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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 wish you could just hit rewind on a math problem? Yeah. That's kinda what we're gonna be diving into today. Right? Inverse functions.
Speaker 2:Right.
Speaker 1:We're gonna be dissecting this illustrative math lesson, and, you know, it's not just about finding those mathematical opposites. Mhmm.
Speaker 2:It
Speaker 1:it helps students actually understand what they represent.
Speaker 2:Mhmm. Yeah. And what's really cool about this lesson is how they bring in these scenarios that are so relatable.
Speaker 1:Right.
Speaker 2:It's not just about abstract equations. It's about making this concept, like, tangible.
Speaker 1:Yeah. I am all about making math relatable.
Speaker 2:Right.
Speaker 1:So walk me through it. Yeah. How does this lesson plan actually set the stage for, like, really understanding inverses?
Speaker 2:Well, it starts with something that I think everyone loves, and that is online shopping.
Speaker 1:Okay.
Speaker 2:So Mhmm. Imagine your students, you know, they're tasked with writing equations to figure out the total cost of buying cookbooks
Speaker 1:Okay.
Speaker 2:From different online stores.
Speaker 1:Right. So far so good.
Speaker 2:Yeah.
Speaker 1:I can see how that's, you know Mhmm. Pretty standard word problem setup. Where does the inverse part actually come in?
Speaker 2:So once they're comfortable, you know, calculating that total cost based on the number of cookbooks, the lesson flips the script.
Speaker 1:Okay.
Speaker 2:Now they're given a set amount of money, and they have to figure out how many cookbooks they could buy.
Speaker 1:Okay.
Speaker 2:So it's like
Speaker 1:So instead of going from cookbooks to cost Yes. They're going from cost to cookbooks.
Speaker 2:Exactly.
Speaker 1:It's like the reversing the whole flow of the equation.
Speaker 2:Yes. Exactly. That's the heart of an inverse function. Right? Mhmm.
Speaker 2:It's undoing what the original function did.
Speaker 1:Okay.
Speaker 2:And then it takes this concept even further with a classic, converting Celsius and Fahrenheit.
Speaker 1:Oh, I remember wrestling with that formula in school. Right. It always just felt like this random string of numbers and operation.
Speaker 2:And this lesson doesn't just throw the formula at
Speaker 1:them. Mhmm.
Speaker 2:It guides them to actually derive the inverse function themselves.
Speaker 1:Okay.
Speaker 2:Which is huge Right. Because it shows them that they can have a dedicated formula for converting from Fahrenheit to Celsius rather than always having to rearrange that original equation.
Speaker 1:Talk about working smarter, not harder. Right. I can see how that would be really empowering for students to have that kinda algebraic agency.
Speaker 2:Totally. And just when they think they've got it Uh-oh. The lesson's like, boom. Rankine scale.
Speaker 1:Wait. Another temperature scale?
Speaker 2:Weather ones.
Speaker 1:Are we trying to give our students a meteorological meltdown here?
Speaker 2:No. Not at all.
Speaker 1:Okay.
Speaker 2:It's strategic. Right. Okay. So now they have these two new equations
Speaker 1:Mhmm.
Speaker 2:And they're challenged to prove that those equations, just by looking at the structure, are inverses of each other.
Speaker 1:It's like being a mathematical detective Yes. Uncovering these hidden relationships.
Speaker 2:Exactly.
Speaker 1:I love that. What other tricks does this lesson have up its sleeve?
Speaker 2:Well, one activity that really stands out is called custom mugs.
Speaker 1:Okay.
Speaker 2:And it uses this really neat info gap structure.
Speaker 1:Okay.
Speaker 2:So students work in pairs, but each student only gets, like, half of the information about this scenario involving buying custom mugs.
Speaker 1:Okay. So they have to work together to kind of crack the case of the custom mugs.
Speaker 2:Yes.
Speaker 1:I'm intrigued. How does that tie back into inverse functions? One student might know the original price per mug and the total budget.
Speaker 2:Right.
Speaker 1:While their partner knows the total number of mugs purchased and the budget
Speaker 2:That's good.
Speaker 1:To solve the problem, they need to use both the original function and its
Speaker 2:inverse
Speaker 1:Mhmm. Which really highlights the practical application of each.
Speaker 2:And it's a brilliant way, I think, to illustrate that both, you know, the original function and its inverse Mhmm. Each provide valuable information
Speaker 1:Right.
Speaker 2:Depending on what you're actually trying to find out.
Speaker 1:Okay.
