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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 feel like your students kinda get lost, you know, in the weeds when they're learning about systems of linear equations?
Speaker 2:Ollie.
Speaker 1:Like, they're just, crunching numbers without really seeing the bigger picture?
Speaker 2:Yeah. Yeah. Yeah. For sure.
Speaker 1:Well, get ready to dive deep because we're tackling lesson 17 from Illustrative Mathematics. And trust me, this deep dive is gonna, like, totally change your teaching games. You'll walk away with some, some really cool insights. And I swear, you'll be hearing those moments from your students left and right.
Speaker 2:I love that. You know, what's really brilliant about this lesson is that it goes way beyond just the, the mechanics of solving for x and y.
Speaker 1:Right.
Speaker 2:It definitely covers that, of course, but it really, really encourages students to, like, to grapple with those deeper concepts. You know? Like, why some systems have one solution or no solutions at all or even infinitely many. So we're we're talking
Speaker 1:about moving beyond just plugging and chugging.
Speaker 2:Exactly.
Speaker 1:We want that deep understanding. Yeah. What are some of the, the key takeaways from this lesson, would you say, that that are really gonna help my students kind of, like, master this topic?
Speaker 2:Okay. So one of the big things is understanding that even a really simple looking system can actually have a lot of conceptual, depth. Okay. Take the warm up activity, for example, you know, a curious system. It it presents this this very straightforward system, but guess what?
Speaker 2:It it beautifully demonstrates this concept of infinite solutions.
Speaker 1:Oh, I'm intrigued already.
Speaker 2:Right. And you'll see how this lesson, it cleverly uses that curious system to kind of showcase all these different solution strategies. Right? So your students might try, like, substitution, and then boom. They'll see how those variables just magically cancel out.
Speaker 1:Oh, I love that.
Speaker 2:Which just clearly shows them, you know, hey. There are infinite solutions here.
Speaker 1:I can already imagine the moments happening in the classroom.
Speaker 2:Exactly. And the best part is the lesson doesn't stop there. Right? It keeps building on that idea with the next activity. What's the deal?
Speaker 2:So Picture this.
Speaker 1:Okay. I'm picturing it.
Speaker 2:You present your students with this this real life scenario. Yeah. We're talking about pool passes and gym memberships. Right?
Speaker 1:Now you've got my attention.
Speaker 2:Right.
Speaker 1:I love that it's relatable. My students can totally connect to that.
Speaker 2:Exactly. It's all about making those connections. Right? Yeah. So you've got this family.
Speaker 2:They're buying a bunch of pool passes. They're buying gym memberships. Right? And they're paying one price. What?
Speaker 2:And then you've got this individual, and they're buying just a few passes, maybe a membership, and they're and they're buying just a few passes, maybe a membership, and they're paying a different price. Okay. Yeah. Classic word problem setup. Totally.
Speaker 2:It's a classic. And, of course, your students,
Speaker 1:their brains are gonna be like, okay.
Speaker 2:I need to figure out how much each pass costs, how much each membership costs.
Speaker 1:Yeah. They're gonna jump right into problem solving mode.
Speaker 2:Exactly. But here's the thing. This specific scenario, it leads to a system with absolutely no solutions.
Speaker 1:Wait. No solutions?
Speaker 2:No solutions.
Speaker 1:Now I'm really intrigued. Tell me more.
Speaker 2:It's like a real world math mystery.
Speaker 1:I love it. So how does the lesson how does it actually, like, bring that to life for the students?
Speaker 2:So students might try to tackle it algebraically. Right?
Speaker 1:Right.
Speaker 2:And they might bump into a contradiction. You know, they'll do their staring at something like like 5. Right. Right. Which obviously doesn't make sense.
Speaker 2:Or they might decide to graph the equations. Right.
Speaker 1:And what are they gonna
Speaker 2:see? Those lines are parallel Oh, parallel lines. Never intersecting like ships passing in the night.
Speaker 1:I love that visual. It really drives home the point of no solution.
