2:Yes. The multiplication mindset.
Speaker 1:The multiplication mindset. Okay.
Speaker 2:It's really important.
Speaker 1:This is huge for understanding exponential growth.
Speaker 2:Yes.
Speaker 1:This is, like, setting the stage. We gotta understand this now.
Speaker 2:Right.
Speaker 1:So I'm excited to see where they go with that.
Speaker 2:Yes. We'll get there. Right. But before we do
Speaker 1:Okay.
Speaker 2:Let's move on to the next activity in the lesson Let's do it. Which is
Speaker 1:taxes and sales. Taxes and sales. Okay. Now we're talking.
Speaker 2:Which I know is something that
Speaker 1:Real world real world, we actually have to deal with.
Speaker 2:Come on. Now so they're like
Speaker 1:So what's the scenario?
Speaker 2:Okay. So say you wanna buy a car. Okay. And it's $12,000.
Speaker 1:$12. Right?
Speaker 2:But there's sales tax.
Speaker 1:Yeah.
Speaker 2:Of course. Right? Gotta have sales tax.
Speaker 1:It's 8%.
Speaker 2:8%. Alright.
Speaker 1:So they're saying there's 2 ways we can think about this.
Speaker 2:Okay. Hit me.
Speaker 1:We can do the whole, like, 12,000 Mhmm.
Speaker 2:Plus Plus 0.08 times 12,000.
Speaker 1:Times 12,000.
Speaker 2:Right. The classic one.
Speaker 1:Or we can just do 1.08
Speaker 2:I see where you're going.
Speaker 1:Times 12,000.
Speaker 2:Times 12,000. There it is again.
Speaker 1:Right. So we're seeing this, like, pattern here.
Speaker 2:Okay. So, like, we were saying that multiplication mindset.
Speaker 1:Yeah. Right? So, again, I'm like, okay. But Yeah.
Speaker 2:So what's the big deal? Why? Why is that one multiplication so important?
Speaker 1:Why is this important?
Speaker 2:I mean, obviously, it's a little bit faster.
Speaker 1:Right.
Speaker 2:But why else? Why do we care?
Speaker 1:Why we care?
Speaker 2:What does this allow us to do? Yeah. Okay. Okay. I think I see where they're going with this.
Speaker 1:Okay.
Speaker 2:This is really important because what if we have multiple percentage changes? Mhmm. What if we have you know, what if they tack on, like, a dealer markup
Speaker 1:Right. Exactly.
Speaker 2:On top of that 8%.
Speaker 1:Like, what if
Speaker 2:What if there's also a discount?
Speaker 1:Right. Yeah. Yeah.
Speaker 2:Suddenly, we're talking about multiple percentage changes.
Speaker 1:Right.
Speaker 2:And if we're thinking about it this way, it's just multiplication, multiplication, multiplication.
Speaker 1:Yeah. Right. So it becomes like a chain?
Speaker 2:Exactly. It's like a chain.
Speaker 1:Okay. It's
Speaker 2:like a chain reaction.
Speaker 1:Gotcha.
Speaker 2:You know? And I think that's where this idea of exponential growth is gonna come in.
Speaker 1:Right.
Speaker 2:Right? Because we're talking about things changing Uh-huh. As a percentage over and over and over again.
Speaker 1:Okay.
Speaker 2:I'm picking up what they're putting down now.
Speaker 1:Yes. Yes. I think I am too. Alright. So this is really setting the stage for
Speaker 2:Yeah. Big things to come. Yeah. We've got scooters.
Speaker 1:Or bigger ideas.
Speaker 2:We've got taxes. We've got cars.
Speaker 1:Yes.
Speaker 2:I mean Yeah. Come on. This is exciting stuff.
Speaker 1:I know. I'm excited. Alright. Let's take a quick break Okay. And then we'll come back.
Speaker 2:Sounds good.
Speaker 1:And talk about more. Alright. So before we get too far into, like, the exponential growth stuff
Speaker 2:Okay.
