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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, everyone. Welcome back. It's time to dive into another math concept, and this one is all about graphing linear inequalities. You know, those tricky things with the greater than and less than symbols. We're taking a close look at this illustrative mathematics lesson plan today to help you guide your students through this often confusing topic.
Speaker 2:And it's a good one. What I really appreciate about this lesson is how it helps students make that leap from one variable inequalities to those involving 2 variables. It's a big step.
Speaker 1:It is a big step. So let's break it down a bit. For those of us who might need a little refresher, what makes this shift to 2 variable inequality so significant?
Speaker 2:Well, it's all about perspective. With 1 variable inequality, you're working with a number line. Right? And your solution is a section of that line. But when you bring in another variable, suddenly you're not confined to a single line anymore.
Speaker 2:You're working in the two dimensional world of the coordinate plane.
Speaker 1:We're venturing into new dimensions here. I can already see how that would change things.
Speaker 2:Exactly. Instead of segments on a line, we're now talking about entire regions of the coordinate plane called half planes. These half planes are all the points, all those x y pairs that make the inequality true.
Speaker 1:Okay. So instead of just finding points on a line, we're shading in whole chunks of the graph.
Speaker 2:You got it. It's like coloring in all the right answers. And this lesson uses a great example to illustrate this. They introduce the inequality x+y7.
Speaker 1:Okay. X+y is less than or equal to 7. I remember that from algebra class, I think.
Speaker 2:And now we're not just looking for values where x+y equals 7, We want all the pairs of x and y that make that whole expression less than or equal to 7.
Speaker 1:So there are lots of different combinations that would work.
Speaker 2:Infinitely many, in fact. And they all fall within a specific region on the graph. So how do you think the lesson plan helps students wrap their heads around this idea of a solution region?
Speaker 1:Well, it looks like they start with a math talk, which I always appreciate. Getting students talking about the math is so important.
Speaker 2:It is. And this math talk focuses on numerical reasoning. They have students plugging in different values for x and y just to see if they make the inequality true or not. It's all about building that foundational understanding before they even get to the graphing.
Speaker 1:It's like laying the groundwork before building a house. You need that solid foundation. And then I see the next activity is called solutions and not solutions. That sounds very straightforward, which I like.
Speaker 2:And brilliantly simple at the same time because it doesn't just tell students the rule. It lets them discover it themselves through exploration. You take an example, inequality, like, let's say, 2xy12.
Speaker 1:2x minus y is greater than 12. Okay.
Speaker 2:And they start plotting points. Some that make the inequality true and some that don't.
Speaker 1:I see. So, they're testing the waters, so to speak.
Speaker 2:Exactly. And as they plot these points, they start to see a pattern emerge. They realize that all the solutions seem to be clustered on one side of the line, while all the non solutions are on the other side. This is where that moment happens. And they organically arrive at the idea that one whole side of the line represents the solution region.
Speaker 1:Oh, and another thing I love about this activity is how it incorporates that pull the class routine. Have you seen that before?
Speaker 2:Oh, yeah. It's a great way to gauge student understanding and get them talking to each other. But more importantly, it emphasizes that there's not always one right way to approach a problem.
Speaker 1:Exactly. And it values those different strategies that students bring to the table. Yeah. It's all about celebrating that diversity of thought in the classroom, which is so important in math. Right?
Speaker 2:It really is. And it's a good reminder for us teachers too. There's often more than one path to the right answer in math. But speaking of paths and directions, I'm curious about how this lesson handles a common misconception. I've seen students struggle with, you know, that idea that the direction of the inequality symbol always tells you which way to shade?
Speaker 2:Yes. The infamous greater than means shade above trap.
Speaker 1:Exactly. It seems so intuitive at first glance.
Speaker 2:It does, doesn't it? But that's why I appreciate how this lesson plan addresses it head on. They don't shy away from those potential pitfalls. In fact, they use them as learning opportunities. They even include a specific counterexample, like 2xy5.
Speaker 1:So even though it's a greater than inequality, you don't necessarily shade above the line.
Speaker 2:Exactly. This particular example forces students to slow down and realize that they can't just rely on that visual shortcut. They actually have to test points or really think critically about what the inequality is telling them. I love that. It's like being a math
Speaker 1:detective, gathering evidence to determine where the solution region lies. Speaking of things that might require some detective work, how does the lesson address those tricky vertical and horizontal lines? Those always seem to trip students up.
Speaker 2:Right. Those can be a bit of a head scratcher. But again, the lesson plan takes a very insightful approach. Instead of just throwing rules at the students, they encourage teachers to guide them towards recognizing patterns.
Speaker 1:Patterns in the coordinates.
