Essential IM

An AI-generated short discussion of an Illustrative Mathematics lesson to help educators prepare to teach it. 

The episode is intended to cover: 

  • The big mathematical ideas in the lesson
  • The main activities students do
  • How to make it interesting for young people
  • Possible misconceptions and how to deal with them.

What is Essential IM?

Lesson by lesson podcasts for teachers of Illustrative Mathematics®.

(Based on IM 9-12 Math™ by Illustrative Mathematics®, available at www.illustrativemathematics.org.)

Speaker 1:

Alright. Ready to head back to high school for a bit? Don't worry. No pop quizzes this time. I promise.

Speaker 2:

Thank goodness for that. Right.

Speaker 1:

Right. But seriously, today, we're taking a deep dive into a high school algebra lesson plan about perfect squares. But it's not really about the math, you know. It's more about, like, peeking behind the curtain to see how teachers break down complex topics and make them actually understandable.

Speaker 2:

Yeah. It's kind of like reverse engineering the learning process, which I find absolutely fascinating. Me too.

Speaker 1:

And the best part, the stuff we're gonna uncover today applies to learning anything, not just algebra. Right? Mhmm. Like, learning a new language, picking up a new skill. Yeah.

Speaker 1:

It's all connected. Absolutely. At its core,

Speaker 2:

learning is about recognizing patterns, making connections, building upon foundational concepts. And this lesson plan, well, let's just say it does a masterful job at all of those things.

Speaker 1:

I like how you said masterful there because it's really is like looking at a well crafted recipe.

Speaker 2:

Oh, okay.

Speaker 1:

Okay. So we've got our lesson plan. We're ready to roll up our sleeves and get our hands dirty. But first, we gotta answer the big question. What's the big idea here?

Speaker 1:

Why perfect squares? What do teachers want their students to walk away with after this lesson?

Speaker 2:

Well, the lesson plan itself highlights 2 main goals, and this is where it gets really interesting. 1st, they want students to truly grasp what a perfect square expression actually is, not just how to solve for it.

Speaker 1:

Okay. I'm sensing a but coming.

Speaker 2:

But, and it's a big but. They also want students to understand why it matters. They're not just teaching them to jump through hoops. They're showing them that these concepts have real world applications.

Speaker 1:

Ah, so it's not just about memorizing formulas. It's about understanding the why behind them.

Speaker 2:

Exactly. It's about connecting those dots and seeing the bigger picture. And you know what? This ties into a crucial aspect of effective learning, motivation.

Speaker 1:

Okay. Say more about that. I'm intrigued.

Speaker 2:

Think about it. When you understand the why, when you see how a concept connects to something bigger, something meaningful to you, it's not just rote memorization anymore. It clicks.

Speaker 1:

Right. Yeah. You're right. Like, when you were a kid and you learned that 2 quarters equals 5 dimes and then it hits you that you could buy 2 candies instead of just 1 Yeah. That kinda click.

Speaker 2:

Exactly. It's those little moments that make learning stick. And this lesson plan, well, it's practically overflowing with those moments, but it's not all sunshine and rainbows. You know? Learning can be messy too.

Speaker 1:

Oh, tell me about it. So how does this lesson plan set students up for success even when things get a little tricky?

Speaker 2:

Well, remember how we talked about this lesson plan being like a well crafted recipe? Well, like any good recipe, it starts with a little prep work. They ease the students in with a warm up activity designed to get those mental gears turning.

Speaker 1:

Sneaky. I like it. What's on the menu for this mathematical appetizer?

Speaker 2:

It's a series of equations that, at first glance, might seem like they're getting progressively harder. But here's the catch, they're all built on the same fundamental structure.

Speaker 1:

Okay. I'm trying to picture this. Can you give us a, for instance, story?

Speaker 2:

Sure. Imagine an equation like this. Parenthesis x+parenthesissquaredequals9.

Speaker 1:

Parenthesisx plus 2. Got it. Right.

Speaker 2:

So it might look a little intimidating, but if you break it down, it's really just saying something squared equals something else squared. And the warm up presents a bunch of these just with slightly different numbers.

Speaker 1:

So it's like they're getting a crash course in pattern recognition without even realizing it.

Speaker 2:

Exactly. By the time the term perfect square is even introduced, it already feels familiar, which makes a huge difference.

Speaker 1:

Oh, totally. Like, oh, I've seen this before. I can do this as opposed to what in the world am I looking at?

Speaker 2:

Precisely. Familiarity breeds confidence. And speaking of confidence, this is where the lesson plan really starts to connect those dots we were talking about earlier. Remember that second goal, the why behind perfect squares?

Speaker 1:

Oh, right. It's all coming together now. So how do they make that connection?

Speaker 2:

Well, think about how you calculate the area of a square. You square the length of one of its sides. Right? The lesson taps into that familiar concept to make perfect squares feel less abstract.

Speaker 1:

I get it. It's like, remember those squares you learned about in elementary school? Well, guess what? They're back and ready to make your life easier.

Speaker 2:

You got it. And just like knowing your multiplication facts makes more advanced math a breeze, recognizing perfect squares becomes a valuable tool in a student's algebra toolkit.

Speaker 1:

It's all about building fluency. Right? Instead of getting bogged down by the mechanics, students can focus on the bigger picture. It's like, imagine trying to have a conversation in a new language while constantly looking up every single word in the dictionary. It should be exhausting.

Speaker 2:

Yeah. Perfect analogy. And speaking of building fluency, the next activity in this lesson plan dives even deeper into the heart of perfect squares.

