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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.)
Ready to dig into some algebra? Get your shovels ready because this deep dive is all about maximizing gardens.
Speaker 2:Sounds fun.
Speaker 1:We're tackling a high school lesson plan and let me tell you it's not your average trip to the blackboard.
Speaker 2:Okay. I'm intrigued.
Speaker 1:We're diving deep into illustrative math's lesson plan all about finding the absolute biggest garden possible within a fixed perimeter. Think of it like this. You got a limited amount of fencing and you wanna become the king or queen of carrots, the sultan of squash, the emperor of eggplants. You get the idea.
Speaker 2:Maximizing efficiency. It's a classic math problem.
Speaker 1:Exactly. And this deep dive is gonna unpack how this lesson helps guide students to those moments. You know, when the concept of quadratic relationships just clicks. What I really love about this lesson plan is it starts by building on what students already know
Speaker 2:Which is?
Speaker 1:Linear and exponential relationships.
Speaker 2:Right. Those nice predictable patterns of change. It specifically aligns with standard hsflea.one if you wanna be technical about it.
Speaker 1:Love the technical stuff. So give us an example of a linear relationship, you know, just for our listeners who might be a little rusty.
Speaker 2:Sure. Imagine you're taking a leisurely stroll at a constant speed. The distance you cover, that's a linear relationship to the time you spend walking.
Speaker 1:Right. Walk twice as long, go twice as far. Simple stuff. So what about exponential relationships? Those always seem a bit more, well, explosive.
Speaker 2:They do, don't they? Think about the spread of a viral video. Each person shares it with a few others and, boom, suddenly it's everywhere. That's not changing at a steady linear rate.
Speaker 1:That's for sure. And that's where our store of the show comes in.
Speaker 2:Quadratic relationships.
Speaker 1:You got it. Quadratic relationships are like the plot twist in the world of functions. They don't just go up or down in a straight line.
Speaker 2:They like to keep things interesting.
Speaker 1:They do. So instead of a straight line or a constant upward curve, they increase and then decrease or vice versa, creating a unique u shaped curve when you graph them.
Speaker 2:They're the roller coasters of the math world.
Speaker 1:Okay. So how does this lesson plan set the stage for understanding these quirky quadratic curves?
Speaker 2:It starts with an activity called notice and wonder. They give the students 3 tables, and each one represents a different pattern of change, linear, exponential, and, of course, our friend, the quadratic.
Speaker 1:So the students become like mathematical detectives, right, Looking for clues in the tables.
Speaker 2:Exactly. They might analyze, for example, how the area of a rectangle changes as you increase the length of one side.
Speaker 1:While keeping the perimeter the same, I'm guessing.
Speaker 2:You got it. That's the key here.
Speaker 1:So what should students be noticing in that quadratic table specifically? What's the big giveaway?
Speaker 2:The numbers in the quadratic table, they're a little less predictable than the others.
Speaker 1:They like to keep us on our toes.
Speaker 2:They do. The numbers might increase for a bit, then hit a peak, and then get this they start to decrease again.
Speaker 1:Creating that classic curve.
Speaker 2:Precisely. And it's a pattern that might not be super obvious just by looking at the numbers themselves, you know, at first glance.
Speaker 1:Like a hidden code.
Speaker 2:Exactly. And that's what makes it so great. It sparks curiosity. They get a taste of the mystery, and it makes them wanna dive in and figure it out.
Speaker 1:It's like they're on the trail of something interesting, but not quite sure what it is yet. So how does the lesson take that initial spark of curiosity and turn it into, you know, a full blown moment?
Speaker 2:Well, it's time to ditch the tables and break out the virtual tools because it's time for the measuring a garden activity.
Speaker 1:I knew we'd get to the gardening part eventually.
Speaker 2:This is where things get hands on. Imagine the students get 50 meters of virtual fencing and their challenge, design a rectangular garden with the maximum area.
Speaker 1:But there's a catch. Right.
Speaker 2:Always. They have to keep that perimeter fixed at 50 meters. No cheating.
Speaker 1:So it's all about finding that sweet spot, that perfect balance between length and width. I can already see students sketching out their gardens, maybe even pulling out some rulers and measuring tapes.
Speaker 2:Well, they don't need actual measuring tapes since it's virtual, but you get the idea. And that's what's so fantastic about this activity. It's all about exploration. Students can experiment with all sorts of lengths and widths, calculate the areas, see how even small changes can make a big difference.
Speaker 1:Oh, I bet some of them will get really into it. They might even create spreadsheets, you know, to keep track of all their data.
Speaker 2:Now you're thinking like a true math enthusiast
Speaker 1:I know.
Speaker 2:And you're spot on. This activity is all about encouraging them to reason abstractly and quantitatively, which aligns perfectly with mathematical practice standard m p 2. But here's the real kicker. One of the biggest moments comes when students realize that maximizing area isn't about making one side of the rectangle super long?
Speaker 1:It's not. But that seems intuitive, doesn't it? The longer the side, the bigger the area.
Speaker 2:You'd think so. Right. But there's a catch. To keep that perimeter fixed, a super long length means a teeny tiny width, and that shrinks the overall area.
Speaker 1:I see. It's all about balance.
Speaker 2:Precisely. And in fact, the lesson plan even points out that with that 50 meter perimeter, a square, you know, where all sides are equal with sides of 12.5 meters, actually gives you the biggest possible area.
