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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.)
Have you ever, like, started a project and just felt totally overwhelmed by all the limitations?
Speaker 2:Oh, yeah.
Speaker 1:Like, trying to plan, say, a class party, but you've got a super tight budget
Speaker 2:Right.
Speaker 1:And not a lot of time. Mhmm. And let's be real. You've got some picky eaters in the mix.
Speaker 2:It's like classic real world constraints just totally colliding.
Speaker 1:Exactly. And it turns out there's, like, a whole branch of math Right. Designed to, you know, help us navigate those situations. And that's kinda what today's deep dive is all about.
Speaker 2:Yeah.
Speaker 1:We're gonna be taking a look at this illustrative math lesson
Speaker 2:Okay.
Speaker 1:On systems of inequalities.
Speaker 2:Okay.
Speaker 1:And let me tell you, it is bursting with really clever ways to help students not just solve for x, but to actually make smart decisions, you know, when they're facing multiple constraints. And that's what's so cool about this lesson. It's not, like, just about shading
Speaker 2:in areas on a graph. Right. It
Speaker 1:inequalities in
Speaker 2:a system true at the same time. And I think even more importantly, it helps students really understand how those solutions actually play out in, like, you know, practical scenarios.
Speaker 1:Yeah. It's like the lesson suddenly gives you x-ray vision or something Uh-huh. Into all the workable options.
Speaker 2:Yeah.
Speaker 1:And this is where the concept of the overlapping region on a graph becomes so powerful. Right?
Speaker 2:Totally. Totally. Because that overlapping region, that's visually showing us all the viable solutions
Speaker 1:Okay.
Speaker 2:All the choices that satisfy every single constraint.
Speaker 1:It's like a cheat sheet, basically.
Speaker 2:Yeah. Yeah. Exactly.
Speaker 1:Okay. So let's dive into, like, how this lesson actually unfolds.
Speaker 2:Okay.
Speaker 1:It starts, I think, with a brain teaser. Mhmm. Like a riddle that seems, like, impossible to solve at least at first.
Speaker 2:And this is where it gets really clever because that riddle, it's setting the stage.
Speaker 1:Okay.
Speaker 2:It's demonstrating that systems of equations, they often have very specific solutions.
Speaker 1:Right.
Speaker 2:It's not just any number will do. So it piques students' curiosity right away.
Speaker 1:I like that. Yeah. Okay. So then the lesson shifts gears a little bit.
Speaker 2:Okay.
Speaker 1:It goes into a quilting activity. Yeah. And this is where students, I feel like, really start to actually grasp the power of systems of inequality.
Speaker 2:Totally. And you see the activity highlights how tough it can be to find those, like, perfect solutions
Speaker 1:Right.
Speaker 2:If you're just, like, looking at separate graphs.
Speaker 1:Yeah. Just eyeballing it.
Speaker 2:It's like we need a more systematic approach. Yeah. And that's where systems of inequalities come in.
Speaker 1:And the lesson doesn't even stop at quilting. It goes into these familiar scenarios
Speaker 2:Right.
Speaker 1:That students actually encounter in their own lives.
Speaker 2:Yeah. Yeah.
Speaker 1:So we're talking, like, managing a bank account, maybe buying concert tickets Sure. Even putting together those, like, tempting snack packages
Speaker 2:Right.
Speaker 1:For, you know, that class party we were talking about earlier.
Speaker 2:Exactly. It helps them connect to these abstract mathematical concepts to, you know, concrete situations
Speaker 1:Right.
Speaker 2:Things they can relate to. And they start to see how systems of inequalities are actually practical tools
Speaker 1:Yeah.
Speaker 2:For making decisions in all sorts of different contexts.
Speaker 1:Love it. And hold on. We can't forget about those visual learners out there.
Speaker 2:Of course not.
Speaker 1:Because this lesson incorporates a scavenger hunt.
Speaker 2:Oh, fun.
Speaker 1:Which sounds like pure fun, but it also subtly reinforces this idea that the solution is the entire overlapping region
Speaker 2:Yes.
Speaker 1:On the graph, not just a single point.
Speaker 2:And that is so important for students to get. It's not just about finding any solution. It's about understanding the full range of possibilities Right. Within those constraints.
Speaker 1:Okay.
Speaker 2:And this is often where we see some common misconceptions pop up.
Speaker 1:Let's unpack those misconceptions a bit. Like, what are some of the things that students might stumble over when it comes to systems of inequalities?
Speaker 2:I think one of the big ones is, like, really getting what a system solution actually means
Speaker 1:True.
Speaker 2:In real world terms. Right?
Speaker 1:Yeah.
Speaker 2:It's not enough to just be able to, like, find a point on the graph. Students need to be able to say what that point actually represents
Speaker 1:Right. Right.
Speaker 2:In the context of the problem.
Speaker 1:Though it's about making that connection between the kinda the abstract math Yeah. And, like, what it actually means It it in the real world.