Speaker 2:And I also think the info gap structure is great Mhmm. Because it really pushes students to develop those communication skills. Right. They've got to explain their thinking. They have to listen carefully to their partners to fill in those gaps.
Speaker 1:It's like a mathematical puzzle where communication is key.
Speaker 2:Totally.
Speaker 1:Now I'm curious. What are some common stumbling blocks that teachers should be prepared for when they're teaching inverse functions? Because I'll admit, algebra and I weren't always on the best of terms back in the day.
Speaker 2:Yeah. You're not alone there.
Speaker 1:Okay. Good.
Speaker 2:One thing the lesson really emphasizes is that teachers should not shy away from reviewing
Speaker 1:Mhmm. You know Right.
Speaker 2:Potentially tricky algebraic maneuvers.
Speaker 1:Mhmm.
Speaker 2:So for instance
Speaker 1:Okay.
Speaker 2:When you're solving for c in that Celsius Fahrenheit formula Right. You encounter fractions. You have the distributive property, and those can throw some students off.
Speaker 1:Fractions, the perennial thorn in the side of algebra students.
Speaker 2:Exactly. And it's easy to assume that they've got those foundational skills down pat.
Speaker 1:Mhmm.
Speaker 2:But I think this lesson smartly points out that a quick refresher can make all the difference.
Speaker 1:Absolutely.
Speaker 2:Another potential pitfall is confusing an inverse function with a reciprocal.
Speaker 1:Oh, yeah. Right. It's easy to see how students might fall into that trap.
Speaker 2:Totally.
Speaker 1:It's like thinking that to undo a shoelace knot, you just pull on one end harder.
Speaker 2:Right.
Speaker 1:Sometimes you need to, like, loosen things up and retrace your steps to unravel it completely.
Speaker 2:That's such a great analogy.
Speaker 1:Right.
Speaker 2:The lesson does a really nice job of emphasizing that an inverse function reverses the process of the original function.
Speaker 1:Mhmm. It's
Speaker 2:not just about, like, flipping fractions or changing signs. Right?
Speaker 1:Right.
Speaker 2:It's about understanding those underlying operations and how they unwind.
Speaker 1:So how does this lesson plan go beyond just the mechanics of actually finding an inverse
Speaker 2:Okay.
Speaker 1:And ensure that students really grasp the concept?
Speaker 2:Well, one way is by emphasizing the power of visualization.
Speaker 1:Mhmm.
Speaker 2:The lesson really encourages teachers to bring in graphs.
Speaker 1:Okay.
Speaker 2:Because when students can see that relationship between the inputs and outputs Mhmm. And how they essentially, like, swap on the graph of an inverse function
Speaker 1:Yeah. Yeah.
Speaker 2:That can be a real
Speaker 1:moment. Like seeing the function reflected in a mirror. Yes. The input becomes the output and vice versa.
Speaker 2:Precisely. And, you know, the lesson doesn't just stop at the mathematical representation.
Speaker 1:Right.
Speaker 2:Remember those real world connections we were talking about?
Speaker 1:Yes.
Speaker 2:The tables and seats active. It's a fantastic example of this.
Speaker 1:Okay.
Speaker 2:Imagine you're setting up hexagonal tables for a party.
Speaker 1:Okay.
Speaker 2:And you need to figure out how many people can sit at a certain number of tables.
Speaker 1:Alright. This is this sounds like my kind of party planning challenge.
Speaker 2:Right.
Speaker 1:I'm invested. What happens
Speaker 2:next? So first, they come up with a function to model that situation. Right?
Speaker 1:Right.
Speaker 2:Pretty straightforward.
Speaker 1:Okay.
Speaker 2:But then comes the twist.
Speaker 1:What's the twist?
Speaker 2:They have to determine the inverse function.
Speaker 1:So instead of figuring out how many people can fit at a given number of tables
Speaker 2:Yes.
Speaker 1:They're figuring out how many tables they need for a specific number of guests.
Speaker 2:You got it. But here's where it gets even more interesting.
Speaker 1:Okay.
Speaker 2:Mathematically, you can end up with fractional answers. Right. You might need 2.3 tables Mhmm. For a certain number of guests.
Speaker 1:But in the real world, you can have 0.3 of a table.
Speaker 2:Exactly. And this activity so brilliantly highlights that
Speaker 1:Right.
Speaker 2:The importance of domain and range in context. Right? Mhmm. You can't have half a table.
Speaker 1:Right.
