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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. Today, we're gonna do a deep dive into teaching parabolas and quadratic equations. Okay. And we're using this algebra 1612 lesson teacher guide to help us.
Speaker 2:Thank you.
Speaker 1:And, it's really amazing how these curves
Speaker 2:Yeah.
Speaker 1:Show up, like, everywhere, right, from bridges to the path of, like, a football.
Speaker 2:Absolutely.
Speaker 1:So let's unpack how to make this concept really click for students.
Speaker 2:What I think is really clever about this lesson
Speaker 1:Mhmm.
Speaker 2:Is that it takes something that's familiar
Speaker 1:Yeah.
Speaker 2:To students, like graphing linear equations Right. And it uses that as, like, a springboard.
Speaker 1:Right. Okay.
Speaker 2:So instead of just Bimper Slope, now we're diving into the world of, like, a, b, and c.
Speaker 1:Right. And it really emphasizes that this isn't just about, you know, plotting points on a graph.
Speaker 2:Mhmm.
Speaker 1:It's about understanding the relationship between the equation itself
Speaker 2:Yeah.
Speaker 1:And its visual representation.
Speaker 2:Absolutely.
Speaker 1:You know, it reminds me of that moment when students finally see how like, changing that in a linear equation Yep. Directly impacts the line's steepness.
Speaker 2:Totally.
Speaker 1:Like, I bet you've seen those light bulb moments before.
Speaker 2:Absolutely. Those are the best. Right. That's what it's all about. Yeah.
Speaker 2:This lesson plan really sets the stage for that kind of understanding with quadratics.
Speaker 1:Okay.
Speaker 2:And it's all about giving students the tools to connect those equations
Speaker 1:Yeah.
Speaker 2:Tables and graphs Mhmm. Helping them to see those different representations as, like, facets of the same mathematical idea.
Speaker 1:Okay. So let's get into, like, the nitty gritty.
Speaker 2:Okay.
Speaker 1:The lesson really wants students to understand the standard form of a quadratic equation Mhmm. Which is y, of x plus, b x plus c.
Speaker 2:K.
Speaker 1:And it seems like that a value has, like, a starring role here. It really does. Yeah. Yeah. The a value is all about the value is all
Speaker 2:about the parabola's direction Okay. And its shape. Mhmm. So a positive a gives you a parabola that opens upwards Okay. And a negative a means it opens downwards.
Speaker 2:So a positive
Speaker 1:a means
Speaker 2:we're dealing with a parabola that looks like
Speaker 1:a smile
Speaker 2:Yeah.
Speaker 1:And a negative one is more of a frown.
Speaker 2:You got it.
Speaker 1:Gotcha. Yep. So what else does it affect other than just the direction?
Speaker 2:A also controls the steepness of that smile or frown.
Speaker 1:K.
Speaker 2:So a larger a value means a much narrower and steeper parabola.
Speaker 1:Mhmm.
Speaker 2:While a smaller a value creates a wider
Speaker 1:Okay.
Speaker 2:More relaxed curve.
Speaker 1:So it's like a dictates how dramatic the parabola is.
Speaker 2:Totally.
Speaker 1:Whether it's like a subtle curve or a sharp climb or something, that makes sense.
Speaker 2:Right.
Speaker 1:But what about the c value then?
Speaker 2:Okay.
Speaker 1:What role does that play in shaping our parabola?
Speaker 2:So c is all about positioning that parabola on the graph. Okay. So just like with the y intercept in linear
Speaker 1:Mhmm.
Speaker 2:Equations
Speaker 1:Mhmm.
Speaker 2:The c value shifts the entire parabola
Speaker 1:Okay.
Speaker 2:Up or down the y axis.
Speaker 1:That's really helpful. Yeah. So a sets the basic shape. Mhmm. And then c determines where that shape sits vertically.
Speaker 2:Exactly.
Speaker 1:Because, like, c is choosing the parabola's starting point.
Speaker 2:I like that.
Speaker 1:Now we're cooking. This is starting to feel like a recipe for parabolas.
Speaker 2:Yes.
Speaker 1:So we've talked about the theory, but, like, how does this lesson plan actually get students to internalize all of these concepts?
Speaker 2:Well, that's where things get really exciting. Right?
Speaker 1:Mhmm.
Speaker 2:Because the lesson plan outlines some really brilliant hands on activities.
Speaker 1:Okay.
Speaker 2:For example, there's one called Quadratic Graphs Galore.
Speaker 1:Okay.
Speaker 2:And it encourages students to use graphing technology Mhmm. Like Desmos.
Speaker 1:Yeah.
Speaker 2:And so they can experiment with changing the a and c values
Speaker 1:That's right.
Speaker 2:And instantly see how the parabola, like, transforms on the screen.
Speaker 1:Oh, wow.
Speaker 2:Like, the dynamics. So they move that slider, and they can watch that parabola change.
Speaker 1:Oh, that's cool. Yeah. So it's like a dynamic puzzle almost That's okay. Watching those connections come to life.
Speaker 2:Absolutely.
Speaker 1:I love that visual element.
Speaker 2:Yeah.
Speaker 1:Speaking of visuals, though, what about those crucial x intercepts, you know, the points where the parabola crosses the x axis?
Speaker 2:Right.
Speaker 1:Those seem really important for students to grasp. Yeah. How does the lesson plan address that?
Speaker 2:So it offers this great optional activity called what do these tables reveal?
