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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.)
Alright, math wizards. Let's dive into something a bit deeper than just, like, you know, crunching numbers. We've got this illustrative math algebra 1 curriculum. Right? Got that.
Speaker 1:Lesson 9, which variable to solve for? Part 2. It's like sneakily powerful stuff. You know what I mean?
Speaker 2:It really is. You see, we often teach equations as a way to, like, find an answer, you know, but this lesson really flips the script. Mhmm. It's about recognizing that equations are flexible. We can rearrange them to highlight any variable we want.
Speaker 1:Okay. I'm intrigued. So it's like choosing which tool to use from, like, a toolbox depending on the job at hand. But why is this flexibility so important for students to grasp? Why is that important?
Speaker 2:Because it gets to the heart of what algebra really is. It's the study of relationships between quantities. When students can just isolate any variable, they're not just solving, they're expressing one quantity as a function of others.
Speaker 1:Hold on. Functions, didn't we leave those back in precalculus?
Speaker 2:We did. But this lesson lays the groundwork. Think about it. Solving for, say, y in terms of x is essentially defining a function. It's saying, if you tell me x, I can tell you y because they're bound by this relationship.
Speaker 1:Woah. I see it now. It's like that input output idea they learn early on, but way more powerful, and this lesson helps them master that manipulation.
Speaker 2:Exactly. It starts with a warm up using a formula about shapes. You know, those 3 d ones Mhmm. With the fancy name? Platonic solids, like a cube or a pyramid where all the sides are the same triangle.
Speaker 2:They have a relationship between the number of faces, vertices, and edges.
Speaker 1:Right. Right. I vaguely remember that from geometry somewhere back there.
Speaker 2:Well, instead of giving them all the values to plug in, this activity has them rearrange the formula first, gets them comfortable with the process of manipulation, not just the answer.
Speaker 1:So it's like stretching before a workout, getting those algebraic muscles warmed up, then what? What happens next?
Speaker 2:Then we set sail for real world applications with the cargo shipping activity. Imagine a cargo ship with a weight limit. Students need to figure out how many cars and trucks they can fit. Pretty standard word problem stuff. Right?
Speaker 1:Yeah. Yeah. Sounds familiar. Where does the solving for any variable bit come in? How does that work?
Speaker 2:Well, after they write the initial equation representing the weight constraint, they're asked to solve for the number of cars then the number of trucks. This shows them how those different forms of the equation are like having different lenses to view the problem.
Speaker 1:Okay. I'm starting to see how this could click for them. Instead of just plugging in numbers, they're building a deeper understanding of the relationship itself. But wouldn't some students just say, why bother rearranging? I can plug in the numbers and get the answer either way.
Speaker 1:No. When they say that?
Speaker 2:You're absolutely right. That's a misconception we have to address head on. We need to help them see the bigger picture. Sure, in this specific scenario, plugging in might be quicker. But imagine if the weight limit changes, or they get a different shipment of vehicles.
Speaker 2:Suddenly, having those presolved equations is like having a formula ready to go. It's efficiency. It's elegance. It's setting them up for success with more complex problems down the road.
Speaker 1:It's like building a robot to do the repetitive work for you. Right? And speaking of more complex problems, I noticed there's another activity, streets and staffing. What's that about?
Speaker 2:That one dives into city budgeting, a topic I'm sure many students find marginally less exciting than cargo ships until they realize it affects their daily lives.
Speaker 1:Right. Potholes in summer jobs. Suddenly, budgeting matters.
Speaker 2:Exactly. This activity emphasizes the strategic choice of which form of the equation is most useful depending on what you already know. If you know how many miles of road you can resurface, you use one version. But if the city council says, this is how many workers we can afford, well, then you need a different version.
Speaker 1:So it's not just about blindly solving. It's about understanding the context and choosing the right tool for the job. This is brilliant. I bet some teachers are out there thinking, my students barely remember how to solve for one variable, let alone choosing the most useful form. What do you say to them?
Speaker 2:That's completely valid. It highlights the importance of gauging where students are at. Maybe they need a little refresher on the mechanics of algebraic manipulation before diving into these nuanced applications. And that's the beauty of a well designed curriculum, isn't it? It often builds in those opportunities for spiraling back to foundational skills.
