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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. So have you ever looked at, like, a bridge? You know? Uh-huh. One of those really long bridges Yeah.
Speaker 1:Or like a satellite dish
Speaker 2:Mhmm.
Speaker 1:And and wondered, like, about the curves. Right. You know how they have that, like, kind of elegant curve to them? Yeah. Yeah.
Speaker 1:They they give them, like, strength and focus?
Speaker 2:Mhmm.
Speaker 1:You might be surprised to know that those shapes, they actually they come from the world of math. Mhmm. Specifically, parabolas
Speaker 2:Right. Right.
Speaker 1:Generated by, you know, at at their at their core, quadratic equations.
Speaker 2:Of course. Yeah.
Speaker 1:And lucky for you, today's deep dive, we're gonna unpack an illustrative mathematics lesson plan.
Speaker 2:Okay.
Speaker 1:That that makes even, I think, like, seasoned mathletes, you know Yeah. Appreciate the beauty of these curves.
Speaker 2:Oh, for sure. For sure. What I think is so fascinating about this particular lesson plan is how it it takes students beyond just, like, plugging in numbers
Speaker 1:Right.
Speaker 2:Into a formula.
Speaker 1:Right.
Speaker 2:You know? Yeah. We're talking about empowering them to look at a quadratic equation and instantly picture, like, in their mind's eye
Speaker 1:Oh, wow.
Speaker 2:That parabola.
Speaker 1:Wow.
Speaker 2:That graceful curve.
Speaker 1:Yeah.
Speaker 2:Understanding how each coefficient influences, like, its position, its shape, you know, all of that.
Speaker 1:So we've got excerpts from the lesson plan itself.
Speaker 2:Okay.
Speaker 1:And then we also have some some presenter notes as well Right. That we're gonna use to to guide our exploration today.
Speaker 2:Excellent.
Speaker 1:Our mission, to equip you with the knowledge to not just understand this lesson, but to teach it.
Speaker 2:I love it.
Speaker 1:Or let's be honest. Maybe just impress your friends with your parabola prowess.
Speaker 2:There you go. That's right. Because who doesn't wanna do
Speaker 1:that? Right.
Speaker 2:Right.
Speaker 1:Exactly. Exactly.
Speaker 2:So imagine you're a student. Right?
Speaker 1:Okay.
Speaker 2:And you're seeing a quadratic equation for the first time. It's it's a little bit like deciphering a code.
Speaker 1:Oh, yeah. Yeah.
Speaker 2:Right?
Speaker 1:Yeah.
Speaker 2:And this lesson cleverly breaks down that code starting with the b coefficient Okay. In the standard form. So that's, like, that's the b x part of the equation Oh, yeah. X plus b x plus c.
Speaker 1:It's like it's like a secret lever Yes. That controls the parabola.
Speaker 2:Yes.
Speaker 1:So you shift that b, and then the parabola is like it it, like, dances across the graph.
Speaker 2:Exactly. Exactly. And and the lesson plan really aims to make that connection crystal clear. It also tackles another another key concept, helping students go from seeing a parabola
Speaker 1:Okay.
Speaker 2:To actually writing out its equation
Speaker 1:Oh, no.
Speaker 2:In both standard and factored forms. Talk about, like, bridging the gap between these abstract equations
Speaker 1:Yeah.
Speaker 2:And then those beautiful curves we see.
Speaker 1:In the real world.
Speaker 2:In the real world. Exactly.
Speaker 1:You imagine the moments when students see that connection?
Speaker 2:Oh, absolutely.
Speaker 1:It's like the equation comes alive.
Speaker 2:It does. It comes alive.
Speaker 1:Speaking of bringing things to life, I love how the lesson uses graphing technology. Yes. Specifically mentioning Desmos.
Speaker 2:Absolutely. I mean, Desmos is fantastic because it allows students to visually experiment
Speaker 1:Right.
Speaker 2:With different values for b so they can see firsthand how changing that one coefficient can make the parabola do really cool things. Right. It can it can make it slide sideways Uh-huh. Like a dancer gracefully moving across the stage
Speaker 1:Right. Right.
Speaker 2:And then leap up or down.
Speaker 1:Yeah. You know? Wow.
Speaker 2:That's that's the magic of be at work.
Speaker 1:And that kind of visual manipulation, I mean, that can be a game changer, I think, for a lot of students.
Speaker 2:Absolutely. Right. Especially when things start to get a little more complex. Right?
Speaker 1:Right.
Speaker 2:So so let's talk about how the lesson guides students through this exploration.