Speaker 1:Let's talk about something a little bit more, I don't know, down to earth Yeah. A lot more fun, which is discounts. Okay. Everyone loves
Speaker 2:a discount. Come on. Who doesn't love a good sale? Exactly. So remembering back to our wheels activity with the scooter Okay.
Speaker 2:Yeah. And the bike.
Speaker 1:Mhmm. Let's say that scooter is now on
Speaker 2:sale for 35% off.
Speaker 1:Okay. Okay. So my inner bargain shopper is, like, perking up right now.
Speaker 2:How are you gonna figure that out? What are you gonna do?
Speaker 1:Well, I mean, I would probably figure out what 35% of a $160 is and then subtract it from the
Speaker 2:total sick. Right. The old school way. But remember that multiplication mindset.
Speaker 1:Here we go. Here we go.
Speaker 2:We don't have to subtract.
Speaker 1:Okay.
Speaker 2:We can multiply that 160
Speaker 1:Okay.
Speaker 2:By, in this case, 0.65
Speaker 1:Hold on.
Speaker 2:To get our answer.
Speaker 1:Where'd that come from? Okay. Where'd you get 0.65?
Speaker 2:So if it's 35% off
Speaker 1:Right.
Speaker 2:That means we're paying 65%. Exactly.
Speaker 1:Okay.
Speaker 2:So we could just multiply by 0.65.
Speaker 1:Gotcha. Gotcha.
Speaker 2:Right? Okay. Instead of doing 2 separate things.
Speaker 1:It's so funny because I I never would have Yeah.
Speaker 2:It's not intuitive. Right?
Speaker 1:No. It's not intuitive.
Speaker 2:It's really not.
Speaker 1:But it's like yeah. Of course. That makes sense.
Speaker 2:Once you see it
Speaker 1:Yeah.
Speaker 2:It's like, oh, okay. Okay.
Speaker 1:Like, I get it. I get it.
Speaker 2:It makes sense.
Speaker 1:So this is, again, not just a shortcut.
Speaker 2:This
Speaker 1:is, like, tying into this bigger picture.
Speaker 2:The bigger picture. Yeah.
Speaker 1:Right. Yeah. So we were talking about earlier
Speaker 2:This is where the exponential growth comes in.
Speaker 1:Okay. Okay. Tell me more.
Speaker 2:So, basically, any time Yeah. We're multiplying by the same factor over and over again.
Speaker 1:Okay.
Speaker 2:Like, 1.20 for that 20% increase or 0.85 for a 15% decrease. Mhmm. We're essentially modeling what's called exponential change.
Speaker 1:Yeah. Okay. So we've gone from
Speaker 2:I know. Right?
Speaker 1:Scooters to exponential growth.
Speaker 2:It's a big jump.
Speaker 1:It's wild.
Speaker 2:But it's all connected, man.
Speaker 1:It's all connected.
Speaker 2:It's all connected.
Speaker 1:Okay. So, like, how? Give me an example.
Speaker 2:Okay. So think about compound interest. Right? It's like the magic of finance
Speaker 1:Right.
Speaker 2:Where your money is making
Speaker 1:Money on money.
Speaker 2:Money on money. Exactly. Yeah. Because it's growing
Speaker 1:Right.
Speaker 2:On the interest that it's already earned. Mhmm. That's exponential change.
Speaker 1:Okay.
Speaker 2:Or what about, like Okay. Population growth. Right?
Speaker 1:Right.
Speaker 2:Like, let's say you have a population of rabbits. Okay. And they're growing by I don't know.
Speaker 1:Like, 10% a year?
Speaker 2:10% every year. Okay. That's exponential.
Speaker 1:Gotcha.
Speaker 2:Right.
Speaker 1:Okay.
Speaker 2:Because it's 10% on top of the previous year's 10%.
Speaker 1:So it's like It
Speaker 2:just keeps building.
Speaker 1:It builds.
Speaker 2:Yeah.