Speaker 2:Mhmm. Teachers could even turn it into a quick review activity.
Speaker 1:Yeah. Like
Speaker 2:Okay. Everyone remember how to isolate x. Let's do a few practice problems together before we tackle this budget scenario.
Speaker 1:Right. Because sometimes a little review is all it takes to jog those memory muscles. But, you know, I can also see some students getting bogged down by the abstractness of it all. How do we keep them grounded, especially when the equations themselves start getting longer and more complex? How do we handle that?
Speaker 2:Absolutely. That's where connecting back to the concrete context Mhmm. Becomes even more crucial. The lesson plan actually suggests some great follow-up questions for the streets and staffing activity.
Speaker 1:Okay.
Speaker 2:Things like, what if the department resurfaces 16 miles of road? How many workers can they hire? Or what if they can't afford to hire anyone new? What happens to the roads?
Speaker 1:I like that. Suddenly it's not just about x's and y's, it's about real world consequences.
Speaker 2:Exactly. And by encouraging students to talk through their thinking, we can help them make those connections explicit. Instead of just saying divide both sides by 7.5, they can say, we're dividing the total budget by the cost per mile of resurfacing to see how many miles we can afford. Yeah. It's about narrating that thought process and anchoring the abstract to the tangible.
Speaker 1:It's like the difference between memorizing a recipe and actually understanding why each ingredient matters.
Speaker 2:Right. I like that analogy. And, you know, speaking of ingredients, sometimes even experienced cooks need a little reminder on basic techniques. The same goes for algebra. While this lesson focuses on the higher level concept of choosing the most useful form of an equation, we can't forget about those fundamental algebraic moves.
Speaker 1:Right. Like remembering to apply an operation to both sides of the equation. Yes. We're dealing with those pesky negative signs.
Speaker 2:Exactly. It's easy to get so caught up in the bigger picture that we forget those little details that can trip students up. So
Speaker 1:it sounds like it's all about finding that balance. Right? Helping students master those foundational skills and empowering them to think critically about why and how they're using those skills.
Speaker 2:You've hit the nail on the head. Yeah. It's about moving beyond the plug and chug mentality
Speaker 1:Mhmm.
Speaker 2:And fostering a deeper understanding of the power and flexibility of algebraic thinking.
Speaker 1:It's like we're giving them the keys to the whole mathematical car, not just teaching them how to drive in 1st gear.
Speaker 2:I love that analogy. And you know what else can be a powerful tool for illustrating these concepts? Spreadsheets.
Speaker 1:Oh, you know, I was just thinking about that. It's like a digital sandbox for exploring equations. Change one value and see how everything else gets affected.
Speaker 2:Exactly. It brings that dynamic relationship to life right before their eyes. Plus, it suddenly introduces them to the idea that this kind of equation manipulation is what makes spreadsheet so powerful in the real world.
Speaker 1:It's all connected. We've covered so much ground in this deep dive. We've talked about the importance of flexibility, connecting to concrete examples, and even revisiting those essential algebraic moves. As we wrap things up, what's that one big takeaway you wanna leave our listeners with, something to really make those light bulbs go off?
Speaker 2:You know, when I think about this lesson, the word that keeps coming to mind is, perspective. Because at its core, that's what we're teaching students to do to shift their perspective to to see that there are multiple ways to represent the same mathematical relationship.
Speaker 1:Oh, I like that. Multiple representations, different lenses, it all ties together.
Speaker 2:Yep. Right. And as they move on to more complex math quadratic equations, systems of equations, even calculus, that ability to shift perspective becomes invaluable. So encourage them to play with equations, to not be afraid to rearrange things and see what happens.
Speaker 1:It's like that old saying, there's more than one way to skin a cat, but for
Speaker 2:algebra? Exactly. And who knows? Maybe this lesson will spark a lifelong love of equations and the power they hold, or at the very least help them ace that next algebra test.
Speaker 1:Hear. Hear. A huge thank you to the authors of Illustrative Math for this thought provoking lesson. And to all you math wizards out there, keep those pencils sharp and those minds curious. Until next time on the Deep Dive.