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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 feel like you're back in algebra class? Like, those x squared plus something, x plus something are staring you down again.
Speaker 2:Yeah. I know exactly what you mean.
Speaker 1:Today, we're tackling those quadratic expressions head on.
Speaker 2:Yeah.
Speaker 1:But don't worry. We're going deeper than just memorizing formulas.
Speaker 2:Right.
Speaker 1:We're gonna uncover the why behind the what
Speaker 2:I like it.
Speaker 1:So you can impress your friends with your newfound math knowledge.
Speaker 2:I'm all about that.
Speaker 1:We're focusing on standard and factored forms.
Speaker 2:Which are super fundamental, by the way.
Speaker 1:Oh, absolutely.
Speaker 2:Like, they're everywhere in math.
Speaker 1:You can
Speaker 2:see them in algebra, obviously. Right. But also graphing functions.
Speaker 1:Mhmm.
Speaker 2:Even physics pops up too.
Speaker 1:Wow. So it's like they're the secret code to the universe.
Speaker 2:You could see that.
Speaker 1:To guide our deep dive today, we've got our hands on an actual teacher's guide for a lesson on quadratic expressions. Oh, cool.
Speaker 2:So it's like we're peeking into the teacher's prep work here.
Speaker 1:Exactly.
Speaker 2:I was wondering what went on behind the scenes there.
Speaker 1:Right. So this lesson plan kicks off with a clear goal. Students need to master the language of standard form and factored form when it comes to quadratics.
Speaker 2:And when we say master, we don't just mean memorizing the definitions.
Speaker 1:Right. It's gotta be deeper than that.
Speaker 2:It's about understanding what those forms represent
Speaker 1:Yeah.
Speaker 2:And how they connect to the bigger picture of quadratic equations.
Speaker 1:So it's like the difference between recognizing a famous landmark and actually understanding its historical significance.
Speaker 2:Exactly. This lesson wants students to be able to explain why these forms matter.
Speaker 1:Oh, wow.
Speaker 2:How are they different? Almost like giving a TED talk on quadratic expressions.
Speaker 1:No pressure, students. Oh. But the lesson then jumps into some hands on activities
Speaker 2:Love that.
Speaker 1:To help build that deep understanding.
Speaker 2:Because they're not just abstract definitions.
Speaker 1:Right.
Speaker 2:Give students a chance to really visualize and manipulate them in a tangible way.
Speaker 1:Yeah. I like that.
Speaker 2:Make it real.
Speaker 1:The first activity is called math talk, opposites attract.
Speaker 2:Okay. Intriguing.
Speaker 1:But before we jump into that Yeah. Let's make sure we're all on the same page about what standard and factored forms actually look like.
Speaker 2:Good idea.
Speaker 1:So in standard form, a quadratic expression follows this structure, x plus a b x plus c.
Speaker 2:Right. Where a, b, and c are those constants
Speaker 1:Yes.
Speaker 2:And a can't be 0.
Speaker 1:Important detail.
Speaker 2:Very important.
Speaker 1:Otherwise, it wouldn't be quadratic.
Speaker 2:Exactly.
Speaker 1:So it's like the official uniform of a quadratic expression.
Speaker 2:I like that analogy.
Speaker 1:Now factored form Yeah. It's a little different.
Speaker 2:It represents the expression of a product of 2 binomials. Oh. Those expressions with 2 terms. Right.
Speaker 1:Like x plus 2.
Speaker 2:Exactly. Or 2 by 1, something like that.
Speaker 1:So instead of that x plus b x plus c structure, we might see something like x plus 3 by 1.
Speaker 2:Precisely.
Speaker 1:So they're both describing the same mathematical relationship.
Speaker 2:Just in different outfits.
Speaker 1:I like it. Different outfits for different occasions.
Speaker 2:Exactly. Each one has its own advantages depending on the situation.
Speaker 1:Okay. So now that we've clarified what those forms look like Yep. Let's unpack this opposites attract warm up activity. Any guesses what it might involve just from the name?
Speaker 2:Well, just taking a stab here. Yeah. It's gotta have something to do with the idea that subtracting a number is the same as adding its opposite.
Speaker 1:Oh, interesting. So, like, if we had the expression by 4 Yeah. We could rewrite that as x plus 1 of 4.
Speaker 2:Exactly. It seems like a small change Yeah. But it's huge for working with those negative numbers.
Speaker 1:Which we see all the time in
Speaker 2:quadratics. All the
Speaker 1:time. It's all about building those connections early on. Right? Yes. This warm up gets
Speaker 2:those
Speaker 1:foundational pieces in place. Nice.
