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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.)
Systems of equations, remember those. You know, the ones that might have made algebra a little, interesting back in the day? Well, get ready to tackle them like a pro. Today's deep dive, making you a rock star at teaching solving systems by substitution.
Speaker 2:We're diving right into lesson 13. Yeah. Solving systems by substitution from this algebra curriculum. Getting right to it.
Speaker 1:And we're not just, you know, reminiscing about high school math. We're giving you, the listener, the tools to make this lesson really stick with your students. So substitution, not just some random procedure. Right? What's the big picture?
Speaker 2:It's about giving students this powerful tool, a tool that helps them solve problems with 2 unknowns. And let me tell you, these unknowns, they're everywhere.
Speaker 1:Okay. I'm seeing the real life connection here. Yeah. But how does this lesson, how does it really make it stick with students?
Speaker 2:This curriculum, it lays it out beautifully. It shows you how to help students recognize when substitution is the way to go. Like, can you isolate a variable easily in an equation? If you can, bam, substitution time.
Speaker 1:So it's all about strategy, not just blindly following steps. Love it. What else does this lesson really hit home?
Speaker 2:Flexibility. It shows there are multiple ways to do substitution because, let's face it, problems don't always come in a nice, neat package. And, of course, it makes sure they can actually solve and check those solutions right because a tool's no good if you don't know how to use it.
Speaker 1:Right. Okay. Let's jump into this lesson. What's the first thing that grabs our attention?
Speaker 2:Is it a match? The opening activity. It's brilliant, really. Instead of going straight into equations, it connects graphs and systems of equations.
Speaker 1:Intriguing. I have feeling it's more than just a simple matching game.
Speaker 2:Absolutely. It's like a warm up for their brains, you know. It gets them thinking about graphing in a new way. Remember those special cases? Horizontal and vertical lines like yp5 or x equal as 2, knowing those equations is a game changer here.
Speaker 1:So simple, but so clever. Exactly.
Speaker 2:They're forced to think about slope, intercepts, how do those things on the graph connect to the solutions. And the best part, they do all of this without even touching a graphing calculator.
Speaker 1:That's some seriously sneaky, in a good way, lesson design. Okay. They're warmed up. They're thinking graphically. What's next?
Speaker 2:Time to put those substitution skills to the test. Four systems is where it's at. But, and here's the clever part, even though these systems are practically begging to be solved by substitution, the curriculum encourages teachers to notice if students are still stuck on graphing.
Speaker 1:Interesting. Why is that so important?
Speaker 2:Because it's a golden opportunity to talk about efficiency, about precision. It's like saying, hey. You could walk to the store, but wouldn't a bike be way faster? Yeah. Gets them thinking critically about their choices.
Speaker 1:And speaking of choices, I see the lesson actually provides different ways to solve the same system in this activity.
Speaker 2:Yes. It really drives home the point that there isn't always one right way in math. Plus, it's so rewarding when students have that moment where they realize, wait, that way was so much faster.
Speaker 1:Oh, I can practically hear those light bulbs going off in the classroom. Okay. So we've got a warm up connecting glass and systems, then some hands on solving, emphasizing flexibility and efficiency. What's next on this learning journey?
Speaker 2:Now it's time to really see what they've learned. 13.3. What about now? More practice. Sure.
Speaker 2:But this time, they have to solve without graphing as a crutch.
Speaker 1:Uh-oh. The training wheels are off. What could possibly go wrong?
Speaker 2:Well, the curriculum points out a common pitfall, forgetting those parentheses when substituting an expression. It's easy to overlook, but it can really mess things up.
Speaker 1:I can totally see that happening. Even seasoned math folks like us make that mistake sometimes. So how does the lesson suggest handling that, you know, so our listeners can help their students avoid that trap?
Speaker 2:It encourages teachers to use numerical examples, you know, really show why those parentheses matter. Let's say you've got 2x plus 3. If you just substitute x, include no parentheses, you get 2 times 5 plus 3. That's 13.
Speaker 1:Okay.
Speaker 2:But use those parentheses correctly, and you've got 2 times 8, which is 16.
Speaker 1:Big difference. Huge. It's like those parentheses. They're little hugs keeping everything together, you know, in order. I'm seeing why this lesson really, really emphasizes them.
Speaker 1:Okay. So we've got the setup. We've got the practice. We even have a common pitfall. What about the final activity?
