Ever get that, like, pit in your stomach feeling when your phone's about to die? And you're nowhere near a charger. Ugh, the worst.

The struggle is real. But you know what? Algebra can actually help us understand that whole battery life thing a little better.

Okay. Now you've got my attention. I know we're diving into a lesson plan about using functions to model battery power, but how does algebra help us out when we're about to hit 0%?

Well, it's not that algebra will magically give you more battery but it can help you make sense of how battery power changes over time. And that's what this deep dive is all about. We've got our hands on some really interesting stuff for this one, excerpts from the actual lesson plan, and even some notes from the developers.

Oh, insider access. I love it. So walk us through this lesson. What makes it so special?

It's all about making math relatable. Instead of just equations on a whiteboard, this lesson taps into something students use every single day, their phones. They learn how functions can actually represent that relationship between time and battery power, whether it's charging or draining.

So they're figuring out the math behind something they already get just intuitively.

Exactly. And the lesson goes even further, digging into average rate of change.

So, like, not just seeing a number, but understanding how quickly or slowly something's changing over time.

You got it. And to get things started, there's a

warm up activity all about, you guess it, their own devices. Students think about how long

it takes to charge their phone, how long a full charge lasts, that sort of thing.

Such a smart way to ease them in. They're already thinking about the concepts without even realizing they're about to do some serious math.

Right. Then, they jump into the first main activity called charging a phone.

Now, picture this. Instead of a boring table of numbers, students get a

visual an image of a phone screen just like the one they're probably used to looking at.

Okay. I'm intrigued. What do they have to do with this phone screen image?

They get a starting battery percentage and have to predict when it'll hit that glorious a 100% charge.

Sounds like a job for average rate of change, but I'm guessing not every student will solve this the same way.

You're absolutely right. Some might go with a linear model, figuring the phone charges at a steady rate. But others might get a little more creative with their functions, maybe thinking about how their own phones tend to charge. And that's what makes this lesson plan so great. Right?

It lets students try different approaches, even spark some debate about which models are closest to reality.

It's like they're really digging into how math plays out in the real world. Love that. But what comes next? How does the lesson challenge them even more?

Okay. So activity 2 is called how long will it last. Now they're switching gears from charging to draining, which you think is just the flip side, but here's where things get really interesting. They get a little dose of reality.

A dose of reality. Are we talking about those moments when your phone dies way faster than you thought it would?

Exactly. Remember those mock up phone screens? Yeah. Well, they make a comeback. Students make their predictions based on that first image, but then, and this is crucial, they get a follow-up.

They see how the battery actually drained.

So it's not just about making a model. It's about testing it against real data. I bet that leads to some good moments.

Oh, absolutely. Chances are that real world battery drain won't perfectly match their predictions. Right? And that's where the learning explodes because now they have to figure out why their models were off and how to make them even better.

That is so smart. They're not just learning formulas. They're learning how to think like problem solvers. Speaking of which, I'm curious, did the developers anticipate any places where students might get tripped up during this activity?

They did. One common stumbling block is all about time, specifically, using it as a reference point. Those phone screens, they show time elapsed since the last charge, like 9 hours ago or 8 hours left.

I see where you're going with this. If they're not careful, they might mix up when time starts in their calculations.

Exactly. And that can throw everything off. That's why the lesson really stresses the importance of helping students find a solid starting point, like when the battery was at a 100% and measure everything from there.

Give them a clear starting line for that mathematical marathon. Okay. So time as a reference point. What was the other misconception they highlighted?

The other biggie is assuming that battery charging or draining is always linear, happening at the same rate.

Oh, tell me about it. My phone charges way faster when it's at, like, 10% compared to when it's almost full?

Precisely. And you know what? The developers included this really cool optional activity where students actually track the charging of their own phones. They can plot that data and quickly see that batteries don't always follow those neat straight lines.

That's amazing. Giving them the chance to see that those real world situations don't always fit into those neat little boxes we expect. Love it.

Right. And it opens the door to conversations about why that might be, you know, diving into the actual science behind how batteries work.

Okay. So they're busting myths about linear charging, but it doesn't stop there, does it? The lesson plan mentioned something about piecewise functions. I feel like that's where things get extra interesting.

You're telling me piecewise functions can be a little mind bending, but they're a perfect fit here. Think about it. Battery charging, it often happens in stages. Right? Like Oh.

That initial fast charge, then it slows down, might even hit a plateau.

Yeah. Definitely noticed that with my phone. So instead of one function being the right answer, this is about choosing the best model for each stage of the process.

Exactly. It's getting students to recognize that real world scenarios, they often need more complex nuanced models.

So fascinating. But I have to ask, are we really expecting high schoolers to grasp piecewise functions from this one lesson? That seems like a lot.

Well, the lesson plan doesn't go too deep into actually using piecewise functions, that are you ready for more section. It's more like a little nudge, a way to spark curiosity and get them thinking beyond the immediate lesson.

I love that. Like, hey, this is just the beginning. There's a whole world of mathematical modeling out there.

Exactly. And that's the beauty of a well designed lesson, isn't it? It's not just about conveying information. It's about planting those seeds of curiosity.

It's giving them the tools to ask those what if questions and then go find the answers themselves. So powerful.

Couldn't agree more. And speaking of powerful lessons, I think we should give a huge shout out to the authors of Illustrative Math for putting this incredible resource together.

Absolutely. This deep dive has been so eye opening. I feel like our listeners are walking away with so much to think about and even some practical strategies to use in their own classrooms.

Right. It's a great reminder that math is everywhere, even if something is ordinary as our phones. And those everyday connections, those can be the most powerful teaching tools of all.

So true. A huge thank you to our listeners for joining us on this deep dive. We'll be back soon with another fascinating exploration into the world around us. Until then, keep those mathematical minds buzzing.