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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 completing the square. Let's be real. It can be a little intimidating to teach. Right?
Speaker 2:Yeah. It definitely has that reputation for teachers and students.
Speaker 1:Totally. Like, you see it on the lesson plan and think, oh, boy. Here we go again.
Speaker 2:Right.
Speaker 1:So that's what we're diving deep into this time. We've got the illustrative math lesson plan here ready to break it down.
Speaker 2:We're gonna make sure that by the end of this, you don't just feel like you can teach completing the square.
Speaker 1:But that you actually feel good about it?
Speaker 2:Exactly. Confident, prepared,
Speaker 1:all of that. Love it. Okay. So let's start big picture. What are the goals of this lesson, mathematically speaking?
Speaker 2:So it's really about moving students beyond just memorizing steps.
Speaker 1:Okay. I like that because it's easy to get stuck there.
Speaker 2:It is. But the goal is for them to really get the why behind completing the square, to generalize it no matter what crazy quadratic equation you throw at them.
Speaker 1:Right. Because those equations can get pretty messy.
Speaker 2:Oh, absolutely. And then, of course, the other big goal is being able to actually solve those equations, especially the ones where the coefficients of the squared term isn't 1.
Speaker 1:Yeah. Those are the ones that always seem to trip students up.
Speaker 2:For sure. And that's why this lesson is so good. It really focuses on the relationship between the coefficients in the standard form and the factored form.
Speaker 1:Okay. I'm intrigued. How does it actually go about doing that?
Speaker 2:Well, it starts with this activity called perfect squares in two forms. Okay. And it uses a common student error as a starting point, which I love.
Speaker 1:Oh, I'm all about turning mistakes into learning moments. So what's the error?
Speaker 2:It's this hypothetical student, Elena, and she's trying to expand the expression, 2x +1 squared.
Speaker 1:Okay.
Speaker 2:But instead of thinking about it step by step, she just distributes the exponent.
Speaker 1:Oh, yes. The classic. Gets them every time.
Speaker 2:Right. So, of course, she ends up with 4x squared plus 1 totally missing that middle term.
Speaker 1:But I bet this lesson doesn't just say, nope wrong and move on, does it?
Speaker 2:You know it. This is where it gets good. Instead of just saying Elena's wrong, the activity has students analyze her thinking. They figure out where she went wrong and why.
Speaker 1:And that why is crucial.
Speaker 2:Absolutely. That's where those visual representations come in. In.
Speaker 1:Diagrams maybe? Always a good visual.
Speaker 2:Exactly. By actually drawing out the expansion, students get to see how that middle term emerges and why it matters for a perfect square trinomial.
Speaker 1:It takes something that could be really abstract and makes it concrete.
Speaker 2:Exactly. And once they've had that moment with Elena's mistake, they move on to activity 2, which is called perfect in a different way.
Speaker 1:Okay. A little different this time.
Speaker 2:A little bit. Instead of expanding to get the standard form, now they're starting with expressions in factored form and going backward to rewrite them in standard form.
Speaker 1:So it's all about reinforcing that connection between the two forms, showing them how it works from different directions. And that's where it gets really cool. Right? Because now we're letting students kind of forge their own path a little bit.
Speaker 2:Exactly. And this is one of the things I love about this activity is that it doesn't force students to use just one method for rewriting those expressions.
Speaker 1:They have options.
Speaker 2:They do. Some of them might really like that visual approach, you know, like using those rectangular diagrams.
Speaker 1:Right. To represent the multiplication.
Speaker 2:Yeah. While other students might be more comfortable with the algebra using the distributive property.
Speaker 1:So it's like all roads lead to the same understanding.
Speaker 2:Exactly. It's about making those connections between the factored form and the standard form. And as they're working through this, they start to notice patterns in the coefficients.
Speaker 1:Which is huge. Yeah. Recognizing those patterns is, like, half the battle sometimes. You know?
Speaker 2:It makes all the difference. Yeah. Like, they might realize that the constant term in that standard form, it's always the square of half the coefficient of the x term from the factored form.
Speaker 1:Oh, yeah. That's a good one.
Speaker 2:Right. And that leads us perfectly into activity 3, which is called when all the stars align. This one is fun. They get to be like math detectives.
Speaker 1:Oh, I like that. Okay. So what's the mystery? What we investigating?
