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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.)
Hey, everyone, and welcome to another deep dive. You know how sometimes you feel like you're just surrounded by information, but you just can't connect the dots?
Speaker 2:Yeah.
Speaker 1:Well, that's where functions come in. Right. We're, like, these little well, they're not so little, but
Speaker 2:Yeah.
Speaker 1:They're these, like, unsung heroes of understanding how things are related.
Speaker 2:Exactly.
Speaker 1:And so today, we're going way beyond just, you know, defining functions, and we're diving into this fascinating world of, like, comparing them.
Speaker 2:Yeah. It's like upgrading from a magnifying glass to, like, a high powered microscope. Oh,
Speaker 1:I like that.
Speaker 2:Comparing functions lets us analyze trends, make predictions, uncover these hidden relationships in data.
Speaker 1:And, you know, for all you teachers out there listening in
Speaker 2:Absolutely.
Speaker 1:We're gonna break down how to make this concept really click for your algebra students.
Speaker 2:For sure.
Speaker 1:We're talking about population changes, the rise of cell phones, even the popularity of TV shows.
Speaker 2:All of
Speaker 1:this is, like, fair game for comparing functions.
Speaker 2:Absolutely.
Speaker 1:And you're gonna walk away with some clear explanations, some engaging example. Yeah.
Speaker 2:And maybe even some moments of your own.
Speaker 1:Well, it all starts with being able to visualize it.
Speaker 2:Okay.
Speaker 1:So just imagine presenting your students with this graph, and it charts the population of Baltimore and Cleveland throughout, like, the entire 20th century.
Speaker 2:Okay. I think I can picture those lines.
Speaker 1:Yeah.
Speaker 2:But what exactly am I looking for?
Speaker 1:Those lines, they kinda become more than just these, like, abstract squiggles on a page.
Speaker 2:Right.
Speaker 1:Students can actually pinpoint where they intersect.
Speaker 2:Okay.
Speaker 1:And that shows us when those populations were roughly equal.
Speaker 2:Okay.
Speaker 1:And, actually, it's kinda surprising it happened twice in the 20th century.
Speaker 2:Wow.
Speaker 1:Students can then see where one city's population might surpass the others Right. And they can analyze all those overall trends.
Speaker 2:Okay.
Speaker 1:Those periods of growth decline or maybe even just stagnation.
Speaker 2:So we're kind of, like, turning our students into data detectives. Exactly. We're searching for those key points
Speaker 1:Yeah.
Speaker 2:Those intersections, those shifts. All of
Speaker 1:it.
Speaker 2:And then they can kind of unlock the story that's hidden within the data precisely. Yeah. And that whole visual understanding Yeah. That becomes the foundation for doing even deeper analysis.
Speaker 1:Yeah.
Speaker 2:For instance, remember those intersection points we talked about? Yeah. Those are really important for being able to solve equations where two functions have the exact same output.
Speaker 1:Okay.
Speaker 2:So it gives students a really concrete way to see why those solutions actually matter. Yeah. But, you know, that visual understanding, that's really just the beginning.
Speaker 1:Okay.
Speaker 2:To, like, really unlock those deeper insights Yeah. We gotta bring in the language of mathematics Yeah. Function notation.
Speaker 1:Okay. So it's like we're giving our students, like, this secret decoder ring
Speaker 2:Exactly.
Speaker 1:So they can unlock even more information from these graphs.
Speaker 2:I love it. So let's say we're looking at that same population graph.
Speaker 1:Okay.
Speaker 2:Instead of just saying, you know, when were the populations roughly equal?
Speaker 1:Right.
Speaker 2:Function notation lets us ask for what value of x does f x equal g x.
Speaker 1:Okay. I see how that's way more precise, but I need your help to connect the dots.
Speaker 2:Sure.
Speaker 1:Why is that precision so important for students to get?
Speaker 2:Well, it lets us move beyond just estimations.
Speaker 1:Okay.
Speaker 2:We can actually pinpoint those exact values.
Speaker 1:Okay.
Speaker 2:And one of the examples our source material uses, which is really clever
Speaker 1:Yeah.
Speaker 2:Is they talk about landline versus cell phone use over time.
Speaker 1:Okay.
Speaker 2:And so function notation could allow us to pinpoint the exact year
Speaker 1:Okay.
Speaker 2:When cell phone use surpassed landlines.
Speaker 1:Gotcha.
Speaker 2:And it would give us this precise marker of this huge technological shift.
Speaker 1:Oh, okay. I'm seeing it. So we're not just, like, eyeballing it on the graph anymore. We're, like, zeroing in on these very specific points.
Speaker 2:I love it.
Speaker 1:And, you know, speaking of that landline versus cell phone example
Speaker 2:Yeah.
Speaker 1:It really brings this whole idea of comparing functions to life.
Speaker 2:Absolutely. It's like this classic
Speaker 1:Right.
Speaker 2:Case of one technology basically being replaced by another.
Speaker 1:Right.
Speaker 2:And what's even more fascinating is the lesson then encourages the students to look at the average rate of change.
Speaker 1:Okay. So we're going, like, a step beyond just seeing that one line is higher than the other at, like, a certain point? Now we're talking about the speed of the change.
