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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.)
Ever find yourself scrolling through your social media feed like, woah. How this topic blow up so fast? Or maybe you're hitting the gym, tracking your progress wondering, am I actually getting fitter? That, my friend, is average rate of change in action, and we're diving deep into it today.
Speaker 2:It's wild. Right? Average rate of change gives us this way to measure those trends, those shifts. And it's not just like, oh, I feel like this is happening.
Speaker 1:Right. It's more than a gut feeling. It's, like, instead of just saying, purr, it's getting colder out here, we can say, hold on. The temperature dropped an average of 5 degrees every hour.
Speaker 2:Exactly. And that shift from it's getting colder to it dropped 5 degrees per hour, that's what gives us the power to really analyze and predict.
Speaker 1:So we're going from this sort of general feeling to a precise measurement. Okay. So we're focusing on a lesson plan for teachers today. But, honestly, whether you're a teacher or just curious about, you know, how the world works, this deep dive is for you. Understanding how we measure change is relevant to all of us.
Speaker 1:Right? Mhmm. And this particular lesson plan uses something we all experience to hook those students those chilly winter days.
Speaker 2:Smart move. Tapping into that shared experience of, ugh, it's freezing. Plus, it cleverly connects to what students already know about slope.
Speaker 1:Right. Like that steep downhill on a roller coaster. You just know it's about to get real. It's that same visual intuition, but now we're applying it to temperature changes. And they don't just throw it all out there at once either.
Speaker 1:This lesson plan, temperature changes.
Speaker 2:And they don't just throw it all
Speaker 1:out there at once either. This lesson plan introduces
Speaker 2:these 2 students, Tyler and May,
Speaker 1:who have different
Speaker 2:opinions on whether the temperature dropped faster between certain times. Such a good way to spark that curiosity, that debate. Oh, I love that. Makes it a puzzle.
Speaker 1:Like, okay. Who was right? And how do we even figure that out using math?
Speaker 2:Exactly. And that's where paying attention to those units comes in, like degrees Fahrenheit per hour.
Speaker 1:Units. They always seem so small, but I have a feeling they make a big difference for students trying to wrap their heads around this stuff.
Speaker 2:Huge. It's not just a random number anymore. Right? It's literally saying, for every hour that ticks by, boom, the temperature dropped this many degrees. That's powerful.
Speaker 1:It grounds the math in a real world phenomenon. It's not just theoretical anymore.
Speaker 2:Exactly.
Speaker 1:Okay. So we've set the stage with this relatable scenario about winter, sparked some curiosity about who's right. Students are starting to get a feel for what average rate of change means. Mhmm. But how do we actually calculate it?
Speaker 1:Let's find out. So how does this lesson plan actually tackle the math part? I mean, calculating average rate of change sounds a little intimidating.
Speaker 2:You'd think so. Right? But this lesson breaks it down in such a smart way, very accessible, even fun. There's this activity, drop some more, and it lets students try out different strategies. It's not just about plugging numbers into a formula, which, let's be honest
Speaker 1:Yeah. Sometimes formulas just make my eyes glaze over. So tell me more about these different approaches.
Speaker 2:Okay. So one way is to look at, like, the hourly changes in temperature and find the average. It's like saying, alright. It went down 2 degrees this hour, then 3 degrees the next, then 4.
Speaker 1:So on average, it dropped 3 degrees per hour. I see how this helps students get a feel for what average rate of change really means.
Speaker 2:Exactly. It's about how much it changes over a period of time. Now another approach they highlight is calculating the total temperature change and dividing that by the total time.
Speaker 1:So if it dropped a total of, say, 15 degrees over 5 hours, hours, boom, 3 degrees per hour average rate of change. This is making sense.
Speaker 2:You got it. And here's where it gets really neat. They connect this idea to something students already know slope.
Speaker 1:Right. Like, how steep a line is on a graph. I remember learning about that, but I never would connected it to average rate of change.
Speaker 2:It's such a brilliant link, and they do it so smoothly. Students start to see that finding the average rate of change is basically like finding the slope between two points on a graph.
Speaker 1:That's so cool. So it's not just a brand new thing. They're building on what they already know.
Speaker 2:Exactly. And speaking of connections, they do introduce the actual formula,
Speaker 1:but Perm rule intensifies.
Speaker 2:Average rate of change equals the change in the output divided by the change in the input.
Speaker 1:Having that formula is handy, especially as things get trickier.
Speaker 2:Definitely. But it's not just a jumble of symbols. You know? It represents something they've already started to understand. But get this.
