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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.)
Have you ever, like, tried to explain why 2 equations that look totally different could actually mean the same thing? It's kinda tricky. Right? And that's exactly what we're looking at today. We're diving into equivalent equations using this really interesting illustrative math lesson plan that's designed for algebra students.
Speaker 2:Yeah. And what I find interesting about this lesson is it's not just about getting the same answer from 2 different looking equations. You know? It really helps students grasp that equivalent equations. They show the same relationship between quantities just written in different ways, and that's pretty neat.
Speaker 1:So it's, like, more about understanding the meaning behind the math, not just, like, the steps.
Speaker 2:Exactly. It's getting to that why behind the what, and that's really what helps this concept stick with students.
Speaker 1:I like that. So how does this lesson actually help students, you know, really get equivalent equations?
Speaker 2:Well, they start by, sparking some curiosity with this activity called 2 expressions. They're given these, well, kinda unusual expressions, nothing too crazy, but different enough to make them think. Right? And then they try these expressions out using different values for x.
Speaker 1:So I'm guessing the catch is that these different looking expressions will actually give them the same answers.
Speaker 2:You got it. It's this, hands on approach that allows students to experience equivalence before they even really know what it is. They're, like, seeing a connection between the expressions even if they can't quite put it into words yet.
Speaker 1:Oh, that's cool. It's like setting the stage for that moment. So then how do they build on that initial spark?
Speaker 2:Well, that's where much ado about ages comes in. In this activity, students actually write equations that show age relationships within families, like the older sibling is twice the age of the younger sibling plus 3 years.
Speaker 1:Oh, I like that. That's relatable. I can see kids getting into that.
Speaker 2:Right. And as they're working through it, they start to realize that multiple equations, even ones that look completely different, can describe that same age relationship.
Speaker 1:Yeah. That makes sense.
Speaker 2:Which is huge because it shows that equivalence isn't about how an equation looks, but about the underlying math relationship it represents.
Speaker 1:It's like saying hello in English and, in French. Different words, same meaning.
Speaker 2:Exactly. And to really make this idea stick, the lesson moves into an activity called what's acceptable. This time, we're talking about, you know, buying jeans with a discount and sales tax. Relatable. Right?
Speaker 2:Who
Speaker 1:doesn't love a good sale? But how do we go from jeans to equivalent equations?
Speaker 2:Well, the students are given this equation that models the cost of the jeans after the discount and tax, and then they're given a bunch of other equations. Some are equivalent to that original equation and some aren't. And their job is to figure out which is which and, more importantly, why.
Speaker 1:Okay. So this is where they really have to think critically.
Speaker 2:Exactly. They're analyzing the changes made to the original equation to see if those changes maintain the equation's balance or if they mess up the solution.
Speaker 1:So what exactly are those legal moves?
Speaker 2:Good question. These legal moves are like the operations you can do to both sides of an equation without changing the answer. You know, things like adding or subtracting the same value from both sides or maybe multiplying or dividing by the same non zero value. Those are super important for manipulating equations while keeping them equivalent.
Speaker 1:Right. It's like a set of rules for, like, equation gymnastics. You can move things around, but only in very specific ways. This feels like a super important point for teachers to emphasize. I can see how some common misconceptions might pop up around this idea.
Speaker 2:Oh, absolutely. One of the biggest ones is that equivalent equations have to look alike after they're simplified.
Speaker 1:Makes sense.
Speaker 2:Right. Students might get thrown off if they simplify one equation down to, like, x 3 and the other one ends up as 2x+4 is 10 even though they actually represent the same thing.
Speaker 1:That's a really good point. Students, they often want those nice clean answers, so it's important to show them that it goes deeper than just the final look of an equation.
Speaker 2:For sure. And another common misconception is that any operation, if you do it to both sides, will keep things equivalent.
Speaker 1:Yes. The allure of I can do anything I want as long as I do it to both sides. Easy to forget that some operations can really mess things up.
Speaker 2:Right. That's why emphasizing that concept of balance is so important. We want students to picture an equation like a balanced scale, where every operation either keeps that balance or throws it off and changes the answer.
Speaker 1:Yeah. It's like those old timey scales, you know, with the 2 trays and the little weights. If you add or take away the same weight from both sides, it stays balanced. But if you change just one side, whoops, everything's off.
Speaker 2:Exactly. Visuals like that are so helpful, especially for students who are still, like, grasping these abstract ideas.
Speaker 1:Absolutely. And, you know, another thing that can really help is making a clear link between what students do when they're, you know, solving for x and the steps involved in actually creating equivalent equations. Oh, okay. Interesting. So how do we connect those dots for them?
Speaker 2:Well, think about it. When students are solving for x, they're usually just trying to get x by itself. Right?
Speaker 1:Right.
Speaker 2:But each step they take, whether it's adding something, dividing by something, they're essentially making a brand new equation that's equivalent to the one before it.
Speaker 1:So it's like they're building this chain of equivalent equations, and each step gets them closer to that final x equals answer.
Speaker 2:Exactly. And by pointing that out, teachers can help students see that solving for x isn't just about getting that final answer. It's about understanding how to maintain equivalence along the way.
Speaker 1:It's like they're not just following the rules, they're understanding the why behind them.
Speaker 2:Exactly. And that deeper understanding is what allows them to use these concepts for, you know, harder math problems down the road.
Speaker 1:That makes sense. Now we've talked about some common misconceptions, and we've touched on some helpful tips for teachers. But before we wrap things up, I'd love to talk about the bigger picture for a sec. Why is understanding equivalent equations so important for students in the long run?
Speaker 2:Well, it's important to remember that this isn't just about one single concept. Right? This lesson is really about building a foundation for a deeper understanding of all of algebra and even beyond.
Speaker 1:So it's a stepping stone.
Speaker 2:Exactly. When students truly get this idea of equivalence, they're better able to tackle tougher algebraic manipulations, systems of equations, even higher level math like functions and calculus. They begin to see how all these different mathematical ideas are connected.
Speaker 1:It's like they've unlocked a secret level in their math brains.
Speaker 2:Yeah. You could definitely say that. And as they, you know, move forward in math, they can take those, legal moves they learned and apply them to all sorts of new challenges.
Speaker 1:That's really cool to think about, all the doors this opens for them. Now before we finish up this deep dive, I wanna leave everyone with a little something to think about. Got it. As you're, you know, going through this lesson with your students, think
Speaker 2:about how this idea
Speaker 1:of equivalence, how it goes beyond just equations? Like, how could you use it to show them connections between different parts of math?
Speaker 2:Oh, that's a great point. You could easily tie this into geometry. Right? Imagine giving students, say, equations for the perimeter or the area of a rectangle, but written in different ways using variables for length and width.
Speaker 1:So they could see how those equations, even though they look different, are actually equivalent because they're all talking about the same geometric idea.
Speaker 2:Exactly. It shows that equivalence isn't just stuck in one little corner of math. It's everywhere. And the more we can help students see those connections, the more it all makes sense to them.
Speaker 1:I like that. It's all about showing them the bigger picture of math.
Speaker 2:Yes. Well, I
Speaker 1:think we've given our listeners a lot to think about today, some really good stuff to help students understand the power of equivalent equations. A big thank you to the authors at Illustrative Math for these awesome lesson materials. And to all our listeners, thanks for joining us on the deep dive. Until next time, keep exploring and keep asking those what if questions. You never know what you might discover.