Speaker 2:You can't have negative guests. Right. Hopefully not.
Speaker 1:That would be a different kind of party.
Speaker 2:It pushes them to think beyond just the calculation and consider those practical limitations.
Speaker 1:That's such a valuable lesson.
Speaker 2:Right.
Speaker 1:And not just in math class.
Speaker 2:Totally.
Speaker 1:Sometimes we need a little dose of real world practicality to balance out our theoretical calculations.
Speaker 2:I love that.
Speaker 1:Speaking of practicality
Speaker 2:Yes.
Speaker 1:What are some actionable tips for teachers to help their students avoid those conceptual potholes that we've been talking about?
Speaker 2:One thing that really resonated with me in this lesson plan was the emphasis on discussion. Okay. You know, it's not enough to just lecture and expect students to absorb everything.
Speaker 1:Right.
Speaker 2:Creating an environment where they feel comfortable asking questions
Speaker 1:Mhmm.
Speaker 2:Even if they seem silly
Speaker 1:Right.
Speaker 2:And sharing their thought processes even if they're unsure. Right. It's crucial.
Speaker 1:It's about fostering those moments where the light bulb goes on and they make those connections for themselves. But to get there, they need that safe space to, like, voice their questions Mhmm. Even make mistakes.
Speaker 2:Absolutely. That's where the real learning happens.
Speaker 1:Right.
Speaker 2:When they feel empowered to really grapple with the concepts and make them their own.
Speaker 1:Yes.
Speaker 2:And this lesson really emphasizes the importance of creating that kind of learning environment.
Speaker 1:Yeah. It sounds like this illustrative math lesson is really onto something
Speaker 2:Yeah.
Speaker 1:With its approach. I especially love how it doesn't shy away from those real world constraints. Like Mhmm. You can't have a fraction of a table.
Speaker 2:Right.
Speaker 1:You know? It's those kinds of connections that make math feel less abstract and more relevant
Speaker 2:Yeah.
Speaker 1:To students' lives.
Speaker 2:Absolutely. It's about showing them that math isn't just confined to text books.
Speaker 1:Mhmm.
Speaker 2:It's a tool for understanding, you know Right. Navigating the world around us.
Speaker 1:So as we're wrapping up our deep dive into inverse functions
Speaker 2:Okay.
Speaker 1:What are some key takeaways you hope our listeners, those busy teachers like themselves
Speaker 2:Yes.
Speaker 1:Will take back to their classrooms?
Speaker 2:I think the biggest takeaway is that power of making this concept, right
Speaker 1:Mhmm.
Speaker 2:Of inverse functions both concrete and relevant.
Speaker 1:Right.
Speaker 2:We talk about all these clever scenarios in this lesson. You know?
Speaker 1:Yes.
Speaker 2:From the online shopping to the party planning. Right. To bring this idea to life. But it's also about encouraging teachers
Speaker 1:Mhmm.
Speaker 2:To find those connections
Speaker 1:Yeah.
Speaker 2:That resonate with their own students' interests.
Speaker 1:Because when students can see how a concept applies to their own world
Speaker 2:Right.
Speaker 1:It's so much more likely to stick with them.
Speaker 2:Absolutely.
Speaker 1:It's like the difference between memorizing a formula Mhmm. And actually understanding why it works and how to use it.
Speaker 2:Exactly. And this lesson provides such a rich toolbox for teachers from the emphasis on visualization
Speaker 1:Mhmm.
Speaker 2:To that info gap activity that Yeah. Encourages collaboration.
Speaker 1:Right.
Speaker 2:There are so many ways to engage students and help them really master this fundamental algebraic concept.
Speaker 1:It sounds like the creators of this illustrative math lesson have really outdone themselves
Speaker 2:Yeah.
Speaker 1:Providing this road map for teaching inverse functions in a way that's both, like, effective
Speaker 2:Mhmm.
Speaker 1:And engaging. Yeah. A huge thank you to them for giving us so much to think about.
Speaker 2:Yes. Absolutely.
Speaker 1:And to our listeners, we hope this deep dive has sparked some new ideas Mhmm. And strategies that you can bring back to your own classrooms.
Speaker 2:Yes.
Speaker 1:Remember, helping students grasp the elegance and utility of inverse functions
Speaker 2:Mhmm.
Speaker 1:That's what makes teaching algebra so rewarding.
Speaker 2:I agree.
Speaker 1:On that note Yes. It's time for us to sign off. Until next time.
Speaker 2:Okay.
Speaker 1:Happy teaching.