Speaker 2:Okay. So this time, they're given expressions in standard form, and they have to figure out what the constant term would need to be to make it a perfect square trinomial.
Speaker 1:So they're kinda working backward in a way.
Speaker 2:Yes. Exactly. They're completing the square, but they're really focusing on that process of figuring out what's missing.
Speaker 1:And that's where that moment happens. Right?
Speaker 2:Yes. When they realize they can use that pattern we were just talking about from the last activity to solve this one.
Speaker 1:That connection between the coefficient of the x term and the constant.
Speaker 2:Exactly. And that is what makes all the difference when those coefficients get more complicated, especially when you're dealing with a coefficient of x squared that isn't a perfect square.
Speaker 1:Right. Because that's when it gets really, really tricky.
Speaker 2:It does. And that's where that optional activity 4 comes in, putting stars into alignment. It's like giving them a chance to try out different tools in their mathematical toolbox.
Speaker 1:I like that.
Speaker 2:Right. They get to experiment with factoring, substitution, completing the square, all on the same equation.
Speaker 1:So they can see what works best in different situations.
Speaker 2:Exactly. Because sometimes factoring might be the easiest way, but other times it might be way too messy. And this activity highlights why the quadratic formula is so powerful.
Speaker 1:Because it always works.
Speaker 2:Always. No matter how crazy the coefficients are.
Speaker 1:Love it. Okay. So we've talked about the moments, but let's be real, there are also those moments where students are gonna get stuck. What are some common things that trip students up with completing the square?
Speaker 2:Yeah. So one of the big ones is just understanding how the standard form and the factored form relate to each other, especially when that leading coefficient isn't 1.
Speaker 1:It's like trying to see how 2 different puzzles actually fit together to make one bigger picture.
Speaker 2:That's a great way to put it. And just like with those puzzles, sometimes you need to break it down a bit.
Speaker 1:Yeah. Sometimes you need to see the pieces before you can see how they fit together.
Speaker 2:Right. And this is where revisiting those earlier activities can be really helpful, like, reminding students how they moved back and forth between those two forms.
Speaker 1:Give them that foundation.
Speaker 2:Exactly. And if they're still struggling, there's this scaffolding strategy in the lesson plan that I really like. Oh,
Speaker 1:tell me more about it.
Speaker 2:Okay. So let's say you're working with 2x +3 squared. Instead of trying to expand it with that coefficient of 2 right away, you can bring in another variable.
Speaker 1:Like a placeholder.
Speaker 2:Yeah. Exactly. Like, you could use a and rewrite it as a plus 3 squared where a is just representing 2 x for now.
Speaker 1:So it's kinda like we're simplifying it temporarily.
Speaker 2:Right. And then once they've expanded a +3 squared, they can substitute the substitute the 2x back in for a. Makes it less overwhelming.
Speaker 1:Exactly. It breaks it down into more manageable steps.
Speaker 2:I love that. Okay. So other than that,
Speaker 1:what else do students tend to get tripped up on? Another big one is that constant term when they're completing
Speaker 2:the square, especially, you guessed it, when the coefficient of x squared isn't a perfect square.
Speaker 1:Yeah. Because then it's not as simple as just, like, taking half of the middle term and squaring it.
Speaker 2:Exactly. It's easy for students to lose track of the y if they're just trying to follow a formula. So,
Speaker 1:again, it goes back to that understanding those connections. Right?
Speaker 2:That relationship between the coefficient of the x term and the constant term.
Speaker 1:Not just memorizing.
Speaker 2:Right. And I think that's what makes this lesson plan so great. It really does emphasize those connections, those moments over just memorizing steps.
Speaker 1:Absolutely. It's about making completing the square less of a chore and more of, like, a puzzle to figure out.
Speaker 2:Exactly.
Speaker 1:Awesome. Well, I think this has been a really insightful deep dive. Any final thoughts for our listeners?
Speaker 2:I think the biggest takeaway is just that completing the square doesn't have to be the scary, dreaded topic. With a little planning and by really focusing on helping students understand the why, it can actually be a really engaging lesson.
Speaker 1:I love that. Empowering both the teachers and the students. And a huge thank you to the authors of Illustrative Math for this fantastic lesson plan. For more deep dives into all sorts of fascinating topics, don't forget to subscribe to our show and check out our archives. Until next time.