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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 equations.
Speaker 2:The equations. Oh, yeah.
Speaker 1:But hold on to your calculators because we're not just crunching numbers here.
Speaker 2:No way.
Speaker 1:We're diving deep into how to make this stuff really stick for our students.
Speaker 2:Sounds good to me.
Speaker 1:And we've got a fantastic guide this time around, an illustrative math lesson plan all about, you guessed it, equations and their solutions.
Speaker 2:I like where this is going.
Speaker 1:Me too. This lesson is all about making sure our students don't just solve for x like robots Right. But that they actually get what a solution means.
Speaker 2:Exactly. It's about understanding, not just memorizing.
Speaker 1:Okay. So let's break down this lesson plan.
Speaker 2:Let's do it.
Speaker 1:It starts with a warm up activity all about granola bars.
Speaker 2:Granola bars.
Speaker 1:You heard that right. Now before you think we've gone off topic, this activity is cleverly designed to ease students back into thinking about solutions in a relatable way.
Speaker 2:Okay. I'm
Speaker 1:intrigued. So there's this scenario involving a granola bar's nutritional information, and students have to figure out which part of the equation actually represents the solution. And what I found really interesting is that it's not just about plugging in numbers and spitting out an answer.
Speaker 2:It's about thinking critically.
Speaker 1:Precisely. Students have to explain why certain numbers just can't be solutions.
Speaker 2:I see. They have to justify their reasoning.
Speaker 1:Exactly. And that's where the real learning magic happens.
Speaker 2:Absolutely. When students have to articulate their thinking, that's when those moments happen.
Speaker 1:Okay. So we've got granola bars, which, by the way, are making me a little hungry.
Speaker 2:Me too.
Speaker 1:But before we take a snack break, let's talk about Jada and her weekend end job.
Speaker 2:Okay. Tell me more.
Speaker 1:So in this activity called weekend earnings, Jada is trying to figure out how much money she'll make working at a bookstore.
Speaker 2:Sounds familiar.
Speaker 1:Right. It's that classic real world algebra problem.
Speaker 2:We've all been there.
Speaker 1:And this activity does a great job of introducing students to the idea of writing equations with one variable.
Speaker 2:And those real world constraints, they make it relatable.
Speaker 1:Exactly. Jada's hourly wage, the bus fare, her desired earnings, it all comes together to form this mathematical story problem.
Speaker 2:And I'm guessing different students will approach solving it in different ways.
Speaker 1:You know it. Some might start plugging in numbers, while while others might jump right into using algebra.
Speaker 2:And that's the beauty of it. It allows for flexibility and different problem solving styles.
Speaker 1:I love that. It highlights that there isn't just one right way to find solution.
Speaker 2:Which is such an important lesson for students to learn.
Speaker 1:Okay. So we've covered granola bars, weekend jobs.
Speaker 2:It's been a wild ride so far.
Speaker 1:Now let's move on to something that might sound a bit scarier, calories from protein and fat.
Speaker 2:Uh-oh. Are we counting calories now?
Speaker 1:You know, the title alone makes me wanna reach for a bag of chips.
Speaker 2:Same here.
Speaker 1:But I promise, this activity is about way more than just watching our waistlines.
Speaker 2:Okay. I'm listening.
Speaker 1:This one dives into a really important algebraic concept, equations with 2 variables.
Speaker 2:Two variables. Now things are getting interesting.
Speaker 1:And get this. It introduces students to the mind blowing idea that a single equation can actually have multiple solutions.
Speaker 2:Wait a minute. Multiple solutions? Isn't there usually just one right answer in math?
Speaker 1:That's what's so cool about this. It challenges that one right answer mentality that students often get stuck in.
Speaker 2:Okay. I'm starting to see why this is a big deal.
Speaker 1:So instead of just crunching numbers, this activity encourages students to think more flexibly and creatively.
Speaker 2:That's awesome. It's like opening up a whole new world of possibilities.
Speaker 1:Exactly. And it helps them see that there isn't always one right way to approach a problem.
Speaker 2:Which is a valuable lesson for, well, pretty much everything in life.
Speaker 1:Okay. So we've got granola bars, snacks, weekend jobs, and now we're talking about multiple solutions.
Speaker 2:My brain is definitely getting a workout.
Speaker 1:Mine too. And that's why we do these deep dives to uncover these moments that can help our students thrive.
Speaker 2:I couldn't agree more. It's all about empowering them to become confident problem solvers.
Speaker 1:So before we move on to those common misconceptions that teachers might encounter, I'm curious why is this concept of multiple solutions so important for students to grasp at this stage?
Speaker 2:Oh, that's a great question. It's all about expanding their problem solving toolkit. When students encounter situations with multiple solutions, it forces them to think more flexibly and creatively. They start to see that there isn't always one right way to approach a problem.
Speaker 1:And that can be incredibly empowering for them. Right?
Speaker 2:Absolutely. It opens up a whole new way of thinking about math and problem solving in general.
Speaker 1:Okay. So we've got granola bars, snacks, weekend jobs, and multiple solutions.
Speaker 2:We've covered a lot of ground.
Speaker 1:We have. And speaking of covering ground, let's shift gears for a moment.
Speaker 2:Okay. Sounds good.
Speaker 1:And talk about those times when a student is just really struggling to grasp this whole equation thing.
Speaker 2:Yeah. Those moments can be tough.
Speaker 1:They really can be.