Speaker 2:Exactly. For example, with a vertical line, students might notice that the x coordinate stays the same for every single point, no matter what the coordinate is. And that pattern recognition helps them understand why the equation of a vertical line is always x equals a constant.
Speaker 1:Oh, I see. So it's not about memorizing some arbitrary rule. It's about seeing the underlying structure, which is much more powerful. I'm really appreciating how this lesson plan is structured. It seems to anticipate those common student struggles and provide teachers with a road map for addressing them.
Speaker 2:Absolutely. And speaking of road maps, it also provides some really valuable tools for reinforcing those key concepts. One of my favorites is the set of discussion questions they suggest using during the synthesis phase of the lesson.
Speaker 1:So when students are pulling everything together Exactly. These questions are so cleverly crafted because they prompt students to articulate
Speaker 2:their It's about
Speaker 1:It's about bringing those implicit assumptions to the surface and making them explicit, making sure they're not just blindly following rules. Right?
Speaker 2:Precisely. It's about fostering that metacognitive awareness. Having students think about their own thinking and verbalizing their thought process is such a powerful way to do that.
Speaker 1:It's like that old saying, if you can't explain it simply, you don't understand it well enough or something like that.
Speaker 2:Exactly. And this ties into another important aspect of this lesson plan, the emphasis on visual representations. We're talking about graphing after all.
Speaker 1:It's about making those abstract concepts visual and concrete.
Speaker 2:Exactly. The coordinate plane becomes this amazing tool for students to actually see the solutions, to see the relationship between the variables, and make sense of the inequality in a tangible way.
Speaker 1:And let's be honest. There's something so satisfying about a well shaded solution region even for us teachers.
Speaker 2:Absolutely. But speaking of satisfaction Mhmm. I think we've covered a lot of ground here, wouldn't you say?
Speaker 1:I agree. I feel much more prepared to tackle this topic with my students now. Having explored this lesson plan in-depth like this we have. We have. But you know me, I always love a little something extra to keep those teacher brains buzzing.
Speaker 1:So before we officially wrap things up, do you have one of your signature thought provoking questions for our listeners?
Speaker 2:I do. I do. And it builds directly on what we've been discussing Mhmm. About graphing a single linear inequality. But what if we took it a step further and we wanted to introduce the idea of systems of inequalities?
Speaker 1:Oh, systems of inequalities. Now that's where things get really interesting, visually and conceptually.
Speaker 2:Absolutely. Instead of just one solution region, now we're looking for the overlapping region that satisfies all the inequalities in the system.
Speaker 1:It's like we're layering those shaded regions on top of each other to find that sweet spot where all the solutions intersect.
Speaker 2:Exactly. And that visual representation is so powerful. Yeah. Because students can see how the solution to the system. It's not just a line or even a single region anymore.
Speaker 2:It's that specific area where all the conditions are met simultaneously.
Speaker 1:It's like the ultimate solution, the one that satisfies everything.
Speaker 2:Exactly. And it's a great way to extend their thinking, to connect it to those real world situations where you often have multiple constraints at play.
Speaker 1:Oh, absolutely. Think about things like optimization problems or resource allocation, even something like figuring out the best time to plant a garden. Based on sunlight and temperature ranges, systems of inequalities are everywhere.
Speaker 2:They really are. And that's something I think we often forget to convey to students, the real world relevance of these mathematical concepts.
Speaker 1:It's so important to make those connections for them, to show them that math isn't just some abstract thing. It's a tool for understanding and navigating the world around us. Exactly. Well, I think that's a fantastic note to end
Speaker 2:on, a
Speaker 1:challenge for our listeners to ponder. As they head back to their classrooms, how can they adapt this lesson on linear inequalities to introduce this exciting world of systems of inequalities?
Speaker 2:It's a journey worth taking, and it all starts with building that strong foundation with single inequalities, which is exactly what this deep dive has been all about.
Speaker 1:Couldn't have said it better myself. So to wrap things up, we've journeyed through the core concepts of this illustrative mathematics lesson plan, explored those cleverly designed activities, and even tackled some common student misconceptions head on.
Speaker 2:We've given you the tools and insights you need to teach this topic with confidence and clarity.
Speaker 1:And a newfound appreciation for the power of inequalities. Remember to guide your students to see those connections between the algebraic representations and their visual counterparts on the coordinate plane.
Speaker 2:And don't underestimate the power of testing points, encouraging your students to articulate their reasoning and using those visual representations to solidify understanding.
Speaker 1:Visuals are key. And with that, we'll leave you to delve into the fascinating world of linear inequalities with your students. Happy graphing, everyone.
Speaker 2:Happy graphing.
Speaker 1:And a huge thank you to the authors of Illustrative Math for this insightful lesson plan. Until next time.