Ever feel like some things in life, they just spiral out of control? Like, you leave a banana out for 2 seconds and suddenly it's a science experiment.

You're telling me, my kids, they could grow mold on like a doorknob. I swear.

Right. But what if we could take that ew factor, that rapid change, and actually use it to teach? That's what we're diving into today with a little help from, you guessed it, illustrative math.

And get this. We're talking exponential functions. Buckle up.

This deep dive, we're tackling lesson 8 from their algebra curriculum, exponential situations as functions.

Because let's be real. Functions can be one of those things that make students hit the math breaks.

Totally. But this lesson, it's clever. It starts by making sure everyone's really clear on what a function is, like, no shaky definitions, just a solid understanding.

Which is huge because if they don't get that foundation, everything else is gonna feel all wobbly.

Exactly. We're talking about recognizing functions everywhere, tables, descriptions, you name it, and then being able to write them out all nice and neat using function notation.

Like, taking those word problems from to uh-huh. I can totally write an equation for that.

Now imagine you're a student maybe a little nervous about this whole function thing, and the lesson opens with rainfall in Las Vegas.

I'm already intrigued. Rain in the desert. Tell me more.

Right. It's brilliant. They use this real world graph of rainfall in Vegas. And immediately, your brain's like, wait. Rainfall can be a function.

Suddenly, you're invested. You gotta figure out how that works.

Exactly. And that's when they've got you hooked. You're thinking about functions even if you don't realize it yet. But it gets even better. Hold on to your hats because things are about to get a little funky.

Oh, I love where this is going. We're going full science experiment, aren't we?

You know it. We're talking moldy bread, people. Alright.

Now this is something everyone can get behind. Nothing says exponential growth quite like a piece of bread that's been sitting out for a little too long.

Okay. But seriously, mold and math, how does that work?

This, my friend, is where things get really fun. Remember how we talked about representing functions in multiple ways? Well, this activity, it's got it all. Tables, graphs, and even equations, all centered around everyone's favorite fungus.

So instead of just being told about exponential functions, students get to see them in action up close and personal.

Precisely. It's all about making those abstract concepts concrete, relatable, even a little bit gross if we're being honest. Because, hey, sometimes a little you factor can be a powerful learning tool.

Okay. I'm starting to see how this could really click for students, but let's be real. Even with the most exciting activities, there are always a few potential roadblocks. What are some things teachers should anticipate?

You're absolutely right. No lesson plan is foolproof. What might seem obvious to us, like how time plays a crucial role in calculating that mold growth can actually be surprisingly tricky for students to wrap their heads around.

Okay. So how do we help them connect the dots? Make time a less abstract concept in this context?

Well, one thing we gotta make sure of is that they're comfy with the whole idea of continuous versus discrete graphs.

Right. Because we're not talking about mold magically appearing overnight. It's a gradual process. Right.

Exactly. It's not just jumping from one point to another. It's a smooth ongoing thing.

So how do you make sure they get that?

Visuals, my friend. Mhmm. Visuals are your best friend here. Think timelines. Think diagrams.

Yeah. Anything that lets them actually see the continuous growth happening.

So we're talking about bringing the visual element into understanding these continuous graphs, but what else can trip students up with this whole moldy bread idea?

Well, there's this tricky thing about exponential growth. Right? Yeah. Can't just keep doubling forever.

Right. Like, at some point, the mold runs out of bread. It's not gonna take over the whole world, thankfully.

Exactly. And that's where we gotta talk about domain, that realistic range for our function. Mold might be a little too good at growing sometimes, but it's not gonna violate the laws of nature.

So we need to make sure students are thinking about those boundaries. Right?

Absolutely. This is where you can bring in those real world connections, get them thinking critically about the limitations of math models.

Which is so important. Math isn't just about equations in a vacuum. It's about understanding how those equations actually apply to the real world.

My 100%. And that's something this lesson does really well. It's not just about the numbers. It's about the thinking behind the numbers.

Speaking of which, once students have gotten their hands dirty with the moldy bread, there's another activity in this lesson plan that's really gonna solidify their understanding of exponential functions.

Right. We're moving on from the bread basket to, well, a whole bunch of other cool stuff.

I love it. Out with the mold, in with the new. Okay. So tell me more. What's this next activity all about?

It's called functionally speaking, And this is where things get really interesting. We're talking bacteria, car depreciation, even algae blooms.

Okay. Hold on. I'm getting flashbacks to my high school biology class, in a good way, of course. But seriously, how does all of that tie into exponential functions?

Oh, it's beautiful, really. You see this activity, it's like a greatest hits compilation of real world situations that just stream exponential function.

Okay. I'm intrigued. So walk me through it. How does it work?

