Ever feel like sometime, you know, when you're teaching, it's like you're just throwing equations at your students and hoping something sticks. Mhmm. Well, today, we're diving into a lesson plan, and it's from illustrative mathematics. And it tackles a topic that, honestly, I think even the most confident math teacher might sweat a little bit about teaching. Yeah.

Systems of equations. Big one. But here's the thing about this particular lesson plan. It is not your average plug and chug kind of approach.

Yeah. No. What's really cool about this is that it really doesn't just tell the students or even show the students how. It guides them really carefully to the why Mhmm. Which is amazing.

And it recognizes that, you know, as teachers, we're not just trying to build their, you know, skills in algebra. We're really trying to foster that mathematical reasoning that's gonna help them, like, way beyond us.

Absolutely.

Way beyond the classroom.

Okay. So let's unpack this a little bit. Right off the bat, they set this scenario, and it's one that I think every teacher, like, regardless of what you teach, can relate to.

Mhmm.

Buying classroom supplies. Oh, yeah.

You've got your calculators. You've got your measuring tapes. Like, it is a universal struggle trying to get the best deal, stay within your budget.

Absolutely. And I think by grounding it in something that is so relatable Mhmm. It just makes it so much more engaging. All of a sudden, you're not finding x and y. You're finding okay.

Well, what's the price of the calculator if I buy this many? You

know? Exactly. Exactly.

It just it just makes sense.

And that's where this gets, like, really interesting because then they have this activity called, is it still true? Yes. Which at first glance, we're like, this seems almost deceptively simple.

Right.

Students, they start with a basic equation, and they're asked to just add or subtract different amounts from both sides

Mhmm.

And just observe if it stays true.

Right. And that's and that's that's the beauty of it is that it's such a simple concept, but it lays the groundwork for such a bigger idea.

It's like they're planting a seed, and they don't even realize it.

Yes.

And that seed is gonna grow into this, like, deep understanding of why elimination works. Exactly. Because think about it. If you're going to allow yourself to add or subtract equations, That whole principle of equality that you just demonstrated there

Mhmm.

That is the thing that makes it okay to do. Yes. It's not just some, like, arbitrary rule. It's like we have to maintain that balance.

Like a seesaw. Right. Exactly. Like, you gotta have equal weight on both sides for it to work.

Yeah.

Okay. So they've got this principle of equality kind of established. Right? Mhmm. They played with it.

What comes next?

So then they've got that classroom supplies activity Right. Which is where that scenario that we were just talking about, the teacher buying the calculators and the measuring tapes comes into play. And so they're presented with the system of equations that actually, like, represents that. And I love how in this next part, this is where they really start to push on, like,

okay. So what does the solution actually represent? Right. You know, it's not just x and y.

It's not just some arbitrary x and y. It's like, what would the price of a calculator be? What would the price of a measuring tape be? Yep. Like, they're starting to see it in a context that they understand.

Exactly. And then building on that idea of keeping things balanced Mhmm. That's when they introduce this idea of adding the equations together.

Right.

And again, they tie it back to that context. Right? Like, oh, if each of these equations represents a separate purchase that you made Right. Adding them together represents your total purchase.

Exactly.

So, again, it's not just like, hey. Add the equations together because I said so.

Right.

It's like, no. Look. This makes sense.

There's a reason.

This makes sense in this context that you get. Yeah. Okay. So they've laid the groundwork. They've made the connection to the real world.

What kind of activities do they use to kind of really solidify this understanding?

So that's where they bring in that a bunch of systems activity

Oh, yeah.

Where they actually get some hands on practice.

So, you know Tend to, like, actually do

the thing. Exactly. They get to apply what they've learned about elimination. They get to

Yeah.

Tackle, like, a variety of systems. Yeah. But I and this is what I think is so clever. One of the systems is, like, intentionally a little trickier than the others.

Oh, interesting.

The variables don't cancel out easily. It's not just a matter of simple addition or subtraction.

So it's like they hit a roadblock on purpose.

In a way, yeah. I mean, it serves a couple of purposes. One, I think it reinforces the idea that, you know, not every system can be solved instantly with, like, this one simple trick.

Right. Right.

Sometimes you need more.

You might need some more tools in your toolbox.

Exactly. And I think even more importantly than that, it sets the stage for, like, that next lesson. Right?

Okay.

Where they're gonna learn some of those more advanced techniques. It's like, now we've created this need.

Yeah. I like that. So it's not just about, like, here's the answer. It's like and here's the thing that, like, makes you hungry to learn more.

Yes. Exactly. Exactly. So that kind of brings us to the putting new equations to work activity

Okay.

Which brings in a fresh scenario, I think it's got, like, hot cocoa and pretzels. You know?

Because, I mean, who doesn't love a good snack break, especially during algebra?

Especially yeah. But it's more than just to, like, you know, satisfy their cravings. It's actually it reinforces this idea of adding equations together.

Right.

But then it asks us this question, like, okay. Mhmm. Does adding the equations always help you solve the system?

Oh, that's good.

Alright. So it's like, are we being strategic about this?

Like Yeah. Is this the right tool for this particular job?

Exactly. Because just like in the real world, there's not always one right answer, one right way to do something. Mhmm. It's about choosing the best strategy for the situation at hand.

Okay. So let's talk about some of those, like, potential pitfalls. Right? Yeah. Like, those common misconceptions that students might develop Yes.

When they're learning about systems of equations because this is always my favorite part.

Like Oh, me too.

It's like that moment of, like, oh, this is why they're struggling.