Speaker 1:A square. Who would have thought?
Speaker 2:It's one of those things that seems counterintuitive at first, but once you see it in action, it just clicks.
Speaker 1:So they've played with the numbers. They've gotten their hands virtually dirty with some garden design. What's the next step in this quadratic adventure?
Speaker 2:It's time to take those patterns and bring them to life visually speaking, of course. And how do we do that?
Speaker 1:I'm guessing it's time to break out the graph paper.
Speaker 2:You know it. In the plotting the measurements of the garden activity, students get to graph the relationship between the length of one side of the rectangle and its corresponding area. Remember those tables from the notice in wonder activity?
Speaker 1:Those they had as plain detective.
Speaker 2:Those are the ones now those numbers get to shine on the graph.
Speaker 1:And this is where that famous quadratic curve makes its grand appearance. Right?
Speaker 2:Get ready for the big reveal. As students plot their data points, what do they see?
Speaker 1:Don't tell me. Let me guess. It's not a straight line, is it?
Speaker 2:Nope. And it's not just any curve either. It's that unmistakable u shape. You know, it can be facing upwards or downwards, but it's always that classic quadratic curve.
Speaker 1:That's so cool. And it's a visual representation of how the area changes. Right?
Speaker 2:Exactly. As the length increases, the area initially goes up, hits its peak at that maximum point, and then what happens?
Speaker 1:And starts to shrink again because that width is getting squeezed.
Speaker 2:You got it. The lesson plan encourages teachers to nudge their students to really make that connection between what they're seeing on the graph and what they observed in that hands on garden activity. It's all coming together.
Speaker 1:It's all about connecting the dots from those hands on experiences to the more abstract world of graphs and equations. Yeah. But let's be real. Learning new math concepts always comes with its own set of challenges. True.
Speaker 1:True. What are some common misconceptions that teachers should watch out for when they're guiding their students through this whole quadratic exploration? What trips students up?
Speaker 2:Well, you know, one common one is all about squares. Squares.
Speaker 1:What about them?
Speaker 2:Some students might initially dismiss squares as a possible solution.
Speaker 1:Really? Why is that?
Speaker 2:Well, they might think, wait a minute. A square? That's not a rectangle.
Speaker 1:I see. They forget that a square is just a special type of rectangle, one where all sides are equal.
Speaker 2:Exactly. It fits the definition perfectly.
Speaker 1:So it's a good reminder for teachers to be on the lookout for that, to reinforce that squares are indeed rectangles. Are there any other common stumbling blocks? What else might trip students up?
Speaker 2:Well, another one has to do with, well, thinking inside the box literally. Students might limit themselves to only using whole numbers for the sides of their gardens.
Speaker 1:So they might say, okay. Aside like the 5 meters, that works. But 5.5 meters? No way. That's just crazy talk.
Speaker 2:Something like that. Yeah. They might overlook those decimals or fractions thinking those aren't allowed.
Speaker 1:Like, they're playing a video game where you can only move in full block increments. No smooth moves allowed. So how do we help them break free from that whole number thinking?
Speaker 2:Well, the lesson plan suggests encouraging students to think beyond those whole numbers, to explore those in between measurements. Teachers can ask questions like, what if the side could be 4.25 meters? What would happen to the area then?
Speaker 1:It's all about encouraging them to be more flexible, more open to those less conventional measurements.
Speaker 2:Exactly. After all, in the real world, things aren't always so neat and tidy. You can't always build a fence using only whole number lengths.
Speaker 1:Exactly. Sometimes you need a bit of a fraction to get the job done. Now I'm curious about one more thing. The material mentions that even when students spot that pattern of increase and decrease in the area, they might have trouble explaining why it's happening.
Speaker 2:Yes. They see the pattern but might not fully grasp the why.
Speaker 1:Right. So how can teachers help bridge that gap, connecting the dots between those numbers and that deeper understanding?
Speaker 2:That's where that visual representation, that graph really comes into play. Encourage students to really look at it and think about what's happening to the width of that rectangle as the length grows.
Speaker 1:As one side gets longer to keep that same perimeter, the other side has to shrink.
Speaker 2:Precisely. And eventually, that shrinking width starts to eat away at the overall area even though the length is increasing. The moment is when they see those two pieces fitting together that relationship between length, width, and how it all affects the area.
Speaker 1:It's about understanding that balance, that interplay between the different dimensions. But we've covered a lot of ground in this deep dive, haven't we?
Speaker 2:We really have. From spotting patterns in tables to designing gardens to understanding how it all connects to those quadratic curves on a graph.
Speaker 1:And what I appreciate about this lesson plan is that it doesn't shy away from those moments. It guides students towards them, helps them build that deeper understanding of quadratic relationships, and it emphasizes that this is just the beginning. Right?
Speaker 2:Absolutely. This lesson lays the groundwork for tackling even more complex algebra down the road.
Speaker 1:It's all connected.
Speaker 2:It really is.
Speaker 1:And a big thank you to the authors of Illustrative Math for creating such a well structured and engaging lesson plan. It's a fantastic resource for teachers.
Speaker 2:I couldn't agree more.
Speaker 1:Well, that's it for this deep dive into the world of maximizing gardens and quadratic relationships. We hope you've enjoyed the journey as much as we have.