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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.)
Okay, teachers. How about this? Let's dive into completing the square. And, yeah, I know. I know.
Speaker 1:Some of you out there, you might be having some, maybe some flashbacks to high school math class, but trust me on this one, this deep dive, way more fun than anything you remember from back then.
Speaker 2:We're gonna make absolutely sure that you are so ready to go, so equipped to tackle this whole lesson with your algebra 1 students, especially especially when it comes to those, you know, those tricky rational numbers.
Speaker 1:Exactly. We are diving into completing the square part 2. This is from the illustrative mathematics algebra 1 curriculum, and you're gonna be so relieved to hear we're not just dealing with those nice and neat integers here.
Speaker 2:Nope. Not this time. This lesson, it takes things a step further. It's challenging those students to really apply completing the square, but to quadratic equations with, you guessed it, rational numbers.
Speaker 1:Rational numbers. And for those of us who maybe need a little bit of a refresher, rational numbers, that just means any number that we can express is a fraction. So we're talking fractions, decimals, the whole thing. But why is this why is this step up in difficulty? Why is it so important for our students?
Speaker 2:You know, because, let's face it, the real world, it rarely gives us those picture perfect whole numbers. Think about it. You're calculating the arc of a baseball. You're designing a ramp, a curved ramp for a skateboard park. You're even trying to figure out how much fabric you need if you're gonna sew a circular pillow.
Speaker 2:You know? These situations, they don't usually involve those nice, neat integers.
Speaker 1:That's a really good point. I never even thought about it like that. So by kind of, like, introducing these messier rational numbers now, are we kinda, like, giving our students the tools to tackle more complex, more realistic problems down the road?
Speaker 2:Precisely. What this lesson is all about is building their comfort, their flexibility with numbers, and going beyond just those basics. And speaking of building a strong foundation, those illustrative mathematics folks, they start this lesson off with this really, really kind of brilliant warm up activity. It's called math talk.
Speaker 1:Right. And it seems, I don't know, maybe deceptively simple if you just kinda glance at it. Yeah. Can you tell us a bit more about why they chose this this type of warm up?
Speaker 2:Absolutely. So this math talk, it focuses on 2 key things. Number 1, getting those students fluent with fractions. And then also having them solve some of those simpler equations but doing it mentally. And now, this might seem like a far cry from the, you know, all the complexities of completing the square, but here's where it gets really, really clever.
Speaker 1:Okay. I'm very intrigued. Tell me more. Tell me more.
Speaker 2:So by activating that prior knowledge that they have, right, of fractions in mental math, the lesson is subtly, subtly making connections to these more advanced concepts that are gonna be coming up. Students, they might not even realize that those connections are being built, but trust me, they are. It's like laying the groundwork for, you know, when you're building a skyscraper, you might not see it in the finished product, but without it, the whole thing, it just crumbles.
Speaker 1:I love that analogy. It's so easy to just kind of rush through those warm ups. But you're right, they are laying the foundation for everything that comes after. It's like that saying, give a man a fish, and you feed him for a day. Teach a man to fish, and he'll probably still struggle with fractions unless he had a really good warm up.
Speaker 2:You might be on to something there. You're definitely right about the importance of a good warm up. So with that really strong foundation in place, the lesson then dives into the main event, solving some harder equations. And this this is where things get really, really interesting.
Speaker 1:And by interesting, you mean this is where we unleash all of those fractions we are talking about.
Speaker 2:Good.
Speaker 1:I'm looking at activity 13.2 right now, and some of these equations, they are not messing around.
Speaker 2:You're telling me. Oh. Take equation 2, for instance. We have 12xecuent zones plus 2 equals 514 x.
Speaker 1:Mhmm.
Speaker 2:And it is just a perfect example of the challenges this lesson throws at students.
Speaker 1:Yeah. That's definitely not as straightforward as some of those equations that we might have tackled in that first part of the lesson. What are some specific hurdles that students might encounter here?
Speaker 2:Well, I'll tell you. For starters, this equation, it isn't set up for the 0 product property, which a lot of students might be used to relying on. They'll have to rearrange the terms to get it into that standard form, x plus bx plus c equals 0. And that's before they can even think about completing the square. And then they have that you know, they've got that pesky 12 coefficient right on the x 4th term.
