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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.)
Alright, everyone. Get ready. Because today, we are taking a deep dive into quadratic functions.
Speaker 2:Quadratics.
Speaker 1:Yeah. You know, those equations that businesses use every day to, like, figure out how to make the most money possible.
Speaker 2:It's one of those math concepts that really shows how practical and relevant math can be even if it seems kinda abstract at first.
Speaker 1:Totally. And lucky for us, illustrative math already has a fantastic lesson plan on this very topic. So we're gonna dig into that today.
Speaker 2:Love their stuff. Stuff. Always so well thought out.
Speaker 1:Okay. So right off the bat, the lesson plan lays out 3 big learning goals they're aiming for. First up, they wanna make sure students really get how to pick the right domain for their quadratic function because in the real world, not every number on that x axis makes sense.
Speaker 2:Exactly. You can't have negative prices for a product. Right? Yeah. Or sell half a movie download.
Speaker 1:Yeah. Yeah. Yeah. I guess not. But that's what makes this so important.
Speaker 1:Right? Helping students see those limitations, like, how the math has to reflect reality. Okay. So that's goal number 1. Goal number 2 is all about those mysterious vertices and zeros and what they really tell us.
Speaker 2:Those are the points where the magic happens. The vertex, especially, it's like that sweet spot businesses are always searching for.
Speaker 1:Because it shows the price point that'll bring in the big bucks. Right?
Speaker 2:Yeah. Exactly. And then the zeros, well, they might seem less glamorous. But understanding those points where the revenue hits 0, that's crucial too.
Speaker 1:Yeah. Gotta avoid those financial pitfalls. Okay. So the 3rd big goal is where things get really interesting. This is where students level up and become quadratic modeling masters.
Speaker 2:It's about taking those real world situations like pricing strategies for a business and translating them into the language of quadratic functions, you know, equations and graphs and all that.
Speaker 1:Basically, giving students a powerful tool they can use to analyze and hopefully conquer the world one parabola at a time.
Speaker 2:I love it.
Speaker 1:To get things rolling, this lesson kicks off with an activity called which one doesn't belong. You know, like that game where you have to spot the odd one out. Clap. Yeah. So in this case, we've got 4 graphs.
Speaker 1:A constant function, a linear function, a discrete function, and, of course, the star of the show, our quadratic function.
Speaker 2:And students gotta put on their thinking caps and figure out why that quadratic function is the odd man out, which is a really clever way to get them focused on the unique features of Quadratix right from the get go.
Speaker 1:Exactly. It's like a warm up for their brains getting them ready to tackle the main event.
Speaker 2:And speaking of the main event, let's talk about the activity at the heart of this lesson. What price to charge? This is where students get to step into the shoes of a business and really see those quadratic functions in action.
Speaker 1:I love this part. It's all about finding that sweet spot, that perfect price point that brings in the most revenue.
Speaker 2:Exactly. So the scenario they're given is that a company is trying to figure out the best price to charge for downloadable movies. To help them out, they get a table that shows different prices per download and the predicted number of downloads they get at each price.
Speaker 1:But here's the thing. They don't just get the revenue handed to them. They've gotta calculate it themselves. Right? Price times number of downloads.
Speaker 2:Right. It's subtle, but it reinforces that revenue formula without making it all about memorization.
Speaker 1:It keeps them engaged. And then, of course, because we love a good visual, they get to plot all that data on a graph.
Speaker 2:And that's when it usually clicks for them when they see that beautiful parabola emerge.
Speaker 1:That moment.
Speaker 2:Exactly. But the activity doesn't stop there. They're challenged to actually prove that this relationship between price and revenue is in fact quadratic.
Speaker 1:So they have to flex those mathematical muscles a bit.
Speaker 2:They do. They can either manipulate equations or look for patterns in the revenue they calculate it, kind of like a little mathematical detective work.
Speaker 1:I love that. And then once they've cracked the code, they get to put on their CEO hats and recommend the best price to maximize those profits.
Speaker 2:It's a really cool way to bring the whole concept to life. But, you know, this lesson plan doesn't just stop at revenue. It goes even deeper with the next activity, domain, vertex, and zeros, which revisits some familial quadratic functions, but adds a new layer of analysis.
Speaker 1:Oh, I'm intrigued. So what kind of familiar functions are we talking about here?
Speaker 2:Well, for example, they look at the classic problem of finding the maximum area of a rectangle when you have a fixed perimeter.
Speaker 1:Okay. I remember that one. Definitely brings back some math class memories.
Speaker 2:Right. And they also analyze those visually appealing growth patterns, like the number of squares in a growing sequence.
Speaker 1:Oh, those are fun. It's amazing how many different things can be modeled with quadratic functions once you start looking for them.
Speaker 2:It's true. And let's not forget the classic example of a falling object. That one can also be represented with quadratic function. The lesson plan does a great job of showing just how versatile these functions can be.
Speaker 1:So they've got all these different scenarios, but then what's the common thread? What are they asked to do with each of these functions?
Speaker 2:That's a great question. So for each one, they have to answer 3 key questions. 1st, they need to figure out the appropriate domain, keeping in mind the context of the problem. Right.
Speaker 1:Because like we talked about earlier, not every number on that x axis is gonna make sense in the real world.
