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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. So we're diving into quadratic equations today, and specifically, a lesson plan about solving them by reasoning.
Speaker 2:Sounds kinda tricky. Most students I've had just wanna memorize the quadratic formula.
Speaker 1:Well and that's understandable to a point. Right? I mean, it's a powerful tool. Yeah. But this lesson plan seems to get at something even more fundamental.
Speaker 1:It's like before you even think about formulas, how can you actually reason through these problems?
Speaker 2:Yeah. And, honestly, that's something I really appreciate about this approach. It tackles that misconception head on. You know, the one where students think equations only ever have one solution.
Speaker 1:Oh, I know exactly what you mean. It's like that saying, the more you learn, the more you realize you don't know.
Speaker 2:Mhmm.
Speaker 1:They think they've got equations figured out, but quadratics can throw a real curve ball.
Speaker 2:For sure. And the lesson starts with a really simple example. X set sorry equals 4.
Speaker 1:Okay. Yeah. I see where they're going with this. Most students, they'll blurt out x equals 2 Yeah. And think they're done.
Speaker 2:Exactly. And that's the perfect opportunity to say, hold on. What about negative 2? Because, of course, negative 2 squared is also 4. It's a really elegant way to show them that quadratics can have 2 solutions.
Speaker 1:It's such a simple illustration, but it really makes you stop and think. And it seems like this lesson is all about encouraging that kind of thinking. Right? It's not just about finding the answer, but understanding the why behind it.
Speaker 2:Absolutely. And that's what I love about this approach. It emphasizes reasoning over rote memorization. Instead of just plugging numbers into a formula, it encourages those moments where students actually see the relationships and patterns for themselves.
Speaker 1:Yeah. And you can see that in the activities they've outlined. Like, there's one where students have to figure out what number, when squared, and then multiply by 3 equals 432. It's almost like a riddle.
Speaker 2:It is. And by working through it, students start to develop that intuitive understanding of squares and square roots. They're not just memorizing a formula. They're building a foundation for solving more complex problems down the road.
Speaker 1:It's like giving them the tools to think like mathematicians, to approach problems strategically and logically, and that's so much more valuable than just memorizing a bunch of steps.
Speaker 2:And that's where this lesson plan really shines. You know, it goes beyond just the basic concepts and actually digs into those common misconceptions. It gives you strategies to address them head on.
Speaker 1:Which so important because those misconceptions, they can really trip students up. It's like they're hitting those invisible potholes in the road.
Speaker 2:Exactly. And one of the big ones they address is that mistake with parentheses and exponents. You know, like, when students see 2 x weight and they just wanna simplify it to 2 x weight.
Speaker 1:Oh, yeah. I've seen that a 1000000 times.
Speaker 2:Yeah.
Speaker 1:And it's kind of understandable. Right? It feels like it should be that simple.
Speaker 2:It does. And that's why it's so important to slow down and really break it down. The lesson plan suggests having students actually write it out, 2 x, 2 x. That visual can be a game changer.
Speaker 1:It's those little details that can make such a big difference. When they see it written out like that, it clicks. Oh, it's 2 times x times 2 times x, which is 4 by set, not 2 x.
Speaker 2:Exactly. And another tricky one that they tackle is that idea that squaring is always reversible.
Speaker 1:Okay. I think I see where you're going with this, but can you give an example?
Speaker 2:Sure. Let's say you square negative 3. You get buying. Right? But then when you take the square root of 9, what's the first thing that pops into your head?
Speaker 2:It's usually 3. Right?
Speaker 1:Yeah. It's like our brains naturally go to the positive root.
Speaker 2:Exactly. And we kind of forget about that original negative 3. But when you're solving quadratic equations, you need both solutions, both the positive and the negative.
Speaker 1:That makes a lot of sense. It's like that missing piece of the puzzle that could really throw them off.
Speaker 2:Right. And the lesson plan encourages you to be really explicit about that. It's not just about finding one answer. It's about understanding all the possible solutions.
Speaker 1:So it's giving them a more complete picture of how quadratic equations actually work.
Speaker 2:And a more solid foundation for tackling those more challenging problems where those misconceptions can really cause trouble, like that one we were looking at earlier. The x plus one wefterex equals 144.
Speaker 1:Right. Where it's not just a simple x plus term anymore. You've got that x plus one in there Yeah. Which can make things a bit trickier.
Speaker 2:And this is where that reasoning approach gets really interesting. I think most students, their instinct is to just expand that whole thing out. You know, multiply x plus 1 by itself.
Speaker 1:Yeah. I could see that. It's like the first thing we learned to do with these kinds of expressions. Right?
Speaker 2:Exactly. But this lesson encourages you to take a step back and think about it a little differently. Instead of immediately jumping into all that multiplication, what if you ask your students, okay. So what squared equals 144?
Speaker 1:I see where you're going with this. It's about making that connection. If x plus one equals 144, then x plus one itself has to be either 12 or negative 12.
Speaker 2:Exactly. It's all about guiding them to see those relationships, and then they can solve for x pretty easily from there. If x plus 1 is 12, then x is 11. And if x plus 1 is negative 12, then x is negative 13. Boom.
Speaker 2:Two solutions, and we didn't even have to use the quadratic formula.
Speaker 1:It's so elegant, and it really drives home that concept of 2 solutions, especially in this case where they're not just opposites of each other, which I think is really valuable.
Speaker 2:Absolutely. It shows them that quadratic equations can have these more nuanced solutions and that this reasoning approach, it can help them find those solutions in a really intuitive way.
Speaker 1:It's like they're building those problem solving muscles, learning to think flexibly and strategically.
Speaker 2:Exactly. And that's gonna serve them well as they move on to more and more complex math. Because this method, this solving by reasoning, it's a great foundation, but it's not always the most efficient way to solve every quadratic equation.
Speaker 1:Right. Like, what if the numbers aren't so neat and tidy? What if you can't just easily figure out what squared gives you that constant term?
Speaker 2:Exactly. And that's where those other tools come in, like factoring and the quadratic formula.
Speaker 1:So it's almost like this lesson is showing them the why behind those tools.
Speaker 2:Mhmm.
Speaker 1:Like, you could build a house with just hand tools, but power tools exist for a reason.
Speaker 2:That's a great analogy. And this lesson, it helps students see that connection. It's about understanding the underlying concepts so they can choose the right tool for the job and solve any quadratic equation that comes their way.
Speaker 1:Well, this has been fantastic. We've really unpacked so much about this approach to teaching quadratic equations. I love how it emphasizes reasoning, tackles those common misconceptions head on, and really empowers students to understand the why behind the what.
Speaker 2:Couldn't have said it better myself. It's all about building that deeper understanding.
Speaker 1:Absolutely. And giving them the tools to become confident, adaptable problem solvers. So to the authors of illustrative math, thank you for these fantastic resources. This is a really insightful approach to teaching a topic that can often be a stumbling block for students. And to all our listeners out there, as you dive into quadratic equations with your students, remember it's not just about memorizing formulas, it's about nurturing that mathematical curiosity.
Speaker 1:Until next time, happy teaching everyone.