Speaker 1:Okay.
Speaker 2:It all starts with a simple warm up. Right?
Speaker 1:It does. It does. This warm up focuses on converting between the standard and factored forms of quadratic equation.
Speaker 2:Mhmm.
Speaker 1:And it might seem basic on the surface Okay. But it really primes students' brains to recognize those important patterns Right. In those equations
Speaker 2:Yeah.
Speaker 1:Which sets the stage for deeper understanding later on.
Speaker 2:It's like laying the foundation for a skyscraper. Right?
Speaker 1:Exactly.
Speaker 2:You need that strong base to support everything that comes after. Exactly. Couldn't have said it better myself.
Speaker 1:Then comes the equivalent expressions section
Speaker 2:Yes.
Speaker 1:Which kinda subtly sets the stage for, like, the main event.
Speaker 2:Yes.
Speaker 1:Understanding how that b coefficient influences the parabola's every move.
Speaker 2:Precisely. And that's where section 13.2, what about the linear term, takes center stage.
Speaker 1:Okay.
Speaker 2:This section, it encourages students to experiment
Speaker 1:Mhmm.
Speaker 2:With changing the b value in various equations Mhmm. And then observe how the parabola dances across the screen in response. Right. They discover that b isn't just about, like, a simple left or right shift
Speaker 1:Right.
Speaker 2:But it's a combination Mhmm. Of horizontal and vertical movement.
Speaker 1:Which, if we're being honest, that can be a bit of a mind bender.
Speaker 2:It can be. It can be. It's like trying to explain to someone how to rub your stomach and pat your head at the same time.
Speaker 1:Right.
Speaker 2:It takes practice and a good guide to get it right.
Speaker 1:Right. Exactly. And that's what makes this lesson plan so valuable. It anticipates those moments Yeah. And provides a clear path to understanding.
Speaker 1:Yeah. But before we get too far ahead, let's take a quick pause to acknowledge those common stumbling blocks
Speaker 2:Okay.
Speaker 1:That, you know, even the most enthusiastic students might encounter.
Speaker 2:Right. Right.
Speaker 1:After all, a little bit of foresight goes a long way in the world of teaching and learning.
Speaker 2:Absolutely. It's true. Yeah. And sometimes those her moments
Speaker 1:Yeah.
Speaker 2:Are are really, like, the stepping stones
Speaker 1:Right.
Speaker 2:To the most satisfying moments.
Speaker 1:Totally. Totally.
Speaker 2:This lesson plan seems to have a real knack for anticipating those hurdles.
Speaker 1:It's like the authors have a 6th sense, you know
Speaker 2:Right.
Speaker 1:For where students might stumble.
Speaker 2:Uh-huh.
Speaker 1:But instead of letting them, like, fall flat on their face, they've built in these clever detours
Speaker 2:Yes.
Speaker 1:That lead to even deeper understanding.
Speaker 2:Precisely.
Speaker 1:Okay.
Speaker 2:Take section 13.3, writing equations to match graphs, for instance. K. Here, the lesson flips the script. Oh. Instead of starting with an equation and plotting the graph, students are presented with graphs and challenged to basically play math detectives
Speaker 1:Hello.
Speaker 2:Working backward to deduce the equation.
Speaker 1:It's like giving them the evidence and saying, alright, detectives. Tell me what happened here.
Speaker 2:Exactly.
Speaker 1:A quadratic equation form.
Speaker 2:Exactly.
Speaker 1:I love that.
Speaker 2:And the lesson plan provides helpful clues along the way, including the coordinates for 3 points on each graph. Okay. This allows students to, like, test their equations
Speaker 1:Uh-huh.
Speaker 2:And fine tune their detective work until they, you know Yeah. Crack the case.
Speaker 1:I love it.
Speaker 2:Yeah.
Speaker 1:Brilliant.
Speaker 2:It's a really cool approach.
Speaker 1:It is. It's a brilliant way to reinforce the idea that there's this two way street, right
Speaker 2:Yes.
Speaker 1:Between equations and graphs.
Speaker 2:Absolutely.
Speaker 1:They each tell a story about the parabola.
Speaker 2:Yes.
Speaker 1:And speaking of stories, did you did you notice how the lesson plan weaves in examples from the real world?
Speaker 2:Yes.
Speaker 1:It's not just about these, like, abstract equations floating in space.
Speaker 2:Right.
Speaker 1:It's about connecting those equations to the shapes we see every single day.
