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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, and welcome back. We are about to take a deep dive into something that I think every algebra teacher out there knows, solving systems of equations. By elimination
Speaker 2:Oh, yeah.
Speaker 1:We're gonna be cracking open lesson 14, solving systems by elimination, part 1. This is from Illustrative Mathematics' algebra curriculum. I always love digging into their stuff.
Speaker 2:They're great.
Speaker 1:They do such a good job.
Speaker 2:Yeah.
Speaker 1:And I'm really excited to see how they tackle this topic because it can be a tricky one.
Speaker 2:You know, I always appreciate illustrative math really seems to go beyond just, like, the how to of math. Yeah. They really dig into the why.
Speaker 1:Yes. Absolutely.
Speaker 2:And I think that that's so important for students.
Speaker 1:Absolutely. It's not just about memorizing steps.
Speaker 2:Right.
Speaker 1:Right? It's like, why does this work? And speaking of why something works, this lesson starts off with a cool visual, balanced hangers. Oh, cool. Have you ever seen that before?
Speaker 2:I have. Yeah. The first time I saw that I don't remember if it was in this lesson or somewhere else, but it really clicked for me. It's just such a clever way to take something that's kind of an abstract concept
Speaker 1:Mhmm.
Speaker 2:And make it really concrete and visual for kids.
Speaker 1:And connect it to something they already understand. Yeah. Like, they've seen a balance before.
Speaker 2:Exactly. They get that. Yeah. So, like, imagine a hanger with 2 squares and a circle on one side Yeah. And then 4 squares on the other side.
Speaker 2:That represents the equation, 2x+y equals 4x.
Speaker 1:Oh, wow. Okay.
Speaker 2:The concern.
Speaker 1:That makes it so visual. Yeah. I love that. That. And then to show adding equations together, they could have another hanger with, let's say, 3 circles and a square on one side and a triangle on the other.
Speaker 1:So that's like y+x equals 3y.
Speaker 2:Exactly. And then the big moment, I think, is when they show how combining the shapes on the hangers Right. Is like adding the equations together. You're not changing the balance. You're just representing it in a different way.
Speaker 1:Oh, I see where you're going with this. Okay. So if you combine the shapes from those two hangers, you'd have 3 squares, a circle, and a triangle on one side, and 4 squares and 3 circles on the other.
Speaker 2:And that new combined hanger still represents a balanced equation, just like adding the original equations together would. It's
Speaker 1:brilliant. Right? Yeah.
Speaker 2:It's so clever.
Speaker 1:It's so simple but effective.
Speaker 2:Right.
Speaker 1:Like, they're sneaking in some really high level math understanding there Absolutely. Without the students even realizing it.
Speaker 2:Right.
Speaker 1:Okay. So they've got these amazing visuals, but, of course, they don't stop there. The lesson then moves into actually using elimination to solve systems.
Speaker 2:And they do a great job of scaffolding it. They start with examples where the coefficients of one variable are already opposites.
Speaker 1:So, like, if you have a 2 x in one equation and then NIDA two x in the other.
Speaker 2:Exactly. Okay. Students see that immediate elimination
Speaker 1:Mhmm.
Speaker 2:Which can really help them build confidence with the method, and then they actually get to analyze a fictional student's work. Yeah. I think his name is Diego. They have to figure out if he's using elimination correctly.
Speaker 1:I love when they incorporate student work because it's so helpful for teachers to see how students might approach a problem
Speaker 2:Totally.
Speaker 1:And where those potential pitfalls might be. For sure. And speaking of pitfalls, what are some common misconceptions that students have with elimination?
Speaker 2:Oh, there are a few that come to mind right away. One of the big ones is forgetting to consider the signs of the coefficients.
Speaker 1:You mean, like, they see a 2 x and a 2 x and just automatically subtract without thinking about whether 1 or both of them should be negative.
Speaker 2:Exactly. They're so focused on finding those matching coefficients that the positive and negative signs kinda get lost in the shuffle.
Speaker 1:Yeah. That's such an easy mistake to make. I've seen that happen in my classroom so many times. And, of course, that one little mistake can throw off the entire solution.
Speaker 2:Totally. Another tricky part for students, I think, can be connecting the algebraic solution back to what it actually represents graphically.
Speaker 1:That's so important, though. It's not enough to just solve for x and y. Right. They need to understand that those values actually represent a point where the graphs of those equations intersect.
Speaker 2:Right. And this lesson does a nice job of emphasizing that connection. It talks about how adding or subtracting equations is essentially creating a new equation that must also be true at that same point of intersection.
Speaker 1:You know, I'm not sure all of my students would totally grasp that. What are some ways that we can really help them to see that connection?
Speaker 2:Visualizing it is key. I like to actually graph the original equations.
Speaker 1:Mhmm.
Speaker 2:And they graph the new equation they create when they add or subtract. And seeing those three lines intersect at the same point can be a really powerful moment for students.
Speaker 1:Like, woah. It all comes back to that same point.
Speaker 2:I love that because then it really reinforces that the solution isn't just some random answer they got. Right. It has a real visual meaning. Exactly. Now the lesson also touches on the fact that sometimes you can't just add or subtract the equations right away.
Speaker 1:Right.
Speaker 2:You might need to multiply one of them first Yeah. Which I think for some students, that's where it starts to get a little more complex.
Speaker 1:Yeah.
Speaker 2:It does. And this lesson doesn't dive too deeply into that in this particular part. But it definitely plants the seed for that idea. And it encourages students to think strategically about when elimination is the most efficient method and when it might make more sense to use a different strategy like substitution.
