Subscribe
Share
Share
Embed
Lesson by lesson podcasts for teachers of Illustrative Mathematics®.
(Based on IM 9-12 Math™ by Illustrative Mathematics®, available at www.illustrativemathematics.org.)
Welcome to the deep dive where we really dig into those lesson plans on your desk. Today, we're gonna tackle something even tougher than, you know, trigonometry Tuesday.
Speaker 2:Oh, tell me about it.
Speaker 1:Quadratic equations. Yeah. Specifically, we're looking at illustrative mathematics algebra curriculum lesson 8, equivalent quadratic expressions.
Speaker 2:Now this is where things get really interesting. Right? I mean, we're not just, like, solving for x anymore. We're giving teachers tools to help students really understand what's going on behind those equations.
Speaker 1:Right. Because let's face it. Teaching quadratics the old school way, you know, just memorizing formulas, it rarely leads to those moments. Yeah. But this lesson this lesson does something different.
Speaker 1:It blends visual learning with that more, abstract algebra in a way that just clicks for a lot of students.
Speaker 2:It's like the difference between, you know, just looking at a map and actually getting out there and walking the terrain. Mhmm. This lesson lets students walk through the concepts using, get this, area diagrams. Remember those?
Speaker 1:Like those rectangle drawings we all did back in elementary school for multiplication.
Speaker 2:Exactly. They're back, but this time, they've got that quadratic power behind them.
Speaker 1:Okay. So let's unpack that a bit. The lesson plan assumes students already have some familiarity with quadratic functions. So how does this specific lesson build on that? Sometimes, I'll be honest, my curriculum planning feels a bit like I'm building a house of cards.
Speaker 2:Oh, you and every teacher I know. Right? But this this isn't a house of cards situation. The curriculum build upon what students have already learned in a really clever way. You see, in earlier lessons, they encountered quadratic functions in different forms, sometimes as products like, you know, x plus 2, x plus 3, and other times as sums like x x plus 5, x plus 6.
Speaker 1:And this lesson is about connecting those 2. Yeah. Bridging that gap so students see they're basically just different outfits for the same mathematical party guest.
Speaker 2:Exactly. Lesson 8 helps them see how those forms are equivalent, which is huge because it lays the groundwork for manipulating and really understanding those quadratic expressions later on, you know, in more complex scenarios.
Speaker 1:Yeah.
Speaker 2:And they'll need that foundation.
Speaker 1:It's all about making those connections, those mental bridges so the understanding really sticks. But speaking of things that stick, and let's be real here, the lesson plan kinda glosses over some hurdles students might trip over. Like, what about those area diagrams? Those can be tough even for teachers, especially when you start throwing in variables.
Speaker 2:Oh, absolutely. And the teacher's guide actually points this out. Decomposing a rectangular diagram into sub rectangles with size like, you know, x plus 2 or Yeah. X plus 5, That's not always intuitive for students.
Speaker 1:So how do we get them over that hurdle? We don't need them thinking math as just some kind of magic trick where the answer just appears.
Speaker 2:Well, here's a strategy. Start simple. Don't jump right into variables. Begin with purely numerical examples when you're first introducing that area model.
Speaker 1:Okay. So, like, if we're working with 12 times 7, we don't just calculate the product.
Speaker 2:Right. We represent it visually as the area of a rectangle.
Speaker 1:So instead of saying 12 times 7 equals 80 4, we're drawing it out.
Speaker 2:You got it. And then we break it down even further. So divide that 12 into 10 and 2 and the 7 into 5 and 2. And then you could visually divide that rectangle into smaller rectangles. One's gonna represent 10 times 5, another one, 10 times 2, and so on.
Speaker 1:I see where you're going with this. It's like they're literally seeing the distributive property in action.
Speaker 2:Exactly. They calculate those smaller areas, add them up, and bam. It's the same answer as 12 times 7, but they've actually visualized how that distributive property works. Then once they got it with numbers, that's when you sprinkle in the variables, but always connecting back to those visuals.
Speaker 1:It's like we're building those bridges, right, between the concrete and the abstract. And speaking of things that can feel a little abstract, the teacher's guide also points out another potential pitfall, those squared terms, like x plus 2 squared. Right. I can already see some students just squaring the x and the 2 separately and, you know, thinking they're done.
Speaker 2:Oh, yeah. And who can blame them for trying? I mean, it seems logical if you don't, you know, think about it too hard. But that's where those little misconceptions can really take root.
Speaker 1:So how do we make sure they don't go down that rabbit hole?
Speaker 2:It's all about laying that good foundation. Right? Yeah. Encourage them to slow down right out x plus 2 squared as x plus 2x plus 2 before they even think about expanding anything.
Speaker 1:It seems like such a small step, but Huge difference. Yeah. It's huge because it forces them to see that they're multiplying that entire expression by itself, not just each term individually. It's like it's like reminding them to actually read the recipe before they start, you know, tossing ingredients into the mixing bowl.
