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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.)
Everyone, welcome back.
Speaker 2:Back with you again.
Speaker 1:You ready to dive into another deep dive?
Speaker 2:Always.
Speaker 1:Alright. Fantastic. So today, we are tackling something that I think a lot of people are kinda scared of, both teachers and students.
Speaker 2:Yeah.
Speaker 1:Graphs.
Speaker 2:Graphs. Yeah.
Speaker 1:And we're looking at lesson 6, features of graphs.
Speaker 2:Yeah.
Speaker 1:From illustrative math, algebra 1 curriculum.
Speaker 2:Yeah. And this is not just about, like, how to plot points on a graph Right. But how to get students to really, like, have that moment Mhmm. Where they see that these squiggly lines and these peaks and dips on a graph actually mean something in the real world.
Speaker 1:Absolutely. So what are we hoping that teachers will take away from this deep dive to make this lesson really sing for their students?
Speaker 2:Yeah. I think you said it. It's how to make it sing, how to make it come alive for students. So we're talking about, like, how do we connect, this abstract idea of function notation, f of x, with, like, the visual language of graphs. Mhmm.
Speaker 2:Right. How do we help students interpret, like, intercepts, maxims, and minimums in a way that, like, connects to something they experience in their own world?
Speaker 1:And speaking of connecting, we have to talk about vocabulary with you because it's one thing to kinda get the concept. It's a whole other thing to be able to use the words like horizontal intercept, vertical intersect, maximum, minimum.
Speaker 2:Exactly. Yeah. I mean, that's the thing about math sometimes is that the vocabulary can be intimidating.
Speaker 1:Right.
Speaker 2:But the nice thing about this lesson is that it really builds on prior knowledge.
Speaker 1:Okay.
Speaker 2:Students have seen intercepts before.
Speaker 1:Right.
Speaker 2:They probably have an intuitive sense of what a maximum and a minimum is. Yeah. But now we're giving them the framework to formalize it.
Speaker 1:I like it. I like it. So let's talk about the first activity in this lesson. It's called walking home. Sounds pretty basic.
Speaker 2:It does.
Speaker 1:But I'm sure it's more than meets the eye.
Speaker 2:There's always more. Right?
Speaker 1:Right.
Speaker 2:So this activity uses something that all students are familiar with. Right? Walking home from school or to school, and it uses that to really solidify, what can be kind of an intimidating concept, which is function notation.
Speaker 1:Okay.
Speaker 2:So imagine Diego, our hypothetical student here
Speaker 1:Okay.
Speaker 2:And he's walking home from school, and we can actually represent his distance from home at any given time using function notation.
Speaker 1:Okay.
Speaker 2:So for example, we could say a fast of 0, meaning the input is 0 Mhmm. Represents how far Diego is from home at time 0 when he starts walking.
Speaker 1:So we're taking something as, like, every day as walking home from school
Speaker 2:Right.
Speaker 1:And using that to show how functions play out in the real world.
Speaker 2:Exactly. And as students work through this activity, they'll start to see how that downward slope on the graph represents the fact that Diego is getting closer to home as time passes.
Speaker 1:Yeah. It's all about making those connections. Right? Absolutely. So we go from a leisurely stroll home from school to something a a little more high flying, should we say.
Speaker 2:Okay.
Speaker 1:A toy rocket and a drone.
Speaker 2:I like it.
Speaker 1:And I always love when we can bring in visuals, especially when they're this exciting. But tell me, what is it about this activity that makes it really work?
Speaker 2:Yeah. You're right. Visuals are so powerful, and this activity definitely leans into that.
Speaker 1:Mhmm.
Speaker 2:But I think what's really interesting here is how they can actually use visuals, like a graph of a toy rocket or a drone's flight path, to actually expose a common misconception about graphs. Oh. And that has to do with the axes.
Speaker 1:The axis. Yeah. Yeah. What is it about those axis that can really kinda throw students for a loop?