Speaker 2:It does. Right. Yeah. Because it's not just that, oh, we haven't found the solution yet.
Speaker 1:Right.
Speaker 2:It's that fundamentally, those lines are never gonna meet.
Speaker 1:There is no solution out there to be found.
Speaker 2:Exactly.
Speaker 1:This is great. I'm already seeing how this lesson can really help students connect those, you know, sometimes abstract equations to, like, real tangible scenarios.
Speaker 2:Yes. And that's what it's all about. And just when they think they've got it, we're gonna hit them with the card sort sorting systems activity.
Speaker 1:Oh, card sorts.
Speaker 2:Always a hit in the classroom. Right?
Speaker 1:Yeah. Always. My students love those. So what makes this card sort, like, extra special?
Speaker 2:Okay. So imagine this. Your students, they have all these systems of equations. Right? K.
Speaker 2:And their job is to sort them Sure. Into 3 piles. One solution, no solutions, or infinite solutions. Okay. Sounds pretty straightforward so far.
Speaker 2:But here's the twist. They have to do it without actually solving for f's and y.
Speaker 1:Woah. Hold on. How are they supposed to sort them if they're not finding the actual solutions?
Speaker 2:That's where the real detective work comes in. They're gonna have to put on their detective hats, look at those coefficients, those constants, and try to spot the clues.
Speaker 1:Okay. So they're looking for patterns. Right? Like, those telltale signs that might indicate parallel lines or maybe equations that are, like, secretly the same, just dressed up differently.
Speaker 2:I got it. It's all about analyzing that structure.
Speaker 1:And that aligns perfectly with mathematical practice 7. Right? Looking for and making use of structure.
Speaker 2:Bingo. And here's a little pro tip. When you get to the card sort activity, think about having your students work in pairs.
Speaker 1:Oh, I like that.
Speaker 2:Because the debates and discussions they're gonna have about those coefficients, about those patterns, it's pure gold.
Speaker 1:I bet. Collaboration is key.
Speaker 2:Absolutely. And then if they're up for a challenge, there's this are you ready for more section. It's where they get to take those systems they've already sorted and actually try to, like, tweak them
Speaker 1:Tweak them how?
Speaker 2:To actually change the number of solutions. So maybe they start with a system that has one solution, and they have to figure out, okay. How can I change this coefficient or this constant to make it have no solutions?
Speaker 1:Oh, that's a whole other level of thinking. They have to really, really understand how those coefficients and constants interact.
Speaker 2:Exactly. They go from solving the mystery to becoming, like, mathematical architects, designing systems with these very specific properties.
Speaker 1:Designing systems. I like that. It's like they're taking ownership of the math. Instead of just being given equations, they're, like, creating them.
Speaker 2:Exactly. And that's what leads to that deeper understanding.
Speaker 1:And that deeper understanding is what's gonna help them when they hit those those inevitable roadblocks. Yeah. Right? Because let's face it. Roadblocks are part of the learning process.
Speaker 2:Oh, for sure. And, you know, speaking of roadblocks, we should probably talk about some of the common misconceptions that tend to trip students up with systems of equations. Because if we can anticipate those misconceptions, we can help our students navigate those tricky spots. Right?
Speaker 1:Absolutely. Let's do that. What are some of the things that, in your experience, tend to really throw students off when it comes to systems of equations? And more importantly, how can we as teachers help them?
Speaker 2:Absolutely. Let's do that. What are some of the things that, in your experience, tend to really throw students off when it comes to systems of equations? And more importantly, how can we as teachers help them?
Speaker 1:You're speaking my language. Yeah. One classic misconception is that students might, like, fixate on the literal meaning of infinite solutions.
Speaker 2:Okay.
Speaker 1:They think it means they have
Speaker 2:to actually find every single solution, which is, you know. Impossible.
Speaker 1:Exactly. It's like trying to count to infinity. It can't be done. Right. It's, like, mathematically mind blowing.
Speaker 2:Totally. And that's where I think those visual representations, they can really be a game changer. Okay. I like that. So instead of just talking about infinite solutions in the abstract, show them the graph.