Speaker 2:I love that about it. And it you know, for example, remember those bridges and satellite dishes we were talking about earlier?
Speaker 1:Right. Exactly.
Speaker 2:The lesson plan helps students see how those parabolic shapes Yeah. Aren't just, like, aesthetically pleasing. You know? Right. They're they're mathematically sound.
Speaker 1:It's like it's like you suddenly start noticing parabolas everywhere Everywhere. In nature, in architecture.
Speaker 2:That's right.
Speaker 1:Even for the arc of a basketball shot.
Speaker 2:Right? Absolutely.
Speaker 1:It makes you appreciate the power of math
Speaker 2:It does.
Speaker 1:To explain the world around us.
Speaker 2:It does. It really does.
Speaker 1:Yeah.
Speaker 2:And and the lesson plan doesn't shy away from the more nuanced aspects of quadratic equations either.
Speaker 1:Okay.
Speaker 2:For example, it highlights the simple graph of yhxsort. Mhmm. Which
Speaker 1:is a
Speaker 2:parabola Okay. Sitting neatly with its vertex at the origin.
Speaker 1:Okay.
Speaker 2:But then it throws in a twist.
Speaker 1:Okay.
Speaker 2:What happens if the equation is y xsort plus 4?
Speaker 1:Oh, good question.
Speaker 2:Right.
Speaker 1:That's that's where those vertical shifts come into play again.
Speaker 2:Exactly. Okay. Okay. Okay. Adding that constant term, the plus 4
Speaker 1:Right.
Speaker 2:Doesn't change the parabola's shape.
Speaker 1:Okay.
Speaker 2:It simply lifts the entire curve upward.
Speaker 1:Interesting.
Speaker 2:And the lesson encourages teachers to let students discover this through experimentation on Desmos
Speaker 1:Oh, I love that.
Speaker 2:Which makes the learning so much more impactful.
Speaker 1:Because it's one thing to be told that a constant term shifts the graph vertically.
Speaker 2:Uh-huh.
Speaker 1:It's another thing to see it happen before your very eyes.
Speaker 2:Absolutely. And and, you know Yeah. It's it's that kind of, like, visual learning Right. That can really solidify these concepts for students.
Speaker 1:Totally. Totally. And I and I love how the lesson synthesis section Yes. Encourages teachers to to prompt students to compare and contrast
Speaker 2:Yes.
Speaker 1:The standard and factored forms
Speaker 2:Yes.
Speaker 1:Asking them to think critically about the advantages and disadvantages of each Yes. When sketching graphs.
Speaker 2:Absolutely. It's about moving beyond rote memorization Yes. To a deeper, more flexible understanding Totally. Of the tools that they have at their disposal.
Speaker 1:Yeah. Yeah. Totally.
Speaker 2:And speaking of tools
Speaker 1:Okay.
Speaker 2:Let's not forget about those potential stumbling blocks that we mentioned earlier.
Speaker 1:Right. Right. Because we we've we've talked about how that Ebex term
Speaker 2:Yes.
Speaker 1:The linear term can be a bit tricky
Speaker 2:Yes.
Speaker 1:Especially when it comes to understanding how it impacts both horizontal and vertical movement.
Speaker 2:It's true. And the lesson plan acknowledges that this is a common area where students need extra support.
Speaker 1:Right. Right.
Speaker 2:But it also provides teachers with some clear strategies
Speaker 1:Okay.
Speaker 2:To help students navigate this challenge.
Speaker 1:Because let's be honest. We've all been there.
Speaker 2:Oh, absolutely.
Speaker 1:Staring at an equation, wondering, like, how on earth does this translate to that curve on the graph?
Speaker 2:Exactly.
Speaker 1:You know, it's it could be a bit like trying to read a foreign language without a dictionary.
Speaker 2:Precisely. Precisely. And this lesson plan acts as that dictionary.
Speaker 1:Okay.
Speaker 2:Decoding the language of quadratic equations
Speaker 1:Oh, wait.
Speaker 2:And making it accessible to everyone.
Speaker 1:And it's not just about understanding the language. It's about being able to speak it, to write it, to manipulate it with confidence.
Speaker 2:That's right.
Speaker 1:So so let's let's delve into some of those specific strategies this this lesson plan provides Yeah. To to help teachers guide their students towards fluency in the language of parabolas. It's like you're you're equipping these students with, like, a a map and a compass Yeah. You know Yeah. To navigate this this sometimes treacherous terrain of quadratic equations.