Speaker 2:Precisely. And you'll notice this lesson really emphasizes that distributive property.
Speaker 1:Mhmm.
Speaker 2:It's more than just a tool. It's, you know, the backbone of expanding those quadratic expressions.
Speaker 1:Yeah. And this isn't just a one and done situation either. We're giving them a skill that they're gonna use again and again and again throughout their math journey.
Speaker 2:Absolutely. This lesson helps them internalize that distributed property. You know, instead of just memorizing something by rote and connecting it with those visual area models. Yeah. That's what makes it come alive.
Speaker 2:Yeah. Helps them really grasp why it works.
Speaker 1:It's all about understanding the why behind the what, which speaking of makes me think about that practice problem in the teacher's guide, the one that asks students to explain why exponential expressions, you know, eventually surpass quadratic expressions and value. It's almost like that mathematical footrace where the tortoise eventually outruns the hare.
Speaker 2:Oh, that's such a good problem because it really pushes those students to think beyond just plotting points on a graph. They have to actually analyze the long term behavior of different types of functions and understand those inherent properties.
Speaker 1:It's a key distinction between quadratic and exponential growth that I've noticed trips students up a lot. They need to see that even if a quadratic expression seems to be, you know, growing really rapidly at first, that exponential expression is it's like that marathon runner hitting their stride. Slowly but surely, they're gaining ground. Plus, it connects back to their earlier work on exponential functions. Always a win.
Speaker 1:Speaking of making connections, I was really drawn to the are you ready for more section in this lesson.
Speaker 2:Oh, me too. It's like a breath of fresh air for those curious minds that wanna kind of, you know, go beyond the lesson's core concepts. That challenge about arranging the squares and rectangles into a larger rectangle. That's so smart. It's like they took the area model concept and turned it into a puzzle.
Speaker 1:It's fun. It reinforces that idea that math isn't just about, you know, memorizing formulas and equations. There's real creativity involved. There's exploration.
Speaker 2:Yeah.
Speaker 1:And it sets the stage for even deeper learning, which I love.
Speaker 2:It encourages that more open ended, playful approach to mathematical thinking. Mhmm. And speaking of open ended, did you see that question they leave the students with? The one about extending the area model to represent even more complex expressions.
Speaker 1:Oh, yeah. Now that's how you plant a seed for future exploration. It's like we're handing them a shovel and saying, go see what you can dig up.
Speaker 2:It challenges them to think about the limitations of the models they've learned so far Yeah. And sparks that realization that, you know, even in math, there's always more to learn, more to understand.
Speaker 1:It really encourages them to think like, you know, mathematicians always do always questioning, exploring, pushing those boundaries. But, before we get too carried away with all this mathematical excitement, I did wanna touch on those learning targets for this lesson. You know, the teacher's guide is very clear that students should be able to rewrite those quadratic expressions in different forms using, using both area diagrams and the distributive property.
Speaker 2:And that's really the heart of it, isn't it? Giving those students multiple tools for their for their math toolboxes. Because, yes, some students, they might be visual learners who really they really thrive with those area diagrams, while others might gravitate towards that more, you know, abstract approach of the distributed property. And this lesson, it validates both.
Speaker 1:And it goes beyond just presenting those tools in isolation. Right? It shows how they're connected. Like, that visual representation really reinforces the algebraic understanding
Speaker 2:Exactly.
Speaker 1:And vice versa. It's like they're seeing the same mathematical truth, but from different angles, which gives them that more well rounded understanding.
Speaker 2:And that's so powerful. Mhmm. Because when students can approach a concept from multiple perspectives like that, it really becomes a part of, you know, a part of their understanding. It's not just something they memorize for
Speaker 1:a test. Absolutely. You know, one of the biggest takeaways for me from this whole deep dive is how important it is to be really explicit in connecting those visual representations to those, to those algebraic expressions. We we can't just assume those connections are gonna magically click for every student.
Speaker 2:Oh, no. You gotta you gotta really build those bridges.
Speaker 1:Right. We have to be the bridge builders. We're the ones guiding them through it, pointing out the connections, highlighting that why behind the what.
Speaker 2:Absolutely. You
Speaker 1:know, it's a good reminder that our job as teachers, it's about more than just delivering content. It's about facilitating that real understanding, making those connections, and and honestly, helping our students see the elegance and the beauty that's inherent in mathematics even even in something like quadratic expressions.
Speaker 2:Couldn't agree more. And, you know, a huge thank you to the team at Illustrative Math for developing such a a well structured and insightful lesson.
Speaker 1:Absolutely. And to all our listeners out there, we hope this deep dive into those equivalent quadratic expressions has given you some fresh perspectives and some strategies to make teaching this lesson a really rewarding experience for you and for your students. Until next time, keep those mathematical fires burning bright.