Speaker 2:So students can tackle more complex stuff later on. So how
Speaker 1:does this opposite
Speaker 2:to track warm up actually play out in the
Speaker 1:classroom? Imagine the teacher throwing out a rapid fire the classroom?
Speaker 2:Imagine the teacher throwing out a rapid fire round of addition and subtraction problems.
Speaker 1:Okay.
Speaker 2:Both positive and negative numbers. Oh. The catch, students have to solve them mentally.
Speaker 1:No calculators allowed.
Speaker 2:No calculators.
Speaker 1:It's like a mental math boot camp.
Speaker 2:Exactly. Forces those students to confront how positive and negative numbers interact Yeah. Especially within addition and subtraction.
Speaker 1:Training their brains to just, like, instantly deal with quadratics.
Speaker 2:Yeah. It's like building those reflexes.
Speaker 1:Okay. So once they've got those mental math skills sharpened
Speaker 2:Right.
Speaker 1:The lesson shifts to activity 2, which is called finding products of differences.
Speaker 2:Okay. This is where things start to get really interesting.
Speaker 1:This is where we see factored form in action. Exactly. So students are gonna be working with expressions like by 2x+5. Okay. And they have to multiply those out.
Speaker 2:Right. To get that standard form.
Speaker 1:Exactly.
Speaker 2:And they use a couple of things here.
Speaker 1:Yeah.
Speaker 2:Visual representations Yeah. Which are always helpful.
Speaker 1:Always.
Speaker 2:And the distributive property.
Speaker 1:Ah, the distributive property.
Speaker 2:Yeah. Yeah.
Speaker 1:Can we talk about that for a second? Sure. Because I feel like that's something that trips people up.
Speaker 2:It can be a little tricky.
Speaker 1:Especially with these binomials.
Speaker 2:Yeah. So, basically, the distributive property tells us that Yeah. You're multiplying a sum by a number.
Speaker 1:Okay.
Speaker 2:You multiply each part of that sum separately.
Speaker 1:So, like, if we had 3 times 2+4 Exactly.
Speaker 2:You'd multiply the 3 by the 2 Okay. And then the 3 by the 4.
Speaker 1:Got it.
Speaker 2:So it becomes 3 by 2 +3 by 4.
Speaker 1:Which is 18.
Speaker 2:Exactly.
Speaker 1:Okay. I see how that works with numbers, but how does that apply to these binomials we're seeing?
Speaker 2:It's the same idea just with variables in the mix. Okay. So let let's say we have x plus 2 by 3.
Speaker 1:Right.
Speaker 2:We treat that first binomial, x plus 2, as our number.
Speaker 1:Okay. So we're distributing that whole thing
Speaker 2:Exactly.
Speaker 1:To each term in the second binomial.
Speaker 2:You got it.
Speaker 1:So first, we multiply x plus 2 by x.
Speaker 2:Yeah. Which gives us x plus 2.
Speaker 1:Right. And then x plus 2 by Madison 3 Exactly. Which is manifest 3x plus 2.
Speaker 2:Perfect.
Speaker 1:So now we have x x plus 2, 3x plus 2.
Speaker 2:Now we're gonna distribute again.
Speaker 1:Oh, right. Because we still have those parentheses.
Speaker 2:Parentheses agree.
Speaker 1:Okay. So for x x plus 2 Right. We get x x plus 2 x. Right. And then for metaf3x plus 2
Speaker 2:We get meta 3 by 6.
Speaker 1:So our expression is now x x plus 2 by 3 by 6.
Speaker 2:Looking good. And last step is to combine those, like, terms.
Speaker 1:Oh, right. We gotta simplify 8 x.
Speaker 2:So the final expression in standard form is x x sought by 6.
Speaker 1:Wow. The distributive property is like magic.
Speaker 2:It's a lifesaver with these kinds of problems.
Speaker 1:It really breaks it down step by step.
Speaker 2:And speaking of breaking things down
Speaker 1:Yes.
Speaker 2:The lesson plan actually encourages teachers to use those rectangular diagram
Speaker 1:Oh, yeah. I love those.
Speaker 2:To visualize the distributive property.
Speaker 1:Because it's not just about the numbers.
Speaker 2:Right. So I've seen the connections.
Speaker 1:Making it visual.
Speaker 2:Exactly.
Speaker 1:So you're connecting that multiplication of binomials to something like finding the area of a rectangle.
Speaker 2:Which students are already familiar with.
Speaker 1:Exactly. It's like they're building on what they already know.
Speaker 2:That's the key to good teaching right there.
Speaker 1:I love how this lesson plan emphasizes using multiple representations.
Speaker 2:Oh, absolutely. It's so important for students to have those different ways of accessing the information.
