Okay. So, like, imagine trying to create, the perfect trail mix.

Oh, absolutely.

You know, delicious, obviously. Right. But also, like, hitting certain goals, maybe for calories or protein.

Yeah.

Trying to keep the sugar content under control.

Right.

Turns out that's actually a real world example of something called systems of Inequalities

That's right.

Which is what today's deep dive is all about.

So are you intrigued? I'm intrigued. Let's dig in.

Let's do it. So we are diving into a lesson plan today called modeling with systems of inequalities in 2 variables. And I know that you, our listener, are about to teach this.

Yes.

So we really wanna give you the insights that you would find valuable.

Absolutely. The goal here is to help teachers equip students to not only just interpret existing models of inequalities, but to actually build them

Yeah.

From scratch Yeah. Which is a pretty high level math skill.

It is.

But this lesson uses a really clever approach to make it stick.

Yeah. I love how it starts with a warm up, which I think is such a smart move. Just to make sure that everybody's on the same page with the basics.

Crucial. Like, laying the foundation before you start building the house.

Yes.

It reminds students about greater than less than symbols, how they work Right. All of that before they jump into the real world application.

Exactly. And speaking of those applications, there are 2 really cool activities.

Yes.

The first one is called a solution to which inequalities, and it gives students these premade trail mix recipes.

Okay.

But each recipe comes with a set of inequalities and a graph.

Interesting. So

they have to kind of put on their detective hats

Ah, okay.

And analyze the recipe the in inequalities of the graph and see how they all connect.

I see.

So, for example, an inequality might represent the maximum amount of sugar allowed in the mix. Right. And then they'd have to see how that's reflected on the graph.

I see. So it's like they're decoding the mathematical DNA of a trail mix.

You really are. Yeah. And then the second activity, design your own trail mix, takes it up a notch. This is where students become mathematical chefs Yeah. Because they get to choose their own ingredients.

I love this.

They set their own nutritional goals, and then they use inequalities to see if their dream trail mix is actually possible. That's fantastic.

And the lesson plan even suggests using Desmos Yes. Which is that online graphing calculator so students can visually see how changing the amounts of different ingredients affects whether or not it meets their criteria.

And so much more engaging than just solving an equation in the abstract.

Absolutely.

Because they can see it.

Yes.

They can manipulate the variables and see what happens.

And it happens in real time

Yes.

Which I think is really cool.

Yeah. And it brings math to life in a really cool way. But, you know, beyond the technology, what I really like about this lesson is that it encourages students to think like mathematicians.

Exactly. It's

not just plugging in numbers.

It's not just rote memorization.

Right.

They're learning to model a real world scenario.

Yes.

Analyze the data, use logic to find solutions. I mean, these are skills that go way beyond the classroom.

Absolutely. And the lesson plan even suggests doing things like a gallery walk so students can share their trail mixes with each other and see how other students approached it.

Oh, that's fun.

It's all about that collaboration.

Collaborative learning. That's great. Yeah. I love that.

So we have this awesome lesson plan, but let's be real. Teaching isn't always easy peasy. No. So let's talk about some potential pitfalls that we might encounter.

Okay.

And one thing that stood out to me is this idea of, like, more than 2 constraints at once Yeah. In some of these activities because, you know, for students who are used to, like, simpler systems

Right.

That could be a little overwhelming.

Yeah. It's like juggling. Right? You start with 2 balls Right. Then you add more as you get the hang of it.

Right. So, you

know, maybe start with just 2 inequalities and then gradually introduce more complex scenarios as students gain confidence.

That makes a lot of sense. Another potential stumbling block I see is plotting those inequalities with decimals. It's so easy to, like, make a tiny little mistake.

So easy, especially by hand.

Especially by hand. Right. And if they're doing it on a graph

Oh, yeah. One misplaced decimal point can throw the whole thing off.

Exactly. That's where technology like Desmos can really be a game changer because it just takes care of all those precise calculations.

Absolutely.

And then the students can focus on the bigger picture.

Yes. Like, what's actually happening with the inequalities in the graph?

Exactly. And it's so much more fun than just erasing and re redrawing those lines over and over again.

Right. Right.

But even with the best tools, students might need some guidance.

Oh, absolutely.

It is.

Yeah. You gotta explicitly teach them how to use Desmos for graphing inequalities.

Okay.

Show them how to put in the equations, adjust the viewing window, make sure they know what those shaded regions mean.

It's like giving them the keys to the car, but making sure they know how to drive.

Precisely. Yes.

Right? Providing clear instructions, I think, is gonna be really helpful

Yes.

And checkpoints so they can, like, make sure they're on the right track.

Yes.

