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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.)
Hi, everybody. Welcome back. Today, we are taking a deep dive into something that might seem a little familiar, especially if you remember high school math class.
Speaker 2:Oh, yeah. I think we've all been there.
Speaker 1:We are talking about inverse functions. But don't worry. This is not about going back to algebra class. This is about showing you how this really cool math concept can be used to understand and even predict things in the real world.
Speaker 2:I think sometimes people have this idea that math is just about memorizing formulas and solving equations, but it's so much more than that.
Speaker 1:It really is. It's a way of thinking, a way of seeing patterns and relationships in everything around us.
Speaker 2:Exactly. And that's what we're going to explore today. We're taking a look at a lesson plan from Illustrative Math.
Speaker 1:They always have such great stuff.
Speaker 2:They do. And this lesson does a brilliant job of explaining inverse functions in a way that is both engaging and easy to understand. Plus, they use examples that are actually relevant to our lives.
Speaker 1:Which is key. So what are we looking at today? What is the kind of crux of this deep dive?
Speaker 2:Well, the lesson focuses on two main things. 1st, how to find the inverse of a linear function.
Speaker 1:Okay.
Speaker 2:And second, and this is the really cool part, how to use that skill to make sense of real world data and solve problems.
Speaker 1:So it is one thing to understand the math on paper, but it is a whole other thing to see how it plays out in the real world.
Speaker 2:Exactly. And to make it even more interesting, they use examples like, get this, water draining from a tank k. And the rise of cell phone use over time.
Speaker 1:So they're taking something that could be kinda abstract, this idea of inverse functions, and grounding it in scenarios that we can all relate to, which I love.
Speaker 2:Yeah.
Speaker 1:So let's start with this water tank example.
Speaker 2:Okay.
Speaker 1:Because I think it's a really clear illustration of how inverse functions work.
Speaker 2:It is.
Speaker 1:So pick her this. You have a water tank and it's slowly emptying out. Now imagine we have this handy function, this equation, that tells us how much water is left in the tank at any given time.
Speaker 2:And this might seem overly simplistic, but this kind of function where one thing changes predictably over time, we see this everywhere.
Speaker 1:Oh, yeah. I mean, think about it.
Speaker 2:Fuel in your car.
Speaker 1:Your phone battery.
Speaker 2:Exactly. Things that deplete at fairly predictable rates.
Speaker 1:Right. So this water tank example, it's like the starting point. Right?
Speaker 2:Right.
Speaker 1:It helps us understand the core concept before we start applying it to more complex stuff.
Speaker 2:Exactly. It gives you a foundation. And in the lesson, they provide you with a specific function that describes how the water level in the tank changes over time. They give you the starting amount of water. They tell you how fast it's draining.
Speaker 1:So far so good.
Speaker 2:Right. Pretty straightforward. But here's where it gets interesting. What ish instead of wanting to know how much water is left after a certain amount of time, we wanna know how long it took for the tank to reach a certain level.
Speaker 1:It's like we're reversing the question.
Speaker 2:Exactly. And that's precisely where inverse functions come in. They give us the power to flip the script, to essentially rewind the process.
Speaker 1:So instead of using the time to figure out the water level
Speaker 2:Right.
Speaker 1:We're using the water level to figure out the time.
Speaker 2:Exactly. We're working backwards. And the coolest part is that there's a way to do this mathematically. We can find the inverse of that original function, the one that told us the water level based on time, and use it to answer this new question.
Speaker 1:So the original function gives us water level given the input of time Right. And the inverse function gives us time given a certain water level.
Speaker 2:Exactly. It's like a reverse lookup.
Speaker 1:I love it. And the lesson walks you through the steps of how to find that inverse function. But what I think is particularly clever is how they then connect this to something even more relatable, cell phones.
Speaker 2:Yes. And I love this example because it takes what we just learned about the water tank, which is a pretty straightforward concept, and applies it to something that is more complex and arguably more interesting.
Speaker 1:Because let's be real, cell phones are like an extension of ourselves these days. They're not just a convenience. They're practically a necessity.
