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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.)
Hey, everyone. Ready to dive into some algebra with me?
Speaker 2:Algebra. I'm in. Today, we're taking
Speaker 1:a look at how equations can model the world around us.
Speaker 2:Oh, intriguing.
Speaker 1:From the price of blueberries to the structure of ancient geometric shapes.
Speaker 2:Okay. You've got my attention.
Speaker 1:It's based on some cool teacher materials about writing equations. So
Speaker 2:Teacher materials.
Speaker 1:If you've ever wondered how math applies to real life, buckle up.
Speaker 2:Yes. Even something like buying blueberries has hidden mathematical structures underneath. Exactly. I'm interested to see how these lessons help students uncover that.
Speaker 1:Me too. So this lesson plan called Writing Equations to Model Relationships
Speaker 2:Catchy title.
Speaker 1:Kicks off with a really cool concept, a math talk about percentages.
Speaker 2:Okay.
Speaker 1:Have you ever heard of a math talk?
Speaker 2:Math talks are such a valuable tool for getting students engaged.
Speaker 1:Oh, definitely.
Speaker 2:It's not just about finding the right answer. It's about encouraging different approaches and really digging into the why behind the math. Right. And in this case, they're using the number 200 as a starting point and having students calculate different percentages of it. Like, okay.
Speaker 2:25% of 200, that's doable. Right?
Speaker 1:True.
Speaker 2:But then they throw in things like 12% or even o from a percent.
Speaker 1:Oh, that's trickier.
Speaker 2:I mean, who even thinks about fractions of a percent?
Speaker 1:That's the beauty of it. It pushes students to think flexibly about percentages, maybe converting to fractions, maybe using decimals.
Speaker 2:Right.
Speaker 1:Maybe even realizing that 1 percent of 200 is just 2, which can be a shortcut for other calculations.
Speaker 2:Oh, clever.
Speaker 1:It's all about building that foundational fluency.
Speaker 2:Okay. That makes sense, getting those mental math muscles warmed up. Mhmm. But where do we go from there?
Speaker 1:Well, once students have those percentage skills down, the lesson plan takes a brilliant turn applying them to real world scenarios.
Speaker 2:Oh, I like when things get real world.
Speaker 1:We're talking buying blueberries, some are earnings, even the slightly terrifying world of car prices.
Speaker 2:Oh, now those are relatable. I'm guessing this is where those equations start coming into play.
Speaker 1:Absolutely. Each scenario cleverly introduces the idea of using variables, you know, those mysterious letters in our equation.
Speaker 2:I do. I remember those.
Speaker 1:They start with specific numbers, then bam, variables.
Speaker 2:Okay. So walk me through one of these scenarios. How do they connect, say, blueberries and equations? Let's take
Speaker 1:blueberries. Imagine a student is figuring out the cost of, let's say, £3 of blueberries at $4.99 per pound.
Speaker 2:Okay. I'm imagining.
Speaker 1:They could write a simple equation for that. But then imagine we swap out those specific numbers with variables. Instead of £3, it becomes £0.
Speaker 2:And for any number of pounds.
Speaker 1:Exactly. And instead of 4.99, it's p of dollars per pound.
Speaker 2:So suddenly, the equation isn't just about one blueberry purchase, it's about every blueberry purchase ever.
Speaker 1:Exactly. It clicks that equations can represent an infinite number of possibilities
Speaker 2:Wow.
Speaker 1:Not just one calculation.
Speaker 2:That is powerful.
Speaker 1:And, you know, this simple example actually links to a core economic concept, supply and demand.
Speaker 2:Woah. Wait a minute, really? Tell me more about that.
Speaker 1:Think about it. If the price p of blueberries drops, we can use our equation to see how the quantity n that someone might buy could change.
Speaker 2:Oh, that makes sense.
Speaker 1:That's algebra modeling real world behavior.
Speaker 2:That's incredible. Who knew blueberries could be so deep? Yep. So we've got blueberries, but what about those summer jobs?
Speaker 1:Right. So with summer earnings, the lesson ups the ante by comparing 2 people's earnings, May and Noah. Okay. Again, it starts with concrete numbers. Maybe May earns $45 more than Noah over the summer.
Speaker 2:Gotcha.
Speaker 1:But then they switch it up using variables to represent their earnings.
Speaker 2:Okay. I'm seeing a pattern here. Start specific, then generalize with variables.
Speaker 1:Living a magic trick.
Speaker 2:It is like a magic trick. You're right.
Speaker 1:And just like a magic trick, there's a bit of subtle misdirection happening here.
Speaker 2:Really?
Speaker 1:See, when you're comparing May and Noah's earnings, you're implicitly dealing with quantities that change in relation to each other.
Speaker 2:How so?
Speaker 1:May's earnings depend on how much Noah earns plus that extra amount.
Speaker 2:Ah, so it's not just about single variables anymore. It's about how they relate to each other.
Speaker 1:Frank. Getting pretty sophisticated.
