You know, like, when we talk about, one thing depending on another in math? Yeah. Like, how the area of a circle, it totally depends on the radius. Right?

Right.

So function notation, it's like our way of writing that relationship, but, like, in a neat and concise way. You know? Yeah. And, today's deep dive, it's all about giving you the tools to teach this to your algebra students.

Absolutely. And we're gonna be, dissecting a lesson plan from illustrative mathematics.

Okay.

It's an open educational resource for, really high quality math instruction.

Jeez. Yeah.

And specifically, this plan helps algebra one students unlock the power of function notation.

Unlocking the power. I like that.

Yeah.

That makes it sound like there's more to it than just plugging in numbers.

Definitely.

And speaking of plugging in, this lesson plan, it kicks off with an activity. Right? Notice and wonder.

Right. It's, it presents these two tables of values, but there's no context.

Yeah.

No labels, just numbers. And this is where it gets interesting. Instead of saying, hey. These are functions, it just asks them to notice any patterns and then wonder about how those numbers, how they relate to each other.

Oh. So it's like setting the stage for those moments, letting students sort of figure it out on their own.

Yeah. Exactly. And it actually aligns with, constructivist learning theory, which basically says students learn best when they're actively building their own understanding.

That makes a lot of sense. So by holding back on those those labels, like input, output, it's almost like it's kind of pushing them to think differently about what the table represents.

Exactly. It is kind of genius in a way.

It is. It is.

It sets the stage for a smoother introduction to that formal notation, which can be really abstract for students seeing it for the first time.

Speaking of that formal notation, the next activity, 4 functions, jumps right in, doesn't

it? It does.

It has students match these verbal descriptions to their function notation.

Yes. And this is where I think things really start to click for students. They begin to see how those symbols, in the notation, they directly translate to the steps you'd take to calculate an output from an input.

Okay. So it's not just memorizing a formula.

Yeah.

It's about understanding the process. Exactly. So, like, if the rule is multiply the input by 3 then subtract 7, they would then connect that to f of x equals 3x minus 7.

Exactly. And this is where, you know, as teachers, we have to be ready for those common,

For sure. There's always gonna be misconceptions.

Right. Students might mix up the order of operations or not totally get what that x represents.

Yeah. Like, it's not just about getting the order right.

It's also understanding, like you said, what that x represents Mhmm. That it could be any number, Right? Depending on what we're talking about.

Yeah. It's like that x is like this tiny little placeholder for, like, a whole bunch of numbers.

Exactly. And the lesson does a good job of showing that by giving different kinds of examples.

Right. It's not just solving for x. It's like, well, what does x even mean in this case?

Exactly. And that leads into the next activity so well, rules for area and perimeter. Okay. This is where it goes from just those abstract functions to something more concrete.

Okay. So we're getting into the real world a little bit now.

Yes. So this activity is great for a couple of reasons, like instead of just those symbols, now students use function notation to describe something real, like how big a square is or the area based on its sides.

Oh, I like that. So it's not just f of x equals this plus that, you know? It's like, okay, that represents like an actual shape.

Exactly. And here's the really cool part. It doesn't just stop at writing out the function. It actually has them make a table

Okay.

Where they try different side lengths

Okay.

And see how the area changes.

Oh, so they're actually, like, seeing it in action.

Exactly. And then it goes a step further and encourages teachers to have them graph it.

Oh, wow.

So now it's not just numbers anymore. They're actually seeing visually how side length and area relate on a graph.

So they're going from words to symbols to tables to graphs all in one activity.

Yeah. It's pretty amazing. That's a

lot of ways to show a function.

It is. And I think seeing them all together really helps students understand that it's all connected, that they're all just different ways of showing the same idea. You know?

Yeah. It's like almost like learning different languages. Once you understand them all, you can choose whichever one makes the most sense in that moment.

Yes. Exactly. And this lesson does such a good job of setting them up to think that way.