Speaker 2:Well, I think it's easy to assume that the axis always represent the same thing. Like, you always have distance on the x axis or whatever. Right. But that's not always the case. And so in this activity Yeah.
Speaker 2:Students might see, like, the horizontal axis and think, oh, that's gotta be horizontal distance.
Speaker 1:Right.
Speaker 2:But in this case, it's actually time.
Speaker 1:Okay.
Speaker 2:So it's not showing the rocket moving horizontally across the graph. It's actually showing how its height is changing.
Speaker 1:Okay. So it's really important On the yeah. To pay attention to those labels
Speaker 2:Always check your labels. Yeah.
Speaker 1:On those axes. Absolutely. Yeah. Yeah. So how can we help students kinda make sure that they're not falling into that trap of assuming what the axis represent?
Speaker 2:Yeah. I think it's always helpful to kind of bring them back to the scenario, like, really ground them in what we're actually measuring here.
Speaker 1:Okay.
Speaker 2:So we might ask them, like, okay. If this axis represents time and this axis represents the rocket's height, how is the rocket's height changing every second?
Speaker 1:Right.
Speaker 2:What would it look like on the graph if the rocket just stayed at the same height for a couple seconds?
Speaker 1:I like that.
Speaker 2:Yeah. Just kind of getting them
Speaker 1:Really anchoring it back in.
Speaker 2:Thinking about it concretely. Yeah.
Speaker 1:In the actual scenario. Yeah. Okay. Yeah. So we've talked about those axes, those tricky axes, but what other important graph features does this activity really highlight?
Speaker 2:Yeah. So this is where we get to dive into maximums and minimums.
Speaker 1:Oh, yeah.
Speaker 2:And in particular, this toy rocket and a drone activity is really good for illustrating the difference between relative and absolute maximums and minimums.
Speaker 1:Oh, that's a good one. Yeah. I can see how that would that could really kinda trip some students up.
Speaker 2:For sure.
Speaker 1:How how can we really make that distinction clear for them?
Speaker 2:Yeah. So I like to use the example of a roller coaster ride. So think about the highest point on the entire roller coaster ride.
Speaker 1:Okay.
Speaker 2:That's your absolute maximum. Right?
Speaker 1:The tippy top.
Speaker 2:The highest of the high points. Yeah.
Speaker 1:Yeah.
Speaker 2:But along the way, there might be some smaller hills. Right?
Speaker 1:Right.
Speaker 2:And those would be relative maximums because they're high points, just not the highest point.
Speaker 1:Okay.
Speaker 2:And the same, obviously, applies for minimums.
Speaker 1:So it's like the absolute maximum is king of the hill.
Speaker 2:Right.
Speaker 1:And then all those relative ones are just like
Speaker 2:Like the local champions.
Speaker 1:Local champions. Yeah. I like it. Exactly. I'm gonna use that one.
Speaker 2:Yeah. Feel free.
Speaker 1:Okay. That's good. So we're looking at those maximums, those minimums.
Speaker 2:Yep. And as students are looking at those, they're also gonna start to notice where the graph is increasing, where it's decreasing, where it levels off.
Speaker 1:Yeah.
Speaker 2:And that's where we get into those intervals of increase, decrease, and constancy.
Speaker 1:Okay. And that all just flows so nicely in this toy rocket and a drone activity.
Speaker 2:It does. It flows really nicely.
Speaker 1:Yeah. It's great. Yeah. So let's move on to our last activity here, which I have to say, it sounds like the most exciting of the bunch. Okay.
Speaker 1:We're talking about the jump activity. We're going bungee jumping.
Speaker 2:Nice. Very nice.
Speaker 1:What makes this one so special? Okay. Budgie jumping. This is, like, the perfect way to get kids thinking about, like On a graph. Maximums and minimums on a graph.
Speaker 1:Right?
Speaker 2:It's so visual. Right.
Speaker 1:Yeah.
Speaker 2:And this activity, the jump, it brings in all the vocabulary we've been talking about. Yeah. Maximum, minimum, vertical intercept, horizontal intercept.
