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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 just see equations as these abstract strings of numbers and letters? Like, they're just something to memorize and not something that actually, like, means something.
Speaker 2:Yeah. It's tough to make it feel relevant sometimes.
Speaker 1:Right. So today's deep dive is all about flipping that script. We're looking at how illustrative mathematics, specifically in lesson 5, helps students see real life dilemmas, like actual problems they might encounter through the lens of graphs.
Speaker 2:It's amazing how a few lines and points can suddenly make those decisions, like,
Speaker 1:actually making a choice feel so much clearer. Right. We're tackling a lesson called equations in their graphs. And it's not just about plotting points. It's about understanding what those points actually represent in a practical sense.
Speaker 1:Like, imagine trying to explain budgeting to someone who just isn't getting it, and then, bam, you show them a graph.
Speaker 2:It's like a light bulb moment for a lot of students. Suddenly, it clicks. It's not just an equation. It's a way to actually visualize all the possible choices they could make and the consequences of each choice.
Speaker 1:Okay. Let's untack the big ideas at play here. One of the coolest things this lesson drives home is that every single point on a graph, like every single dot, is actually a solution to the equation it represents.
Speaker 2:And this is where I think that activity, the snacks in bulk one, really shines. It takes something relatable that kids understand, like figuring out how many almonds and figs you can buy with a certain amount of money and turns it into a graphing exercise.
Speaker 1:So instead of just blindly connecting dots on a grid, students start to see, oh, this point means I can get £2 of almonds and £3 of figs, and it still fits my budget. Like, it actually means something.
Speaker 2:Exactly. Each point becomes less about just coordinates on a grid and more about a real decision they could make with real implications.
Speaker 1:Which brings us to another crucial idea the lesson emphasizes. Graphs aren't just about possibilities, they're about constraints and limitations too.
Speaker 2:I like to think about how this changes the conversation from here's everything you could possibly do to here's where things get interesting. Like, here are the choices that actually make sense. What are the trade offs we have to consider within these boundaries?
Speaker 1:Oh, I love that. It's like, yeah, I could technically spend all my money on figs, but this graph is showing me I'd have 0 almonds if I did that. Maybe not the best plan.
Speaker 2:And that's such a valuable real world lesson. Right? We're constantly working within constraints. Time, money, resources, and graphs provide a framework for understanding those limitations visually.
Speaker 1:And this is where I think the graph it act activity is so smart. It introduces students to graphing technology not as a way to make pretty pictures.
Speaker 2:But as a powerful problem solving tool, it's like, okay. You can write down the equation for how much water is draining from a tank, but wouldn't it be cool to actually see when it'll be empty?
Speaker 1:And that's where, for me as a teacher, you can almost see those gears turning in their heads and that light bulb moment for the student.
Speaker 2:Absolutely. They go from, I'm just plotting these silly points to, I'm using this graph to, like, actually make a prediction about what's gonna happen.
Speaker 1:Now I know from experience that sometimes our students get tripped up by misconceptions. You know, those little misunderstandings that can really throw them off.
Speaker 2:For sure. And with this lesson, one of the big ones is that they start thinking that every single point on a graph is a valid solution.
Speaker 1:Right. Like, it doesn't matter what the graph is actually showing.
Speaker 2:Exactly. And that's not always the case. Like, in that snacks example, the graph might show a point that's, you know, negative pounds of almonds.
Speaker 1:Which obviously you can't buy at the store.
Speaker 2:Exactly. You can't have negative almonds. So how do we make sure students don't fall into that trap that they're not just blindly trusting every point on the graph?
Speaker 1:It's like we need to teach them to use their common sense too.
Speaker 2:Yeah. Context is key. We need to really emphasize to them that while an equation might have infinite solutions, in the real world, situations often impose limits.
Speaker 1:So we'd explain that even though some points technically, like mathematically fall on the line, they just don't make sense in this particular scenario.