Speaker 2:Exactly. Yeah. Like, for example, in that concert ticket scenario we were talking about, a solution could be, like, a combo of how many adult tickets and how many student tickets
Speaker 1:Mhmm.
Speaker 2:You can buy and stay within a certain budget.
Speaker 1:Okay.
Speaker 2:But students need to go beyond just, like, oh, here are the numbers on the graph. Right. They need to be able to say, okay. This means we can buy this many adult tickets and this many student tickets, and we're still good.
Speaker 1:Yeah. Yeah. It's like making that leap from, like
Speaker 2:Totally.
Speaker 1:Points on a graph Yeah. To, like, actually what you're gonna do.
Speaker 2:Real decisions. Right? Yeah. Yeah.
Speaker 1:Exactly. Yeah. Okay. Another tricky one is, understanding that some systems don't even have any solutions Oh, interesting. Which can be kinda weird for students at first.
Speaker 1:Right? Because they're so used to, like Joy. Here's the equation. Solve for
Speaker 2:x. Right. There's always
Speaker 1:an answer. Yeah.
Speaker 2:There's always an answer. But, like, in real life, sometimes those constraints
Speaker 1:just clash.
Speaker 2:Yeah. And there's no magic combination that's gonna make everyone
Speaker 1:happy.
Speaker 2:Right. Like, imagine trying to find a solution where, say, one inequality is a budget limit of, like, $100.
Speaker 1:Okay.
Speaker 2:But then another constraint is that you have to spend at least $150.
Speaker 1:You'd be in trouble.
Speaker 2:You'd be stuck. Right? Mathematically impossible. Yeah.
Speaker 1:You'd be like, wait a minute.
Speaker 2:And that, I think, is a really valuable lesson.
Speaker 1:Okay.
Speaker 2:Because sometimes realizing that things are just incompatible Yeah. Is just as important as finding a solution when there is one. You know?
Speaker 1:It's like knowing when to, like, cut your losses or something.
Speaker 2:Right. Exactly.
Speaker 1:This just isn't gonna work.
Speaker 2:It's not gonna work.
Speaker 1:And that's real life.
Speaker 2:Are you at all?
Speaker 1:It's not just math class.
Speaker 2:It's about being able to navigate those no solution situations.
Speaker 1:Right. Because you're gonna hit them
Speaker 2:Oh, yeah.
Speaker 1:Not just on a test
Speaker 2:All the time.
Speaker 1:But, like, in your actual life.
Speaker 2:Exactly.
Speaker 1:Okay. So shifting gears a little bit Okay. For our listeners who are, you know, thinking about teaching this lesson Yeah. What are some, like, actionable things to keep in mind?
Speaker 2:Well, I think it's super important to really emphasize how much this lesson builds on what students already know
Speaker 1:Okay.
Speaker 2:About, like, just single inequality.
Speaker 1:Right. Because they've already learned that.
Speaker 2:They have. Yeah.
Speaker 1:So it's like connecting those dots.
Speaker 2:Exactly. Connect those dots because they know how to graph a single inequality.
Speaker 1:Right.
Speaker 2:They get what the shaded area means.
Speaker 1:Right.
Speaker 2:Now it's like, okay. Let's combine a bunch of these.
Speaker 1:Yeah. Yeah. Take it up a notch.
Speaker 2:And see what happens in that overlapping part.
Speaker 1:Love it. Love it.
Speaker 2:And and vocab is huge here.
Speaker 1:Vocabulary.
Speaker 2:Yeah. The lesson plan has this great glossary of terms.
Speaker 1:Oh, nice.
Speaker 2:So everyone's on the same page.
Speaker 1:That's always helpful. It
Speaker 2:is because then there's no wait. What does that word mean?
Speaker 1:Right. Exactly. Exact We're
Speaker 2:all speaking the same language.
Speaker 1:Okay. Love it.
Speaker 2:Yeah.
Speaker 1:So we've got connecting it to prior knowledge.
Speaker 2:Yes.
Speaker 1:We've got making sure our vocabulary is solid.
Speaker 2:Yes.
Speaker 1:What else?
Speaker 2:Oh, and for those, like, real moments Yeah. Using graphing technology is amazing.
Speaker 1:Okay. Like what?
Speaker 2:Desmos, things like that.
Speaker 1:Gotcha.
Speaker 2:Because being able to actually visualize those overlapping regions
Speaker 1:Right.
Speaker 2:It's huge.
Speaker 1:Totally.
Speaker 2:And then students can play around with it. You know?
Speaker 1:So it's not just, like, static on the page.
Speaker 2:Exactly. It's dynamic.
Speaker 1:Yeah. They can, like, mess with the equation a little bit and see what happens.
Speaker 2:Totally. They can change the inequality, see how the solution set changes. It's awesome.
Speaker 1:I love that. I love that.
Speaker 2:Yeah. It's like giving them a front row seat to how these systems actually work.
Speaker 1:It's like they get to, like, be in the driver's seat almost.
