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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.)
Alright. So have you ever been up late grading algebra tests
Speaker 2:Mhmm.
Speaker 1:And you wanna figure out how many points each question is worth? Yeah. You could do the long division every single time Right. Or you could be cool and rearrange that formula to have the points per question right there. Yeah.
Speaker 1:That is kinda what we're diving into today. We're looking at lesson 8 from illustrative mathematics.
Speaker 2:Okay.
Speaker 1:And it's all about helping teachers like you demystify sorry.
Speaker 2:Into, like, the mechanics of solving for a variable. Right. It really starts with making sure students understand, like, the why.
Speaker 1:Yeah. Because it's not just about finding an answer. Right. It's revealing the relationship between variables.
Speaker 2:Yeah. Exactly. And the lesson plan uses a really great example of calculating the area of a parallelogram. Mhmm. So we all have that formula, like, ingrained in us.
Speaker 1:Yeah.
Speaker 2:Area equals base times height.
Speaker 1:Yeah.
Speaker 2:But imagine you're a student, and this time, you know the area and, you know, the height, but you need to find the base.
Speaker 1:That's where rearranging the formula to solve for base becomes really important.
Speaker 2:Exactly.
Speaker 1:Yeah. And useful. Yeah. I am already seeing some moments happening in the classroom.
Speaker 2:Yeah. And they have this really clever activity Great. Called post parade cleanup to illustrate that.
Speaker 1:Post parade cleanup. Yes. Alright. Tell me more.
Speaker 2:So picture this. There's been a parade k. And now they're at 2 miles of confetti and what have you to clean up. Okay. And students are given the challenge of figuring out how much road each volunteer would need to clean based on how many volunteers there are.
Speaker 1:Oh, so if you have more volunteers, each person cleans a shorter section.
Speaker 2:Exactly.
Speaker 1:But if you only have a couple of volunteers Mhmm. They've got more ground to cover.
Speaker 2:Exactly. And it really highlights how being able to rearrange an equation, in this case, relating the total distance Uh-huh. The number of volunteers, and then the length clean per volunteer, how rearranging that equation makes it, you know, much easier.
Speaker 1:So it's like having a Swiss army knife of equations.
Speaker 2:Exactly. You can
Speaker 1:flip it around to the tool that you need.
Speaker 2:And speaking of flipping things around
Speaker 1:Yeah.
Speaker 2:This activity also sets the stage for another key idea in the lesson, equivalent equations. You know, it can be tricky for students to kinda grasp that an equation can be written in several different ways and still be true.
Speaker 1:Right. It's like saying Yeah. I have $5
Speaker 2:Yeah.
Speaker 1:Or I have a $5 bill.
Speaker 2:Yeah.
Speaker 1:It looks different. It sounds
Speaker 2:different. Exactly.
Speaker 1:But it's the same amount of money.
Speaker 2:Exactly.
Speaker 1:Yeah. And so How
Speaker 2:does this lesson plan illustrate that for teachers?
Speaker 1:Well, it really emphasizes that you can manipulate equations. You can add or subtract things to both sides. You can multiply, divide things on both sides. Mhmm. And it doesn't actually change the relationship between the variables.
Speaker 2:Right.
Speaker 1:It just allows you to kind of use that equation in a different way to find a different answer.
Speaker 2:It's like rearranging furniture.
Speaker 1:Yeah.
Speaker 2:Your furniture's still the
Speaker 1:same. Exactly.
Speaker 2:You just moved it around to be more functional for the space.
Speaker 1:Exactly. And they have another really fun activity.
Speaker 2:Called filling and emptying tanks.
Speaker 1:Filling and emptying tanks.
Speaker 2:Yes.
Speaker 1:This sounds a little more exciting.
Speaker 2:So imagine 2 water tanks
Speaker 1:Okay.
Speaker 2:That are filling or draining at different rates.
Speaker 1:Okay.
Speaker 2:And students are given scenarios like this tank started with this much water.
Speaker 1:Uh-huh.
Speaker 2:It's filling at this rate.
Speaker 1:Okay.
Speaker 2:How much water will be in it after this many minutes?
Speaker 1:So they have to first put that word problem into an equation Yeah. And then they have to rearrange it.
Speaker 2:Exactly. And then they might need to figure out how long will it take Right. For that tank to reach this volume.
Speaker 1:Okay.
Speaker 2:So then they have to rearrange it again to solve for time instead of volume.
Speaker 1:Oh, it's like they're little engineers.
Speaker 2:Exactly.
Speaker 1:And I know from experience that even with great activities like this, there are always some misconceptions.
Speaker 2:Yeah. What are some of the things that
Speaker 1:that we watch out for?
Speaker 2:Yeah. So a really common one that I see is that students think they always have to solve for the variable before they plug in any values.
Speaker 1:Oh, so they think it's like a strict order of operations.
Speaker 2:Exactly. And this lesson plan does a really nice job of subtly showing teachers how to guide their students to see that it's not always the most efficient way.
Speaker 1:Right. Because sometimes it's just like, if you ask me to multiply 15 by 0
Speaker 2:Right.
Speaker 1:I don't have to go through my timetables for that.
Speaker 2:Right.
Speaker 1:I just know the answer.
Speaker 2:Exactly.
Speaker 1:Sometimes it's just obvious. Yeah. And that leads into the other misconception, which is Yeah. Students having a hard time making that connection Yeah. Between the word problem Uh-huh.
Speaker 1:And the equation.
Speaker 2:Especially with something abstract like this, filling and emptying tanks.
Speaker 1:Right. It's not immediately apparent how that connects to the equation.