Speaker 1:Okay.
Speaker 2:And, basically, students calculate the a values for different x values.
Speaker 1:Mhmm.
Speaker 2:And in doing so, they see firsthand how those a and c values that we were just talking about
Speaker 1:Right.
Speaker 2:Directly influence where that parabola hits that x axis.
Speaker 1:Ah, so they're building a deeper understanding of the relationship between, like
Speaker 2:The equation.
Speaker 1:Components and then the visual representation of those x intercepts.
Speaker 2:Absolutely.
Speaker 1:It sounds like this lesson plan is all about connecting those dots for those students.
Speaker 2:Yeah. It provides a framework for students to develop a more intuitive understanding
Speaker 1:Right.
Speaker 2:Of these quadratic equations. Okay. But importantly, it doesn't shy away from addressing some of the common misconceptions.
Speaker 1:Which, let's be honest, can sometimes be, like, the most important part of teaching a new concept. For sure. So what are some of those potential stumbling blocks that the lesson highlights?
Speaker 2:Okay. And
Speaker 1:then how does it recommend navigating those?
Speaker 2:So one very common misconception is that students will confuse the factors in factored form
Speaker 1:Okay.
Speaker 2:With the x intercepts directly.
Speaker 1:Oh, interesting.
Speaker 2:So that they might see something like x plus 1 Alright. And assume the x intercept is at positive one on the graph.
Speaker 1:Yeah. You just pull it right out.
Speaker 2:Right. And they forget about that necessary step of setting that factored form equal to 0 and actually solving for x.
Speaker 1:Oh, that sneaky sign change.
Speaker 2:It's okay.
Speaker 1:It gets me sometimes even still.
Speaker 2:Right.
Speaker 1:But it's great that the lesson emphasizes that solving for x is really key here Okay. Not just pulling a value straight from that factored form. Yeah. What other misconceptions do they discuss?
Speaker 2:Yeah. It's like they say sometimes you have to make the mistake to truly understand the rule.
Speaker 1:Right.
Speaker 2:So important to address those common missteps.
Speaker 1:Absolutely.
Speaker 2:Are there any other areas where students tend to get tripped up?
Speaker 1:Yeah. So the lesson plan also points out that students might assume that a positive a value k. Always means that the parabola's vertex
Speaker 2:Mhmm.
Speaker 1:Is gonna be above the x axis.
Speaker 2:Right. Even though a positive a means we're dealing with a parabola that curves upwards.
Speaker 1:That entire curve could be shifted downwards depending on the c value.
Speaker 2:Exactly.
Speaker 1:Right.
Speaker 2:It's a great opportunity to remind students that c really does play a role Yeah. In positioning that whole parabola on the graph. Right. So even a happy upward facing parabola might find itself, like, dipping below the x axis.
Speaker 1:That's a fantastic visual to keep in mind.
Speaker 2:Right.
Speaker 1:We've really broken down the roles of a and c today.
Speaker 2:Yeah.
Speaker 1:But there's still that mysterious b
Speaker 2:Right.
Speaker 1:In our standard form.
Speaker 2:Yeah.
Speaker 1:Y o x x plus b x plus c.
Speaker 2:Yeah.
Speaker 1:I'm curious. What can you tell us about b?
Speaker 2:Well, so unlike a and c, which have these, like Yeah. Really direct visual interpretations
Speaker 1:Right.
Speaker 2:B is a little bit more subtle.
Speaker 1:Alright.
Speaker 2:It doesn't just, like, stretch or shift the parabola on its own. Mhmm. B kinda works behind the scenes
Speaker 1:Okay.
Speaker 2:To influence the parabola's position
Speaker 1:I understand.
Speaker 2:In a more nuanced way.
Speaker 1:Gotcha.
Speaker 2:So it's responsible for those, like, horizontal shifts
Speaker 1:Oh, okay.
Speaker 2:We see in some parabolas Right. Like moving them left or right on the graph.
Speaker 1:So b is, like, the quiet collaborator
Speaker 2:Yes.
Speaker 1:Working in tandem with a and c to fine tune that parabolas' position.
Speaker 2:Exactly.
Speaker 1:Fascinating. And it sounds like the lesson plan intentionally leaves b as a little bit of a cliffhanger.
Speaker 2:It does. The authors pose this very thought provoking question to the students Okay. Which is, how do you think the b value might affect the parabolas graph?
Speaker 1:Oh, that's cool.
Speaker 2:That's brilliant. Leaving that open ended question encourages students to go beyond just rote memorization
Speaker 1:Right.
Speaker 2:And delve into their own exploration.
Speaker 1:I love that. Yeah. And this deep dive has been incredibly insightful.
Speaker 2:It's been fun.
Speaker 1:I think we've gone beyond just the mechanics of graphing these quadratic equations.
Speaker 2:For sure.
Speaker 1:We've really explored the why behind those graceful curves
Speaker 2:Yeah.
Speaker 1:And all those common pitfalls that students might encounter along the way.
Speaker 2:Absolutely.
Speaker 1:And armed with these insights, I bet our listeners are gonna be amazing guides for their students.
Speaker 2:I hope so.
Speaker 1:Helping them see those light bulb moments click.
Speaker 2:That's the goal.
Speaker 1:Right. Awesome. Well, a huge thank you to the authors of Illustrative Math
Speaker 2:Yeah.
Speaker 1:For this insightful lesson plan.