Speaker 2:Yes. Exactly.
Speaker 1:Just, like, observe.
Speaker 2:It's active, not passive. Exactly.
Speaker 1:Yeah. Okay. So we've got the graphing technology. Yeah. Anything else in terms of, like
Speaker 2:Oh, and you know what I love? This lesson doesn't just stop at, like, find a solution.
Speaker 1:Okay.
Speaker 2:It goes a step further. Oh. It encourages students to think about, like, what is the best solution Oh, interesting. Given, you know, the specific situation.
Speaker 1:Okay. So it's like
Speaker 2:It's not just about finding any combination of numbers that works. Right. It's about thinking critically.
Speaker 1:So, like, going back to our concert ticket example. Yeah. It's not just about, like
Speaker 2:Finding any way to buy tickets within the budget.
Speaker 1:Right. It's like how many chaperones do we need to bring?
Speaker 2:Exactly. Or, like, how many students even wanna go?
Speaker 1:Right. Right.
Speaker 2:It adds this whole other layer Literally. Of real world decision making.
Speaker 1:It's, like, strategic.
Speaker 2:It is.
Speaker 1:It's like thinking through all the different angles.
Speaker 2:It's not just about, like, here's one right answer, and we're done.
Speaker 1:Right. Exactly. Exactly. It's more nuanced.
Speaker 2:Yeah. It's more like how you actually have to make decisions.
Speaker 1:Now one thing I wanted to highlight
Speaker 2:Okay.
Speaker 1:Is that section in the lesson plan, are you ready for more? Oh, yeah. Yeah. Tell me about that.
Speaker 2:It is pure gold.
Speaker 1:Okay. Why? What makes it so great? So what's so special about this are you ready for more section? It sounds like it's pushing students beyond just the basics.
Speaker 2:Yeah. Totally. It challenges students to kind of become, like, math creators
Speaker 1:Okay.
Speaker 2:Instead of just solvers.
Speaker 1:Interesting.
Speaker 2:They have to actually design their own w n systems of inequalities.
Speaker 1:Oh, wow.
Speaker 2:But there's a twist.
Speaker 1:Okay. I'd like a good twist. What is it?
Speaker 2:The twist is they have to create a system that only has one solution. Oh, that's So it's not just about, like
Speaker 1:Finding any solution.
Speaker 2:Finding any solution. It's about Right. Really understanding how those equations and graphs work together.
Speaker 1:So they're not just finding the answer. Right. They're, like, building the puzzle.
Speaker 2:Exactly.
Speaker 1:I love that.
Speaker 2:It takes their learning to a whole new level.
Speaker 1:It's like they've kind of graduated from solving Yeah. To, like, really mastering it.
Speaker 2:Totally. Yeah. I love it. And, you know, thinking back on our deep dive today
Speaker 1:Yeah.
Speaker 2:I think this lesson really shows how cool it is.
Speaker 1:Okay. To connect math to the real world.
Speaker 2:Yeah. It's not just, like, it's
Speaker 1:It's not just symbols and equations. You know?
Speaker 2:That it's, like, actually useful.
Speaker 1:It's about giving students the tools to make decisions
Speaker 2:Right.
Speaker 1:Every day.
Speaker 2:Like, their own decisions.
Speaker 1:Yeah. Whether it's, like, planning that party
Speaker 2:Right.
Speaker 1:Or managing their money or even just, like
Speaker 2:Picking out the best deal at the store.
Speaker 1:Exactly.
Speaker 2:It's math in action, and I think that's what makes this lesson plan Totally. So engaging and effective.
Speaker 1:It really helps students see why math matters.
Speaker 2:Right. It's not just
Speaker 1:It's not just in your textbook.
Speaker 2:Yeah. It's like, you're gonna use this.
Speaker 1:It's all around them.
Speaker 2:Exactly.
Speaker 1:Exactly. What we want. Right?
Speaker 2:Yes.
Speaker 1:We want them to see the math.
Speaker 2:Not be scared of it. To
Speaker 1:see the patterns, make good choices.
Speaker 2:Yeah. Those are skills that, you know, for life go way beyond the classroom.
Speaker 1:Totally.
Speaker 2:Love it. Yeah. Well, this has been so fun.
Speaker 1:Oh, it's been great.
Speaker 2:Systems of inequalities.
Speaker 1:Who knew?
Speaker 2:Who knew? Right. So interesting.
Speaker 1:All those moments. Yeah. Real practical strategies.
Speaker 2:For sure.
Speaker 1:So big thanks to Illustrative Math
Speaker 2:Yes. Thank you.
Speaker 1:For this awesome lesson plan.
Speaker 2:They're amazing.
Speaker 1:And to our listeners, thanks for joining us.
Speaker 2:Yeah. Thanks for being here.
Speaker 1:Keep those minds curious.
Speaker 2:Keep asking those questions.
Speaker 1:And we'll see you next time.
Speaker 2:See you later.