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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.)
Ever feel like your brain's stuck back in algebra class? You're staring at equations, and you just wanna, like, run away.
Speaker 2:It can be a little intimidating Right. Really.
Speaker 1:But what if there's a way to help our students actually enjoy the process of working with equations?
Speaker 2:Oh, I like the sound of that.
Speaker 1:Right. And luckily for us, there are some brilliant minds out there who are tackling this very challenge. We're about to deep dive into a lesson plan from illustrative mathematics called explaining steps for rewriting equations.
Speaker 2:I'm familiar with that one. It's designed for algebra 1.
Speaker 1:You got it. And it's all about helping those students understand not just the how of solving equations, you know, the mechanics of it all Right. But the why behind those crucial steps, why those steps actually work.
Speaker 2:It's kinda like you can memorize a recipe, but until you understand why each ingredient is there Exactly. You won't really be a master chef. Right? This deep dive is all about giving teachers the tools and insights they need to make this lesson really resonate with their students.
Speaker 1:Because let's face it, who hasn't memorized the steps for solving equations without truly getting it?
Speaker 2:Oh, I've been there. It's all about building that solid foundation and this lesson zeros in on 3 major goals for students. Understanding why certain operations create equivalent equations
Speaker 1:Okay.
Speaker 2:Why dividing by a variable is a big no no serious math trouble.
Speaker 1:Right.
Speaker 2:Finally, how to spot those equations that just don't have any solutions? Talk about frustrating.
Speaker 1:Yeah. Those can be tricky. So how does this lesson hook students right from the get go? What's the attention grabber?
Speaker 2:They ease into it with this really clever warm up activity called math talk. Could it be 0?
Speaker 1:Okay. I'm intrigued. Tell me more.
Speaker 2:So, basically, students are presented with a variety of equations Mhmm. And they had to figure out if 0 could be a solution.
Speaker 1:Interesting. So instead of just jumping into solving for x, they have to think more conceptually about how substituting 0 for x would affect affect the whole equation.
Speaker 2:Exactly. Take an equation like 3x +8. We'll use 8. Now students might be tempted to just dive in and start shuffling things around.
Speaker 1:Right. Get that x all by itself.
Speaker 2:Exactly. Yeah. But if they pause for a moment and think, okay. What happens if x is 0? They'll see.
Speaker 1:Oh, I see where you're going with this. Yeah. The equation actually works if x is 0.
Speaker 2:Exactly. It highlights that core concept of what it means for 0 to be a solution.
Speaker 1:That's such a good point. I love how this warm up activity encourages that deeper understanding of what a solution really represents before they even start doing all the algebraic manipulation.
Speaker 2:Exactly. And that understanding flows beautifully into the next part of the lesson, which focuses on why certain algebraic moves are allowed when you're trying to create equivalent equations, those equations that have the same solution.
Speaker 1:So this is where we go beyond memorizing steps and get into those all important why questions. Like, why can we add the same number to both sides of an equation and still maintain that beautiful balance?
Speaker 2:Exactly. And to really hammer home that point, this lesson has an activity called and I bet you can guess the name.
Speaker 1:Explaining acceptable moves.
Speaker 2:You got it.
Speaker 1:Okay. That sounds promising. How does it work?
Speaker 2:Students work in pairs, and each pair gets a list of equations. Some of those pairs, they demonstrate valid algebraic moves
Speaker 1:Okay.
Speaker 2:While others, well, they don't.
Speaker 1:Sneaky. I like it.
Speaker 2:And the students have to analyze each of these pairs and explain why the equations are or aren't equivalent. It's all about making them justify their thinking.
Speaker 1:I love that. It's not enough to just know the answer. You have to be able to explain it.
Speaker 2:Exactly.
Speaker 1:That's where the real understanding kicks in. It's, like, they become active rule makers, not just passive rule followers. So walk us through an example here. What kind of equations might these students be grappling with?
Speaker 2:Okay. So let's say a pair gets the equations 2 times the quantity x plus 3 equals 16 and 2x plus 6 equals 16. They might recognize, oh, the distributive property allows us to go from that first equation to the second one. They're equivalent.
Speaker 1:Right. It's just that distributive property in action multiplying that 2 into those parentheses. But then what about a pair like 2 times the quantity x plus 3 equals 16 and 2x plus 3 equals 18? At first glance, it looks like maybe they just added 2 to both sides.
Speaker 2:Right. It has that feel.
Speaker 1:But something feels off.
Speaker 2:And that's where the real moment happens. They're gonna realize that this move doesn't maintain that crucial equivalence because only part of that left side got doubled.
