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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 thinking, do I have enough for this? Like, maybe you're eyeing that new phone, but aren't sure if your bank account can handle it.
Speaker 2:Or how about deciding how many friends you can invite to the concert based on how much you wanna spend on tickets?
Speaker 1:Exactly. We're constantly weighing those how much questions, and today's deep dive is all about giving teachers the tools to explore those scenarios with their students all through the lens of inequalities. We're cracking open lesson 23 from illustrative maths algebra curriculum.
Speaker 2:And get this. They actually use those relatable real world situations like bank accounts and concert tickets to bring the math to life.
Speaker 1:It's brilliant, really. Instead of just throwing abstract symbols at students, they're showing them how math connects to their everyday day lives. So what's the game plan for this lesson? What are the big ideas teachers should be aiming for?
Speaker 2:The heart of it is about giving students the power to represent those how much situations, those limitations using the language of inequalities. And it's not just about writing them down, it's about understanding what they represent both graphically and numerically.
Speaker 1:So, like, instead of just saying x has to be greater than 10, they're visualizing what that actually means on a number line or even a coordinate plane.
Speaker 2:Exactly. And they're not just passively looking at graphs either. This lesson encourages them to use technology, specifically Desmos, to really explore those solutions dynamically.
Speaker 1:Desmos. Okay. I've heard whispers about this magical math tool. For those who are new to the game, give us the lowdown. What's so special about Desmos?
Speaker 2:Well, picture this. A free online graphing calculator where students can not only visualize inequalities, but actually manipulate them in real time. They can change the parameters, see how those changes impact the solution region on the graph. It's incredibly powerful for visual learners.
Speaker 1:So instead of just seeing a static image in a textbook, they're interacting with the math, making it come alive.
Speaker 2:Exactly. Desmos takes those greater than, less than symbols and transforms them from something abstract into something tangible and, dare I say, fun.
Speaker 1:Fun and math in the same sentence. You're speaking my language. But even with awesome tools like Desmos, there's still a finesse to teaching this material effectively. Right?
Speaker 2:For sure. There are always nuances, little things that can make a big difference in how well students grasp these concepts.
Speaker 1:Okay. So let's roll up our sleeves and get into the nitty gritty of this lesson. What are some key activities and insights that teachers should be aware of?
Speaker 2:So this lesson dives right into problem solving with inequalities in 2 variables, and they've got some really cool scenarios laid out. We're talking bank accounts, concert tickets, advertising packages, things kids can actually wrap their heads around.
Speaker 1:And isn't that key? Making it relatable. Okay. So to make this super concrete, should we pick 1 and break it down?
Speaker 2:Definitely.
Speaker 1:How about those concert tickets? Always a hit with the algebra crowd.
Speaker 2:Excellent choice. Alright. So picture this. You're a student. You got this problem in front of you, and it says a band wants to make at least, let's say, $1,000 from their concert.
Speaker 1:Making that cash. I'm already hooked on this problem.
Speaker 2:Right. And they're selling tickets at different prices, maybe, like, $20 for floor seats and 15 for the balcony. So the big question is, how many of each type of ticket do they need to sell to hit their goal, that $1,000?
Speaker 1:Okay. So it's not just about finding one answer. It's about figuring out all the different combinations that'll work.
Speaker 2:You got it. And that's where inequalities come in. See, in a classic algebra problem, you might be solving for a single value of x. But here, we're dealing with a range of possibilities. There could be multiple combinations of floor and balcony tickets that get them to their goal.
Speaker 1:I'm seeing why this is such a great way to teach inequalities. It's not just theoretical. It's like, here's how this math actually plays out in a real life scenario.
Speaker 2:Exactly. And the lesson guides students through each step. First, they've gotta define their variables.
Speaker 1:So they're figuring out, okay, what are the unknowns we're trying to solve for?
Speaker 2:Exactly. And in this case, it would be the number of floor tickets, maybe they use f for that, and the number of balcony tickets, maybe a b.
Speaker 1:Got it. So they're assigning those variables to represent the real world quantities. Then what?
Speaker 2:Then comes the fun part, translating the word problem into an actual inequality. This is where they'll use those ticket prices, $20 for floor, 15 for balcony, and that target earnings of $1,000 to set up an inequality that represents all the possible combinations.
Speaker 1:So they're taking those concrete numbers and turning them into that abstract math language of greater than or equal to. Right?
Speaker 2:Yep. And this part can be tricky for some students, like research shows that a lot of kids struggle with inequalities that don't perfectly fit that 2 variables mold. What if the problem throws in a curveball like a limit on how many balcony tickets they can sell?
Speaker 1:Right. Because that adds another layer of complexity.
Speaker 2:Exactly. That's why illustrative math emphasizes that teachers should be ready to break things down, maybe revisit simpler examples like just x is greater than 5 to solidify that foundational understanding before jumping into the deep end with these two variable inequalities.