So imagine this. You've got these scenarios. Right? Like a population of bacteria doubling every hour or a car losing a certain percentage of its value every year.

Okay. I'm seeing the exponential connection here. But what are the students actually doing with these scenarios?

Great question. They're not just observing this time. They're really getting into the nitty gritty of why these situations are exponential. And that means identifying those independent and dependent variables, which, let's be honest, can be a bit of a head scratcher for some students.

Oh, tell me about it. It's like which one is the chicken and which one is the egg. Sometimes it all gets jumbled in my brain.

It happens to the best of us, but that's where clear explanations and real life examples are so crucial, and this activity provides a great framework for that.

So instead of just saying independent variable and dependent variable, we're talking about cause and effect. Right? Like, in the car example, the age of the car is gonna influence its value.

Exactly. The older the car, the lower the value. It's all about helping students see those relationships, those cause and effect connections, and then being able to translate that into the language of math, which means dot, you guessed it, dot function notation.

Which, let's be real, can feel like a whole other language sometimes.

Oh, for sure. It's like the secret code of algebra. But once students crack that code, it's like a whole new world of math opens up to them.

Totally. And speaking of opening up new worlds, this lesson plan doesn't just stop at the basics. There's also an optional activity that's sure to challenge even the most eager learners.

Right. We're talking deciding on a graphing window. And trust me, it's more exciting than it sounds. This activity really takes things to the next level by connecting to the whole world of mathematical modeling.

Okay. I have to admit, that title doesn't exactly scream excitement, but I trust you. So break it down for me. What makes this activity so challenging?

It's all about perspective, really. See, graphing these exponential functions, it can be kinda tricky because they can grow or decay so quickly. One minute you're looking at a flat line, the next it's shooting off the chart. This activity forces students to really wrestle with that, to think critically about how they're representing that growth or decay.

So it's like, how do you choose the right zoom level on a camera to capture the most important parts of the action?

Perfect analogy. It's all about finding that sweet spot, that Goldilocks window that tells the most accurate and insightful story.

I love that, finding that just right view. Okay. So we've covered a lot of ground here from mold experiments to card appreciation to finding the perfect graphing window. What are some key takeaways for teachers who are about to embark on this exponential adventure with their own students?

Well, I'd say the biggest takeaway is don't be afraid to get a little creative. This lesson plan is a great example of how you can take something as simple as a piece of bread and turn it into a powerful learning experience.

Right. Who knew mold could be so educational?

Seriously. But it's not just about the mold, is it? It's about those moments, those connections students make when they see abstract concepts come to life.

And this lesson does such a great job of providing those opportunities from those hands on mold observations to those thought provoking graphing challenges. It's all about making those connections.

Absolutely. And speaking of connections, we gotta talk about cause and effect. Helping students understand that relationship between independent and dependent variables, that's huge.

It's like the key to unlocking the whole concept of functions.

Right. And once they see that, once they can look at a situation and say, okay, this is causing that to change, then they can start to make sense of all those equations and graphs and things.

It's about giving them the tools to not just solve problems, but to actually understand the why behind the how.

Exactly. And that's what makes this lesson plan so powerful. It's not just about memorizing formulas. It's about developing those critical thinking skills, that ability to analyze a situation, to identify patterns, to make predictions.

Skills that will serve them well no matter what they choose to do in life.

A 100%. Now we've talked a lot about the big picture stuff, but let's not forget about the practicalities of teaching this lesson. One thing that might trip some teachers up is that whole concept of time as a variable.

Right. Because time, it's always moving, always changing. How do we capture that on a graph? How do we help students make sense of that?

Well, one thing I always tell teachers is don't be afraid to slow things down. Instead of just looking at the overall growth of the mold, have students zoom in on those early stages.

So almost like they're watching a time lapse video in super slow motion.

Exactly. Encourage them to think about what's happening hour by hour or even minute by minute. How much is the mold growing in a tiny increment of time?

And how does that tiny increment of growth contribute to the overall exponential pattern?

Exactly. It's all about connecting those small scale changes to that big picture idea of exponential growth.

And speaking of big picture ideas, we can't forget about the importance of real world connections. This lesson does such a great job of grounding those abstract math concepts in relatable examples.

Totally. And I think that's one of the most important things for teachers to remember. Always be on the lookout for those real world connections. Whether it's mold, bacteria, or something completely different, finding ways to make math relevant to students' lives, that's what's gonna make it stick.

So true. And on that note, I think it's time to wrap up this exponential extravaganza.

This has been fun. I always love a good deep dive into illustrative math.

Me too. Big thanks to the authors of Illustrative Math for these incredible resources. And to our listeners, thank you for joining us for another deep dive. Until next time. Happy teaching.