Speaker 2:This lesson plan really emphasizes the visual representation.
Speaker 1:Oh, I
Speaker 2:love that. So, like, actually drawing out the tanks Yeah. Marking the water level at different times.
Speaker 1:I love that.
Speaker 2:And even, like, acting it out Yeah. If they're feeling really adventurous.
Speaker 1:That's such a good idea.
Speaker 2:It really brings that equation to life.
Speaker 1:It does. Yeah. And maybe instead of just saying distance equals rate times time, maybe you say
Speaker 2:Yeah. Like, you're taking a field trip to the zoo Yes. 60 miles away.
Speaker 1:I've done that calculation so many times.
Speaker 2:And you're going 40 miles per hour. How long will it take you to get there?
Speaker 1:And suddenly Yeah. It seems applicable to their lives.
Speaker 2:Right. And when students see the connections to their own lives Yeah. They're more likely to engage with the material.
Speaker 1:Absolutely. And not. So we've explored some great activities. Yeah. Some common misconceptions or hurdles that students might encounter.
Speaker 2:Yeah. As we go into the teacher takeaways
Speaker 1:Yes.
Speaker 2:What are some of the things
Speaker 1:What are some key points
Speaker 2:Yeah.
Speaker 1:That you think are worth emphasizing for teachers with this lesson?
Speaker 2:I think what's really important about this lesson is that it really emphasizes that deeper understanding of why we solve for a variable Yes. Not just how.
Speaker 1:Instead of just the steps.
Speaker 2:Right. Because when students understand the purpose Right. Behind the mechanics, it stops feeling like a bunch of arbitrary rules
Speaker 1:Yes. I love that.
Speaker 2:That they have to memorize.
Speaker 1:It becomes a problem solving toolkit.
Speaker 2:Exactly. And this lesson provides teachers with such creative and engaging activities That they
Speaker 1:can adapt. To guide them on that journey. And personalized for their own classroom. Absolutely.
Speaker 2:Yeah. And, you know,
Speaker 1:when we
Speaker 2:empower teachers Yes. To bring their own unique flare to these concepts, that's when the magic really happens.
Speaker 1:Like, we're giving them the recipe
Speaker 2:Yes.
Speaker 1:Or encouraging them to add their own spices.
Speaker 2:Exactly.
Speaker 1:Which brings us to our final thought for this deep dive.
Speaker 2:Okay.
Speaker 1:What's one area in your life Okay. Outside of the classroom Okay. Where rearranging an equation or a formula Oh. Would be surprisingly helpful.
Speaker 2:You know, I use proportions all the time when I'm cooking.
Speaker 1:Okay.
Speaker 2:You know, if I only have half the amount of sugar that a recipe calls for Yeah. Knowing how to adjust the other ingredients proportionally Yeah. Is key to not having a baking
Speaker 1:disaster. That's a good one. What about you? It makes me think about, like, calculating how much paint you need to cover a wall.
Speaker 2:Oh, yeah.
Speaker 1:And suddenly, you need to know the area formula.
Speaker 2:It's all around us
Speaker 1:Yeah.
Speaker 2:If we look for it. Yeah. And I think by helping our students see those connections
Speaker 1:Yeah.
Speaker 2:We can really help them appreciate Yes. The elegance and power of algebra.
Speaker 1:I love it. A huge thank you to the brilliant minds at Illustrative Mathematics
Speaker 2:Yes.
Speaker 1:For this lesson plan.
Speaker 2:Thank you.
Speaker 1:And thank you for joining us on this deep dive.
Speaker 2:Thanks for having me.
Speaker 1:Keep those moments coming. Oh. I think what's really important about this lesson is that it emphasizes that deeper understanding of why we solve for a variable.
Speaker 2:Right. Not just the how to.
Speaker 1:Not just the steps. Yeah.
Speaker 2:Because when students understand the purpose behind the mechanics Yes. It stops feeling like a bunch of arbitrary rules that they have to memorize.
Speaker 1:It becomes a problem solving toolkit.
Speaker 2:Exactly. And this lesson provides teachers with Yeah. Such creative and engaging activities
Speaker 1:That they can adapt.
Speaker 2:To guide them on that journey.
Speaker 1:And personalized for their classroom.
Speaker 2:Absolutely.
Speaker 1:Yeah.
Speaker 2:And, you know, when we empower teachers to bring their own unique flair to these concepts, that's when the magic really happens.
Speaker 1:It's like we're giving them the recipe
Speaker 2:Yes.
Speaker 1:But we're encouraging them to add their own spices.
Speaker 2:Exactly.
Speaker 1:Which brings us to our final thought for this deep dive.
Speaker 2:Okay.
Speaker 1:What's one area in your life outside of the classroom where rearranging an equation or a formula would be surprisingly helpful?
Speaker 2:You know, I use proportions all the time when I'm cooking.
Speaker 1:Okay.
Speaker 2:If I only have half the amount of sugar that a recipe calls for
Speaker 1:Right.
Speaker 2:Knowing how to adjust the other ingredients proportionally is key to not having a baking disaster.
Speaker 1:That's a really good one. Yeah. It makes me think about, like, calculating how much paint you need to cover a wall.
Speaker 2:Oh, yeah.
Speaker 1:You suddenly need that area formula.
Speaker 2:It's all around us Yeah. If we look for it.
Speaker 1:Yeah. And maybe by helping our students see those connections, we can really help them appreciate the elegance and power of algebra. Exact A huge thank you to the brilliant minds at Illustrative Mathematics for this lesson plan. And to our listeners, thank you for joining us on this deep dive. Keep those moments coming.