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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 some students just get equations like boom, instant connection, while others are kinda scratching their heads?
Speaker 2:Yeah. It's like a light bulb moment. Some kids have it, some kids not so much.
Speaker 1:Well, today, we're really diving deep into this lesson plan. It's from illustrative mathematics, And it's called Writing Equations to Model Relationships, part 2.
Speaker 2:Catchy.
Speaker 1:Right? But seriously, this plan could be a game changer for helping all L students understand those connections between, like, the real world and the world of algebra.
Speaker 2:It's not just about, you know, plugging in numbers. It's about getting the concept behind it.
Speaker 1:Exactly. So we'll break it all down, the activities, the stuff that can trip students up, and hopefully, give you some new ideas to use in your own classroom.
Speaker 2:Sounds like a plan.
Speaker 1:Okay. So what makes this particular lesson so interesting?
Speaker 2:Well, it doesn't just throw a bunch of formulas and equations at students and say, go solve.
Speaker 1:Which can be intimidating.
Speaker 2:Totally. Instead, it guides them step by step to really understand the relationships between numbers first. Then they're ready to express that relationship, that understanding in the language of Albert.
Speaker 1:So it's like building a bridge. Yeah. First, you have to have the solid ground, then you can start building out.
Speaker 2:Exactly. And this lesson plan uses these really cool activities, engaging activities to help students build that bridge from, you know, concrete to abstract.
Speaker 1:Love it when a plan is concrete. So give us an example. What's one of these bridge building activities?
Speaker 2:Okay. So there's one called finding a relationship. Students get a table of values right, and the challenge is they have to figure out the pattern hiding in those numbers.
Speaker 1:Oh, I love a good pattern, hon. But isn't there just one right answer?
Speaker 2:That's where it gets interesting. The lesson actually emphasizes exploring different strategies, not just finding a single correct solution.
Speaker 1:So multiple paths to the same destination.
Speaker 2:Exactly. And that's super valuable because it lets students see that math isn't always about just one right way. There can be flexibility and creativity in how you approach a problem.
Speaker 1:Okay. So walk us through this. What does this look like in action?
Speaker 2:Sure. Take this table from the lesson plan. One column is the side link of a square. The other is the area.
Speaker 1:Okay. Classic squares. Got it.
Speaker 2:So you might have a student who notices that the area is increasing, but it's going up by odd numbers, like plus 3, then plus 5, then plus 7.
Speaker 1:Interesting. I can see that.
Speaker 2:Right. But then another student might look at it and say, wait. The area, it's just the side length squared. Like, 1 times 1 is 1, 2 times 2 is 4, 3 times 3 is 9, and so on.
Speaker 1:Oh, wow. Two totally different ways of seeing the same relationship.
Speaker 2:Exactly. And that's the beauty of it. This activity lets us, as teachers, see how our students think, how they process information, which is invaluable for tailoring our teaching. It's like a window into their mathematical minds.
Speaker 1:Love it. Tailoring is key, and I bet those classroom discussions get super interesting when students share all these different approaches.
Speaker 2:Oh, absolutely. And that leads perfectly into the next activity, something about 400. This one bumps it up a notch. Students have to describe the relationships they find, but they gotta use both words, a and d equations.
Speaker 1:Oh, getting all articulate with our math. I like it. What kind of relationships are we talking about here?
Speaker 2:So there's this really relatable example, the meters from home and meters from school scenario.
Speaker 1:Okay. Every student's daily journey.
Speaker 2:Right. So imagine a student, they're walking to school, total distance, 400 meters.
Speaker 1:Got it. A nice little walk.
Speaker 2:The table might show how the meters from home increases, you know, as they're walking, while the meters from school decreases.
Speaker 1:Makes sense. Getting closer to 1, farther from the other.
Speaker 2:Now some students might see this as just simple addition. The two distances always add up to 400 meters. Right? Yeah.
Speaker 1:Makes sense.
Speaker 2:But then you might have others who focus on the difference between those two distances, like how much farther is school than home at any given point, and they start to see a pattern in how that difference changes.
Speaker 1:So many layers to uncover. It's not just about spitting out the equation. It's about understanding why that equation works, being able to explain it, like you said, in different ways.
Speaker 2:Exactly. It's about building that deep understanding, that communication. And that's where those classroom discussions become like gold. You know? Students explain their thinking.
Speaker 2:They challenge each other, and boom, suddenly the light bulbs start going off left and right.
Speaker 1:Love those classroom light bulb moments.
Speaker 2:Okay.
Speaker 1:Okay. So we've got these foundational activities building that bridge. Right? What's next? Where where does this lesson take us?
Speaker 2:Well, it gets even more intriguing. Activity 3 is called what are the relationships? And this is where they up the complexity.