Speaker 2:And to complete the square to do it effectively, they'll need to get rid of that fraction, which means, you know, involve some careful multiplication of all the terms in the equation.
Speaker 1:Okay. So we've got rearranging terms. We're dealing with fractions. It's a this is a lot. It's like a mental juggling act.
Speaker 1:Any other curve balls you think we should warn our listeners about?
Speaker 2:Well, you know the lesson. It also throws in some decimals just to keep things interesting. Mhmm. It's all about, you know, making sure those students really, really understand that the process of completing the square, it's the same. No matter whether you're working those integers, the fractions, or those decimals, it's all about that deeper, that conceptual understanding.
Speaker 1:So it's really about emphasizing that flexibility, that adaptability. Now I know our listeners, the incredible teachers that they are, they are gonna wanna know, like, what are the best ways that they can support their students through all of this? What advice would you give?
Speaker 2:I think the key is to really break it down. Emphasize those fundamental steps. Remind them. Go back to the beginning. Put that equation in standard form.
Speaker 2:Then if they're dealing with a coefficient on that exocla term, you gotta kinda, like, guide them through that process of factoring out that coefficient. And when those fractions and decimals, when they start to cause trouble, encourage them to take their time. Show their work. Don't be afraid to double check those calculations.
Speaker 1:Little victories along the way.
Speaker 2:Exactly.
Speaker 1:Now I have to say, one of the things I love love about the illustrative mathematics curriculum is it goes beyond just getting the right answer. Right. It's all about developing critical thinking skills.
Speaker 2:Yeah.
Speaker 1:And this lesson has this really cool activity, activity 13.3, where students actually get to analyze errors, but, like, in these worked examples. Tell us a little bit more about the thinking behind that. Oh, I
Speaker 2:love it. It's brilliant. It really, really is. Because by analyzing these, you know, the flawed solutions, students, they get to play math detective, which not only reinforces their understanding of the steps involved, but also helps them to kinda, like, anticipate common pitfalls.
Speaker 1:That's so important. I remember, as a student myself, sometimes I would make the same mistake over and over and over again. And I feel like if I had just taken a minute to really analyze why I was making those mistakes, I probably could've saved myself a lot of headaches.
Speaker 2:Exactly. And that's what makes this activity so powerful. Mhmm. It's not just about spotting where someone went wrong Yeah. But understanding why that error happened in the first place.
Speaker 1:Yeah.
Speaker 2:And then, you know, what could have been done differently.
Speaker 1:So what kind of errors are we even talking about here? Are there any that you tend to see pop up really frequently when students are, you know, tackling completing the square?
Speaker 2:Oh, yeah. One very common error is forgetting to add that same value to both sides of the equation when you're completing the square. Yeah. It's so easy to get caught up in those calculations and forget that very, very fundamental rule of algebra. Whatever you do to one side, you have to do to the other.
Speaker 2:And when you're dealing with fractions and signs, it's even easier to make a slip up.
Speaker 1:Right. It's those little details.
Speaker 2:Exactly. Exactly. Another common mistake is incorrectly factoring that perfect square trinomial that you get you know, you create during that whole process of completing the square.
Speaker 1:I can see that one happening for sure. It's like you're so focused on getting to that perfect square trinomial that you rush through the factoring step, and you're not really thinking.
Speaker 2:Exactly. And then, of course of course, you have those computational errors that can happen when you're working with fractions and decimals. It's really easy to, you know, make a little mistake when you're adding, subtracting, multiplying, dividing, all of those.
Speaker 1:It sounds like a lot for teachers to be mindful of as they're guiding their students through this activity.
Speaker 2:It is. It is. But the great thing is that those illustrative mathematics materials, they provide a really helpful framework, a really helpful way of thinking about those student errors. They categorize them into 4 main types, careless errors, computational errors, you've got gaps in understanding, and then a lack of precision.
Speaker 1:Oh, that's so much more helpful than just, like, you know, right or wrong. Can you break down those those four error categories just a little bit more for us?
