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