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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 some lesson plans are, like, hidden treasure maps instead of, like, you know, straightforward plans?
Speaker 2:Yeah.
Speaker 1:I kinda feel that way about this graphing from the vertex form lesson you sent me. Yeah. It's one of those things where it's like, we all know vertex form is, like, super important for quadratics.
Speaker 2:Right.
Speaker 1:But diving into this algebra 1 6 16 lesson teacher guide 1 dot PDF, I think it's gonna help us, like, find those moments for our students.
Speaker 2:Yeah. And those moments, that's where the learning happens. Right? Like, it's one thing to be able to, like, you know, plug and chug in the formula. Mhmm.
Speaker 2:But to really understand, like, why it works and how that connects to, like, the bigger picture of quadratic function.
Speaker 1:Totally. I remember struggling with this back in the day where I was just like, okay. If I put this number in here and I do this thing, like, I don't really get it, but I'm getting the right answer. Sure. And that never feels good.
Speaker 2:No. It doesn't. And, you know, the beauty of this particular lesson is it really focuses in on those deep understandings. So you're not just plotting points, you're understanding how the equation and the graph are speaking to each other.
Speaker 1:Okay. So we're talking about helping students go beyond simply recognizing standard form and factored form and vertex form, and instead helping them understand, like, when do you use each one as the right tool for the problem?
Speaker 2:Exactly. And this lesson doesn't just tell students that it actually helps them discover it.
Speaker 1:Oh, cool. So how does it do that?
Speaker 2:So one thing that stood out to me when I was looking at the lesson goals is that they emphasize the why behind the math.
Speaker 1:Okay.
Speaker 2:It's clear that this lesson is not about memorizing steps. It's about making connections between the equation and the graph.
Speaker 1:Oh, that's huge. Yeah. And it even explicitly mentions creating a graph rather than just plotting points.
Speaker 2:Right. And when I saw that, I thought that was very intentional. They're really trying to highlight how the vertex form provides a blueprint for creating that parabola.
Speaker 1:Which is so important. Right? Yeah. Because it's that understanding that allows them to then manipulate equations with a purpose, knowing how each change will affect the graph.
Speaker 2:Absolutely. And I think that brings us to the real heart of this lesson, which is helping students visualize how changing the different parts of the vertex form will actually transform that graph.
Speaker 1:Okay. Let's dive into the specifics then. What's the first activity in this treasure map of a lesson plan? So the lesson kicks off with a warm up activity called 16.1, which form to use.
Speaker 2:Right.
Speaker 1:And it seems like just a simple review of the different forms.
Speaker 2:Yeah.
Speaker 1:But I feel like it's more strategic than that.
Speaker 2:Absolutely. This warm up is setting the stage for that whole lesson.
Speaker 1:Okay. Remember how
Speaker 2:we were talking about choosing the right tool for the job? Yeah. That's what this activity is about. Okay.
Speaker 1:It
Speaker 2:gets them thinking that way about the quadratic forms.
Speaker 1:Okay.
Speaker 2:So they're presented with these different situations. Mhmm. And they have to decide, is standard form, factored form, or vertex form gonna be the most helpful here?
Speaker 1:So it's not just about knowing the forms. It's about knowing when to use them.
Speaker 2:Exactly. It's about understanding, like you said, the strengths and weaknesses of each form.
Speaker 1:Okay.
Speaker 2:That way, when they see vertex form being used in the context of graphing, they kind of already have an appreciation for why that's a valuable tool.
Speaker 1:And that leads us perfectly into the heart of the lesson activity 16.2, sharing a vertex. Yes. Tell me more about this one.
Speaker 2:This is where things get really interesting because instead of just giving them the vertex form and saying, like, hey. Graph this.
Speaker 1:Yeah.
Speaker 2:They're presented with a little bit of a puzzle here.
Speaker 1:Okay.
Speaker 2:So they get the vertex. Mhmm. And they're given one other point on the parabola.
Speaker 1:Okay.
Speaker 2:And using that information and their understanding of vertex form, they have to figure out what the equation of that parabola is.
Speaker 1:So they're, like, working backwards using the clues to find the answer.
Speaker 2:Exactly. And it gets even better. What? The activity then asks them to figure out whether that vertex represents a minimum or maximum value without even graphing it.
Speaker 1:Wait. How is that even possible?
Speaker 2:Right. It's all about connecting the dots between the equation and what the graph would look like.
Speaker 1:Okay.
Speaker 2:So remember, in vertex form, a s tells us whether the parabola opens up or down.
Speaker 1:Right.
Speaker 2:So if a is positive, we've got that happy parabola Yeah. With the minimum value at the vertex.
Speaker 1:Right.
Speaker 2:But if a is negative, it flips that parabola upside down.