Speaker 2:It is. And it sets the stage for even more complex algebraic thinking down the road.
Speaker 1:I bet.
Speaker 2:But before we get too far ahead of ourselves, let's talk about the last real world scenario.
Speaker 1:Okay.
Speaker 2:The one that probably fills many people with a sense of dread, car prices.
Speaker 1:Oh, yes. Car prices, the bane of my existence. I can already imagine the equations getting a bit more, shall we say, robust.
Speaker 2:Robust is one word for it. This is where students get to grapple with sales tax, those mysterious dealer fees.
Speaker 1:Don't remind you.
Speaker 2:It's the real deal.
Speaker 1:Okay. So they're not just figuring out the base price. They're adding in, what, like, a 6% sales tax on top of that Yeah. Plus a $120 dealership fee. You got it.
Speaker 1:My head hurts already.
Speaker 2:And that's precisely what makes this example so great. It pushes students to create equations that can handle multiple variables and operations.
Speaker 1:I can see that.
Speaker 2:What if we know the total price, but not the original price of the car?
Speaker 1:Good question.
Speaker 2:Can we rearrange the equation to solve for that unknown?
Speaker 1:Oh, it's like a detective story but with numbers. Exactly. And the tools they develop here, the ability to translate a real world scenario into a flexible adaptable equation. That's powerful stuff.
Speaker 2:It is powerful. You know, I have to admit, when we started this deep dive, I thought we were just gonna be talking about, well, basic algebra.
Speaker 1:Yeah.
Speaker 2:But this is so much more. It's like you're showing me all these hidden connections I never knew existed.
Speaker 1:And believe it or not, we're about to take an even bigger leap. We're going from the everyday math of buying cars to are you ready for this? Yeah. The timeless elegance of Platonic solids.
Speaker 2:Wait. Hold on. Platonic solids?
Speaker 1:You know them.
Speaker 2:Like, those perfectly symmetrical shapes I remember from geometry class. What do those have to do with algebra?
Speaker 1:Everything. Everything. Remember those shapes? The tetrahedron, the cube, the dodecahedron?
Speaker 2:Or do I love geometry?
Speaker 1:They aren't just pretty faces. They hold hidden mathematical relationships. And guess what? We can use equations to unlock those secrets.
Speaker 2:Okay. My mind is officially blown. Please tell me more. How on earth do we connect equations to these 3 d shapes?
Speaker 1:Well, this lesson plan focuses on the relationship between three key elements of a Platonic solid.
Speaker 2:Okay.
Speaker 1:Its faces, its vertices, those No. Pointy corners Yeah. And its edges.
Speaker 2:Right. Right. I remember those.
Speaker 1:Students are given a table with some of these values for a few different shapes, and their challenge is to find the missing values.
Speaker 2:Okay. So it's like a puzzle, but instead of fitting pieces together, they're filling in numbers.
Speaker 1:Precisely. But here's the real kicker. They're not just filling in numbers randomly.
Speaker 2:There's a kicker.
Speaker 1:They're trying to figure out the equation that connects these three elements for any platonic solid.
Speaker 2:So there's, like, the secret code, this elegant equation that unlocks the relationship in any of these shapes. You're killing me. What is it?
Speaker 1:Are you ready for it? Here it is. F+veequals2.
Speaker 2:Wait. Seriously? That's it. It seems so simple, but what does it even mean?
Speaker 1:It means that in any platonic solid, if you add the number of faces, f in vertices, v, and then subtract the number of edges, e, you will always always get 2.
Speaker 2:No way. That blows my mind. You're telling me this equation works for any tetrahedron, any cube, any dodecahedron out there.
Speaker 1:Every single one. It's like a fundamental truth about these shapes.
Speaker 2:Wow.
Speaker 1:A hidden harmony revealed through algebra.
Speaker 2:You know, this is why I love doing these deep dives with you. I come in thinking I know a thing or 2, and then bam, you hit me with knowledge like this. It's incredible.
Speaker 1:And speaking of knowledge, this lesson plan doesn't just throw these mind blowing concepts at students and leave them hanging. It also does a fantastic job of addressing common misconceptions that can trip them up.
Speaker 2:Oh, yeah. We all have those moments where we think we've got it and then oops. What kind of things do students get stuck on when it comes to equations?
Speaker 1:Well, one common stumbling block is mixing up the concepts of very and constant.
Speaker 2:Okay. So, like, sometimes a letter in an equation represents something that changes, and sometimes it represents something that stays the same.
Speaker 1:Exactly. And it's easy to see why that could get confusing.
Speaker 2:Totally. It's like in a recipe knowing which ingredients you can adjust based on your taste and which ones you absolutely can't mess with.
Speaker 1:I like that analogy. And this distinction is so important, especially as students start working with more complex equations.
Speaker 2:So how does the lesson plan help them get a handle on this very versus constant idea?