Speaker 1:Okay. So we're seeing this
Speaker 2:And that's the key.
Speaker 1:In the real world.
Speaker 2:Yeah. This is real world stuff.
Speaker 1:Like, this is actually how things change.
Speaker 2:This is not just some abstract math concept. Right?
Speaker 1:This is, like, how
Speaker 2:This is life.
Speaker 1:This is life. This is how things work.
Speaker 2:Wow.
Speaker 1:It's pretty cool.
Speaker 2:It is really cool.
Speaker 1:Yeah.
Speaker 2:But Okay. Let's be real. Yeah. This lesson plan isn't all sunshine and rainbows.
Speaker 1:No. Of course not.
Speaker 2:It's not just like
Speaker 1:What are they what are they what are they missing?
Speaker 2:Right.
Speaker 1:What are they not telling us?
Speaker 2:Because they do mention Yeah. That there are some common misconceptions
Speaker 1:Right. Right.
Speaker 2:That students and, honestly, probably all of us For sure. Have about percentages. Totally. So what are some of the stumbling blocks?
Speaker 1:Okay. Well, I think one of the biggest ones
Speaker 2:Okay.
Speaker 1:Is knowing when to add or subtract
Speaker 2:Okay.
Speaker 1:Versus when to multiply.
Speaker 2:Right.
Speaker 1:Right? Because it's easy to fall into that trap of thinking And that's
Speaker 2:all, like
Speaker 1:Oh, it's a 10% increase.
Speaker 2:Just add 10%?
Speaker 1:I'm just gonna add 10% to this thing.
Speaker 2:Right. Exactly. Right. But as we've seen, like
Speaker 1:Not so fast. Not so fast. Right.
Speaker 2:It's not always we gotta think about it multiplicatively.
Speaker 1:Right.
Speaker 2:You know?
Speaker 1:Like, with that car and the sales tax
Speaker 2:Exactly.
Speaker 1:We weren't adding 8%.
Speaker 2:Right.
Speaker 1:We were multiplying by 1.08.
Speaker 2:Right. To get that total amount.
Speaker 1:Right. Right. Right.
Speaker 2:Right. So it's not
Speaker 1:It's a little bit counterintuitive sometimes.
Speaker 2:Right. It's a little tricky.
Speaker 1:It's not what you would think at first glance.
Speaker 2:Okay. And then what else? What other pitfalls are there?
Speaker 1:Well, I think another big one is just, like Oh. Not understanding, like, why
Speaker 2:Right.
Speaker 1:The multiplication shortcuts work. Yeah. You know? Like, we kinda glazed over that a little bit.
Speaker 2:I'll be honest. I'm still a little Right. Early on it.
Speaker 1:It's like we're just accepting it as truth. Right. We're like, why?
Speaker 2:Right.
Speaker 1:Why does it work?
Speaker 2:Tell me why.
Speaker 1:And it all comes back to that distributive property.
Speaker 2:Okay.
Speaker 1:Which, you know, again, we could easily just be like, okay. Here's the rule. Just memorize it.
Speaker 2:Right. Right. But if
Speaker 1:we don't understand why it works
Speaker 2:Right.
Speaker 1:Then we're not really learning.
Speaker 2:Then what are we even doing here? Yeah. Exactly. We're just robots at that point.
Speaker 1:Okay. So how do we avoid these pitfalls?
Speaker 2:Okay. That's the $1,000,000 question. Right?
Speaker 1:Right.
Speaker 2:I think it's about asking why.
Speaker 1:Okay.
Speaker 2:Like, don't be afraid to ask why.
Speaker 1:Yeah. Just be curious.
Speaker 2:Be curious. Explore.
Speaker 1:Right.
Speaker 2:Test things out. You know?
Speaker 1:You'll just take things for granted.
Speaker 2:Exactly. Yeah. Don't just accept the formulas.
Speaker 1:Like, really try to understand.
Speaker 2:Run around with it. Yeah. See what happens.
Speaker 1:Okay.
Speaker 2:You know?