Speaker 1:Right. Not the whole thing.
Speaker 2:Exactly.
Speaker 1:They're not equivalent at all.
Speaker 2:And that explaining acceptable moves activity
Speaker 1:Yeah.
Speaker 2:It really forces them to articulate that kind of thinking. Mhmm. It's not enough to just get the right answer.
Speaker 1:They have to justify it.
Speaker 2:Yes.
Speaker 1:And that kind of analytical thinking, it's not just for algebra.
Speaker 2:Oh, absolutely not.
Speaker 1:That's a skill that will serve those students well in any field that requires problem solving and logical reasoning critical thinking at its finest. So we've laid the groundwork with that could it be 0 warm up.
Speaker 2:Yes.
Speaker 1:And now students are flexing those reasoning muscles with this explaining acceptable moves activity. What comes next in this algebraic adventure?
Speaker 2:Alright. So now it's time to get to the really fun part.
Speaker 1:The fun part. Okay. You've got my attention.
Speaker 2:This is where things get a little tricky. This section of the lesson is called when things get tricky, and believe me, this is where those common math gotchas come out to play.
Speaker 1:Oh, I know those gotchas. All too well. They can really trip students up if they're not careful. What kind of tricky situations are we talking about here?
Speaker 2:Well, this section focuses on 2 major things that tend to throw students off. Well Equations that don't have any solutions, and those can be frustrating.
Speaker 1:Oh, yeah. For sure.
Speaker 2:And then that ever so tempting trap of dividing by a variable, which seems like it should be fine, but can really lead to some problems.
Speaker 1:Okay. Those are definitely concepts that can cause some serious head scratching if students aren't careful. So how does the lesson bring these gotchas to light? How do they make it clear to students what's going on?
Speaker 2:They have this other really cool activity it's called, and I bet you can guess.
Speaker 1:Let me guess. It doesn't work.
Speaker 2:You got it.
Speaker 1:Okay. I like where this is going.
Speaker 2:So in this activity, students are presented with different scenarios that, like, at first glance, they look like they follow all the rules of algebra, but they actually lead to these impossible outcomes.
Speaker 1:Oh, it's like a math mystery. They have to put on their detective hats and figure out what went wrong where the logic went off the rails.
Speaker 2:Exactly. It's all about that critical thinking.
Speaker 1:So give me an example. What would one of these head scratching scenarios look like?
Speaker 2:Okay. Picture this. You've got a student who's trying to solve the equation 6 x equals x. So they decide, alright, I'm gonna divide both sides by x.
Speaker 1:Okay. Seems reasonable enough. Follow the rules. Treat both sides equally.
Speaker 2:Right. Seems like a totally legit move. Totally. But here's the catch. What if x happens to be 0?
Speaker 1:Oh, right. You can't divide by 0. It's, like, the ultimate mathematical sin. You can't divide a pizza into 0 slices. It just doesn't make sense.
Speaker 2:Exactly. It's undefined. You get this this mathematical abyss. And the student might think, oh, I've solved for x. But in reality, they've just created this impossible situation.
Speaker 1:It's a trap.
Speaker 2:It is. And that's what this activity highlights so well. Yeah. That even though dividing by a variable can seem like a valid move, it can totally derail the whole solution process, especially if that sneaky 0 is involved.
Speaker 1:It's not just about memorizing those steps. It's about truly understanding the nuances.
Speaker 2:Yes.
Speaker 1:All those potential pitfalls that might be lurking within those seemingly simple rules.
Speaker 2:Exactly. And that's what's so great about this lesson. It helps teachers anticipate these really common student struggles and address them head on.
Speaker 1:So this lesson lays out these tricky concepts, gives students a chance to really grapple with them. It seems like it's really well done. But we both know that even with the most brilliantly designed lesson, those pesky misconceptions, they can still linger. Are there any common areas where students might still get tripped up even after working through this lesson?
Speaker 2:Oh, for sure. One of the big ones is understanding why certain equation transformations are just, like, totally off limits. They might look okay on the surface, but they mess up that delicate balance of an equation.
Speaker 1:Right. Like, maybe trying to add 3 to one side of the equation, but then subtracting 3 from the other side.
Speaker 2:Yeah. I've seen that.
Speaker 1:They might think, hey, I'm changing both sides. I'm keeping it balanced, but no. Yeah.
Speaker 2:It seems balanced, but they miss big picture. Those changes don't maintain that essential equality that's at the heart of every equation. And to combat that, we need to really hit home that idea of balance. And one way to do that is to use visual aids.