Speaker 2:Precisely. Think of it like measuring the speed of a car. Okay. Just like speed tells us how many miles a car is covering per hour
Speaker 1:Right.
Speaker 2:Average rate of change tells us how much a function changes over a specific interval.
Speaker 1:Okay. I'm with you. So in that cell phone example Yeah. We're not just looking at when cell phones overtook landlines Right. But how fast each technology was adopted or, like, phased out.
Speaker 1:Exactly. And this is where it gets really interesting. Okay. So the
Speaker 2:illustrative math lesson Yeah. They highlight that the rate at which landline use decreased
Speaker 1:Mhmm.
Speaker 2:Is shockingly similar to the rate at which cell phone use increased.
Speaker 1:Wait. So as people were getting cell phones, they were ditching their landlines at, like, almost the same pace.
Speaker 2:That's the question. Oh. That comparing functions lets us ask.
Speaker 1:Right.
Speaker 2:It reviews these really cool hidden connections.
Speaker 1:Wow.
Speaker 2:It makes students think critically about cause and effect in the real world.
Speaker 1:Right.
Speaker 2:But, you know, we also have to think about where students might stumble with these concepts.
Speaker 1:Right. Yeah. That's the teacher in you coming out. Because, you know, our listeners are always looking for ways to make these concepts really stick with their students, for sure. So what are some common misconceptions that that students might come across when they're working with comparing functions?
Speaker 2:I'd say one of the biggest hurdles is kind of the abstract nature of it all.
Speaker 1:Right.
Speaker 2:You know, students are just looking at lines on a graph without any, like, real world connection Yeah. It's easy to get lost in all the symbols and equations.
Speaker 1:Yeah. It's like they say, it's all Greek to me. Exactly. You know? We need to bridge that gap
Speaker 2:They do.
Speaker 1:Between the abstract and something relatable.
Speaker 2:For sure. That's why actually visually representing these comparisons Okay. Like in that phone example and always coming back to the real world is so important.
Speaker 1:It's about making it click, showing them that math isn't just something you do in your textbook. It's actually alive in the world around us.
Speaker 2:Exactly.
Speaker 1:What about function notation in and of itself? That seems like it could be another tricking point for students.
Speaker 2:Absolutely. Yeah. That's why you really wanna encourage students to almost, like, translate that notation into, like, everyday language.
Speaker 1:So instead of freaking out when they see f x, e g x, they can just think, like, okay. This just means I gotta figure out where these two things are the same.
Speaker 2:Exactly. It's all about giving them the tools to decode it Right. And to really see the meaning behind the symbols. Yeah. And then they can relate it back to those graphs and those real world situations we've been talking about.
Speaker 1:It's like we're sending them on a mathematical treasure hunt.
Speaker 2:I love it.
Speaker 1:And those function notations are like the clues to help them get to the prize.
Speaker 2:Speaking of uncovering treasures Yeah. I think this deep dive has given us some really valuable things for teachers to think about.
Speaker 1:Alright. So it hit us with those expert insights.
Speaker 2:Okay.
Speaker 1:What are the big takeaways here when it comes to teaching comparing functions?
Speaker 2:Well, first things first. You gotta make those real world connections. Yeah. It doesn't matter if it's population trends, technology adoption, or even something fun like TV show viewership.
Speaker 1:Oh, yeah. The illustrative math lesson even uses that as an example.
Speaker 2:It's brilliant. Make it relatable.
Speaker 1:Right. Because when students see how math is actually playing out in the real world around them Yeah. It's suddenly a lot more exciting than just, like, some random numbers on a page.
Speaker 2:For sure. Yeah. And my second big takeaway
Speaker 1:Yeah.
Speaker 2:Don't be afraid to get creative.
Speaker 1:Okay.
Speaker 2:Have your students come up with their own examples. Yeah. Let them translate those real world scenarios into functions and graphs themselves.
Speaker 1:Right. So we're turning them into mathematical explorers.
Speaker 2:I love it.
Speaker 1:Venturing out and discovering the power of functions in places you would never expect.
Speaker 2:Expect. Precisely.
Speaker 1:Well, we've covered a lot of ground today. We talked about graphs. We decoded function notation.
Speaker 2:We did.
Speaker 1:We even uncovered these really cool hidden connections when it comes to, like, the rise and fall of different technologies.
Speaker 2:Yeah. And most importantly Yeah. We talked about how to give teachers the knowledge and the strategies they need
Speaker 1:Right.
Speaker 2:To make this topic not only accessible but, like, actually fun for their students.
Speaker 1:Couldn't have said it better myself. Huge thank you to our expert today for shedding some light on this really cool area of math.
Speaker 2:It was my pleasure. I always love talking about functions.
Speaker 1:And to all of you listening out there, thank you so much for joining us on another deep dive. But But before we let you go, we want to leave you with a little something to think about. We've seen how comparing functions can reveal all these hidden trends in populations, technology, even entertainment. So what other real world things could we analyze this way?
Speaker 2:That's a great question.
Speaker 1:What stories are just waiting to be told if we look through the lens of functions? Mhmm. Keep those mathematical minds of yours going, and we'll catch you next time on the Deep Dive.