Speaker 2:But there are also opportunities for growth.
Speaker 1:So true. So from your experience, what are some common misconceptions teachers should be prepared to encounter when teaching about equations and solutions.
Speaker 2:Oh, there are definitely a few big ones.
Speaker 1:Like what? Give us the rundown.
Speaker 2:Well, one common one is confusing solving an equation with simply evaluating it.
Speaker 1:Okay. I can see how those 2 could get mixed up in a student's mind.
Speaker 2:Right. Especially when they're first learning about equations.
Speaker 1:So how can teachers help students understand the difference between solving and evaluating?
Speaker 2:It's all about making that distinction super clear and explicit.
Speaker 1:Give me an example.
Speaker 2:Sure. Evaluating an equation is like plugging in a known value for the variable and seeing what you get, like we did with that snack example earlier.
Speaker 1:Right. We were checking if a specific combination of protein and fat match the calorie count.
Speaker 2:Exactly. But solving an equation means figuring out what that unknown value is in the first place.
Speaker 1:And as we've seen, there can be multiple possibilities for that unknown value.
Speaker 2:Precisely.
Speaker 1:Okay. That's super helpful. What's another common misconception teachers might encounter?
Speaker 2:Another big one is connecting those abstract equations back to real world situations.
Speaker 1:Ah, yes. Bridging that gap between the symbols on the page and the actual world around them.
Speaker 2:Exactly. Some students struggle to see the relevance of equations outside of a math textbook.
Speaker 1:So how can teachers help students make those real world connections?
Speaker 2:It's all about rooting those equations in something familiar and engaging to them.
Speaker 1:So instead of just presenting an equation like 2x+11, we could frame it as a real life problem.
Speaker 2:You got it. Like, imagine you're trying to save up for that new video game that costs, say, $60.
Speaker 1:Okay. I like where this is going.
Speaker 2:You've already saved $25, and you earn $5 for every chore you do around the house.
Speaker 1:So the equation 25 +5xequals60 could represent how many chores you need to do to reach your goal.
Speaker 2:Exactly. Suddenly, that abstract equation has a real world purpose and meaning.
Speaker 1:It's not just about solving for x anymore. It's about achieving a tangible goal.
Speaker 2:Right. And that can make all the difference in terms of student engagement and understanding.
Speaker 1:I love that. It's all about finding those hooks that make the math come alive for students.
Speaker 2:Precisely.
Speaker 1:Okay. So we've talked about the importance of distinguishing between solving and evaluating, and we've touched on the power of making those real world connections.
Speaker 2:We're on a roll.
Speaker 1:We are. But before we wrap things up, I wanna circle back to something you mentioned earlier about those moments.
Speaker 2:Those are the best, aren't they?
Speaker 1:They really are. Those moments when a student's face lights up and you can practically see the gears turning in their head.
Speaker 2:It's like witnessing a breakthrough in real time.
Speaker 1:Exactly. So what are some specific strategies teachers can use to help facilitate those moments when teaching about equations and solutions?
Speaker 2:Well, one powerful strategy is to encourage students to visualize the problem.
Speaker 1:Visualize.
Speaker 2:How do you mean? Like, instead of just thinking about an equation as a bunch of symbols, encourage them to draw it out.
Speaker 1:Oh, interesting. So using diagrams, graphs, even just simple drawings to represent the problem.
Speaker 2:Exactly. Visualizing can help students make sense of the relationships between different parts of the equation.
Speaker 1:Right. It's like giving them a different lens to view the problem through.
Speaker 2:Precisely. And when those different representations click, that's when those moments often happen.
Speaker 1:Okay. That makes sense. Are there any other strategies teachers can use to spark those breakthroughs?
Speaker 2:Another great one is to create opportunities for students to explain their thinking to each other.
Speaker 1:Oh, I love that. Peer to peer learning can be so powerful.
Speaker 2:It really can. When students have to explain a concept in their own words, it forces them to really grapple with it.
Speaker 1:And they might even uncover gaps in their own understanding during those conversations.
Speaker 2:Exactly. And let's not forget the power of real world applications.
Speaker 1:We've touched on that a bit already, but I'm always up for more ideas.
Speaker 2:Well, challenge students to come up with their own real world scenarios that could be represented by a given equation.
Speaker 1:Oh, I like that. It's like turning the tables and letting them be the problem creators for a change.
Speaker 2:Exactly. It not only deepens their understanding of the math, but also sparks their creativity in critical thinking skills.
Speaker 1:It's all about making those connections and showing them that math is so much more than just textbook exercises.
Speaker 2:Precisely. It's a tool for understanding and navigating the world around us.
Speaker 1:Well said. I feel like this deep dive has been a real eye opener even for a seasoned math enthusiast like myself.
Speaker 2:Me too. It's a good reminder that even the most familiar concepts can be explored in fresh and engaging ways.
Speaker 1:Absolutely. And that's what it's all about, constantly learning and growing alongside our students.
Speaker 2:Wouldn't agree more.
Speaker 1:Well, a huge thank you to the creators of this illustrative math lesson for providing the inspiration for today's deep dive.
Speaker 2:It's a fantastic resource for bringing those moments to life in the classroom.
Speaker 1:It really is. And, listeners, we wanna hear from you. How are you bringing these concepts to life in your classrooms? What creative strategies are you using to help your students master equations and unlock those moments? Head over to social media and share your experiences and insights.
Speaker 1:Until next time. Happy teaching.