Speaker 2:Right? Have them actually look at it and ask them, what do you notice about all these points on this line?
Speaker 1:Right. Right.
Speaker 2:And suddenly, it clicks. They realize, wait a minute. Every single point on this line represents an x and a y that actually satisfy both equations. It's a beautiful thing.
Speaker 1:It makes infinite feel a little less daunting. You know? It's not this endless, like, scary list of numbers.
Speaker 2:Yeah.
Speaker 1:Yeah. It's just all the points neatly arranged on this line.
Speaker 2:Exactly. And, you know, you can even bring it back to the equations themselves. Okay. When they see that the two equations are actually equivalent, maybe after they do a little simplifying or rearranging, they realize that any solution that works for one equation automatically works for the other Right. Even if there are, you know, infinitely many possibilities.
Speaker 1:So it's not so much about find the answer. It's more about understand the relationship.
Speaker 2:Yes. You got it. That's a great way to put it.
Speaker 1:Okay. That's really helpful. Yeah. So infinite solutions, we've tackled that one. What other misconceptions do you often see?
Speaker 2:Okay. Let's see. Another one that pops up is students mixing up no solutions with 0 solutions. Okay. I could see how that can happen.
Speaker 2:Right. They sound very similar.
Speaker 1:Yeah. They both involve not having, like, a single solution Right. But in different ways.
Speaker 2:Right? You're exactly right. So when we say no solutions, what we really mean is that there is absolutely no combination of x and y that would make both of those equations true at the same time.
Speaker 1:Okay.
Speaker 2:Think parallel lines again. Right. Never intersecting. They exist in the same, like, mathematical universe, but their paths are never gonna cross. Okay.
Speaker 2:So it's not just that we haven't found the solution yet. It's that there's no solution out there to be found. Precisely.
Speaker 1:Okay. Got it.
Speaker 2:Whereas with 0 solutions, we're usually talking about a situation where, like, there could be solutions, but none of them actually fit the specific criteria of the problem we're trying to solve.
Speaker 1:So it's more like we searched everywhere that made sense within the boundaries of this problem, and we just we came up empty.
Speaker 2:Yes. Perfectly said.
Speaker 1:Okay. So to make sure my students don't get tripped up by this, should I, like, give them a bunch of different examples? Show them situations where they might encounter no solutions versus 0 solutions.
Speaker 2:Yes. A 100%. Variety is key. The more examples they see, the more they'll start to internalize that difference.
Speaker 1:Variety is the spice of math class.
Speaker 2:I love that.
Speaker 1:Okay. So we've talked about a lot of really important teaching strategies today. We've talked about embracing multiple solution strategies. We've talked about anticipating those common misconceptions. But before we wrap up this deep dive into lesson 17, I kinda wanna, like, zoom out for a second.
Speaker 1:Sure. We've been deep in the weeds of this specific lesson, but how does it all connect to the bigger picture of, you know, teaching algebra?
Speaker 2:I'm so glad you asked.
Speaker 1:We've been really deep in the weeds of this specific lesson, but how does it all connect to the bigger picture of, you know, teaching algebra?
Speaker 2:I'm so glad you asked. That's such an important thing for math teachers to think about. You know? We don't want to be teaching these lessons in isolation. Right.
Speaker 2:We want them to build towards this, like, deeper, richer understanding of mathematics. Right? Absolutely. And this lesson this lesson on systems of linear equations, it's really laying the foundation for so many other really important concepts.
Speaker 1:Like, what give me some examples. What does this lesson connect to?
Speaker 2:Well, for starters, it connects directly to the idea of functions. Okay. When students are, you know, crashing those systems of equations, what they're really doing is visualizing relationship between 2 linear functions.
Speaker 1:Okay.
Speaker 2:And that idea of functions I mean, it just, like, explodes from there.
Speaker 1:Why?
Speaker 2:It pops up again and again throughout algebra, throughout I mean, it goes on and on.