Speaker 1:Oh, visuals. Love it.
Speaker 2:Yeah. Like, think about a balance scale.
Speaker 1:Oh, I like that. Right. So you could literally show them, like, if you add weight to one side but then take away weight from the other, it's not balanced anymore. The equation's not true.
Speaker 2:Exactly. You could even have them try it out with actual objects, bringing that kinesthetic element.
Speaker 1:I love that. Hands on learning. Make it real for them.
Speaker 2:Trust me. When they can physically feel that imbalance, it just clicks.
Speaker 1:So true. Okay. So we've got those balance issues to watch out for. What other misconceptions tend to pop
Speaker 2:up? Another tricky one is when students think that just rearranging terms within an expression is, like, not allowed. For example, they might think that 2 x plus 5 has to stay in that
Speaker 1:order.
Speaker 2:Right. It can't be written as 5+2x.
Speaker 1:Yeah. Yeah. They're forgetting about the good old commutative property of addition.
Speaker 2:Exactly. We need to remind them that the order you add numbers in doesn't matter, doesn't change the sum.
Speaker 1:Right.
Speaker 2:Think about it this way. It doesn't matter if you put on your socks first or your shoes first.
Speaker 1:You're still wearing both. I'm totally using that one. Yeah. It's such a simple but effective analogy. Okay.
Speaker 1:So we've covered some really common areas where students might still need a little extra guidance. But what else can teachers take away from this deep dive into the world of equations? So this lesson does a really nice job of highlighting that common mistake of dividing both sides of an equation by a variable rope. It seems so intuitive, but it can really lead to those sneaky wrong answers. What other common errors should teachers be on the lookout for?
Speaker 1:Are there other traps students might fall into in the world of equations?
Speaker 2:Oh, there are a few. Yeah. One classic mistake is forgetting to apply an operation to every term in an equation. Like, they might remember to add 5 to one side, but then forget to add it to both terms on the other side.
Speaker 1:It's like they're so focused on one part of the equation that they completely forget about the rest.
Speaker 2:Exactly. They lose sight of that balance we were talking about earlier, and those visual cues can be really helpful here.
Speaker 1:Oh, visuals are always good.
Speaker 2:Right. Even something as simple as drawing a line down that equals sign.
Speaker 1:I like that. A visual reminder. It's like whatever you do on one side of that line, you gotta do to everything on the other side.
Speaker 2:Ex
Speaker 1:No exceptions. Okay. What other pitfalls should we watch out for?
Speaker 2:Misapplying the distributor property. That's another big one.
Speaker 1:Ah, yes. The distributor property.
Speaker 2:Students might multiply the number outside the per parentheses by the first term inside.
Speaker 1:Right.
Speaker 2:But then they completely forget about that second term.
Speaker 1:They're, like, halfway there, and then, bam, they hit a wall.
Speaker 2:It's like they run out of steam.
Speaker 1:So how do we help students avoid those distributive property disasters? What are some strategies?
Speaker 2:Sometimes, it helps to just break it down into smaller, more manageable steps. So instead of tackling that whole distribution at once, have them write out each multiplication separately.
Speaker 1:So instead of going straight from, like, 2 times the quantity x plus 3 to 2x plus 6 Right. They would actually write it out as 2 times x plus 2 times 3.
Speaker 2:Exactly. It seems like a small thing.
Speaker 1:Right.
Speaker 2:But you would be surprised how much such a simple step can really help prevent those careless errors.
Speaker 1:It's all about finding those little scaffolding techniques. This has been such an insightful deep dive. Do you have any final thoughts for our listeners? Maybe something for them to chew on as they head back into the classroom ready to tackle equations with their students.
Speaker 2:You know, we spent a lot of time today talking about common errors and how to fix them.
Speaker 1:Right.
Speaker 2:But I think it's super important to remember that those mistakes, they're just part of the learning process.
Speaker 1:So important to remember that.
Speaker 2:It's not about being perfect. It's about encouraging that growth mindset, that understanding that intelligence and mathematical ability, they're not fixed. They can be developed.
Speaker 1:I love that. It's not about avoiding mistakes. It's about learning from them.
Speaker 2:Exactly. So encourage your students to view those missteps as opportunities for growth.
Speaker 1:I love it. Mistakes are just stepping stones on the path to success. What a powerful message to end on. A huge thank you to the brilliant minds at illustrative math for creating this thought provoking lesson plan. And to all of our amazing listeners out there, keep up the fantastic work you're doing in the classroom.
Speaker 1:We'll be back next time with another deep dive into the world of education. Until then, happy teaching.