Speaker 2:This lesson plan doesn't stop at winter temperatures. They take it a step further with some real world data.
Speaker 1:Okay. Now I'm intrigued. Tell me more.
Speaker 2:Students get to analyze population graphs of California and Texas. Talk about making math relevant.
Speaker 1:That's awesome. Showing them it's not just stuck in textbooks. It's all around us. I bet those graphs spark some interesting conversations.
Speaker 2:Totally. Especially because they focus on specific periods, 1970 to 2010, and 1900 to 2000.
Speaker 1:I'm guessing those periods weren't picked randomly.
Speaker 2:Right. By looking at different time frames, students see how average rate of change can reveal different trends, different stories. It's like using math to understand how things change over time. Plus, working with those real world graphs means they practice estimating coordinates too.
Speaker 1:Which is a good reminder that in the real world, it's not always about getting that perfect answer. Sometimes it's about using what we know to make good estimates. Speaking of which, we've talked about how this lesson introduces average rate of change, connects it to slope, uses real world data. But what about those moments where it just clicks for students? It's like those light bulb moments, right, when a concept really clicks.
Speaker 1:But even with the best explanations, some things can still trip people up. For
Speaker 2:sure. And with average rate of change, there are definitely a few common misconceptions that pop up.
Speaker 1:It's great that this lesson plan actually names a few of those potential pitfalls. Right? Gives teachers a heads up.
Speaker 2:Totally. One of the biggies is mixing up average rate of change with instantaneous rate of change.
Speaker 1:Yeah. I can see how that could happen. It's like thinking the average tells you exactly what's happening at every single moment.
Speaker 2:Exactly. It's easy to forget about the average part. Think of it like a road trip. Your average speed might be 50 miles per hour, but you were probably stuck in traffic, sometimes going way slower.
Speaker 1:And other times, you were cruising.
Speaker 2:Exactly. That average evens out all the ups and downs over the whole trip.
Speaker 1:Right. It smooths everything out. It's
Speaker 2:the same with average rate of change, big picture trend, not necessarily capturing every little zig and zag along the way.
Speaker 1:Makes sense. Now what about that other misconception, something about negative average rate of change?
Speaker 2:Oh, yeah. That one throws people off. They think a negative average rate of change means the thing never increased during that time.
Speaker 1:Oh, interesting. So just because the overall trend is downward, doesn't mean there couldn't have been moments of, like, little upward blip.
Speaker 2:Exactly. Negative average rate of change just means, overall, there was a decrease during that specific interval. But within that time frame, could definitely be some increases happening.
Speaker 1:Wow. I hadn't thought about it like that. It's like a good reminder not to jump to conclusions based on just one piece of information.
Speaker 2:Absolutely. And that's what's so cool about this concept. It makes you think carefully about the details of change.
Speaker 1:So we've got confusing average with instantaneous and then that whole negative average rate of change thing. Was there a third misconception they mentioned?
Speaker 2:Oh, yeah. This one trips students up too. It's thinking you can only compare average rates of change if the time periods are exactly the same length.
Speaker 1:Oh, right. But why wouldn't you be able to compare, say, how much something changed in an hour versus how much it changed over 3 hours?
Speaker 2:You absolutely can. That's where understanding those units is super important. Average rate of change tells you how much something changes per unit of input. So as long as we're clear about our units, the length of the interval isn't as critical as you might think. Comparing the average temperature drop per hour over 2 hours to the average temperature drop per hour over 6 hours is completely valid.
Speaker 1:So it's not about the total time. It's more about the rate of change within that chunk of time, like a ratio.
Speaker 2:Exactly. And the great thing is this lesson plan doesn't just point out these misconceptions. It gives teachers practical strategies, you know, and relatable examples to clear up that confusion.
Speaker 1:Like a toolkit. I love that. Mhmm. You know? And I'm not even a teacher, but I'm finding all of this so insightful.
Speaker 1:We went way beyond just defining average rate of change Yeah. Connected it to real stuff, different ways to calculate it, even anticipated where students might get tripped up.
Speaker 2:It's about giving students this powerful lens to analyze the world. Right? Average rate of change helps make sense of trends, make predictions. Everything from population growth to climate change to even how viral a social media post might become.
Speaker 1:It really is all connected. On that note for everyone listening, as you go about your day, pay attention to those trends around you. What stories are they telling? What happens when we start putting numbers to those changes? Food for thought.
Speaker 1:This has been another awesome deep dive. Big thanks to the authors of illustrative math for this thought provoking lesson plan. Until next time, keep those brains engaged.