Speaker 2:Absolutely. So the lesson brings in this real world data about the percentage of US homes that rely solely on cell phones, meaning they've completely ditched their landlines. And it shows how this percentage has been increasing over time.
Speaker 1:Which we all know intuitively. Right? Like, it's not news that more and more people are going mobile only, but to actually see the data, to see how it could be modeled using a linear function, that's pretty cool.
Speaker 2:Right. And that is the heart of data analysis. It takes what we observe and tries to fit it into a model we can use to understand trends and even predict future behavior, and the lesson does a fantastic job of illustrating that process.
Speaker 1:And it's not just about looking at a bunch of numbers. Right? They encourage students to actually visualize this trend using graphs.
Speaker 2:Yeah. They plot the data points, and then they try to find a linear function that best fits that data. And the cool thing is that this is exactly how analysts in all sorts of fields make sense of the world around us. They look for patterns. They fit functions to those patterns, and that allows them to make predictions about what might happen in the future.
Speaker 2:So you can see how this seemingly simple concept of inverse functions
Speaker 1:Right.
Speaker 2:Can be applied to understand some pretty complex and relevant issues.
Speaker 1:Absolutely. So they've got the water tank example to illustrate the basic concept, and then they hit us with the cell phone data to show how it plays out in the real world. But they don't stop there, do they?
Speaker 2:No. They don't. They take it a step further and challenge students to use what they've learned to make predictions about future cell phone trends, Like, when will a certain percentage of HOMs rely solely on cell phones? It's like a little forecasting exercise. And, of course, this naturally leads them back to the concept of inverse functions, but this time, they're applying it to this real world dataset.
Speaker 1:I love that. It's one thing to understand a concept in theory, but it's a whole other level to be able to apply it to real data and make predictions about the future.
Speaker 2:Absolutely. And that's the power of math.
Speaker 1:It's not just about crunching numbers. It's about using those numbers to understand and navigate the world around us.
Speaker 2:And you know what I find interesting about this particular lesson?
Speaker 1:What's that?
Speaker 2:It's not just about teaching the concept of inverse functions. It's about giving teachers the tools to help their students really grasp this idea.
Speaker 1:Oh, you mean, like, they actually address some of the common challenges that students might face that's smart.
Speaker 2:Exactly. They acknowledge that even with the most engaging examples, algebra can still be tricky.
Speaker 1:It's true. And sometimes it's not even the math itself. It's the way it's presented. You know? Yeah.
Speaker 1:Like, the notation can be intimidating.
Speaker 2:Absolutely. And they highlight that. One of the biggest hurdles they point out is function notation.
Speaker 1:Oh, yeah. That's a big one. The language of math can sometimes be the most confusing part.
Speaker 2:Right. Like, students might totally get the concept of a function, but then they see f x, and it throws them for a loop.
Speaker 1:It's like a secret code they haven't learned yet even if the underlying idea is something they understand.
Speaker 2:Exactly. And the lesson emphasizes that teachers need to be mindful of this. It's not enough to just teach the math. You have to teach the language of math as well.
Speaker 1:That makes sense. So that's one hurdle, function notation. What other challenges do they identify?
Speaker 2:Well, the other big one is the flipping concept itself.
Speaker 1:Yeah. That whole input output swap. It's like you're looking at a mirror image, but you have to reorient yourself.
Speaker 2:Exactly. And for some students, that can be a real mind bender. They get stuck thinking about the relationship between the variables in one direction, and it's tough to shift gears and think about it in reverse.
Speaker 1:It's like driving in England. Suddenly, everything is on the wrong side. Well, not wrong. Just different.
Speaker 2:Right. You know how to drive, but you have to rewire your brain a little to adapt to the new
Speaker 1:layout. Exactly. But the good news is that this lesson plan doesn't just leave teachers hanging with these potential problems.
Speaker 2:No. They actually offer some really practical strategies for overcoming these hurdles, which I think is really helpful.
Speaker 1:So what are some of their tips? Like, how do you help students decode the language of f x or wrap their heads around this whole flipping business?
Speaker 2:Well, one of the key things they stress is the importance of making connections to concrete examples.