Speaker 2:Precisely. A negative weight of almonds isn't just wrong. It's nonsensical. It doesn't even make sense to think about. And helping students grasp that distinction that there's a difference between what's mathematically possible and what's actually realistic is so crucial.
Speaker 1:Now another misconception I've encountered is that graphing is just about, well, plotting points.
Speaker 2:Yes. The connect the dots mentality where they're just going through the motions without really understanding the connection back to the equation or the problem itself.
Speaker 1:Yeah. And that's where I see things really fall apart. They can plot the points, but they don't get what it means. So how do we ensure students aren't just going through the motions, that they're really thinking critically about what that graph represents?
Speaker 2:By making them think critically about the graph, like, we can challenge them to use it to actually answer specific questions about the situation, not just plot it and walk away.
Speaker 1:Like what? Give me an example.
Speaker 2:Okay. Let's go back to that leaky tank example. We could ask, according to the graph, when will the tank be exactly half full?
Speaker 1:Oh, I see what you mean. That forces them to really examine the graph and think about what it's telling them. They have to engage with the graph as a problem solving tool, not just a drawing.
Speaker 2:Exactly. It's no longer just a visual representation of an equation. It's a source of information that can help them make predictions and decisions about the scenario.
Speaker 1:This is where I get really excited as a teacher. When students see this connection for the first time, when they realize that the graph is actually telling them something useful, you can practically see the light bulbs going off. It's like they suddenly understand the why behind the math.
Speaker 2:Absolutely. And that's what makes it stick with them long after the lesson is over. It's not just about memorizing formulas anymore. It's about understanding how those formulas can be used to solve real problems.
Speaker 1:It's so rewarding when you see those connections click in their minds. So for those of you listening who are about to teach this lesson, what are some key takeaways, like the big picture things you'd emphasize?
Speaker 2:I think the biggest thing is to not be afraid to push your students beyond the mechanics of just plotting points on a graph.
Speaker 1:Like, don't just let them get away with plotting the points and saying, okay. I'm done.
Speaker 2:Right. Make them think. Ask them, what does this specific point on the graph actually represent in this problem? Why does this point matter? What information is this point giving us?
Speaker 1:It's about moving beyond just the how and really digging into the why. Right? Like, why are we even doing this?
Speaker 2:Exactly. When we connect these abstract concepts like equations and graphs to actual real world examples, that's when the learning becomes so much more engaging, relatable, and ultimately memorable for our students.
Speaker 1:Like, they finally see the point of it all.
Speaker 2:Exactly. And one thing you could do is even have students create their own scenarios, like, come up with their own problems and then graph them. That makes it even more real for them.
Speaker 1:Oh, I love that. It allows them to be creative, but also really cements that understanding of how the math actually applies to their lives.
Speaker 2:It's powerful stuff.
Speaker 1:It really is. Now before we wrap up today's deep dive, I wanna leave everyone with something to kinda chew on. We've been totally laser focused on linear equations and their graphs today. But what if we zoom out for a moment? You know, look at the bigger picture.
Speaker 2:I think I know where you're going with this. Go on.
Speaker 1:What if we take those core concepts points as solutions? Graph showing limitations using graphing technology as a problem solving tool. And we think about how they might apply to more complex relationships, like stuff that's not just a straight line. Like, what about curves?
Speaker 2:That's such a good point. It's easy to get caught up in the specifics of linear equations, but those core concepts, those big ideas have implications for so many other areas of math and, frankly, just life in general.
Speaker 1:It's like this lesson is just the tip of the iceberg. You know? It plants a seed for a whole new way of thinking about problems and relationships.
Speaker 2:Absolutely. It opens up a world of possibilities for our students to explore.
Speaker 1:And on that note, we'll leave you to ponder those possibilities. A huge thank you to the authors of Illustrative Mathematics for developing this awesome and thought provoking lesson. And to all of you listening, keep up the amazing work you're doing in the classroom. You guys are rock stars. Until next time.