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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.)
Hey, everyone. Welcome back for another deep dive. So, listeners, can you picture this? It's the 1st day teaching function notation in Algebra 1. And I don't know about you all, but I could practically see the question marks hovering over my students' heads.
Speaker 1:Oh, yeah. But have no fear because today, we're taking a deep dive into a lesson plan from illustrative mathematics called Interpreting and Using Function Notation to hopefully equip you with the tools to, you know, turn those question marks into moments.
Speaker 2:Yeah. You know, it's funny how function notation becomes, like, second nature to, I don't know, us math folks.
Speaker 1:Mhmm.
Speaker 2:But for a lot of students, it's their first real encounter with this, what's the right word, like, this abstract mathematical language.
Speaker 1:Right.
Speaker 2:We're handing them this decoder ring to unlock this whole new world of understanding about relationships and patterns.
Speaker 1:Absolutely. But before we, decode all the secrets of this lesson plan, let's just take a step back for a second. For any listener who maybe isn't an algebra teacher, could you give us a quick, like, jargon free explanation of what function notation actually is?
Speaker 2:Oh, sure. Imagine you have this machine. Right? You put something in, and based on some kind of rule, it spits something out.
Speaker 1:Okay.
Speaker 2:That's essentially what a function does, and function notation is really just a special way we write and talk about those functions. Instead of saying something like y equals 2x plus 1, we say f of x equals 2x plus 1.
Speaker 1:Okay.
Speaker 2:It's like, like, we're giving the function a name, f, and then this really clear way to see what goes in the x inside the parentheses and then what comes out, which is that whole expression, f of x.
Speaker 1:Yeah. It's like a code that not only tells us what the machine does, but also helps us, like, keep track of the input and output
Speaker 2:Exactly. With
Speaker 1:those, like, handy labels. So with that in mind, what are the big moments we want our students to to experience with this lesson?
Speaker 2:I think this lesson does a really nice job of highlighting, let's see. We could say 3 major goals. The first one is that we want them to be able to look at a statement written in function notation and, like, instantly be able to visualize it on a graph. So it's like they can speak both. You know?
Speaker 2:They can speak algebra, and they can speak, you know, the language of graphs Yeah. And can effortlessly translate back and forth.
Speaker 1:It's like they're becoming, like, mathematical bilinguals.
Speaker 2:I love that.
Speaker 1:What's next on our list of moments?
Speaker 2:The second one is we wanna make it real for them. We want them to look at a real world situation. Like, maybe we're talking about the height of a drone over time
Speaker 1:Okay.
Speaker 2:And see how function notation can be used to actually model and understand that relationship.
Speaker 1:So connecting those abstract symbols to something, like, tangible they can wrap their heads around.
Speaker 2:Exactly. That's huge.
Speaker 1:Yeah. And then, finally, we don't just want them to understand it, but we want them to take information that's given to them in function notation and then actually go ahead and make a sketch of a possible graph that represents that relationship. Oh, nice. So from understanding to application. Exactly.
Speaker 1:I like it. So how does this lesson plan go about guiding our students to those moments? What are some of the the key activities that we should be prepared to unpack?
Speaker 2:Well, they start off with this activity called observing a drone, and it presents students with a graph that shows a drone's height over time. Now even though they haven't really formally been introduced to function notation yet, they're already, you know, comparing different points on the graph. They're essentially working with these input output pairs without even realizing it.
Speaker 1:It's like they're dipping their toes into the pool before diving in head first.
Speaker 2:Exactly.
Speaker 1:I remember when I first taught function notation, like, visualizing it on a graph just made it click for so many students. It was amazing.
Speaker 2:And it subtly introduces that format of x, f x, visually before we're throwing all the algebraic notation at them. And then, it moves on to the smartphones activity, which, let's be honest, this is gonna get their attention right.
Speaker 1:Smartphones. Yeah.
Speaker 2:This activity really cleverly uses smartphone ownership data to reinforce those crucial details about units in a really, I think, engaging way.
Speaker 1:Okay. Units. I can already sense this is where some students might get tripped up. Tell me more about how this activity tackles that.
Speaker 2:So the activity might say something like, in 2010, there were 296,600,000 smartphone owners.
Speaker 1:Okay.
Speaker 2:Now the function they're given measures the number of owners, in millions. Okay. So students can't just plug in 296,600,000. They have to understand that that's 296,600,000 in the context of this specific function.
Speaker 1:Those sneaky units. They always seem to pop up where you least expect them.
Speaker 2:I know. Right? And it's not just the units of the output that they have to pay attention to. The input is defined as years after 2000. So for the year 2010, they would need to use an input of 10, not 2010.
Speaker 2:And then, you know, these might seem like small details Right. But they're actually really important. The authors of this lesson really emphasize, you know, that precision. Mhmm. And this activity really forces them to hone that skill.
Speaker 1:Yeah. It's like they're learning to be, precise chefs in the kitchen of algebra, like Yeah. One misplaced ingredient or in this case, digit, and the whole dish is thrown
Speaker 2:off. Exactly. And these activities are really just the beginning. The lesson is packed with all these opportunities for, you know, deeper learning, but it also anticipates those common misconceptions that can sometimes trip students up. You know, speaking of misconceptions, that last point about units in the smartphones activity
Speaker 1:Mhmm.
Speaker 2:That brings up a big one.
Speaker 1:Oh, absolutely. Because it's so easy.
Speaker 2:It's so easy for students to get lost in those large numbers
Speaker 1:Yes.
Speaker 2:Especially when we're talking, like, 1,000,000 or 1,000,000,000. They might see, like, 2,320,000,000 and not fully grasp the magnitude
Speaker 1:Right. Exactly.
Speaker 2:Treating it like it's a few 1,000.