Speaker 1:So it's not just about learning the steps of elimination, but also about knowing when and when not to use
Speaker 2:it. Exactly.
Speaker 1:I love that.
Speaker 2:It's about equipping those students with a toolbox of strategies and helping them develop that mathematical flexibility.
Speaker 1:Yes. I love that. Okay. So we've talked about the visuals, the misconceptions, the importance of strategy. Now before we wrap up, I wanna go back to something you mentioned earlier, that idea of multiplying equations.
Speaker 2:Yeah.
Speaker 1:Remember those balanced hangers?
Speaker 2:Oh, yeah. How could I forget? Such a clever visual.
Speaker 1:I'm wondering, could we modify that analogy to help students understand when they might need to multiply an equation?
Speaker 2:Oh, that's an interesting challenge. Let me think. What if we had hangers with different weight ratios?
Speaker 1:Weight ratios. Oh, I like where you're going with this.
Speaker 2:For example, you could have one hanger where a circle is twice as heavy as a square and another hanger where a circle is 3 times as heavy as a triangle.
Speaker 1:Okay. I'm trying to visualize this. So then students would have to figure out how to adjust the weights on the hangers, meaning the coefficients in the equations. Mhmm. So that when they combine the hangers, they can isolate one type of shape.
Speaker 2:Exactly. Oh. It's like those weight puzzles where you have to figure out, like, how many marbles equal a toy car.
Speaker 1:Oh my gosh. Yes. That's brilliant. Right? It's like sneaking in a little physics to teach algebra.
Speaker 2:Exactly. And just like with those puzzles, students would be developing their problem solving skills and their algebraic reasoning
Speaker 1:Wait.
Speaker 2:Without even realizing it.
Speaker 1:Math disguised as a game. I love it. Right.
Speaker 2:It's all about finding those concrete representations that click with students' learning styles.
Speaker 1:And on that note of creative teaching strategies, I think we've reached the end of our deep dive for part 1 of solving systems by elimination.
Speaker 2:It's been a pleasure unpacking this lesson with you.
Speaker 1:It's so fun. I love these conversations.
Speaker 2:Me too.
Speaker 1:So fascinating. Yeah.
Speaker 2:They are.
Speaker 1:And thank you to all of you for joining us on this journey of pedagogical discovery.
Speaker 2:I mean, I'm sure there are other ways you could think about it.
Speaker 1:Yeah. Definitely. But I really like that visual.
Speaker 2:It's very cool.
Speaker 1:It's so smart. Okay. So we've talked about the balance hangers. We've talked about some of those common student misconceptions. Are there any others that come to mind for you?
Speaker 2:I think those are the big ones. You know, for me, I always like to think back to what did I struggle with when I was learning this? Because chances are, if I struggled with it, my students are probably gonna struggle with it too.
Speaker 1:That's so true. And it makes you more empathetic teacher too.
Speaker 2:Oh, absolutely. Yeah. Because you've been there. Yeah. You remember that feeling of confusion, and you can kind of anticipate that
Speaker 1:Mhmm. And
Speaker 2:try to head it off at the past.
Speaker 1:Absolutely. Okay. So we've got these great visuals. We've talked about misconceptions. What else does this lesson touch on that you think is important?
Speaker 2:Let me see. Well, they talk about how sometimes adding equations is productive, and sometimes it's not. Yeah. And that's a really important distinction for students to make.
Speaker 1:Yeah. Because I think sometimes students get so caught up in just, like, learning the steps of a procedure
Speaker 2:Right.
Speaker 1:That they forget to actually think about, like, what am I trying to accomplish here?
Speaker 2:Exactly.
Speaker 1:And is this the best tool for the job? And
Speaker 2:that's something that I think illustrative math does really well in general Yeah. Because they encourage students to think about what are the different strategies I have available
Speaker 1:Right.
Speaker 2:And when is each strategy going to be the most efficient or the most helpful?
Speaker 1:It's about working smarter, not harder.
Speaker 2:Exactly.
Speaker 1:I love that.
Speaker 2:And this lesson, even though it's focused on elimination, it does a nice job of kinda planting that seed for students to start thinking more strategically about their problem solving.
Speaker 1:I like that a lot. So we're really setting the stage here for them to become more independent math learners.
Speaker 2:Exactly. Which is ultimately the goal. Right?
Speaker 1:Absolutely. Okay. So we've covered a lot of ground here, the visuals, the misconceptions, the importance of strategic thinking. Is there anything else in this lesson that you think we should touch on?
Speaker 2:I think we've hit the highlights.
Speaker 1:Okay. Fantastic. Well, then let's take a quick break, and when we come back, we'll wrap up our deep dive into part 1 of solving systems by elimination. Welcome back, everyone. So we've been taking a deep dive into solving systems of equations.
Speaker 2:By elimination.
Speaker 1:Yes. Using this really interesting lesson from Illustrative Mathematics. And I have to say, I'm so impressed with how they present this topic.
Speaker 2:Me too. I feel like every time we do one of these deep dives, I always come away with some new insight
Speaker 1:Me too.
Speaker 2:Or some new way of thinking about a familiar concept.
Speaker 1:Absolutely. And it really makes me appreciate the work that curriculum developers do.
Speaker 2:Yeah. It's easy to take these lessons for granted, but there's so much thought and care that goes into crafting them.
Speaker 1:It's true. It's really an art form. Well, on that note of appreciation, I think it's time for us to wrap up this deep dive.
Speaker 2:It's been a pleasure as always.
Speaker 1:Likewise. And a huge thank you to all of you for joining us today. We'll see you next time for another exciting deep dive. Into the world of mathematics education