Speaker 1:Mhmm.
Speaker 2:Students are they're seeing them on the graph, but they're also connecting it back to, like, this real world like
Speaker 1:taking those dry terms out of the textbook and Yes. Throwing them off a
Speaker 2:bridge. Exactly.
Speaker 1:Exactly. Like an oomgee cord. That's a
Speaker 2:great way to put it. Yeah. And one of
Speaker 1:the things I really like about this activity
Speaker 2:is that it addresses this misconception that students often have
Speaker 1:Okay.
Speaker 2:About the difference between the maximum of a graph Okay. And the maximum of a function.
Speaker 1:Oh, that's a tricky one.
Speaker 2:It is tricky.
Speaker 1:How does the jump help us make that distinction?
Speaker 2:Okay. So let's picture that bungee jump. Right?
Speaker 1:Okay.
Speaker 2:So the highest point that the jumper reaches
Speaker 1:Yeah.
Speaker 2:That's represented by the maximum point on the graph.
Speaker 1:Okay.
Speaker 2:But the maximum value of the function, that's the actual height of that point.
Speaker 1:Right. So let
Speaker 2:Like, how far are they from the river they're jumping over or whatever?
Speaker 1:Yeah. It's, like, the difference between looking at the point on the graph Right. Versus, like, the actual It's the actual number. Value. Yeah.
Speaker 1:Exactly. Yeah.
Speaker 2:So it gives them a way to kind of visualize
Speaker 1:Yeah.
Speaker 2:What can be kind of a subtle difference.
Speaker 1:Yeah. That's great. And I see here that this activity also includes an are you ready for more challenge.
Speaker 2:It does.
Speaker 1:What can students dig into with that?
Speaker 2:I'm glad you asked because this is where things get even more interesting.
Speaker 1:Okay.
Speaker 2:This challenge asks them to think about, like, how does the length of the bungee cord
Speaker 1:Okay.
Speaker 2:Actually affect the shape of the graph?
Speaker 1:Oh, interesting.
Speaker 2:So now we're not just identifying features on the graph. We're thinking about, like, what factors in the real world actually influence
Speaker 1:How the graph looks.
Speaker 2:How the graph looks. Exactly.
Speaker 1:So it was really building that that deeper understanding.
Speaker 2:Absolutely. Yeah.
Speaker 1:And then one other potential pitfall that this activity highlights for us is this misconception that a graph always shows you. Like Yes. The whole picture of what a function's doing.
Speaker 2:A whole story. Yeah.
Speaker 1:Yeah. The whole story.
Speaker 2:And that's not always the case. Right. So in the jump activity, you're only seeing a portion of the bungee jump. Like, what's happening before and after that? Who knows?
Speaker 1:Right. Are there other peaks? Are there other valleys? What's going on? Exactly.
Speaker 1:Exactly. It's so impertinent. Yeah.
Speaker 2:Yeah. It's about getting them to think beyond the edges of the graph. Like, don't be limited by what you see.
Speaker 1:Yeah. Don't be afraid to ask those questions.
Speaker 2:Exactly.
Speaker 1:Wow. This has been great.
Speaker 2:It has.
Speaker 1:I really I love how all of these activities Mhmm. Not only reinforce those key concepts about graphs Mhmm. But also really help teachers address those common misconceptions.
Speaker 2:Yes. And in such an engaging way, I mean, who doesn't love a good bungee jump or a toy rocket?
Speaker 1:Right. It makes it so much more fun It does. Than just, you know, dry equations on a board.
Speaker 2:Absolutely. It brings it to life.
Speaker 1:Yeah. Fan fantastic. Well, huge thank you. This has been so insightful.
Speaker 2:It's been my pleasure.
Speaker 1:And thank you to the authors of Illustrative Math for this awesome lesson. And to all of you listening
Speaker 2:Yes. Thank you.
Speaker 1:We hope this empowers you to go out and inspire some moments in your classrooms.
Speaker 2:Absolutely.
Speaker 1:Until next time. Happy teaching.