Speaker 2:Exactly. Exactly. And so to tackle, you know, to tackle that DEX hurdle, one thing that the lesson plan really emphasizes is breaking down the movement Okay. Into, like, bite sized pieces. Okay.
Speaker 2:Right?
Speaker 1:Yeah.
Speaker 2:So instead of trying to digest the horizontal and vertical shifts, you know, simultaneously
Speaker 1:Right.
Speaker 2:It encourages, you know, having students focus on one direction at a time. Okay. So imagine, like, sliding the parabola sideways, like you're adjusting a picture on a wall.
Speaker 1:Right.
Speaker 2:That's that's the best at work horizontally. And then, you know, imagine that picture magically floating up or down.
Speaker 1:Okay.
Speaker 2:Right. That's b influencing the vertical position.
Speaker 1:I love how you're you're bringing in these, like, vivid images.
Speaker 2:Yes.
Speaker 1:You know? Because it's it's so much easier, I think, to remember something when you can really visualize it.
Speaker 2:Visuals are key.
Speaker 1:Yeah.
Speaker 2:And this lesson plan really emphasizes that, especially with, you know, tools like Desmos where you can actually see those manipulations happening in real time.
Speaker 1:Right.
Speaker 2:Now remember those equations that lack the linear term?
Speaker 1:Oh, yeah. Yeah.
Speaker 2:The ones that leave students wondering if they've accidentally wandered into a math riddle?
Speaker 1:Right. Right. Those can be real head scratchers.
Speaker 2:They can be real head scratchers. Yeah. And the lesson plan tackles those head on, suggesting a strategy that's all about pattern recognition. So encouraging
Speaker 1:teachers to
Speaker 2:have students graph a series of equations Mhmm. Subtly tweaking the constant term each time. Okay. Right?
Speaker 1:So start
Speaker 2:with y xsus. Mhmm. Then graph yxsus+1 followed by yxx4 and so on. With each graph, students see the parabola shifting along the axis Okay. Revealing the impact of that constant term.
Speaker 1:So it's like it's like they're almost like conducting experiments.
Speaker 2:Exactly. Right.
Speaker 1:And they're, like, uncovering the hidden language of parabolas.
Speaker 2:Exactly. Exactly. About that. And through those experiments, those moments, they emerge organically.
Speaker 1:Yeah.
Speaker 2:Right? Students begin to see that even without that term Right. The constant term holds the key Yeah. To the parabola's vertical position. It's all connected.
Speaker 1:It's all connected. And speaking of connections, like, one thing that I'm struck by
Speaker 2:Yeah.
Speaker 1:With this lesson is how it it it goes beyond just graphing.
Speaker 2:Right.
Speaker 1:Right? It delves into the why Yeah. Behind the how Yes. How encouraging that critical thinking and deeper understanding.
Speaker 2:Absolutely.
Speaker 1:Right? Yeah. It even includes, like, a cool down activity where students flex their newfound knowledge.
Speaker 2:Yes.
Speaker 1:Without relying on technology. It's like it's like taking the training wheels off, you know, and letting them experience that freedom and satisfaction of navigating parabolas on their own.
Speaker 2:I love that. That's a really important point.
Speaker 1:Right. To make sure that they that they really are grasping it. You know? Absolutely. Well, we've we've covered a lot of ground in this deep dive
Speaker 2:We have.
Speaker 1:From unpacking those core goals of of this illustrative mathematics lesson Yes. To exploring those common student hurdles
Speaker 2:Yes.
Speaker 1:And most importantly, equipping our listeners with with strategies to make this elegant math come alive.
Speaker 2:Yes.
Speaker 1:So what are your what are your final thoughts for for our listeners, you know, as they as they embark on their own parabola adventures?
Speaker 2:You know, I would say, remember Yeah. Parabolas aren't just confined to textbooks or graphing calculators. They're everywhere. So as you go about your day today, take a moment
Speaker 1:Yes.
Speaker 2:To appreciate the parabolic curve of a fountain, the arc of a well hit golf ball
Speaker 1:Right.
Speaker 2:Even the shape of a delicious pizza crust fresh out of the oven.
Speaker 1:Oh, what a delicious thought to end on. A huge thank you to the authors of Illustrative Mathematics
Speaker 2:Yes.
Speaker 1:For crafting such an engaging and insightful lesson plan.
Speaker 2:Absolutely.
Speaker 1:And to our listeners, keep exploring, keep questioning, and keep discovering the beauty and wonder of math in the world around you.
Speaker 2:That's right.
Speaker 1:We'll see you in our next deep dive.