Speaker 1:What's that look like?
Speaker 2:13.4, a system to solve. The cooldown gives them one last problem, helps solidify everything. And this one. It really zeros in on choosing the best way to substitute, which honestly often depends on how the system is set up.
Speaker 1:So they're not just learning a procedure. They're developing real problem solving skills. I like it.
Speaker 2:Exactly. And what's really cool, the curriculum, it encourages teachers to collect those student responses, you know, see the different paths they took.
Speaker 1:Oh, that's so valuable. Seeing how your students think, how they approach a problem, you can learn so much from that. Speaking of learning, what about those common struggles, the ones that might not be so obvious? What should teachers what should they be ready for?
Speaker 2:I've got you covered. The notes for Deep Dive Presenters. That document, it highlights some key things. One common misconception, students forget to find the second variable. They solve for the first, and they think they're done.
Speaker 1:Like, they crossed the finish line, but they're only halfway there.
Speaker 2:Right. We gotta remind them, the solution, it's a coordinate pair, an x y duo marking the spot where those lines meet on a graph.
Speaker 1:Oh, visualization. Such a powerful tool in math. Okay. What's another one? Another misconception that might trip students up.
Speaker 2:This one's a real grain twister. Students get confused when the isolated variable, it equals an expression instead of just a number, like it's yx2xplusone instead of just, like, y5.
Speaker 1:Yeah. I can see how that would throw them off. It's like they're substituting a whole jumbled mess instead of something nice and neat. How do teachers how can they help them make sense of that?
Speaker 2:Start simple. Start with simpler examples, then gradually bring in those more complex expressions. The more they see it in action, the more comfortable they'll get.
Speaker 1:Practice makes progress. Right? Okay. We can't we can't have this deep dive without talking about, well, the elephant in the room, or should I say the parenthesis in the equation.
Speaker 2:Oh, you know it. We've mentioned it a few times, but it's important. Forgetting those parenthesis when substitute it's like, I don't know, forgetting your shoes before a marathon. You're gonna trip up.
Speaker 1:So true. And I love how this lesson gives those concrete strategies. What else can teachers do, though? What else can they do to make sure this lesson really clicks with their students?
Speaker 2:That final synthesis activity. Brilliant. It gives students these systems, they look different, and asks, would you use substitution? Why or why not?
Speaker 1:So it's all about making smart decisions, not just blindly using a method.
Speaker 2:Exactly. It forces them to analyze. Look at the system and say, is substitution the best way to go here?
Speaker 1:It's like they get to be the boss. You know? Choose the right tool for the job. I love how this lesson really digs into the when and the why of substitution, not just the how. Exactly.
Speaker 1:It's about empowering those students to really truly solve problems.
Speaker 2:Yeah. And speaking of empowering students, I have one final thought for our listeners.
Speaker 1:Okay.
Speaker 2:Something to think about when they're planning their lessons.
Speaker 1:Oh, I love a good thought provoking question. Hit us with it.
Speaker 2:Okay. So remember, is it a match? That first activity?
Speaker 1:Yeah. The one connecting graphs and systems.
Speaker 2:Right. Right. How could you how could you change it up
Speaker 1:Oh, okay.
Speaker 2:To include systems that would be, like, terrible for substitution.
Speaker 1:Oh, interesting.
Speaker 2:You know?
Speaker 1:Flipping the script. I like it. Yeah. It makes them think about when substitution isn't the best way to go.
Speaker 2:Exactly. It deepens their understanding, helps them develop a more well rounded approach, you know, for tackling systems of equations.
Speaker 1:This has been fantastic. I feel like I've brushed up on my algebra skills, and I've got some great new strategies to use in the classroom.
Speaker 2:That's what we like to hear.
Speaker 1:Right.
Speaker 2:Remember, it's not just about memorizing those procedures.
Speaker 1:Right.
Speaker 2:It's about equipping those students, giving them the tools and the thinking skills they need so they can be confident no matter what kind of system of equations they come across.
Speaker 1:Couldn't have said it better myself. A huge thank you to the authors of Illustrative Math for these amazing materials.
Speaker 2:Absolutely.
Speaker 1:And to you, dear listener, for joining us on this mathematical journey.
Speaker 2:Until next time.
Speaker 1:We'll catch you on our next deep dive where we'll we'll, you'll have to tune in to find out.