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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 have one of those moments where a math concept just clicks and you're like, woah, I see the matrix. Yeah. That's what we're diving into today. The elegant dance of solving systems of equations, and get this.
Speaker 2:Okay.
Speaker 1:We're dissecting a specific lesson plan for algebra 1 students called solving systems by elimination part 3. Think of this deep dive as your backstage pass, not just to remember the how to, but to really feel the why, ready to multiply your way to solutions.
Speaker 2:Let's break it down.
Speaker 1:This lesson's not about throwing formulas at you. It's like, hey. Let's multiply an equation by a constant and see what happens. But don't worry. We're not messing with its core, its graph, just giving it a makeover.
Speaker 1:Then we'll tackle the main event, the elimination method, strategically multiplying 1 or even both equations to make a variable vanish.
Speaker 2:Oof.
Speaker 1:Right.
Speaker 2:I love it.
Speaker 1:And then get ready for it, equivalent systems.
Speaker 2:No. Yes.
Speaker 1:Sounds intense, but it's really about how each step gets us closer to cracking the code.
Speaker 2:What's fascinating is how this seemingly simple lesson, it's like the foundation of algebra. We're not just solving for x and y. We're understanding how equations behave. Yeah. It's like learning the rules of the game.
Speaker 1:Okay. So equivalent equations. What's that all about?
Speaker 2:Imagine a balanced scale. Add or subtract the same weight from both sides.
Speaker 1:Right.
Speaker 2:Still balance. Right? Yeah. Equivalent equations are similar. Multiply, divide, add, subtract.
Speaker 2:As long as it's on both sides, the core solution is untouched. Interesting. They're just making it easier to find.
Speaker 1:This is where it gets super cool. The lesson plan has this activity called graph it
Speaker 2:Yeah.
Speaker 1:Which brings those equivalent equations, like, to life.
Speaker 2:I love that activity.
Speaker 1:You actually get to see it happen.
Speaker 2:Yeah.
Speaker 1:They use the equation, by 4 y or as an example.
Speaker 2:Okay.
Speaker 1:So let's say you take that equation and you multiply both sides by, I don't know, 2. Mhmm. What happens to the graph?
Speaker 2:You would think the line gets steeper, right, or it shifts over.
Speaker 1:Yeah.
Speaker 2:But, no, it stays exactly the same. Woah. It's a visual demonstration of exactly what we're talking about. Right. When you multiply both sides of an equation by the same number, it doesn't change what the solution set is.
Speaker 1:So the relationship between x and y is the same.
Speaker 2:Exactly. We're
Speaker 1:just looking at it through a different lens.
Speaker 2:Precisely. Ready to see how this helps us solve actual systems.
Speaker 1:Oh, yeah.
Speaker 2:Alright. So the lesson then dives into what are called equivalent systems using 2 equations as an example.
Speaker 1:Okay.
Speaker 2:We'll call them equation a and equation b.
Speaker 1:Equivalent systems. Yeah. Sounds like it's kinda scary.
Speaker 2:So think of it this way. We could take one of those equations Okay. And multiply it by some carefully chosen number.
Speaker 1:Mhmm.
Speaker 2:And it allows us to then create a new equation that when you add it to the other equation, one of the variables, poof, disappears.
Speaker 1:Woah. What Isn't that cool? So we're making it simpler.
Speaker 2:It's like this puzzle piece that just fits perfectly to get rid of one of the unknowns.
Speaker 1:And then you're only left with 1 to solve for.
Speaker 2:Exactly. And that's where this idea of equivalent systems comes in.
Speaker 1:Okay.
Speaker 2:Every time we make a new system Yeah. By multiplying, adding, or subtracting equations, We're essentially getting closer to solving the original problem.
Speaker 1:Oh, interesting.
Speaker 2:It's like we're kind of, like, strategically simplifying the problem, but not changing what the solution is.
Speaker 1:Gotcha.
Speaker 2:We're just kinda rewriting it in a way that makes it easier to see the answer.
Speaker 1:Okay. So how do we make sure that all of our systems are equivalent?
Speaker 2:That's where this idea of justifying each step comes in.
Speaker 1:Okay.
Speaker 2:So the lesson really emphasizes that every time you multiply, you're doing it on both sides of the equation Right. Maintaining that balance.
Speaker 1:Right.
Speaker 2:So it's almost like leaving a trail of breadcrumbs
Speaker 1:Yeah.
Speaker 2:So that you can retrace your steps.
Speaker 1:I like that.
Speaker 2:Make sure that you didn't make any mistakes along the way.
Speaker 1:What I love about this lesson plan is it's not just telling you how to do it. Right. They actually give you these activities to really, like, make sure that you get it.
Speaker 2:Absolutely.
Speaker 1:Like, one is called what comes next.
Speaker 2:Oh, that's a fun one.
Speaker 1:And they give them all these equations, and they're all equivalent But they're
Speaker 2:all jumbled up.
Speaker 1:Yes. And you have to figure out what
Speaker 2:order to put them in.
Speaker 1:Like a pug? Yeah. To get to the answer. Exactly. It forces you to really think through, like Yes.
Speaker 2:What's the reasoning behind
Speaker 1:each step?
Speaker 2:And that's so much more beneficial
Speaker 1:Right.
Speaker 2:Than just memorizing a formula or a process.
Speaker 1:Totally. What are some of the, like, things that students might find tricky about this?
Speaker 2:So one thing that students sometimes get tripped up on is if you have to multiply an equation by a fraction to get to
Speaker 1:Oh, yeah.
Speaker 2:To get opposites. You know what I mean?
Speaker 1:Fractions are always a
Speaker 2:little Yeah. They're like, woah. We're dealing with fractions now.
Speaker 1:Right. Exactly.
Speaker 2:But it's the same principle. Right?
Speaker 1:Right. It
Speaker 2:You can multiply by a fraction, and it's still legal.
Speaker 1:Right.
Speaker 2:And sometimes it's the only way or the easiest way to get those opposites.
Speaker 1:Yeah. Yeah. Totally.
Speaker 2:Another thing that students, I think, struggle with sometimes is just this whole idea of equivalent systems.
Speaker 1:Okay.
Speaker 2:They're like, why are we doing this extra step?
Speaker 1:Right. Like, isn't that more work?
Speaker 2:I already have a system of equations. Why do we have to make a new one?
Speaker 1:Right.
Speaker 2:But once they kind of understand that it's really just a tool to make the problem easier.
Speaker 1:Right. It's all about making it easier.
Speaker 2:It clicks a little bit more.
Speaker 1:Okay. This has been awesome. We've unpacked the power of equivalent equations, explored the strategy of elimination, and even tackled those little moments that can trip us up.
Speaker 2:I love it.
Speaker 1:It's incredible how such a seemingly simple concept connects to the very foundation of algebra.
Speaker 2:Absolutely. It's like this underlying structure.
Speaker 1:Right. And before we go, remember, like, are you ready for more section we talked about.
Speaker 2:Yes.
Speaker 1:Think about this. Could you use this method, but in reverse? Could you design your own system of equations, maybe even with a specific solution in mind?
Speaker 2:Well, that's a great challenge.
Speaker 1:That's for you to ponder. Huge shout out to the authors of Illustrative Math for this amazing lesson plan. Until next time. Keep those equations balanced.