Speaker 1:So it's about fostering that curiosity.
Speaker 2:Yeah. And that's what makes learning fun.
Speaker 1:Right.
Speaker 2:You know? Fun. When you're like, oh, that's cool.
Speaker 1:Right. Like that moment
Speaker 2:I get it now.
Speaker 1:Where it finally clicks.
Speaker 2:Yes. Exactly.
Speaker 1:Yes. Yeah. It's so satisfying.
Speaker 2:It is. It really is.
Speaker 1:When you finally, like, get it.
Speaker 2:Like, yes. Yes. Light bulb moment.
Speaker 1:Totally. So And this lesson plan doesn't just, like, leave us hanging with all these formulas and, like
Speaker 2:Right.
Speaker 1:Concepts.
Speaker 2:Yeah.
Speaker 1:They actually wrap it up with this really cool challenge.
Speaker 2:Right. I like it. I like it. Hit me with it.
Speaker 1:Okay. So imagine a school Okay. Where the student population grows
Speaker 2:Mhmm.
Speaker 1:By 8% every year.
Speaker 2:8% a year. Okay. That sounds
Speaker 1:Yeah.
Speaker 2:Sounds pretty good.
Speaker 1:On the surface. Yeah. Yeah. Like, more students, more funding.
Speaker 2:Right. More teachers, potentially.
Speaker 1:Yeah. Exactly. Yeah. But then they're like
Speaker 2:Okay.
Speaker 1:Is this sustainable forever? Oh,
Speaker 2:good question.
Speaker 1:Right.
Speaker 2:Yeah.
Speaker 1:Like, it makes you think.
Speaker 2:So it's not just about the math. Right. It's about the real world.
Speaker 1:Like, what are the real world implications?
Speaker 2:That's yeah.
Speaker 1:Because in the real world
Speaker 2:Right.
Speaker 1:You can't just have Unlimited growth. Unlimited growth forever.
Speaker 2:Right. Exactly. It makes you think about things like
Speaker 1:Right. Like, there's only so many
Speaker 2:Rim sources.
Speaker 1:Resources. Like, how many kids can you fit
Speaker 2:Space.
Speaker 1:In a classroom?
Speaker 2:Exactly.
Speaker 1:Right. Yeah.
Speaker 2:Like, at some point
Speaker 1:Gotta factor all that stuff in.
Speaker 2:Mhmm.
Speaker 1:Right? It's gonna level off.
Speaker 2:Yeah. Yeah. The real world is messy.
Speaker 1:The real world is messy.
Speaker 2:It's not as clean as our math equation.
Speaker 1:It's true. Unfortunately. That's a good point.
Speaker 2:But I think that's a good I'm some go It's a good reminder.
Speaker 1:Reality check.
Speaker 2:Yeah. Reality check.
Speaker 1:That, like
Speaker 2:That the math is only part of the story.
Speaker 1:Right. Exactly.
Speaker 2:Like, it's a good starting point.
Speaker 1:Right.
Speaker 2:But it's not the whole story.
Speaker 1:What are the other factors at play?
Speaker 2:Exactly. Yeah. Well, I think that's a perfect place to maybe
Speaker 1:wrap it up.
Speaker 2:Wrap things up. So big thank you Yes. To the authors Absolutely.
Speaker 1:Of illustrative math For creating this awesome lesson.
Speaker 2:For this awesome lesson plan.
Speaker 1:Yeah.
Speaker 2:And to you guys out there listening
Speaker 1:Yeah.
Speaker 2:Thanks for joining us
Speaker 1:As always.
Speaker 2:On this deep dive.
Speaker 1:Yeah. Thanks for diving deep with us.
Speaker 2:Into the world
Speaker 1:Into the world.
Speaker 2:Of percentage change
Speaker 1:Of percentages.
Speaker 2:And exponential growth.
Speaker 1:It's more exciting than it sounds.
Speaker 2:It is. It really is. Until next time.
Speaker 1:Until next time.