Speaker 1:Right. Because some students might connect better with the visual
Speaker 2:Exactly.
Speaker 1:While others might grasp it more algebraically. It's about meeting those different learning styles.
Speaker 2:Okay. So after students have practiced with the distributive property
Speaker 1:Yep.
Speaker 2:Activity 3 challenges them to identify and classify those quadratic expressions in their different forms.
Speaker 1:Alright. So it's like a spot the quadratic form game.
Speaker 2:Exactly. And this helps to really solidify their understanding of what makes each form unique.
Speaker 1:And here's the thing. Yeah. It's not just about identifying them in isolation.
Speaker 2:Okay. What do you mean?
Speaker 1:Well, the lesson plan encourages teachers to mix it up.
Speaker 2:Okay.
Speaker 1:So students might see expressions in standard form
Speaker 2:Right.
Speaker 1:Factored form
Speaker 2:Mhmm.
Speaker 1:And even some tricky ones that might throw them off.
Speaker 2:So like a quadratic that's missing a term
Speaker 1:Exactly.
Speaker 2:But is still technically in standard form.
Speaker 1:You got it.
Speaker 2:Sneaky.
Speaker 1:It's about pushing students to think critically about those underlying structures.
Speaker 2:It's like a mental workout for their quadratic muscles.
Speaker 1:Exactly. And speaking of potential pitfalls
Speaker 2:Oh, yeah.
Speaker 1:I'm curious about any common misconceptions Okay. Students might have about these forms. Yeah. Like, what kind of things trip them up?
Speaker 2:Well, one common one is thinking a quadratic expression in standard form has to have all three terms.
Speaker 1:Right. It has to be that x plus b x plus c.
Speaker 2:Yeah. They get stuck on that format.
Speaker 1:But we could have b or c be 0 Exactly. And it's still standard form.
Speaker 2:Totally.
Speaker 1:So something like 5 by 6 8.
Speaker 2:Yep.
Speaker 1:That's still standard form even though it doesn't have that x term or a constant term.
Speaker 2:It's like the minimalist version of a standard form quadratic.
Speaker 1:I like it.
Speaker 2:Another misconception Yeah. Is mixing up what standard form and factor form are good for.
Speaker 1:Oh, okay.
Speaker 2:Like, they think they can use them interchangeably for any problem.
Speaker 1:Right. Because they both represent the same thing
Speaker 2:They do.
Speaker 1:Just in different ways.
Speaker 2:But they have different strengths and weaknesses.
Speaker 1:Okay. I see what you mean.
Speaker 2:So standard form. Yeah. It's great for quickly seeing the y intercept of the parabola.
Speaker 1:Oh, right. Because that's just our c value.
Speaker 2:Exactly.
Speaker 1:Whereas factored 4
Speaker 2:Yes.
Speaker 1:That helps us find those x intercepts.
Speaker 2:Exactly. Where the parabola crosses that x axis.
Speaker 1:So it's like choosing the right tool for the job.
Speaker 2:That's a great way to put it.
Speaker 1:You wouldn't use a hammer to tighten a screw Exactly. Unless you were feeling very adventurous that
Speaker 2:day. Or frustrated.
Speaker 1:Okay. So circling back to our deep dive here.
Speaker 2:Yes.
Speaker 1:It's more than just memorizing those rules.
Speaker 2:It's about understanding.
Speaker 1:Right. It's about giving students the tools to think flexibly about those mathematical relationships.
Speaker 2:To approach problems from different angles.
Speaker 1:And to choose the most efficient strategy.
Speaker 2:Exactly. It's like we're equipping them with a mathematical toolbox.
Speaker 1:I love that analogy.
Speaker 2:And this lesson plan. Yeah. It's a great example of how to do that effectively.
Speaker 1:Absolutely. So to wrap things up
Speaker 2:Yes.
Speaker 1:Quadratic expressions might seem like the villains of algebra class sometimes
Speaker 2:Right.
Speaker 1:But they're actually pretty powerful tools.
Speaker 2:Once you learn how to use
Speaker 1:them they
Speaker 2:can unlock a deeper understanding of the mathematical world around us.
Speaker 1:So the next time you encounter a parabola out in the wild
Speaker 2:Okay.
Speaker 1:Whether it's the arc of a basketball shot or the shape of a satellite dish
Speaker 2:I like it.
Speaker 1:Remember those quadratic expressions are hard at work behind the scenes.
Speaker 2:They're the unsung heroes of the math world.
Speaker 1:Absolutely. And a big thank you to Illustrative Math for providing these fantastic materials for our deep dive today.
Speaker 2:They're doing great work.
Speaker 1:Until next time. Happy exploring the fascinating world of math.