So we've got these engaging activities. We've talked about some potential issues. But what I think is really cool is that this lesson goes way beyond trail mix.

It does.

Oh, yeah.

Oh, yeah. It gives them tools to model any situation where they encounter limitations and variables.

Okay.

We're talking about real life problem solving skills here.

Okay. So give me an example.

Yeah.

What kind of real world problems can they tackle with this? Let's think about budgeting. Okay. Okay. So, tackle with this?

Let's think

about budgeting.

Okay. Okay. So you've got

a certain income. Right? Yeah.

You've got various expenses, maybe some savings goals.

It's like

trying to fit all the pieces together without going over budget.

Exactly. And that is a system of

inequalities waiting to be modeled. Each piece of that

It is.

Like, your income or your expenses can be represented as an inequality, and the solution lies in finding the combination that satisfies all of those constraints.

That's such a practical example. I

because it's something they can relate to.

They can. And whether they're managing an allowance or thinking about future finances Right. Like, this is a really important concept.

Absolutely.

Or even time management.

Oh, yes. Right. Big one for students.

They could use these same principles Absolutely. To juggle homework assignments, extracurriculars, maybe even getting enough sleep.

Sleep in there too.

Right.

Yeah. So their available time becomes one set of constraints.

Okay.

And then their desired time for each activity is another set.

Got it.

And by setting up that system, they can see, is this even possible? Yeah. Do I need to make some adjustments?

Right.

Maybe become more efficient with my homework time?

Right.

Maybe I need to cut back on some of those extracurriculars.

It's amazing how this one concept can be applied to so many different things.

It is. And that's what makes it such a valuable tool for them to have because it's not about just finding the right answer on a test. Right. It's about understanding how to approach and solve problems in the real world with limited resources and competing priorities. It's like we're giving them a superpower.

I love that. The ability to make smarter decisions using math. Yes. That's amazing. And that brings us back to this lesson plan.

While it focuses on a 2 ingredient trail mix for simplicity

Right.

It's worth considering how the complexity changes when you introduce even more ingredients.

Yes.

Right?

Like, what happens when we add those chocolate chips?

I was just thinking that. What about those chocolate chips?

Right. What about pretzels for a salty sweet combo? Oh, salty and sweet. Really good. I

combo? Oh, salty and sweet. Pretty good. Suddenly, we're not just dealing with 2 variables anymore.

Right.

Each ingredient adds another layer of complexity to the system. Wow. Right? So with 3 ingredients, they'd need to work with a system of inequalities in 3 variables

Okay.

Which is graphed in 3 dimensions.

Right.

So that gets really interesting.

I was just thinking almonds, raisins, chocolate chips. Now we're talking about a three-dimensional graph.

Three-dimensional graph.

That's getting pretty advanced.

That's a fun challenge, though.

It is. Yeah.

It is. For those students who are ready to take it to the next level.

Exactly.

Yeah.

And it highlights the importance of using the right model for the situation

Yes.

Because it's not always gonna be 2 variables. That's good. Could be more. It could be less. Right.

So we've explored this incredibly versatile lesson plan that not only teaches them about systems of inequalities, but also equips them with these problem solving skills that they can use in all areas of their lives.

It's really amazing, And we even discovered a hidden world of culinary mathematics along the way.

We did. We did. I don't know about you, but I'm craving some trail mix now.

Me too.

But before we go, I wanna give a shout out to the authors of Illustrative Math

Yes.

For creating such an engaging and thought provoking lesson.

Absolutely. Their creativity and dedication to making math accessible and engaging for all learners is just fantastic.

I completely agree. And to you, our listener, we hope that this deep dive has given you some valuable insights and inspiration for teaching this topic.

We hope so. And remember, math isn't just about memorizing formulas. It's about understanding the world around us

Yes.

And using that to make informed decisions.

It really is. Yeah. And who knows? Maybe you'll even inspire your students to create the next viral trail mix sensation.

I love it. Yeah. Now that would be a delicious application of systems of inequalities.

I I love it. Alright. Well, that's it for our deep dive today.

Okay.

We'll catch you next time on the deep dive. It's like we've uncovered this, like, secret layer of math It is. That's, like, all around us.

Yes.

But we just needed the right tools to actually see it.

Yeah. And once they start to recognize those patterns, it can be so empowering for them.

I bet.

It's like they've unlocked a code Right. To understanding how decisions are made and how systems are designed.

So we've talked about how this lesson is way more than just snacks. Right. So let's actually dive a little bit deeper into those real world applications that you mentioned. Yep. You talked about things like budgeting

Yes.

And time management.

Right.

What would those applications actually look like in terms of systems of inequalities?