Speaker 1:Okay.
Speaker 2:And then the vertex becomes the maximum point.
Speaker 1:So just by looking at that a value, they can instantly tell.
Speaker 2:Yes. And I think this activity does a beautiful job of driving that point home. Yeah. It's really emphasizing that it's not just about finding the vertex on a graph. It's about understanding what that vertex represents.
Speaker 1:It's about making those connections.
Speaker 2:Exactly.
Speaker 1:Okay. And this is really cool. And I love how we're, like, getting to these moments. Yeah. That's what it's all about.
Speaker 1:And the learning journey continues with activity 16.3 card sort
Speaker 2:Yeah.
Speaker 1:Matching equations with graphs.
Speaker 2:Yes.
Speaker 1:And so now they're really solidifying that connection between, like, the visual and the equation.
Speaker 2:Exactly. And, again, it goes beyond just, like, pure matching because they have to justify their choices.
Speaker 1:Oh, cool. So they have to explain why.
Speaker 2:Exactly. They
Speaker 1:have to explain why they made those matches.
Speaker 2:Exactly. They're really gonna be able to explain how those a, h, and k values are affecting that graph.
Speaker 1:It's like they're becoming fluent in quadratic.
Speaker 2:Yes. Exactly.
Speaker 1:And then to, like, top it all off, we have activity 16.4, sketching a
Speaker 2:graph. Right.
Speaker 1:And this is where they take everything they've learned
Speaker 2:Yes.
Speaker 1:And they put it into practice.
Speaker 2:It's like their final exam for graphing.
Speaker 1:It is. Mhmm. But it's not really about, like, testing them. Right? It's about giving them a chance to practice what they learn
Speaker 2:Exactly.
Speaker 1:And to build that confidence.
Speaker 2:Yeah. Are there any common misconceptions that we should, like, look out for as teachers?
Speaker 1:Yeah. You know, no matter how well we design a lesson, students are gonna run into some bumps in the road.
Speaker 2:For sure.
Speaker 1:And one that came up in the notes is that sometimes students have a hard time identifying that vertex when the scales on the axis are different.
Speaker 2:Oh, yeah. I've seen that happen. Mhmm. They forget to look at how the scale is different.
Speaker 1:Right. And that can really throw them off. Yeah. So encourage teachers to really emphasize the scale on those graphs. Maybe even have students label it themselves.
Speaker 1:Yeah. That's a good idea. What other things did you notice?
Speaker 2:Another one is understanding how the a value impacts how steep that parabola is.
Speaker 1:Okay.
Speaker 2:Like, why does a bigger a make it steeper and a smaller a make it wider?
Speaker 1:So how can we get that point across?
Speaker 2:Visuals are always super helpful for that.
Speaker 1:Okay.
Speaker 2:So showing them what happens when you graph y equals x squared versus y equals 2x squares.
Speaker 1:Yeah.
Speaker 2:They can really see the difference side by side.
Speaker 1:So show, don't tell. Right.
Speaker 2:Exactly.
Speaker 1:And speaking of transformations, what about that negative a value?
Speaker 2:Oh, yeah. That one can be tricky.
Speaker 1:Because it flips it.
Speaker 2:Right. They forget that it reflects it across the x axis.
Speaker 1:So how can we help them with that?
Speaker 2:You know, I think real world examples are great for this one.
Speaker 1:Okay. Like what?
Speaker 2:So if you think about throwing a ball up in the air Okay. It makes that parabolas shape. Right. But because gravity is pulling it down, it's flipped upside down.
Speaker 1:Oh, that's a good one.
Speaker 2:So connecting it back to something concrete can really help.
Speaker 1:Yeah. And, you know, in all of this talk about graphs and equations, it's easy to forget that, like, the real goal here is to get kids curious about math.
Speaker 2:Definitely.
Speaker 1:And this lesson plan actually ends with this thought provoking question. Suppose we change the equation to y equals by 3bunds plus 2. What stays the same and what changes about the graph?
Speaker 2:Oh, that's a really good question.
Speaker 1:And it's so simple, but it really gets them thinking critically.
Speaker 2:Yeah. It encourages them to keep exploring even after the lesson is over.
Speaker 1:Which is what we want. Right?
Speaker 2:Exactly.
Speaker 1:Well, this has been such a great deep dive. I feel like I have a much better understanding of this lesson plan now.
Speaker 2:Me too.
Speaker 1:A huge thank you to the authors of Illustrative Math for this amazing resource.
Speaker 2:Yes. Thank you.
Speaker 1:And thank you to you out there for joining us on this deep dive into the world of quadratics.
Speaker 2:Thanks, everyone.
Speaker 1:Until next time. Happy teaching.