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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 there. Ever find yourself explaining linear and exponential functions and kinda wishing you had, like, the perfect lesson plan to make it all click? Well, this deep dive is for you. Yeah. Especially for all you teachers out there getting ready to tackle these concepts.
Speaker 1:It's
Speaker 2:a big one.
Speaker 1:We're gonna break down a lesson plan from Illustrative Mathematics. They're algebra 1 curriculum.
Speaker 2:Yeah.
Speaker 1:We're gonna pull out those light bulb moments, get you feeling confident and inspired to teach this stuff.
Speaker 2:Yeah. What I really like about this lesson is how it doesn't just, like, throw these abstract equations at students. It actually, like, builds on what they should already know from earlier grades.
Speaker 1:Okay.
Speaker 2:Like, from their math classes.
Speaker 1:Right.
Speaker 2:Like, stuff like properties of exponents and, you know, just those good old linear equations. It's like building a bridge from familiar territory.
Speaker 1:I love that analogy. And speaking of bridges, it it feels like there's a bigger picture here. Right? It's not just about the equations themselves, but, like, what they allow us to understand about the world around us.
Speaker 2:Totally. Yeah. Like, imagine if you're trying to understand, like, population growth, you know, or how investments can increase over time. Right. Even something like how information spreads online.
Speaker 2:Mhmm. Linear and exponential functions, they give us the tools to actually, like, model and understand these real world things.
Speaker 1:Okay. So we've established the stuff matters. How does the lesson plan actually go about unpacking these, like, crucial concepts for students?
Speaker 2:Well, it lays out some very clear learning objectives right off the bat. So by the end of this lesson, students shouldn't just be able to recognize these functions Mhmm. But they should really understand how they change over equal intervals.
Speaker 1:And that's where things get really interesting. Right? We start to see those, like, fundamental differences Yeah. Between how linear and exponential functions actually behave.
Speaker 2:Exactly. So with linear functions, we're dealing with a constant rate of change, kinda like a car traveling at a steady speed.
Speaker 1:Okay.
Speaker 2:But with exponential functions, hold on tight, Because things can escalate quickly. It's not about adding the same amount each time anymore.
Speaker 1:Right.
Speaker 2:We're multiplying Yeah. Which gives us that, you know, explosive exponential growth.
Speaker 1:That's a great way to put it. So we've got these two distinct types of functions, each with their own, like, personality.
Speaker 2:Totally.
Speaker 1:This lesson wants to give students, like, a front row seat to see them in action.
Speaker 2:Yeah. And it does so by guiding them through these really carefully structured activities that illuminate these, like, key differences.
Speaker 1:Alright. I am so ready to dive into the specifics.
Speaker 2:Thank you.
Speaker 1:Let's take a closer look at how this lesson plan actually tackles these concepts.
Speaker 2:Okay.
Speaker 1:Starting with linear functions, what's the first move?
Speaker 2:Alright. So the lesson plan jumps right into what they call activity 20.2
Speaker 1:Okay.
Speaker 2:Which introduces this specific linear function, f x to x plus 1.
Speaker 1:Okay.
Speaker 2:And then it has the students kinda explore it.
Speaker 1:Ugh. So we're moving beyond the abstract and getting our hands a little dirty with some, like, real math. I'm here for it. What kind of exploration are we talking about? What are students being asked to do with this function?
Speaker 2:So they're basically, like, taking it for a test drive. They plug in different values for x. Okay. And then they get to see what happens to the output, the f x. It's all about recognizing patterns, connecting the dots between the equation and the results.
Speaker 1:Okay. I'm intrigued. What kind of pattern starts to emerge when we start playing around with those x values in a linear function like this? What's the big reveal? So we're plugging in different numbers for x, you know, and observing those outputs.
Speaker 1:Like, we're scientists in a lab. What are we hoping these student scientists will, like, discover?
Speaker 2:We'll start to notice this really cool thing about linear functions. For every unit increase in x, so, like, say we go from an input of 2 to an input of 3 Okay. Okay. The output, the f x, it changes by a constant amount, and it's not random or anything.
Speaker 1:Right.
Speaker 2:It's all determined by the equation.
