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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 wonder who engineers design those cool parabolic shapes?
Speaker 2:It all starts with understanding quadratics.
Speaker 1:And today, we're taking a deep dive into a lesson plan that helps students master a pretty key skill.
Speaker 2:Factoring quadratic expressions.
Speaker 1:Specifically, those with a negative constant term.
Speaker 2:And it's not just about finding the right numbers. It's about helping students grasp the why behind the process. We'll unpack how this lesson empowers teachers to address common misconceptions head on.
Speaker 1:Sounds intriguing. So this lesson plan, it's called rewriting quadratic expressions in factored form, part 2.
Speaker 2:From illustrative math.
Speaker 1:And it aims to help teachers guide students to do what exactly?
Speaker 2:Well, first off, it helps them understand how to multiply a sum and a difference.
Speaker 1:Using a diagram, right, for visualization.
Speaker 2:Yep. Then it dives into taking a quadratic expression in standard form, you know, x o plus play x plus c.
Speaker 1:Worst size negative, of course. Exactly.
Speaker 2:And helps them write it in factored form.
Speaker 1:Okay. So it's about that back and forth between standard form and factored form. But just to make sure everyone's on the same page, can you quickly remind us why standard form is so important in the world of quadratics?
Speaker 2:Sure. Remember that standard form, that x plus b, x plus c structure gives us a clear picture of the coefficients and those coefficients. They're key to understanding the shape, the direction, even the location of the parabola the quadratic represents.
Speaker 1:Right. So without a solid grasp of standard form, students are kinda stuck at square 1.
Speaker 2:Exactly.
Speaker 1:Got it. So the lesson helps students recognize how those numbers and signs in both forms are connected.
Speaker 2:Exactly. And this lesson does a great job of highlighting how the distributive property or FOL, as it's often called, is essential for moving between those forms.
Speaker 1:Now about that whole OIL thing.
Speaker 2:Oh, you wanna do a refresher?
Speaker 1:Maybe just a quick one.
Speaker 2:FOL. Just Just a handy acronym to help us remember the order of operations when we're multiplying binomials. 1st, outer, inner, last.
Speaker 1:Right. F o I l. Okay. That's definitely ringing a bell now.
Speaker 2:It refers to the terms we multiply in a specific order.
Speaker 1:To make sure we've covered all the combinations.
Speaker 2:You got it. It's a lifesaver when you're expanding factored forms back into standard form.
Speaker 1:Okay. That makes sense. So we're connecting the dots between different forms of quadratic expressions, and this lesson gives teachers the tools to help their students do the same. But you mentioned that this lesson focuses specifically on quadratics with a negative constant term. Why is that such a big deal?
Speaker 1:Don't all quadratics behave the same way?
Speaker 2:That's where it gets really interesting. Quadratics with a negative constant term add a layer of complexity. See, they always factor into a sum and a difference.
Speaker 1:And that can be a bit of a head scratcher for students who are used to seeing factors with the same sign.
Speaker 2:Exactly. They're like, wait, the rules just changed. And that's why this lesson is so valuable. It tackles this challenging concept head on.
Speaker 1:This is reminding me of a time when
Speaker 2:You were about to share a story about teaching quadratics.
Speaker 1:Oh, right. This lesson just reminded me of when I first started teaching. I remember explaining negative constant terms, and it felt like I was speaking a foreign language. If only I'd had this lesson plan. Speaking of, what do those activities look like?
Speaker 1:How does this lesson play out in the classroom?
Speaker 2:Well, they've got a great sequence of activities, starting with a warm up on sums and products.
Speaker 1:A warm up. Like, to get those mental math gears turning?
Speaker 2:Exactly. But they don't use just any numbers. They use integers. And that's smart because it really lays the groundwork for understanding those opposite signs in factored form.
Speaker 1:That makes sense. It's like they're introducing the fundamental concept right from the start. So students are warming up with integers, getting comfortable with sums and products. What's next?
Speaker 2:Then comes the main course, the activity on, you guessed it, negative constant terms. Yeah. This is where things get really interesting. Students analyze tables of quadratic expressions.
Speaker 1:Looking for patterns.
Speaker 2:Yeah. And they discover how that negative constant term in standard form always leads to a sum and a difference in the factored form.
