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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.)
Ready for a deep dive today? We're tackling teaching inequalities in one variable. You know, that algebra concept that can sometimes feel like you're speaking a different language.
Speaker 2:Absolutely. It can feel that way for teachers and students alike. That's why we're diving into this illustrative math teacher's guide today. We wanna give you what you need to teach inequalities in a way that helps students really get it.
Speaker 1:Love it. And looking at the guide, it seems to focus on finding any value that makes an inequality true. Is that the right place to start, or are we jumping the gun?
Speaker 2:That's a question a lot of teachers ask. I mean, starting with finding any value might seem a little strange at first, especially if students are used to equations where there's usually one right answer.
Speaker 1:Right. It's a different way of thinking.
Speaker 2:It is, But that's actually why this approach can work so well.
Speaker 1:Okay. Now you've got me curious. Tell me more.
Speaker 2:So think about it. When you start with finding any value, you're shifting the focus away from that one right answer and showing them that inequalities can have a bunch of solutions, a whole range.
Speaker 1:So it's not about the answer, but about understanding there's a whole solution set.
Speaker 2:You got it. And that's a really important shift because it helps students see inequalities as a concept, not just a procedure.
Speaker 1:I like it. And this makes me think of that example teachers use all the time, that sign restaurants have about maximum capacity. They're not saying exactly how many people are allowed inside, just setting a limit, showing a range.
Speaker 2:Perfect example. And that real world connection is something the guide uses too with an activity called orchard.
Speaker 1:Orchard. Tell me more.
Speaker 2:So it gives students a scenario where they're comparing costs for a class trip to 2 different orchards.
Speaker 1:Okay. Sounds fun already.
Speaker 2:Each orchard has different prices. One might have a lower entry fee, but charge more per student, while the other has a higher entry fee, but charges less per student.
Speaker 1:So students have to figure out the better deal depending on how many students are going.
Speaker 2:Exactly. To do that, they naturally start using inequalities maybe even without realizing it at first.
Speaker 1:Because they're comparing expressions, finding that point where one orchard becomes cheaper than the other.
Speaker 2:You got it. That's where the happens. They're not just solving problems on a worksheet. They see how it works in a real situation.
Speaker 1:I love that. But inequalities can still be tricky even for those of us who love math. What are some common problems teachers might run into with this lesson?
Speaker 2:One of the biggest ones is that thing with flipping the inequality symbol.
Speaker 1:Oh, I remember that. I felt like I was just guessing when to flip it.
Speaker 2:It's tempting to give students a bunch of rules to memorize, but that rarely works.
Speaker 1:It's true. You need to understand why you're doing something, not just memorize it.
Speaker 2:Right. The guide emphasizes helping students get the why behind the flip, not just the when.
Speaker 1:Show them the why, not just the how, love it. So how do we do that?
Speaker 2:For instance, when you're multiplying or dividing both sides by a negative number, you're basically reversing the order on the number line.
Speaker 1:So, visually, you're flipping the entire number line, which means the inequality sign has to flip too to keep things accurate.
Speaker 2:Exactly. When students see that, the rule makes sense. It's not just random. It's a logical result of what you're doing.
Speaker 1:That's such a good point. We need to give students the understanding of why, not just tell them to flip the symbol.
Speaker 2:Absolutely. And the guide has some excellent ways to do just that, which we'll get into shortly.
Speaker 1:Awesome. Besides the Orchard activity, what else does this lesson use to really make these concepts stick?
Speaker 2:Well, the guide has this activity called equality and inequality that builds on that connection between inequalities and equations.
Speaker 1:The equality and inequality. Sounds interesting. What's that one all about?
Speaker 2:It's all about making that link between inequalities and equations crystal clear.
Speaker 1:I like it.
Speaker 2:Students get an equation, and then they have to work with an inequality that's really similar to it. Okay.
Speaker 1:So they can see how the 2 are connected.
Speaker 2:Right. It helps them see that the answer to the equation is like a starting point for the inequality solution.
Speaker 1:Okay. So imagine a number line. Right? We solve the equation, get our point, and then the inequality tells us which way to go from there above, below, maybe even just a section.
Speaker 2:Exactly. Solving the equation gives them that crucial starting point, then the inequality sign shows them the rest.
Speaker 1:That's a good moment right there, don't you think? Seeing that connection.
Speaker 2:Absolutely. It's not about memorizing anymore. They can start to figure things out more naturally.