Speaker 1:Love that. Meeting students where they're at. Yeah. Okay. So we've talked about setting up the inequality, but what about actually solving it?
Speaker 1:That's where Desmos comes in. Right?
Speaker 2:100%. Once they've got their inequality set up, they can plug it right into Desmos.
Speaker 1:And boom, the solution appears.
Speaker 2:Well, it's it's a little more nuanced than that. Desmos will show them a graph. And here's the cool part. The solution isn't just a single point on that graph. It's an entire shaded region.
Speaker 2:Every point within that region represents a combination of floor and balcony tickets that would get the band to their $1,000 goal.
Speaker 1:So it's visually representing all those possible solutions? That's gotta be way more impactful than just seeing a bunch of numbers on a page.
Speaker 2:Right. It brings that abstract concept of multiple solutions to life. And it opens the door for even deeper discussions, like, a teacher could ask, okay, we found all these possible solutions, but are some better than others? Which combination would make the band the most money?
Speaker 1:Oh, I like that. It's taking it a step further from just solving the inequality to actually using it to make strategic decisions. Okay. This lesson is full of gems. I'm also really intrigued by this card sort activity they've got.
Speaker 1:Can we talk about that?
Speaker 2:Absolutely. Imagine a bunch of cards spread out on a table. Some have little stories, like that concert scenario. Others have inequalities written out. Some show graphs and some just list potential solutions.
Speaker 2:Students have to work together, talk it out, and match the cards that all represent the same inequality in different forms.
Speaker 1:Oh, that's clever. So it's like they're putting together a puzzle making those connections between the different representations of the same mathematical idea.
Speaker 2:Exactly. And that's where the real moments happen. When they're actively engaged, debating, explaining their reasoning, that's when those concepts really sink in.
Speaker 1:It's not just passively filling out a worksheet. It's about owning those ideas. Right?
Speaker 2:Right. And for teachers, it's like a window into their students' thinking. You can see where they're making connections, where there might be some confusion brewing, and adjust your teaching accordingly.
Speaker 1:A win win for everyone. So to make this card sort activity really sing, any tips for teachers?
Speaker 2:Definitely encourage them to go beyond just matching. Push them to explain why they think those cards go together. Ask them, how does this graph show the same thing as this word problem? Or can you come up with a real life example that fits this inequality?
Speaker 1:It's all about deepening that understanding through discussion and justification. Love it. I'm loving all these practical tips. It seems like illustrative math really knocked it out of the park with this lesson. Anything else teachers should keep in mind as they wrap up this topic with their students?
Speaker 2:You know, one thing that I always emphasize is that math isn't just about finding the right answer. It's about using these tools to make sense of the world around us, and inequalities are a perfect example.
Speaker 1:Right. Because real life is rarely as simple as x equals 5. There are usually multiple possibilities, multiple solutions.
Speaker 2:Exactly. When students really get inequalities, they're not just solving for x. They're understanding limitations, analyzing options, making smart choices. These skills come in handy everywhere.
Speaker 1:Totally. It's like budgeting and planning, even just figuring out how many errands you can squeeze into an afternoon. That's all about inequalities.
Speaker 2:You got it. It's like we're giving them this superpower to decode the hidden math in their everyday lives. And here's a little bonus tip. Once they've tackled those predesigned scenarios in the lesson, why not challenge them to become the problem creators themselves?
Speaker 1:Oh, I like where you're going with this, having them dream up situations that involve inequality.
Speaker 2:Exactly. Tell them, come up with your own real life scenarios. Maybe it's planning a party on a budget or designing a garden with limited space.
Speaker 1:Or figuring out how many hours they need to work to buy that new video game they've been wanting.
Speaker 2:Yes. It not only gets them thinking creatively, but also deepens their understanding of the math behind it.
Speaker 1:And sparks some fun debates. Right? Like, wait a minute, your inequality says you can buy 20 pizzas with $5. Something's not adding up here.
Speaker 2:Exactly. Those are the moments when you know they're really starting to internalize it, seeing the relevance.
Speaker 1:Making those connections. So to wrap things up, if there's one big takeaway from this deep dive, it's that teaching inequalities isn't just about symbols and shaded graphs. It's about giving students a framework for understanding limitations, exploring possibilities, and making informed decisions in a world full of how much questions.
Speaker 2:Beautifully said. And the exciting part is this is just the beginning. Inequalities open the door to so many other amazing math concepts.
Speaker 1:A huge thank you to Illustrative Math for creating such a well designed lesson and for always pushing the boundaries of how we teach and learn math.
Speaker 2:It's been a pleasure diving into this with you.
Speaker 1:And to our listeners, we hope this deep dive has sparked some new ideas and inspired you to see the world through the lens of inequalities. Until next time, keep those mathematical minds buzzing.