So we've talked about the activities, like, what could be tricky and how cool this whole multi representation thing is. But before we finish up with the lesson plan, what about that lesson summary? What are the big takeaways they want teachers to remember?

Well, the summary really emphasizes how important it is for students to be able to connect those different ways of looking at a function. So like we said, the words, the equations, the tables, those graphs, it's not enough to just plug in numbers. Right? They need to understand what it all means and how those pieces fit together.

Yeah. It's like the difference between knowing the ingredients in a recipe and understanding how they work together to make, like, a cake or something.

Perfect analogy.

Okay. So the lesson summary, it's all about making those connections, like how powerful function notation can be for showing relationships. But let's take this deep dive even further. How could we change up this lesson for kids who need more help or kids who are ready for something more challenging?

Yeah. That's where we come in, right, as teachers.

Right.

We gotta meet those different needs in the classroom.

Exactly. So for students who like to learn by doing, what could we do to make this even more hands on?

Well, remember that rules for area and perimeter activity. Yeah. Like, instead of just picturing the squares in their heads, what if we bring in actual square?

Oh, cool.

Different sizes. Let them actually measure the sides, calculate the areas Okay. Even graph the results. We could use a big grid on the floor, have them use string and pushpins.

I love that. It's like that grid becomes a giant function machine.

Exactly. Yeah. And then for students who might need some extra support Yeah. We could break those activities down into smaller steps.

Okay.

So instead of matching the words straight to the function notation

Right.

Maybe start with just matching numbers to function notation Right. Like input output pairs.

Okay.

And then gradually bring in those verbal descriptions once they've got a handle on that.

That makes sense. Just build it up piece by piece.

Exactly.

Now for those students who are ready for a real challenge, what could we do to, like, really make them think?

We could introduce the idea of domain and range.

Oh, interesting.

That gets it how not all inputs make sense in every situation.

Okay.

Like, if we're talking about the side length of a square, we can't have negative numbers. Right?

Right. So you

have this whole other layer of thinking about functions.

Yeah. It's not just plugging anything in anymore.

Exactly. And when we start thinking about connecting this lesson to other math topics

Yeah.

The possibilities are really endless. Like what? Well, we already talked about geometry with area and perimeter stuff. Right. But this could also lead into talking about linear equations, slopes, e intercepts.

Oh, I see what you mean.

Because when they graph those area and perimeter functions, they're basically making lines. They are. So that connection could lead to some really good conversations about what those different representations tell us.

Wow. This has been great. We've covered so much

It has.

From those lesson plan activities to what could be tricky to how we can adapt it for different learners.

Absolutely.

But if there's just one thing you want our listeners to take away from this whole deep dive, what would it be?

I think it's that function notation. Yeah. Might seem kind of abstract at first. Yeah.

It's a lot of symbols.

But it's really a powerful tool for understanding how the world works.

Okay.

It's how we talk about relationships between things, whether it's like the path of basketball or how quickly something's growing. Yeah. It's everywhere.

I like that. So we're not just teaching formulas. We're teaching them to think about the world in a new way.

Exactly.

And on that note, as we wrap up our deep dive into the power of function notation, I wanna leave everyone with something to think about.

Okay. I like it.

Think about something your students do every day, something ordinary, but where you can see a function hiding in plain sight.

Oh, I like that.

How could you use that to teach them about function notation in a way that really clicks?

So many possibilities.

Right. From social media likes to how much a taxi costs, the world is full of functions just waiting to be discovered.

That's a great way to put it.

And to all our amazing listeners out there, the teachers who bring these ideas to life, thank you for joining us. Us.

Yes. Thank you.

We hope this deep dive gave you some fresh perspectives and ideas for your classrooms.

It's been fun.

And, of course, a huge thank you to Illustrative Math for such a thought provoking lesson plan. Until next time, keep exploring, keep asking questions, and keep diving deep.