So let's take budgeting as an example. Let's say a student wants to save up for a new phone. Right. And maybe they have a part time job

Right.

Where they're making, we'll say, $200 a week. Okay. And they have expenses, of course, $50 a week for transportation, $30 for entertainment, whatever it may be.

Makes sense.

And their goal is to save at least $100 each week for this new phone.

Okay. I can see where the system of inequalities is starting to come in here.

Exactly. So their income is one constraint. Right? They can't spend more than they earn.

Right.

Each of those expense categories is another constraint.

Uh-huh.

And then their savings goal is, like, the last constraint.

So they would write out inequalities to represent each of those constraints.

Yes.

Like, total expenses have to be less than or equal to 200 dollars

Exactly.

Or savings has to be greater than or equal to

a $100. Exactly. They can use those to see, you know, am I on track here

Right.

To reach my

goal? It's like they're becoming like financial detectives using math.

Using math to stay on budget.

Yes. Yeah. Exactly.

Yeah.

And then the same principles would apply to time management as well.

Absolutely. Right. Let's say they have 10 hours a week for homework, 5 hours for extracurriculars, and they wanna get 8 hours of sleep Yeah. Each night.

So their available time becomes, like, one set of constraints

Exactly.

And then how much time they actually wanna allocate for each one Yes. Is another.

Exactly.

And they could see if their current schedule is even feasible.

Right. Or if I need to make some adjustments here.

It's amazing how this one concept can be applied to so many different things.

It is. And that's what makes it so valuable for them to have. Yeah. Because it's not about just, like, can you find the right answer on this test?

Right.

It's do you understand how to approach and solve problems in the real world

Right.

With limited resources and competing priorities?

Because that's real life.

That's real life.

Right. That's amazing. Right.

And

that kind of brings us back to the lesson plan.

Okay.

It focuses on a 2 ingredient trail mix Yep. To keep it simple.

Right.

But we were just talking about what happens when you add in more ingredients.

What happens when you add more ingredients?

Right. Like, what about those chocolate chips and pretzels that we

were talking about? Suddenly, it's not just 2 variables anymore.

Right. Now we're talking about

Each one of those adds another layer of complexity I'll move on. System. Right?

Three ingredients. That's a three-dimensional graph.

Three-dimensional graph.

We're getting really advanced here.

That could be a fun challenge, though.

It could be a fun challenge.

For those who are ready to take it to the next level.

Exactly. Yeah. And it really does highlight the importance of choosing the right model

Yes.

For the situation.

Exactly. Because it's not always gonna be 2 variables.

It's not. It could be more.

Could be more. Could be less.

It could be less Right. Depending on

what you're trying to figure out. You've got to adjust.

Yes. Exactly. So we've explored this really versatile lesson plan that teaches them about systems of inequalities.

Mhmm.

But then it also gives them these really important problem solving skills.

And we even discovered a hidden world of culinary mathematics along the way.

We did. I don't know about you, but I'm ready for a snack break.

I am too.

I'm thinking trail mix.

Sounds delicious.

But before we go, I do wanna give another shout out

Yes.

To the authors of Illustrative Math for creating such an engaging

Amazing.

And thought provoking lesson.

It really is a great example of how to take something that could feel very abstract

Right.

And make it relevant to students' lives.

Yes. Absolutely. And to you, our listener, we hope that this deep dive has given you some valuable insights

Yes.

And inspiration for teaching this topic.

We hope you have as much fun with it as we did.

Yes. And remember, math isn't just about memorizing formulas.

No. It's about understanding the world.

It's about understanding the world around us

Yes.

And making informed decisions.

Yes.

And who knows? Maybe you'll even inspire your students to create the next viral trail mix sensation.

Yeah. Wouldn't that be some

I think that would be amazing. Yeah. Alright. We really did. It's like we opened up this whole pantry of possibilities.

Mhmm. And it all started with trail mix.

And that's the beauty of a good lesson plan. Right?

Yes.

It sparks that curiosity. Yeah. It encourages them to explore. And, ultimately, it empowers them to really see the world

Yes.

Through that mathematical lens.

It's really amazing. And to our listener, if you are about to embark on this teaching adventure

Yes.

We hope that you have so much fun with it.

We hope so.

We did. We had a blast.

Yeah. And remember Yeah. Sometimes the best learning happens when you step outside the textbook. Absolutely. To embrace those real world applications

Yes. Of math. Couldn't agree more. So go forth and spread the joy of systems of inequalities

Yes.

And maybe even inspire some tasty creations along the way.

Happy teaching.

Happy teaching. Happy snacking. That's it for our deep dive today. We'll catch you next time on the deep dive.