It's huge. Yeah. And I think as educators, we have to be able to anticipate those, like, so that we can address them proactively. And one of the big ones is just understanding the why behind adding equations together. Like, they might be able to do it right.

Mhmm. Mechanically, they might be able to be like, oh, yeah. Yeah. The x's, I have the y's, I have the whatever. But then they might not really get like, why was that a useful thing to do?

It's like they're following a recipe.

Yeah. But

they don't understand the flavors they're trying to create.

Yes. And I think we need to help them see that we're strategically trying to eliminate a variable which is going to make our lives easier. It's gonna make the system easier to solve. It's not just, like, an extra step we're throwing in there.

Right. It's not busy work.

Right.

It's a means to an end.

Exactly. Exactly. And another tricky one is what I call the, like, it's magic kinda misunderstanding. So, like, if they didn't fully grasp that principle of equality that we were talking about

Right. Yeah.

All of a sudden, adding equations together just seems like some arbitrary, like Right.

Like, it's this trick.

Yes.

Like, oh, just add them together. Poof.

Right. Like, where did that come from?

Like, the answer magically appears.

Exactly. And that's why, like, constantly coming back to that, like, we're not changing the solution in some, like, mystical way. We're just rewriting the information in a way that's, like, more usable. Right?

It's like translating a sentence into a different language. Yeah. Like, it means the same thing. We're just expressing it in a way that, like, might make more sense to somebody.

That's a great way to put it. That's a great way to put it. And then I think, finally, there's always the danger of overgeneralization. Right?

Okay.

Where they start to think, oh, elimination is, like, the answer to everything. It's

a one size fits all.

Right. And as we saw with that a bunch of systems activity, it's not always that straightforward.

Right.

Right. Sometimes you gotta pull out some other tools.

So how can we, as teachers, how can we help students kind of, like, avoid those traps? Like, what are some practical takeaways?

Well, one of the biggest ones is encouraging visualization.

Okay.

Because when they can actually see those equations as lines on a graph Mhmm. And understand that, like, oh, the solution is where they intersect. It just makes it so much more concrete for them. It's not just these, like, abstract symbols anymore.

Like, I'm giving you a map instead of just telling you, like, turn left, turn right, go here.

Yes. Exactly. Exactly. And don't be afraid to, like, really connect back to prior knowledge. Like, remind them of what they already understand about keeping equations balanced because that's such a crucial foundation for them to internalize, like, why we're allowed to add or subtract equations in the 1st place.

Right. It's about showing them that, like, math isn't just, like, these isolated islands.

Totally.

It's this, like, interconnected web of, like

It goes together. Yeah. It all goes together. And I think, finally, this one's so important. Never shy away from discussions.

Yes.

Like, encourage them to talk about their reasoning. Ask them, like, why did you choose that strategy? How do you know if your solution makes sense?

Yes. Because it's not just about getting the answer right.

Exactly.

It's about, like, being able to explain your thinking.

And that's where those light bulb moments happen. You know? It's in those discussions where they're like, oh, I get it now.

Yes. It's so true. And, like, giving them that space Yeah. To really, like, talk it out.

It makes such a difference.

Because when we step back and we look at this whole lesson

Yeah.

Right, it's not just, like, a set of activities.

Mhmm.

It is a journey.

It is.

It's like this carefully crafted experience. Right? It takes them from those concrete examples to the abstract reasoning, from simple equations to those more complex systems.

And it really gives them tools to be successful. You know? It's like

It's like they're climbing this algebraic mountain, and this lesson is, like, giving them the gear

Yes.

And the, like, the guide.

You know? To get to the top. And and by focusing on that why alongside the how, like, we're not just giving them the tools. Mhmm. We're giving them the power to, like, actually use them effectively.

Yes. They're not just becoming, like, equation solvers. Right. They're becoming, like, problem solvers.

Yes. Exactly. And that is so much more valuable in the long run because, like, equations are gonna change. The types of problems are gonna change. But if they can, like, approach it with that, like, problem solving lens.

That's gonna last them that's gonna last them way beyond

Exactly.

You know, whatever algebra one class they're in right now.

You got it. You got that's what's so exciting about this.

It is exciting. You know? And I think back to, like, even just my own math education. Like Mhmm. I kinda wish I had had, like, teachers who were approaching it this way.

Right. Right.

Like, it would have made such a difference for me, I think.

It makes a world of difference, and it just shows you, like, we're doing good work here. You know?

Yes. We are.

Like, as math teachers, we're out there spreading the good word. You know? We're trying to make

We are spreading the good word.

Make it work for everybody. It doesn't to be this, like, scary, intimidating thing.

No. Absolutely not.

So as we kind of wrap up this deep dive into systems of equations, like, what's one, like, big takeaway you're hoping that our listeners are gonna walk away with today?

I think the biggest takeaway, honestly, is just to slow down. Like, let the kids really wrestle with the why.

Yes. It's

easy to get caught up in, like, okay. Here's the steps.

Totally.

But, like, that deep understanding, the kind that, like, really sticks with them

Yeah.

That comes from those, like, Yeah. Moments, like, when it finally clicks. And sometimes that takes a minute.

It does. It does. And it takes some productive struggle sometimes, which is, okay. We have to let them, like, sit in it a little bit and Yeah. You know, trust the process, trust that they can get there.

Trust the

process, trust our students. I love it.

Exactly.

This has been fantastic. As always, a huge thank you to Illustrative Math for this lesson plan. And for our listeners, thank you for joining us on this deep dive.

Yes. Thanks, everyone.

Keep those algebraic adventures going. We will catch you next time.