Speaker 1:Bring on the challenge. But hopefully, not too complex. We don't wanna scare them off.
Speaker 2:Don't worry. It's done really cleverly. This is actually how the lesson introduces the idea of inverse variation. But in a subtle way, they don't even use the term explicitly at first.
Speaker 1:Sneaking in those big ideas, I like it. Subtly is key sometimes.
Speaker 2:Right. And the parallelogram example is a perfect illustration of this. It's all about getting students to think beyond those simple straightforward relationships.
Speaker 1:Okay. I'm intrigued. Hit me with this parallelogram example.
Speaker 2:So picture this. You've got a table, and it shows the base and height of different parallelograms. But here's the catch. They all have the same area.
Speaker 1:Okay. Same area, different dimensions. Got it.
Speaker 2:As the base gets bigger, the height has to decrease proportionally to keep that area constant.
Speaker 1:Ah, I see where you're going with this. So it's not as straightforward as one thing goes up, the other goes up kind of relationship.
Speaker 2:Exactly. And that's where some students might get tripped up. They might just see the height decreasing and focus on that, missing the bigger picture that the area is staying the same.
Speaker 1:It's like that saying, you can't see the forest for the trees. They get stuck on one detail and miss the overall pattern.
Speaker 2:Right. And that's a really common misconception, not just with this specific concept, but in math in general. It highlights how important it is to look at the whole picture, the whole dataset to really understand the relationship.
Speaker 1:It's like teaching them to be a little mass detective. Right? They gotta gather all the clues before they can solve the case.
Speaker 2:Exactly. And this lesson plan, it does a fantastic job of pointing out those potential pitfalls, those common misconceptions that teachers should watch out for. It's like having a road map of potential wrong turns, which helps us anticipate where students might struggle, and then we can address those issues head on.
Speaker 1:Okay. Love a good road map. Yeah. So what's one of the first misconceptions we should have on our radar?
Speaker 2:Well, a big one is this assumption that every single relationship has to be written out as a formal equation. Some students, they crave that equation. That's the answer to them. But they might struggle with verbal explanations or visual representations. Like, those aren't mathy enough.
Speaker 1:It's like they're so focused on getting to the right answer that they miss the journey, the exploration of the relationship itself.
Speaker 2:Exactly. And that's where we, as teachers, play such a crucial role. We need to constantly reinforce that there are multiple valid and valuable ways to represent a mathematical relationship.
Speaker 1:So it's about giving them a toolbox. Right? Not just one hammer, but a whole set of tools. Yes. Encourage them to describe the pattern in words, to sketch out a diagram,
Speaker 2:to use whatever tools help them make sense of it. All of these approaches deepen their understanding. And then once they have that solid grasp, the equation, it just naturally falls into place. Like that moment when it just naturally falls into place.
Speaker 1:Like that moment when everything clicks. Yeah. They see the connections. They understand the why behind the what.
Speaker 2:Exactly. And those are the moments we live for as educators. Those moments when the light bulb goes off and you can just see the excitement in their eyes.
Speaker 1:Absolutely. Speaking of light bulb moments, I'm curious to hear more about this idea of inverse variation, how the lesson subtly introduces it, but we'll save that for after a quick break.
Speaker 2:And when they have that moment, it sticks with them. So much better than just memorizing a formula.
Speaker 1:Way more powerful. Now you mentioned misconceptions, those little traps students can fall into. What other tricky spots should we be ready for?
Speaker 2:Right. So remember how we were talking about looking at the whole picture? Well, another common hiccup is when students, you know, they're eager to find the answer, they might jump the gun and misinterpret a pattern by just looking at a tiny piece of the data.
Speaker 1:Like, they see a few trees. Like, they've got the whole forest figured out.
Speaker 2:Exactly. Like that parallelogram example, remember. If they only focus on, say, the first few rows of that table, they might assume the height just keeps decreasing at a steady pace with every increase in the base. Right.
Speaker 1:But that's not the whole story, is it?
Speaker 2:Nope. If they zoom out, look at the entire dataset, they'd see it's proportional, not a simple linear decrease.
Speaker 1:So it's a classic case of don't judge a relationship by the first few data points.
Speaker 2:Exactly. And that's such a valuable teaching opportunity. That's where we can step in and guide them. Right? Encourage them to test their ideas, ask themselves, okay, does this pattern hold up for all the values?
Speaker 2:What if we keep going with this table?
Speaker 1:It's like teaching them to be those scientists, always testing their hypotheses, making sure their conclusions are solid.
Speaker 2:Exactly. Those critical thinking skills, they're gold in any subject, not just math class. Right?
Speaker 1:Totally. Okay. So we've been talking about linear relationships. But math, it's a big world out there.