Speaker 1:Wait. Hold on. Is that constant change? Is that what the slope represents?
Speaker 2:Yeah.
Speaker 1:That 2 inch in in our example, f x equal 2 x plus 1.
Speaker 2:You got it.
Speaker 1:Okay.
Speaker 2:The slope isn't just some abstract number. It's like the engine driving the change in a linear function. So in our example, when x increases by 1, f of x increases by, you guessed it, 2, the value of our slope.
Speaker 1:Okay. That makes sense. When I see it like that, it's very intuitive. But how does this lesson make sure students don't just see this pattern as, like, specific to this one example? How do we help them generalize this whole constant change idea?
Speaker 2:Well, that's where, you know, careful teaching comes in. The lesson really stresses that this link between the slope and that constant change in output, it's not a coincidence.
Speaker 1:Right.
Speaker 2:It's like a core property of all linear functions. Yeah. To hammer this home, it suggests, like, having students experiment with other linear equations.
Speaker 1:Okay.
Speaker 2:They start to see the same pattern over and over, but with different functions. Right. That's how they grasp that underlying principle.
Speaker 1:It's like they're becoming fluent in, like, the language of linear functions.
Speaker 2:Exactly.
Speaker 1:Seeing the patterns, understanding the grammar.
Speaker 2:Yes. And once they've got that linear foundation, the lesson plan takes us to a more, well, dynamic world. Okay. Exponential functions.
Speaker 1:Alright. Things are about to get a little wilder. We're moving from that steady world of adding to the world of multiplication.
Speaker 2:Yeah.
Speaker 1:Where does this lesson plan take us to explore this new territory?
Speaker 2:It takes us to activity 20.3. Students encounter this exponential function, g x, 3 x.
Speaker 1:Okay.
Speaker 2:This is where things get really interesting.
Speaker 1:3 to the power of x. That sounds like it could get big really fast.
Speaker 2:It does. That's the whole point. Yeah. So this time, when that input increases by a constant amount, the output, it doesn't just increase by a constant amount. Okay.
Speaker 2:It's multiplied by a constant factor.
Speaker 1:Okay. Let's unpack that a bit. So we're shifting gears. We're going from addition to, like, straight up multiplication.
Speaker 2:Yeah.
Speaker 1:How does that actually change the game, the relationship between our inputs and outputs?
Speaker 2:So instead of a nice steady incremental change, like we saw with linear functions, we're gonna get more of, like, a a snowball effect. Okay. And the lesson uses some really great examples. So let's say we increase x by 1 in our function, g x equals 3 x.
Speaker 1:Okay.
Speaker 2:Now instead of just adding 3 to the output, we're multiplying that whole output by 3.
Speaker 1:So if g one equals 3, then g 2 would be 9
Speaker 2:Exactly.
Speaker 1:Because we're multiplying by 3 each time.
Speaker 2:Exactly. And if we go to g 3, we multiply by 3 again, so g 3 would be 27. Yeah. You see how things escalate. Yeah.
Speaker 2:That multiplicative effect, that's at the heart of exponential growth.
Speaker 1:It's like each step along the x axis is like another multiplier, just making the output, like, skyrockets. Very different feel from the linear functions.
Speaker 2:For sure. And this pattern, it doesn't just apply to single unit increases in x either. Okay. The lesson makes it clear. If we increase x, say, by 3, we're essentially multiplying g x by 3, but 3 times in a row, which is the same as multiplying by 27 or 3 cubed.
Speaker 2:Yeah. This idea of repeated multiplication, that is key to understanding these exponential functions.
Speaker 1:This is where I imagine some students might get a little tripped up. Right? Because they're tempted to look for those, like, familiar patterns, the addition, the constant differences we saw with linear functions. But exponential growth, it doesn't really care about that, does it?
Speaker 2:You're right. It's a common mistake. And the lesson plan, it gets this. It encourages teachers to emphasize this. Exponential functions, they're all about that repeated multiplication.
Speaker 2:It's not about how much we're adding each time. It's about that multiplying.
Speaker 1:And that distinction, that shift in thinking, it's everything. Right? Yeah. That's what unlocks understanding these exponential functions. But it's not just about knowing the difference between the two types of functions, is it?