Speaker 1:It encourages teachers to really emphasize the meaning behind the variables. For example, remember those blueberry equations?
Speaker 2:Yeah. Or n and p.
Speaker 1:The n for the number of pounds, that can vary depending on how much you wanna buy. But if the price per pound is set at, say, $4.99, that p is a constant in that specific situation. It helps to make those distinctions clear.
Speaker 2:Right. So it's about understanding the context, not just blindly plugging in numbers. What other misconceptions do they address?
Speaker 1:Another tricky area is translating words into equations. It seems straightforward, but you'd be surprised how easy it is to make a mistake, especially when the word problem gets a bit twisty.
Speaker 2:Oh, tell me about it. It's like those if train a leaves Chicago problems that used to haunt my math classes. What kind of mistakes do students make with this word to equation translation?
Speaker 1:Well, they might mix up the order of operations, maybe forgetting those all important parentheses.
Speaker 2:Oh, yeah. Parentheses. Gotta love them.
Speaker 1:Or they might misinterpret the relationship between quantities, like, remember May and Noah's summer earnings. It's easy to accidentally write an equation that says, May's earnings plus $45 equals Noah's earnings when it should be the other way around.
Speaker 2:Yeah. Those little details can really trip you up. It's like accidentally putting salt instead of sugar in your cookies. One tiny mistake, and the whole thing is off.
Speaker 1:Exactly. And that's why the lesson plan encourages teachers to have students test their equations. Have them plug in some actual numbers and see if the equation holds true. It's a simple strategy, but a powerful one for catching those errors early on.
Speaker 2:It's like having a built in error detection system, and I bet that kind of thinking, that habit of checking your work is valuable beyond just math class. Right?
Speaker 1:Absolutely. It's about building those critical thinking skills, that ability to step back and ask, does this make sense?
Speaker 2:So as we wrap up this deep dive into the world of equations, what's the big takeaway for you? What's the one thing you hope our listeners will remember?
Speaker 1:You know, what fascinates me is how this lesson plan reveals the power of abstract thinking. It starts with something tangible, like buying blueberries, but then it guides students to see beyond the concrete to understand that those same mathematical relationships apply to countless situations even to the eternal forms of Platonic solids.
Speaker 2:It's like we've unlocked a secret code to the universe. We started with simple equations, and now we're talking about hidden harmonies and geometry and even economics. It just goes to show there's math everywhere if you know where to look.
Speaker 1:Precisely. And that sense of wonder, that thrill of discovery, that's what makes exploring these concepts so rewarding.
Speaker 2:I couldn't agree more. A huge thank you to the creators of Illustrative Math for this inspiring lesson plan and to you, dear listener, for joining us on this mathematical adventure. We hope this deep dive has sparked your curiosity and maybe even inspired you to find the hidden equations all around you.
Speaker 1:Because whether you're budgeting for groceries, designing a building, or simply marveling at the night sky, algebra is more than just formulas on a page. It's the language of the universe, and it's waiting to be discovered.
Speaker 2:Yeah.
Speaker 1:Another tricky area is translating words into equations. It seems straightforward.
Speaker 2:I bet.
Speaker 1:But you'd be surprised how easy it is to make a mistake.
Speaker 2:Oh, for sure.
Speaker 1:Especially when the word problem gets a bit twisty.
Speaker 2:Oh, tell me about it. It's like those if train a leaves Chicago, problems that used to haunt my math classes. What kind of mistakes do students make with this word to equation translation?
Speaker 1:Well, they might mix up the order of operations.
Speaker 2:Right.
Speaker 1:Maybe forgetting those all important parentheses.
Speaker 2:Oh, yeah. Parentheses. Gotta love them.
Speaker 1:Or they might misinterpret the relationship between quantities. Like, remember May and Noah's summer earnings? It's easy to accidentally write an equation that says, May's earnings plus $45, Errol's Noah's earnings, when it should be the other way around.
Speaker 2:Yeah. Those little details can really trip you up. It's like accidentally putting salt instead of sugar in your cookies. One tiny mistake and the whole thing is off.
Speaker 1:Exactly. And that's why the lesson plan encourages teachers to have students test their equations.
Speaker 2:Oh, that's smart.
Speaker 1:Have them plug in some actual numbers and see if the equation holds true. Oh, I It's a simple strategy, but a powerful one for catching those errors early on.
Speaker 2:It's like having a built in error detection system. And I bet that kind of thinking, that habit of checking your work is valuable beyond just math class. Right?
Speaker 1:Absolutely. It's about building those critical thinking skills.
Speaker 2:Definitely.
Speaker 1:That ability to step back and ask, does this make sense?
Speaker 2:So true. Well, I think we've covered a lot of ground today from blueberries to platonic solids who knew equations could be so fascinating.
Speaker 1:It's all about those hidden connections. Right?
Speaker 2:It really is. A huge thank you to the creators of Illustrative Math for this inspiring lesson plan and to you, dear listener, for joining us on this mathematical adventure.