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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 find yourself wondering if there's more to those algebra classes than meets the
Speaker 2:eye? Mhmm.
Speaker 3:Today's deep dive is for you. Okay. We're cracking open a lesson plan on factoring quadratics, but trust me
Speaker 2:Okay.
Speaker 3:This isn't your average trip back to algebra 101.
Speaker 2:Yeah. What's fascinating about this topic is that it's full of these moments
Speaker 3:Right.
Speaker 2:Where seemingly simple concepts actually unlock a deeper understanding of mathematical structure.
Speaker 3:I'm intrigued. Yeah. What kind of source material are we talking about here?
Speaker 2:We're diving into excerpts from an actual 8th grade algebra lesson plan from illustrative math.
Speaker 1:Okay.
Speaker 2:And get this, they start with something called math talk.
Speaker 3:Okay.
Speaker 2:Using these deceptively simple multiplication problems like 98 times a102.
Speaker 3:Okay. On the surface, that sounds Right. Pretty standard.
Speaker 2:Yeah.
Speaker 3:Where does the deep dive come in?
Speaker 2:That's where the lesson plan really shines.
Speaker 3:Okay.
Speaker 2:It uses these simple problems to get students and us thinking about the distributive property.
Speaker 1:Right.
Speaker 2:Remember that one? It lets us multiply a sum by a number by distributing it across each term.
Speaker 3:Okay. I vaguely remember that from my school days. Right. I'm starting to see where this is going.
Speaker 2:Exactly. So instead of just crunching 98 times a 102 the old fashioned way Right. You could think of it as a 102, a 100+2.
Speaker 3:K.
Speaker 2:This is where we see that difference of squares pattern emerging
Speaker 3:Right.
Speaker 2:Which is key to understanding why certain factoring techniques work.
Speaker 3:Hold on. So you're saying that this simple multiplication problem is secretly a quadratic expression in disguise?
Speaker 1:Precisely.
Speaker 3:Wow.
Speaker 2:And by recognizing this pattern, we can actually simplify the problem and solve it in our heads.
Speaker 3:Right.
Speaker 2:It's like a mental math shortcut that's rooted in some pretty sophisticated algebra.
Speaker 3:That's pretty cool.
Speaker 2:Right.
Speaker 3:It reminds me of those times I'm able to do quick calculations in my head and surprise myself.
Speaker 2:Yeah.
Speaker 3:Now I'm realizing there's probably some serious math underpinning those moments.
Speaker 2:Absolutely. And this lesson plan goes even further. Using this concept as a springboard to explore why multiplying a sum and a difference Mhmm. Like in our example
Speaker 3:Right.
Speaker 2:Always results in a quadratic expression without a linear term. Yeah. And there's no x term. Right. Just an x squared.
Speaker 3:I'm starting to see how this lesson goes beyond just memorizing formulas.
Speaker 2:Right.
Speaker 3:It's about understanding the structure of these expressions.
Speaker 2:And to really drive that point home
Speaker 1:Okay.
Speaker 2:The lesson plan incorporates visual aids.
Speaker 1:Okay.
Speaker 2:Diagrams to make this whole cancellation process crystal clear.
Speaker 3:Diagrams.
Speaker 2:Yeah.
Speaker 3:Okay. Now that's what I call thinking outside the textbook. Yeah. I'm guessing this isn't just about doodling in math class. Right?
Speaker 2:Not at all. Okay. Imagine a rectangle divided into 4 smaller rectangles.
Speaker 3:Okay.
Speaker 2:This is how the lesson visually represents the expansion of factored expressions.
Speaker 3:Okay.
Speaker 2:For instance, x plus 3 and by 3 would be the lengths of your rectangle sides.
Speaker 3:Mhmm.
Speaker 2:Now by calculating the areas of those smaller rectangles in side Right. And then adding them up Okay. You can actually see how the linear term Oh, right. That x term Yeah. Disappears.
Speaker 3:That's incredible. It's like a light bulb moment suddenly connecting those abstract symbols to a concrete image.
Speaker 2:Right.
Speaker 3:I wish they taught me algebra this way.
Speaker 2:Yeah.
Speaker 3:But I have a feeling this whole factoring thing isn't always so neat and tidy. Right?
Speaker 2:You're right.
Speaker 3:Okay.
Speaker 2:The lesson makes sure to address a crucial point. Mhmm. Not all quadratic expressions can be factored so easily.
Speaker 3:Right.
Speaker 2:In fact, there's a common misconception that students often develop Yeah. Thinking any quadratic can be force fit into factored form.
Speaker 3:Yeah. That sounds familiar.
Speaker 2:Yeah.
Speaker 3:I remember trying to jam everything into those neat little parentheses.
Speaker 2:Exactly.
Speaker 3:Right?
Speaker 2:It's like trying to fit a square peg into a round hole. Sometimes it just doesn't work. Right. And that's okay.
Speaker 3:Yeah.
Speaker 2:Recognizing those situations is a key part of mathematical thinking.
Speaker 3:Right.
Speaker 2:This lesson does a great job of using concrete examples to illustrate this point.
Speaker 3:Okay. So give me an example. Okay. What kind of quadratic expression would make me pull my hair out trying to factor it?
Speaker 2:Let's go back to that x set 9 example.
Speaker 3:Okay.
Speaker 2:It fits the difference of squares pattern perfectly.
Speaker 3:Right.
Speaker 2:So it factors nicely into x+3by3. Right. But what about x sub+9? Okay. That's a whole different ballgame.