Speaker 1:It's like this lesson is, like, planting a seed that can grow into this huge mathematical tree.
Speaker 2:Exactly. That's a great analogy. And and don't forget about matrices.
Speaker 1:Oh, what? Matrices.
Speaker 2:Those systems of equations that your students are working with, they can be represented and solved using which is, like, this powerful tool, especially when you get into systems that have, you know, way more than 2 variables.
Speaker 1:So we're not just teaching them to solve for x and y. Yeah. We're kind of, like, opening the door to this whole other way of thinking about math.
Speaker 2:A 100%. And here's the thing. You know, when we really emphasize those connections for kids, when we show them that math isn't just this bunch of, like, random isolated topics, it becomes so much more meaningful for them.
Speaker 1:It's like we're helping them see the forest instead of just getting lost in the
Speaker 2:trees. Exactly.
Speaker 1:And, you know, speaking of seeing the bigger picture, this lesson does such a great job of highlighting the importance of using different representations. Right?
Speaker 2:Absolutely. We've
Speaker 1:got equations. We've got graphs. We've got those real world scenarios. Why is that variety so important, would you say, especially when we're teaching systems of equations?
Speaker 2:Because math is so much more than just, you know, symbols on a page. When students can approach a problem from different angles using graphs, using equations, using their own intuition even, it just, like, deepens their understanding so much more.
Speaker 1:It's like having a whole toolbox full of tools. Right?
Speaker 2:Exactly.
Speaker 1:You're ready for anything. Whatever problem comes your way, you've got the right tool for
Speaker 2:it. Exactly. And let's not forget about those those moments. Right? Sometimes a student might be really struggling with the algebra, with the the manipulating of those equations.
Speaker 2:Right. And then they see the graph, and suddenly it just clicks. Yes. They make a connection that they wouldn't have made otherwise.
Speaker 1:I love those moments. I live for those moments as a teacher.
Speaker 2:The best.
Speaker 1:And you know what else I found really brings on those moments? Letting students share their different solution strategies.
Speaker 2:Oh, so important.
Speaker 1:Because this lesson seems like it's, like, perfectly designed for that. Right?
Speaker 2:Absolutely. Because when students see that there are multiple valid ways to approach a problem, it reinforces that idea that math isn't about just memorizing a procedure.
Speaker 1:Right.
Speaker 2:It's about thinking critically. It's about being flexible. Mhmm. It's about applying what you know in a creative way.
Speaker 1:It's about empowering them to find their own way to the solution. A 100%. And when
Speaker 2:they explain their thinking, you know, when they can articulate And when they explain their thinking, you know, when they can articulate why their method works, that's when you know they've really, really got it. It's like they're not just learning math. They're, like,
Speaker 1:teaching math. They're becoming these mathematical ambassadors. I love that.
Speaker 2:Yes. Spreading the math love.
Speaker 1:Well, I have to say this has been amazing. This deep dive has been so insightful. I feel like I haven't just gained this, like, deeper understanding of this particular lesson on systems of linear equations.
Speaker 2:Mhmm.
Speaker 1:But you've also reminded me of the power of connecting those big mathematical ideas, embracing those multiple representations, and really fostering that classroom culture where students, they feel confident, they feel comfortable sharing their thinking even if it's different from someone else's.
Speaker 2:It's been such a pleasure, like, diving into this with you. It's true. I always say the best way to learn is to teach. And today, I feel like I've learned so much from our conversation.
Speaker 1:Me too. A huge thank you to the authors of Illustrative Math for creating this, like, really well crafted and thought provoking lesson.
Speaker 2:Yes. Thank you.
Speaker 1:And to our listeners, thank you for joining us on this mathematical journey. Remember, teaching systems of linear equations, it's not just about finding those solutions, it's about sparking that curiosity, encouraging that deep thinking, and empowering your students to see the world through this, like, mathematical lens.
Speaker 2:And never be afraid to ask why. Yeah. Both you and your students.
Speaker 1:Words to teach by. Until next time. Happy teaching, everyone.