Speaker 2:Exactly. Then they have to interpret the vertex. What does it represent in this specific scenario? Is it the maximum area, the starting point of the falling object?
Speaker 1:So it's about connecting those abstract points on the graph to the actual meaning within the problem.
Speaker 2:Precisely. And finally, they have to tackle those zeros. What do they tell us in each context?
Speaker 1:So it's really about taking a 360 degree view of the quadratic function and understanding what it all means.
Speaker 2:Absolutely. It's like you're piecing together a puzzle, and each of these elements, the domain, the vertex, the zeros, they all contribute to the bigger picture.
Speaker 1:And speaking of puzzles, this lesson wraps up with a cool down activity that really puts their knowledge to the test.
Speaker 2:It's like a final exam for their quadratic skills. They're given a graph that represents a company's revenue based on the price of their product, and their task is to use everything they've learned to analyze that graph.
Speaker 1:Like, identify different price points, figure out where the maximum revenue is?
Speaker 2:Exactly. And, of course, determine an appropriate domain because we know that's important too. It's a really satisfying way to end the lesson, leaving them with a sense of accomplishment and a deeper appreciation for how these functions play out in real life.
Speaker 1:It's like they get to be financial consultants using their quadratic know how to help a business succeed.
Speaker 2:Exactly. But, of course, we know that even with the best lesson plans, students are bound to hit a few bumps in the road. That's why I appreciate how this one anticipates some common misconceptions and gives teachers strategies to address them head on.
Speaker 1:Looks like they've got our backs.
Speaker 2:Right. One common area where students tend to get tripped up is the difference between zeros and horizontal intercepts. You know those points where the graph crosses the x axis?
Speaker 1:Yeah. I've definitely seen that one. It's like they kinda use those terms interchangeably, but this lesson makes a point of highlighting that subtle but important distinction.
Speaker 2:It's all about precision and language. Totally. So the lesson emphasizes that zeros are those specific input values that make the output 0, while horizontal intercepts are those points where the graph visually crosses that x axis.
Speaker 1:Okay. So it's really about connecting those algebraic and graphical representations, making sure students understand both sides of the coin.
Speaker 2:Precisely. And then, of course, there's the whole challenge of domain limitations. Students might grasp the concept in theory, but when it comes to those real world scenarios
Speaker 1:Yeah. That's where things can get a little fuzzy.
Speaker 2:It's like they hit a wall trying to reconcile those abstract equations with the constraints of the real world.
Speaker 1:Right. Like, you can't have a negative number of products.
Speaker 2:Exactly. That's why the lesson emphasizes clear examples and really encourages teachers to prompt students to think critically about those limitations.
Speaker 1:It's all about asking those what is questions.
Speaker 2:Yes. Can time be negative? Can we have half a customer? Those kinds of things.
Speaker 1:Those are the questions that really get them thinking.
Speaker 2:Absolutely. And, you know, while we're on the topic of potential stumbling blocks, I think it's worth mentioning that sometimes elusive vertex
Speaker 1:Oh, yes. The invisible vertex. How can something that's supposed to be the peak of the parabola, like the star of the show, be invisible?
Speaker 2:Well, it all comes down to the domain again. Sometimes, depending on the context of the problem and those real world limitations we've been talking about, not all quadratic graphs will actually visually display the vertex.
Speaker 1:So it's like the vertex is there mathematically speaking, but it might be hiding off in a region of the graph that's not relevant to the actual scenario.
Speaker 2:Exactly. And that's a really important takeaway for teachers to emphasize to their students. Even if they can't see the vertex on the graph, it still exists. It still has meaning in the context of the problem.
Speaker 1:It's like a phantom vertex always lurking behind the scenes. Okay. So we've talked about some common misconceptions, but what are some key takeaways for teachers who want to make this lesson really sing for their students?
Speaker 2:Well, 1st and foremost, I'd say don't be afraid to really emphasize those visual representations. Graphing those parabolas is so essential.
Speaker 1:Because sometimes a picture really is worth a 1,000 equations.
Speaker 2:Exactly. Seeing that curve, that peak, it really helps drive home the point that we're looking for that maximum point, that sweet spot where revenue is maximum.
Speaker 1:Which makes it feel more real.
Speaker 2:It does. But beyond the graphs, I'd also encourage teachers to go beyond the textbook examples and get students thinking about other real world scenarios where maximizing revenue is key.
Speaker 1:Oh, I love that. Get them applying these concepts to their own lives.
Speaker 2:Exactly. Have them think about setting the price for their next killer app or figuring out how much to charge for those custom sneakers they design.
Speaker 1:It's amazing what a difference those kinds of personal connections can make.
Speaker 2:Absolutely. And finally, don't underestimate the power of discussion. Encourage students to collaborate, to bounce ideas off each other, to justify their reasoning.
Speaker 1:Because two heads are always better than 1 when it comes to conquering quadratic functions. Right?
Speaker 2:Exactly. Collaborative learning can make a world of difference. You know, this deep dive has really highlighted how powerful quadratic functions can be. They're not just some abstract mathematical concept. They're a tool that can be used to analyze and optimize so many different aspects of our world.
Speaker 1:It's like seeing the world through a different lens. And on that note, we'll wrap up this deep dive into the fascinating world of quadratic functions and revenue. A huge thank you to Illustrative Math for the inspiration for this episode. Until next time everyone. Happy teaching.