Speaker 3:So why is x sub+9 a no go for factoring?
Speaker 2:Well, if we were to try and factor that into 2 binomials like we did before
Speaker 3:Right.
Speaker 2:We'd need to find two numbers that multiply to positive 9
Speaker 3:Okay.
Speaker 2:But add up to 0. Right. Think about it. There's no combination of positive or negative numbers that can pull that off.
Speaker 3:You're right. Yeah. I see why that would be a deal breaker for factoring.
Speaker 2:Exactly.
Speaker 3:So, basically, we're not just learning how to factor. We're also learning when not to factor.
Speaker 2:Exactly.
Speaker 3:Uh-huh.
Speaker 2:And that's a big step toward a deeper understanding of mathematical structure.
Speaker 3:Right.
Speaker 2:You're not just memorizing rules. Right. You're starting to see the underlying why.
Speaker 3:I'm starting to see why this deep dive is so important.
Speaker 2:Yeah.
Speaker 3:It's shining a light on those moments that can make math truly click. But before we get too carried away, I do wanna touch on a point that's been lingering in my mind. Why should we actually care about factoring quadratics?
Speaker 2:A question close to my heart. While this specific lesson doesn't dive deeply into real world applications Mhmm. They're all around us.
Speaker 3:I have to admit, when I think of quadratics, I mostly just think back to my algebra textbook.
Speaker 2:I understand. Yeah. But believe it or not, quadratics pop up in all sorts of unexpected places.
Speaker 3:Okay.
Speaker 2:For example, imagine you're a civil engineer designing a bridge.
Speaker 3:Okay.
Speaker 2:The curve of the bridge's arch can often be modeled using a quadratic equation.
Speaker 3:Okay.
Speaker 2:Factoring that equation could help determine the optimal dimensions and materials needed for the bridge to be strong and stable.
Speaker 3:Woah. I had no idea my commute involved that much algebra.
Speaker 2:Right. Yeah. And it's not just engineering.
Speaker 3:Okay.
Speaker 2:Quadratics are also essential in physics computer graphics.
Speaker 3:Okay.
Speaker 2:Even financial models use them.
Speaker 3:Okay. Now my mind is really blown. Yeah. I never realized my bank account was secretly doing algebra.
Speaker 2:It is. And understanding the structure of these equations, how to manipulate them
Speaker 3:Right.
Speaker 2:It builds a level of mathematical intuition that goes way beyond just plugging numbers into a calculator.
Speaker 3:That's a great point.
Speaker 2:Yeah.
Speaker 3:It's like learning a new language. Once you understand the grammar, you can start to really express yourself.
Speaker 2:Precisely. Right? And, yeah, the lesson plan itself even hints at something even more mind blowing.
Speaker 3:Here's where it gets really interesting.
Speaker 2:Yeah.
Speaker 3:Lay it on me.
Speaker 2:Remember how we talked about x set 9 being factorable? Yeah. Well, the lesson mentions that even x set 5, where 5 isn't a perfect square, can be factored.
Speaker 3:Wait. What?
Speaker 2:Yeah.
Speaker 3:How is that even possible?
Speaker 2:Right.
Speaker 3:My algebra teacher definitely never taught me that.
Speaker 2:It requires venturing into the world of irrational numbers, something the lesson plan acknowledges as something to be explored later.
Speaker 3:Irrational numbers. Okay. Now I'm really intrigued.
Speaker 2:Right.
Speaker 3:It's like there's a whole secret level of factoring that we haven't even unlocked yet.
Speaker 2:Exactly. And it highlights a key takeaway from this deep dive. Math is full of surprises.
Speaker 3:Right.
Speaker 2:There's always more to learn, more connections to make.
Speaker 3:Right.
Speaker 2:And even familiar concepts can reveal hidden depths.
Speaker 3:It's like this whole deep dive has opened up a secret passageway in my brain Yeah. Leading to a whole new level of mathematical understanding.
Speaker 2:And to think Yeah. It all started with a simple multiplication problem.
Speaker 3:Exactly. It's amazing what we can uncover when we take the time to really dissect these concepts.
Speaker 2:Right.
Speaker 3:I have to say, this has given me a newfound appreciation for those who teach math.
Speaker 2:Right.
Speaker 3:It's clearly about so much more than just memorizing formulas.
Speaker 2:Absolutely. Right? This lesson plan is a perfect example of how effective teaching can make even the most abstract concepts engaging and accessible.
Speaker 3:A huge thanks to the authors of Illustrative Math for creating such an engaging and insightful lesson plan.
Speaker 2:Absolutely.
Speaker 3:For our listeners, you can find a link to the full lesson plan in our show notes Yeah. In case you wanna take your own deep dive. It's a valuable resource Okay.
Speaker 2:For educators and anyone looking to brush up on their algebra skills Right. Or maybe even conquer some old math phobias.
Speaker 3:And with that, we've reached the end of our deep dive into the vaccinating world of factoring quadratics.
Speaker 2:Yes.
Speaker 3:It turns out there's a lot more to it than meets the eye.
Speaker 2:A lot more. Right. Remember, the next time you encounter a seemingly simple equation
Speaker 3:Okay.
Speaker 2:Take a moment to appreciate the intricate dance of numbers and patterns at play.
Speaker 3:Yeah.
Speaker 2:You might just unlock a whole new level of understanding.
Speaker 3:Until next time. Keep those brains engaged and those moments coming.