Speaker 1:Precisely. It's like they need a way to visualize just how huge e those numbers really are.
Speaker 2:Right. Right.
Speaker 1:So as teachers, you know, we can help them bridge that gap Yeah. By having them create a visual scale.
Speaker 2:Mhmm.
Speaker 1:I imagine, like, a number line. Right? It starts at 0 and has markers for every million, k, then every 10,000,000.
Speaker 2:Okay.
Speaker 1:Every 100,000,000.
Speaker 2:It would be like, like, zooming out on a map or something, helping them see the bigger picture of those numbers instead of just, like, the individual digits.
Speaker 1:Exactly. Yeah. Another strategy is to connect it to something familiar. You know, ask them, like, how many people live in our city? Right.
Speaker 1:And then compare that number to something like the global number of smartphone users.
Speaker 2:Oh, wow.
Speaker 1:And suddenly, those 1,000,000 and billions become a lot more concrete.
Speaker 2:Yeah. It's all about making those abstract concepts, like, relatable
Speaker 1:Yes.
Speaker 2:To their world.
Speaker 1:Exactly.
Speaker 2:Are there any other common misconceptions that this lesson plan helps us kind of, like, anticipate.
Speaker 1:Yeah. There's another big one that I often see come up
Speaker 2:Uh-huh.
Speaker 1:In the boiling water activity
Speaker 2:Right. K.
Speaker 1:Where students need to sketch a graph
Speaker 2:Right.
Speaker 1:Based on data points about water temperature over time.
Speaker 2:Oh, I know this one.
Speaker 1:You know this one. They get so fixated on finding the exact temperature at every single second that they, like, miss the overall trend the graph is supposed to represent.
Speaker 2:Right. They crave that absolute precision.
Speaker 1:Yes.
Speaker 2:But sketching a graph is more about capturing the essence of the function
Speaker 1:Right.
Speaker 2:The general relationship between those variables.
Speaker 1:Yeah. It's like, it's like the difference between, drawing a perfect portrait and, like, sketching a caricature.
Speaker 2:Oh, I like that.
Speaker 1:You know, they both capture something essential, but in different ways.
Speaker 2:What a great analogy. And in this case, you know, we want them to feel comfortable with the idea that there can be multiple correct sketches as long as they reflect the given data points and that overall trend.
Speaker 1:Absolutely. It's about shifting their thinking from, like, find the one right answer to, like, show me you understand the relationship.
Speaker 2:Exactly. Yeah. And that shift in thinking is so important as they, you know, progress to more complex algebraic concepts.
Speaker 1:Right. Right. So to recap, we've covered, like, 2 major potential pitfalls. Mhmm. You know, grappling with the magnitude of large numbers.
Speaker 1:Mhmm. And overcoming that, like, need for absolute precision when sketching graphs. Are there any other, like, teacher takeaways this this deep dive reveals?
Speaker 2:You know, what I find so fascinating is how this lesson plan well, it's focused on, you know, the nuts and bolts of function notation.
Speaker 1:Mhmm.
Speaker 2:It's subtly laying this groundwork for so much more in algebra.
Speaker 1:Oh, okay.
Speaker 2:It's like opening a door to this whole new world of mathematical understanding.
Speaker 1:Okay. I'm intrigued. What's behind that door? What can our students do once they've, like, mastered the basics of function notation?
Speaker 2:Well, think about it.
Speaker 1:Okay.
Speaker 2:Function notation is the key to understanding transformations.
Speaker 1:Right.
Speaker 2:Compositions of functions, inverse functions. It underpins so many more complex concepts.
Speaker 1:Why
Speaker 2:Once they can confidently, you know, read and write in this new language, those conversations become so much richer.
Speaker 1:It's like they've they've learned the alphabet, and they can finally start forming, like, words and sentences. Yes. And pretty soon, they'll be writing, like, whole stories with math.
Speaker 2:That's a fantastic way to put it. And speaking of stories, this lesson plan has certainly given us a lot to think about.
Speaker 1:It really has. And speaking of things that deserve a huge shout out, a big thank you to the brilliant minds at Illustrative Math
Speaker 2:Yes.
Speaker 1:For creating such a a a well structured and engaging lesson plan. So for our listeners who are, you know, about to teach this concept Yep. Any final, like, words of wisdom to send them off feeling ready to to inspire?
Speaker 2:You know, one thing that I find really resonates with my students is when they can see how math is relevant
Speaker 1:Oh, absolutely.
Speaker 2:To their own lives.
Speaker 1:It's one thing to learn about, like, you know Right. Abstract functions.
Speaker 2:Mhmm.
Speaker 1:But it's a whole different ballgame when they realize they can use those functions to, like, model things they actually care about.
Speaker 2:Exactly. So maybe after you've, you know, guided them through the ins and outs of function notation, challenge them to find a scenario Okay. From their own life Got you. That they could represent with the function.
Speaker 1:Oh, I like that.
Speaker 2:Could they track the growth of their savings account over time, or, could they model the arc of their basketball shot? Something like that.
Speaker 1:I love that. It turns them from, like, passive learners into, like, mathematical detectives.
Speaker 2:Yes. Exactly.
Speaker 1:Searching for patterns and relationships in their own worlds.
Speaker 2:And who knows? Yeah. They might even discover that math can be a little bit fun.
Speaker 1:That's the dream, isn't it? To spark that joy of discovery
Speaker 2:It is.
Speaker 1:And show them that math isn't just about, like, memorizing formulas. It's about making sense of, like, the world around us.
Speaker 2:Well said.
Speaker 1:Thanks for joining us on this deep dive into the fascinating world of function notation. Notation. We hope you've walked away with some fresh insights and, practical strategies to empower your students and ignite their mathematical curiosity. Until next time, happy teaching.