Speaker 2:Mhmm.
Speaker 1:How can we take these ideas, these activities, and use them to introduce even more, I don't know, exciting, maybe even slightly scarier relationships? I'm talking quadratics, inverse variations, all that good stuff.
Speaker 2:You read my mind. That's what's so great about this lesson plan. It's like a springboard. It lays the groundwork, and then boom, you can launch into all sorts of cool extensions.
Speaker 1:Okay. I'm ready for lift off. Give me an example.
Speaker 2:Alright. So picture this. You present your students with a table. One column, it's the side length of a square, just like before. But here's the twist.
Speaker 2:The other column, it's not area anymore. It's the volume.
Speaker 1:Oh, we're getting three-dimensional. I like where this is going.
Speaker 2:Right. Now we're talking about cubes, not just squares. They're working with a cubic relationship without even realizing it at first.
Speaker 1:Sneaky. And that makes it so much more accessible. Instead of just throwing the equation at them, they're discovering it, experiencing it.
Speaker 2:Exactly. They could explore how that volume it explodes compared to the side lengths, maybe even build some models, you know, really visualize how that relationship plays out.
Speaker 1:I'm already picturing the sugar cubes and the Play Doh coming out.
Speaker 2:Exactly. Hands on learning for the win.
Speaker 1:Always. Yeah. Okay. But you mentioned inverse variation too. Mhmm.
Speaker 1:I'm not letting you off the hook that easily. How can we bring that into the mix?
Speaker 2:Oh, I've got you covered. And this ties in beautifully with what we were just talking about, getting them to discover the concept on their own.
Speaker 1:I love a good mathematical mystery.
Speaker 2:Okay. Imagine this. You give them a table, and this time, it shows the speed of, let's say, a car and the time it takes to travel a certain distance.
Speaker 1:Okay. Got it. Speed and time. Classic variables.
Speaker 2:Right. So they'll notice as the speed goes up, the time goes down. Makes sense.
Speaker 1:Intuitive. Faster you go, less time it takes.
Speaker 2:But the key here is it's not a simple straightforward decrease. It's not like double the speed, cut the time in half. It's that inverse relationship where the product of the speed and time, it always equals that fixed distance.
Speaker 1:Oh, there it is, sneaking in that inverse variation again.
Speaker 2:Exactly. And by having them play with the data first, analyze the patterns, they're primed to understand that moment when you finally introduce the formal concept.
Speaker 1:It's like those magic trick reveals. Right? The audience is way more impressed when they've been part of the process trying to figure it out themselves.
Speaker 2:A 1000000 times more impressed. It's not just, oh, that's a neat trick. It's, woah, how did that even happen? And suddenly, they're hungry to learn more.
Speaker 1:Exactly. They're invested. They're curious, and they're ready to tackle those more complex ideas because they've built that solid foundation.
Speaker 2:It's about lighting that fire, that love of learning, and this lesson plan it gives you the kindling and the matches.
Speaker 1:Love it. So we've talked about the activities, the common pitfalls, the exciting extensions. What's the big takeaway here? What should teachers keep in mind as they dive into this lesson plan with their students? It's like we're giving them this whole new lens to see the world through.
Speaker 1:All those hidden connections start popping out.
Speaker 2:And it all goes back to those basic building blocks, you know, patterns, different ways of representing things, getting them to really analyze and test their thinking. That's how we raise up these confident math thinkers.
Speaker 1:This lesson plan really does lay it all out so beautifully. It's like step by step. How to take them from those concrete examples to that deeper, more abstract understanding.
Speaker 2:And we can't forget about the power of those moments, you know, when a student finally cracks the code and it just clicks. Makes all the difference.
Speaker 1:Absolutely. That's what makes it all worthwhile, those moments when you see a light bulb go on. Yeah. So how can we, as teachers, set the stage for those breakthroughs?
Speaker 2:Well, it's about creating those opportunities. Right? Designing activities that challenge them just enough, but also provide the right support. Anticipating those common hiccups, those misconceptions so we can address them head on. And above all, nurturing that spirit of curiosity, that love for the puzzle.
Speaker 1:It's like we're not just teaching them math, we're teaching them how to think, how to approach any problem with a curious and analytical mind.
Speaker 2:Exactly. And when they have those tools, who knows what they'll accomplish?
Speaker 1:Couldn't agree more. So big thanks to the brilliant minds over at Illustrative Mathematics for creating such an incredible resource.
Speaker 2:Absolutely game changer.
Speaker 1:And to all of you listening, we hope this deep dive has sparked some new ideas, maybe even a little excitement for exploring those patterns and relationships, those hidden gems in the world of algebra. Until next time, keep those light bulbs shining.