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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 run into a math problem and feel like something's missing? Like, you just need that one tool to make it click.
Speaker 2:Oh, I know that feeling.
Speaker 1:That's where completing the square comes in. It's really elegant, and today, we're going deep on teaching it effectively, and we're getting insights from illustrative mathematics.
Speaker 2:Always a good source.
Speaker 1:They've got this lesson plan designed for teachers, so we're taking a shortcut to easing that lesson on completing the square.
Speaker 2:And we're here to help you go beyond just solving those equations.
Speaker 1:Exactly.
Speaker 2:We'll uncover the core concepts, what makes this technique so powerful, and how to guide your students past those tricky spots.
Speaker 1:Love it. So the lesson plan kicks off by highlighting the structure of perfect square expressions. What's the deal with those?
Speaker 2:Right. It's all about recognizing them
Speaker 1:Right.
Speaker 2:And learning how to manipulate them. But here's the thing. You have to maintain what we call mathematical balance.
Speaker 1:Mathematical balance. So in this context, that means
Speaker 2:It means whatever we add to one side of an equation to create that perfect square, we have to add to the other side as well.
Speaker 1:Makes sense.
Speaker 2:Like a balanced scale. Right? Yeah. Gotta keep both sides equal.
Speaker 1:Totally. And the lesson plan brings this idea of balance to life through the completing the square technique itself. I love how they use activities to walk students through it, like perfect or imperfect, a great way to get them thinking about the characteristics of those special squares.
Speaker 2:Absolutely. And then it moves into building perfect squares. Pushes students beyond just recognizing those squares to actually constructing them from scratch.
Speaker 1:From scratch. I like it.
Speaker 2:Yeah.
Speaker 1:They even have a table with examples. Help students connect the different forms of perfect squares, and I really appreciate that they include those rectangular diagrams. Visual learners, this is for you.
Speaker 2:Oh, those visuals are key. Seeing the square literally being completed, that can really solidify the concept.
Speaker 1:It's like a light bulb moment. And speaking of moments, the lesson plan highlights this key insight. The direct relationship between the coefficient of the linear term and the constant term in a perfect square.
Speaker 2:Right. The source points out that the constant term in a perfect square is always half the linear term coefficient squared. Huge. Game changer for students when they're starting to see the patterns in these expressions.
Speaker 1:It's one of those things that seems so obvious in hindsight. But having it pointed out, especially with visuals, can make a huge difference for students who are just wrapping their heads
Speaker 2:around it. Exactly. And as teachers, recognizing those moments is so important. It lets us see where students might need a little extra help or a different approach to get it.
Speaker 1:Love it. So let's talk about potential hurdles. What are some common misconceptions students might have about completing the square? The lesson plan mentions a few.
Speaker 2:It does. One big one is students assuming the constant term in a perfect square has to be a perfect square itself.
Speaker 1:Oh, I see. So they see a constant term like 7 and think, nope. Can't be part of a perfect square.
Speaker 2:Exactly. And to address this, the lesson plan suggests those rectangular diagrams. Visually breaking it down helps students see why that assumption isn't always right.
Speaker 1:Right. It's about that relationship between the terms that matters, not just the individual values.
Speaker 2:Right. Right. Another tricky part is students mixing up the steps when completing the square. Forgetting to add that squared term to both sides of the equation throws off the balance completely.
Speaker 1:There it is again. That concept of mathematical balance, so crucial.
Speaker 2:Totally. Encouraging students to write out every step clearly, even if it seems a bit much, can be super helpful.
Speaker 1:It's like those dance steps. Yeah. Now practice to get it right.
Speaker 2:Right.
Speaker 1:But speaking of practice, why is this technique so important? What makes completing the square so essential for students to learn?
Speaker 2:Well, it gets them a really versatile tool.
Speaker 1:Okay.
Speaker 2:Like, the lesson plan points out completing the square is super helpful with quadratic equations that you can't factor easily.
Speaker 1:So those times you just can't find those nice, neat factors, completing the square is there for you.
Speaker 2:Exactly. It may not always be the quickest way, but it's consistent.
Speaker 1:Like having that Swiss army knife.
Speaker 2:Right.
Speaker 1:Always reliable, always there when you need it, but it's more than just a way to solve a problem. Right?
Speaker 2:I got it. It's about understanding that mathematical structure, those relationships. Yep. Completing the square makes students think flexibly and see how different concepts connect.
Speaker 1:It shows how math isn't about memorizing formulas.
Speaker 2:Right.
Speaker 1:It's patterns and logic making those connections, and this technique shows that off. Yeah. I know we've been focused on the practical stuff, but the lesson plan talks about some interesting nuances too.
Speaker 2:It does. It points out that while completing the square is super useful, it's not always the most efficient way to solve every single quadratic equation.
Speaker 1:Right. Right.
Speaker 2:Sometimes factoring or other techniques might be faster.
Speaker 1:So it's about knowing the right tool for the job.
Speaker 2:Exactly. That kind of thinking is so important as they move on to more complex math.
Speaker 1:So it's about giving students those decision making skills for problem solving. And I don't know. Thinking about this technique more broadly, it makes you wonder, could this idea of completing something to solve a problem be useful in other math areas?
Speaker 2:Interesting. It's like a broader problem solving thing.
Speaker 1:You know? Yeah. Like, we're finding something that could go way beyond quadratics. It's like planting a seed. Right?
Speaker 1:Could grow in ways you wouldn't expect.
Speaker 2:I like that.
Speaker 1:Maybe there are geometric proofs or even stuff in calculus where this completing the thinking might be useful, something to think about.
Speaker 2:For sure. And that's what's so cool about really digging into these math ideas. We start with one technique, and it opens up all these possibilities and connections.
Speaker 1:It reminds us that teaching math isn't just about, you know, formulas and steps. It's about that curiosity, exploring, helping students think like mathematicians.
Speaker 2:Well said. And this lesson plan from Illustrative Mathematics gives us a great framework for doing that.
Speaker 1:It does. So many good insights, strategies, and those moments. It can really make a difference in the classroom. Yeah. So teachers out there getting ready to teach, completing the square, remember to hit those key concepts, watch out for those tricky spots, and most importantly, have fun with it.
Speaker 2:Definitely. Have fun.
Speaker 1:Big thanks to illustrative mathematics for this awesome lesson plan and to you, our listeners, for diving deep with us into completing the square. Catch you next time.