Speaker 1:Oh, I like that. So instead of just throwing abstract symbols at them, you ground the concepts and things that students already understand.
Speaker 2:Precisely. So for instance, instead of just talking about f x in the abstract, converting temperatures from Celsius to Fahrenheit. Okay. So they can see how a real world conversion works, and then you can show them how that
Speaker 1:same process can be represented mathematically using function notation. Exactly.
Speaker 2:And by using examples that are relevant to their lives, it makes the abstract feel more tangible, less intimidating.
Speaker 1:It's like, hey, you already get this concept. You just didn't know there was fancy math language for it.
Speaker 2:Exactly. And when it comes to that flame problem, the lesson emphasizes the power of visuals.
Speaker 1:Woah. Like graphs.
Speaker 2:Exactly. They suggest having students plot a function and its inverse on the same graph.
Speaker 1:I could see how that would be helpful. Like, you could literally see the relationship between the two, how they mirror each other across that diagonal line.
Speaker 2:Right. It takes something that's very abstract and makes it visual concrete, and that can make a huge difference for students who are struggling to grasp the concept.
Speaker 1:So we've talked about making these concepts stick for students. But I think it's worth asking, why should anyone really care about inverse functions? Like, beyond the classroom, what's the big takeaway for someone who's not planning on teaching algebra or, you know, using it in their everyday life? Why is this concept even important?
Speaker 2:That's a really great question, and it gets at the heart of why we even bother exploring these lesson plans. It's because inverse functions, they aren't just some abstract math thing. They're, like, they're tools. Right? And these tools, they help us understand and even mess with the relationships between things in the real world, which I think is super cool.
Speaker 1:So it's like having this superpower to see how things are connected even when those connections aren't obvious.
Speaker 2:Yes. Exactly. It's like having x-ray vision but for data.
Speaker 1:I love that. So give me an example. How does this play out in real life?
Speaker 2:Well, think about it. We talked about the water tank example, right, and how you can use an inverse function to figure out how long it took for the tank to reach a certain level. You can apply that same principle to all sorts of things.
Speaker 1:Like what?
Speaker 2:Like, imagine you're trying to figure out how long it takes for a certain medicine to reach a certain concentration in your bloodstream. Or maybe you're a marketer, and you're trying to understand the relationship between ad spending and website traffic.
Speaker 1:Okay. I'm seeing it.
Speaker 2:Or you're trying to predict the spread of information online, like how quickly a news story will go viral. All of these things, they involve understanding relationships between variables.
Speaker 1:And often those relationships aren't immediately obvious.
Speaker 2:Right. But if you can figure out the function that describes that relationship
Speaker 1:You can then find its inverse.
Speaker 2:And suddenly you have this whole new level of understanding, and you can start to answer questions that you couldn't before.
Speaker 1:And make better decisions, presumably?
Speaker 2:Exactly. Whether you're a scientist, a business person, or just someone trying to make sense of the world around you, being able to think in terms of inverse functions gives you a huge advantage. So, yeah, they might seem kinda abstract at first Okay. But trust me, they're way more relevant to our lives than you might think.
Speaker 1:That's so cool. It's like that moment when something just clicks, and you see the world in a whole new way.
Speaker 2:Exactly. And the best part is you don't have to be a math whiz to grasp these concepts. This lesson plan is a perfect example. It takes something that could seem intimidating and makes it accessible to anyone.
Speaker 1:I love that. It's like you don't need a PhD in mathematics to appreciate the beauty and the power of math.
Speaker 2:Exactly. And that's what we're all about here. Right? Yeah. Making these ideas accessible and showing people that math is for everyone.
Speaker 1:100%. Well, on that note, I think it's time to wrap up our deep dive into the world of inverse functions. We covered a lot of ground today, but hopefully, you walked away with a better understanding of this powerful concept and how it pops up in some unexpected places. And who knows? Maybe you'll even start seeing inverse functions places.
Speaker 2:And who knows? Maybe you'll even start seeing inverse functions everywhere you look.
Speaker 1:Bet you will.
Speaker 2:Yeah.
Speaker 1:Thanks for joining us.