Speaker 2:Yeah. Absolutely. So careless errors, those are those little slip ups. Right? We all make them.
Speaker 2:I make them. Like, forgetting to copy a sign correctly, maybe misreading a number. Computational errors, those are mistakes that happen, you know, during those actual calculations like adding fractions incorrectly, messing up the decimal place, gaps in understanding. Well, those happen when, you know, a student maybe they don't fully grasp a concept and that leads to more fundamental errors in how they approach a problem.
Speaker 1:Mhmm.
Speaker 2:And then finally, that lack of precision, those are errors that come from maybe not being super careful with rounding, simplifying, or even just expressing their answer in the correct form.
Speaker 1:That framework is so valuable, though, because it helps teachers it lets them go beyond just identifying, okay, this is an error and actually delve into, like, the why, the why behind it. It's like you said. It's about turning those mistakes into learning opportunities.
Speaker 2:Absolutely. Yeah. When teachers can really pinpoint what's at the root of that error, then they can provide much more, you know, targeted support and instruction. Maybe the student maybe they just need a quick refresher on how do we do those fraction operations. Or maybe they need a little bit of help connecting those conceptual dots, like, between different parts of the problem solving process.
Speaker 2:I'm conceptual dots, like, between different parts of the problem solving process.
Speaker 1:Meeting those students right where they are and just giving them the tools they need. So as we're we're wrapping up this deep dive into completing the square, what are the the biggest takeaways? What do you want our listeners to really walk away with?
Speaker 2:1st and foremost, I really want them to feel confident. Completing the square, it's not a magic trick. Right? It doesn't just work on some equations and not others. You know, this is a powerful tool, and it can be applied to any quadratic equation, even those with those rational numbers.
Speaker 1:Like the Swiss army knife of quadratic equation solvers.
Speaker 2:Exactly. And to build on that, I hope teachers will really emphasize those connections, right, between the simpler cases of collating the square, those ones with the integers, and then these more complex examples where we bring in those rational numbers. That process, it's fundamentally the same. It's just, you know, those calculations, they might get a little bit, well, a little bit messier.
Speaker 1:It's all about building those bridges of understanding. Right? From the familiar to the new to the maybe a little bit intimidating at first.
Speaker 2:Precisely. And finally, and I can't I can't overemphasize this enough, fostering a growth mindset in that math classroom. Encourage your students to embrace those errors. See them as opportunities. When they analyze those mistakes, whether it's theirs or someone else's, they're really deepening their understanding in ways that they maybe they couldn't even imagine before.
Speaker 1:Becoming those mathematical detectives.
Speaker 2:Exactly.
Speaker 1:Now before we let our listeners get back to all that lesson planning, I have one last one last question for you. So the illustrative mathematics materials, they do mention at one point that completing the square, it's not always gonna be the most the most efficient way to solve a quadratic equation. What are your thoughts on that?
Speaker 2:Oh, that is such a good point. It's so important too. So while completing the square is really powerful, it can be a little bit, you know, maybe like using a sledgehammer to crack a nut, especially if that equation is really easily factorable. Or if those students, maybe they have some other tools they can use. You know, they've got their toolbox like the quadratic formula.
Speaker 1:So it's important for those teachers to really help their students develop that strategic thinking about, like, when to use what tool.
Speaker 2:Absolutely. Yeah. And as they're learning more, as they're progressing in that mathematical journey, they're gonna encounter more and more tools, more techniques. Right? And part of becoming a really confident capable mathematician is, you know what, knowing which tool is gonna be the best, the most efficient one for that particular job.
Speaker 1:It's about having the whole toolbox and being able to reach in and use the right one.
Speaker 2:I love that. Well, on that note, we'll have to save that quadratic formula deep dive for another day.
Speaker 1:I'm ready for it.
Speaker 2:But for now, a huge, huge thank you to the authors of Illustrative Mathematics because they provide such rich and engaging materials for us to work with. But an even bigger thank you to all of you out there, our amazing listeners, for joining us on this deep dive. Thank you for the amazing work you do every single day to empower your students. We will see you next time.