Speaker 1:But as with anything, deeper understanding can sometimes lead to new misunderstandings. What are some things teachers should watch out for with this activity?
Speaker 2:Good point. And the guide actually mentions one that's really common. Students trying to change those inequality signs without really getting why?
Speaker 1:Oh, I've seen that. Yeah. They just wanna keep the sign the same even if they've divided or multiplied by a negative number.
Speaker 2:It happens all the time. They get caught up in the steps without thinking about why those steps work.
Speaker 1:Right. We don't want them to just be robots blindly following rule.
Speaker 2:Exactly. So teachers really need to emphasize that it's not enough to just get the right answer. Students need to be able to explain how they got there, why their steps make sense, and how each thing they do changes the inequality.
Speaker 1:It's all about that why. Right? We want them to understand the math, not just memorize it.
Speaker 2:Exactly. Now speaking of understanding, I really like this optional activity in the guide called more or less. It focuses on visualizing the solutions, which can be so helpful for some students.
Speaker 1:I'm all about visual approaches to math. It's like unlocking a secret code for those who learn best that way.
Speaker 2:I agree. It's a powerful tool. More or less has students graph the expressions on both sides of the inequality instead of just sticking with the algebra.
Speaker 1:So instead of just symbols and numbers, they're seeing how these things behave on a graph. Right?
Speaker 2:Exactly. They can literally see where one side is bigger, smaller, or equal to the other side, which creates a visual representation of the solution.
Speaker 1:So smart. This could be huge for visual learners, that moment right there on the graph.
Speaker 2:You said it. And the guide recommends using Desmos for this activity, which is a great choice.
Speaker 1:Desmos. I love Desmos. For those who haven't used it, it's this awesome free graphing calculator online, super easy to use, and makes graphing really interactive.
Speaker 2:Exactly. And that interactivity is key here. With Desmos, students can change the value of x with a slider and see how it affects the graphs and the solution in real time.
Speaker 1:It's like bringing the math to life. Seeing those lines move around and the solution change right in front of you, it just clicks in a different way.
Speaker 2:You got it. It makes it way more engaging and helps them see the connections.
Speaker 1:I'm convinced. We've covered so much today from real world examples to visualizing things on a graph. As we start to wrap up this deep dive, what does the guide suggest for closing out this lesson?
Speaker 2:They recommend ending with a simple question. How does solving the equation help with solving the inequality?
Speaker 1:I love it. It brings us right back to where we started.
Speaker 2:Why? It makes them think about how those two things connect.
Speaker 1:Exactly. It's not just about remembering information, but about really understanding it.
Speaker 2:Yes. And being able to explain it in their own words.
Speaker 1:Exactly. So to wrap things up, what are some key takeaways for our listeners who are getting ready to teach this?
Speaker 2:Well, I think this guide really shows how helpful visuals can be, like that number line we talked about. It's such a simple tool.
Speaker 1:It is, but it's so powerful for helping students see that range of solutions.
Speaker 2:Exactly. And don't forget about Desmos. Being able to play with those graphs and see the solutions change can be a game changer.
Speaker 1:Absolutely. It's like making the math come alive.
Speaker 2:Right. Another important thing is to be ready for those common mistakes students make, like that whole flipping the inequality sign thing.
Speaker 1:Oh, yeah. That's a big one.
Speaker 2:It is. But if we know those tricky parts are coming, we can help our students understand the why behind the rules, not just memorize them.
Speaker 1:Give them a map before they get lost in the woods.
Speaker 2:Exactly. And maybe the most important thing of all is to focus on understanding, not just memorizing steps.
Speaker 1:All about those moments. Right?
Speaker 2:Exactly.
Speaker 1:Yeah.
Speaker 2:We want them to get it, not just repeat it.
Speaker 1:Love it. Any final thoughts for our listeners as they dive into teaching inequalities?
Speaker 2:Let me ask you this. How can you connect these ideas to the things your students are into? What are their interests?
Speaker 1:Oh, I like that. Make it real for them.
Speaker 2:Yeah. Yeah. Could you turn that orchard problem into something about video games or use inequalities to analyze sports stats?
Speaker 1:So many possibilities. When students see how math connects to their world, it just clicks.
Speaker 2:Exactly.
Speaker 1:It's been great exploring this with you, and a big thank you to the authors of Illustrative Math for this fantastic resource.
Speaker 2:My pleasure. Until next time.