Speaker 1:It's about seeing them together.
Speaker 2:A 100%. Yeah. And that is where this lesson plan takes us next.
Speaker 1:Okay.
Speaker 2:It doesn't leave the students hanging with these separate pieces.
Speaker 1:I am ready to see it all come together. So let's move to the grand finale in part 3 where these two functions come face to face. Okay. So we've got these linear functions, nice and predictable.
Speaker 2:Yeah.
Speaker 1:Then the exponential functions, way more dramatic.
Speaker 2:Yeah.
Speaker 1:But now it's time for these functions to, like, meet face to face
Speaker 2:Here we go.
Speaker 1:The main event. What does this lesson plan do to bring it all together?
Speaker 2:So it sets up this really cool moment. Right? Part 4 brings back our linear function, hxkx equals 3x plus 2
Speaker 1:Okay.
Speaker 2:And our exponential friend, kx equals 2x. Okay. And get this, it asks students to keep track of how the outputs for both of them change as x increases.
Speaker 1:So we're really putting these functions head to head Head
Speaker 2:to head.
Speaker 1:Seeing how they each react to the same inputs. Yeah. Are we sticking with those single steps or going bigger this time?
Speaker 2:We're gonna do it all. That's starting with those increases of 1 just like before.
Speaker 1:Right.
Speaker 2:So we can directly compare how those constant differences stack up against the constant factors we talked about.
Speaker 1:So with each step, our linear function, h x, is gonna add 3. Right?
Speaker 2:Exactly.
Speaker 1:And our exponential function, k x, that one keeps multiplying its output by 2.
Speaker 2:Yep. And then just to really drive home how different their growth rates are, we kick it up a notch.
Speaker 1:Okay. Here we go.
Speaker 2:We look at what happens when x increases by 5 Okay. And then hold on by 10.
Speaker 1:Oh. Wow. Yeah. I can see where this is going.
Speaker 2:Right.
Speaker 1:Those bigger jumps, that's where the difference between linear and exponential gets really obvious.
Speaker 2:Totally. Our linear function will just keep adding 3 for each of those increases to x. But the exponential one Yeah. Each jump means it's multiplying by 2 and and again and again. We're talking night and day here.
Speaker 1:It's like comparing, I don't know, like, a nice walk in the park Uh-huh.
Speaker 2:To, like, blasting off
Speaker 1:in a rocket. Exactly. And I think seeing those
Speaker 2:drastically different outputs side by side Yeah. It just makes it click. You know? It's not just memorizing. It's understanding.
Speaker 2:Yeah. It's like they can actually picture
Speaker 1:how these functions play out in the real world.
Speaker 2:And that's what makes a good lesson. It's not just about the numbers themselves, but giving students the tools to actually apply these ideas outside of class in real life.
Speaker 1:Absolutely. But speaking of going beyond the textbook, this lesson plan doesn't stop there, does it? There's one last little thought provoking question it leaves us with.
Speaker 2:Oh, it's a good one.
Speaker 1:What's that final nugget of wisdom?
Speaker 2:So remember how we've mostly been dealing with whole number increases to x?
Speaker 1:Yeah. Those steps. What if
Speaker 2:and here's the curve ball. What if x increased by, say, a fraction, like 12? What would happen to the output then?
Speaker 1:Oh, wow. That's a really interesting question. I'm already trying to think about that. Yeah. We've been so focused on those whole number steps.
Speaker 2:Mhmm.
Speaker 1:Like, this lesson is saying, hey, there's a whole world between the steps too. Like, what happens with fractions or decimals even?
Speaker 2:Exactly. There's so much more to discover. And it starts by asking those, like, what if questions.
Speaker 1:I love that. Such a good place to end this deep dive. Huge shout out to illustrative mathematics for this lesson plan.
Speaker 2:For sure. It's a good one.
Speaker 1:Really engaging. Really got us thinking.
Speaker 2:It's a good reminder that math is about way more than just formulas.
Speaker 1:Right. It's about patterns and connections and staying curious.
Speaker 2:Exactly.
Speaker 1:To everyone listening, keep those math brains working. We'll catch you on the next deep dive where we'll explore even more fascinating topics.