Speaker 1:It's all about those moments. Right? Seeing those patterns click into place.
Speaker 2:Precisely. This activity helps students go beyond just plugging numbers into a formula. It gets them to really understand the why behind it. It's like cracking a code.
Speaker 1:Cracking the code of quadratics. And to really drive those connections home, they have another activity focused on the factors of a 100 and Megas 100.
Speaker 2:Right. And choosing a 100 and I guess 100 is brilliant. These are familiar numbers that students can really understand.
Speaker 1:Those are numbers they use all the time, so it makes the concepts more concrete.
Speaker 2:Exactly. And as they work through this activity, they start to see how positive and negative factors impact that b value in the standard form.
Speaker 1:It's like you're highlighting that crucial relationship between the signs of the factors and the resulting sum.
Speaker 2:Exactly. Each activity builds on the one before it. But we all know even with the best lessons, there are always misconceptions that trip students up.
Speaker 1:Definitely. Like, what what are some big ones you've seen with factoring, especially with those negative constant terms?
Speaker 2:Oh, there are a few. One classic one is forgetting that adding a negative is the same as subtracting.
Speaker 1:I've seen that so many times.
Speaker 2:It seems simple, but it can really throw them off when they're trying to factor.
Speaker 1:Totally.
Speaker 2:And then there's mixing up when to add and when to multiply the factors.
Speaker 1:Yes.
Speaker 2:Students might forget that the constant term comes from multiplying the factors, but the coefficient of the linear term comes from adding them.
Speaker 1:Those little details can make all the difference. Speaking of which, what are your thoughts on using visual aids here? Things like algebra tile or area models.
Speaker 2:I'm a huge fan of visual aids. Yeah. Showing is always better than telling. Right?
Speaker 1:Absolutely. Visuals just make it real.
Speaker 2:Exactly. Instead of just telling students how factoring works, we can show them. Algebra tiles, area models, they give a concrete representation of those abstract concepts.
Speaker 1:It's one thing to move numbers around on paper, but when you can see those areas combine or those tiles fitting together, it just clicks. Have you found that certain visuals resonate better with students when it comes to factoring?
Speaker 2:You know, that's a great question. It probably depends on the students and their learning styles. Some students might like the hands on aspect of algebra tiles. Yeah. Others might find area models easier to understand, but the key is to use visuals strategically.
Speaker 1:Right. Don't just throw a visual up there just because it's gotta have a purpose. What are some ways that teachers can use these visuals effectively to really maximize their impact? So how can teachers use visuals effectively to maximize their impact?
Speaker 2:Right. It's about being intentional. Like, you could use algebra tiles to physically model the process of factoring. Students could start with representing the quadratic expression with the tiles.
Speaker 1:And then they could rearrange them, right, to actually see the factored form.
Speaker 2:Exactly. And with area models, teachers could draw a rectangle.
Speaker 1:And then divide it into sections to represent the terms. Right?
Speaker 2:Yep. That helps students visualize the distributive property in action.
Speaker 1:It's all about making those abstract concepts tangible.
Speaker 2:Precisely. Oh, and one final tip for teachers. Don't teach this lesson in isolation.
Speaker 1:What do you mean?
Speaker 2:Well, connect it to previous lessons, the ones where the constant term was positive.
Speaker 1:So students can see the bigger picture.
Speaker 2:Exactly. By connecting it back to what they already know, students will develop a deeper understanding of quadratics as a whole.
Speaker 1:It's like we're building a bridge from one concept to the next. Exactly. We've gone from those basic building blocks of integers and sums and products to confidently factoring quadratics with those tricky negative constant terms. That was quite the deep dive.
Speaker 2:It really was. We uncovered so many great insights.
Speaker 1:For me, the biggest takeaway is the importance of moving beyond memorization. This lesson plan is full of strategies that help teachers guide their students towards a deeper understanding of the why behind factoring.
Speaker 2:I totally agree. It's about helping teachers create those moments in their classrooms, those moments where the light bulb goes off. And, you know, it makes me think about other ways to make these abstract concepts more relatable to students. What about real world examples?
Speaker 1:That's a great question. Something for us all to think about. A big thanks to the authors of Illustrative Math for giving us so much to think about when it comes to teaching factoring. Until next time, keep those deep dives going.