Speaker 1:

So we're diving deeper. What's next on this mathematical treasure hunt?

Speaker 2:

Well, you know how sometimes you see someone you know, but they look different. Maybe they got a new haircut or something.

Speaker 1:

Right. Like, you recognize them, but it takes a second.

Speaker 2:

Exactly. And believe it or not, that's kind of what we're doing with perfect squares in this next activity.

Speaker 1:

Okay. Now I'm really curious. What's going on in this lesson plan?

Speaker 2:

So they're giving students these perfect square expressions, but and this is where it gets good. They're asking them to rewrite them in different forms. It's like a mathematical makeover.

Speaker 1:

A makeover for math problems. I've never heard it put that way before, but I like it.

Speaker 2:

Right. So it's not just about recognizing a perfect square in its, let's say, natural habitat. It's about being able to spot it even when it's wearing a disguise.

Speaker 1:

I'm starting to see why this lesson plan is so effective. It's like they're training their students to be mathematical detectives.

Speaker 2:

Exactly. They're learning to see through those and recognize those underlying patterns, which is, like, the superpower of algebra.

Speaker 1:

Superpower of algebra. I'm putting that on a t shirt. Okay. So give me an example of one of these mathematical makeovers.

Speaker 2:

Alright. So let's say you have the expression, parenthesis x plus 3 close parenthesis squared. Right? It's a perfect square in its factored form.

Speaker 1:

Got it.

Speaker 2:

But then they might ask students to rewrite it in its standard form, which would be x squared plus 6x+9. Same perfect square, just a different outfit.

Speaker 1:

So it's like they're taking those perfect squares for a spin in the wardrobe department.

Speaker 2:

Precisely. And the more outfits they see those perfect squares in, the easier it becomes to recognize them in the wild, so to speak.

Speaker 1:

It's like once you've seen a tiger in the wild, in the zoo, and in a nature documentary, you're gonna be able to spot that tiger anywhere no matter what it's wearing or what it's doing.

Speaker 2:

Right. You're not gonna mistake it for a house cat anytime soon.

Speaker 1:

Exactly. But recognizing those perfect squares, that's just step 1. Right? What's the ultimate goal?

Speaker 2:

You got it. All this preparation, all this pattern recognition, it's all leading up to the main event, solving equations with perfect squares.

Speaker 1:

It's like all the roads in this lesson plan are leading to Rome.

Speaker 2:

And let me tell you, this lesson plan does not disappoint when it comes to that final destination.

Speaker 1:

Okay. Now you really have my attention. What makes it so special?

Speaker 2:

Well, a lot of lesson plans, they'll show you one way to solve a type of equation and call it a day, but this one, this one's different.

Speaker 1:

Oh, so don't leave me hanging.

Speaker 2:

This lesson plan, it introduces not one, but two distinct methods for solving equations with perfect square.

Speaker 1:

Oh, giving students options. I like it. It's like having a culinary school that teaches you both French and Italian cooking.

Speaker 2:

I love that analogy. It's not about limiting students to one rigid approach. It's about empowering them to find the method that resonates most with their learning style.

Speaker 1:

It's like saying, hey. You can get to the top of this mountain using either this winding path or that steep climb. Choose your own adventure.

Speaker 2:

Exactly. And that kind of flexibility, it's so important in math. It helps students develop problem solving skills that go way beyond just memorizing formulas.

Speaker 1:

Like, they're not just learning algebra. They're learning how to think like mathematicians.

Speaker 2:

Yeah. Which is, let's face it, a pretty valuable skill to have no matter what you end up doing in life.

Speaker 1:

Couldn't agree more. But let's be real. Even with the best lesson plan in the world, there are always gonna be those little things that trip students up. So let's talk about those common misconceptions, shall we? Those little potholes on the road to perfect square mastery.

Speaker 2:

You've hit on another strength of this lesson plan. Yeah. It doesn't just present the information. It anticipates those common stumbling blocks and offers teachers strategies for addressing them head on.

Speaker 1:

Like having a guidebook that not only shows you the best hiking trails, but also warns you about poison ivy and grumpy bears.

Speaker 2:

Exactly. Forewarned is forearmed as they say.

Speaker 1:

So what kind of poison ivy are we dealing with when it comes to perfect squares? What are some of the most common misconceptions teachers should be prepared for?

Speaker 2:

Well, one that comes up a lot is confusing a perfect square number with a perfect square expression.

Speaker 1:

Okay. Help me out here. What's the difference again? I'm having a bit of a brain fart.

Speaker 2:

No worries. It's easy to get those terms mixed up. Think of it like this. A perfect square number is like a single LEGO brick, while a perfect square expression is like a structure built with those bricks.

Speaker 1:

Oh, okay. I see. So 25 is a perfect square number because it's the result of 5 times 5. Right? Yeah.

Speaker 1:

But parenthesis x plus 5 close parenthesis squared, that's a perfect square expression.

Speaker 2:

Exactly. And the lesson plan emphasizes the importance of helping students distinguish between those 2 because, you know, while they both involve perfect squares, they behave a little differently in algebraic equations.

Speaker 1:

It's like the difference between knowing the definition of a word and knowing how to use it in a sentence.

Speaker 2:

Precisely. And speaking of sentences, let's not forget those pesky

Speaker 1:

Oh, fractions. Those always seem to cause a bit of a stir.

Speaker 2:

They do, don't they? But the truth is fractions and perfect squares, they can totally be friends.

Speaker 1:

They could be friends. I like how you think, but I can see why fractions might throw some students for a loop.

Speaker 2:

Right. It's like, hey, I just got used to dealing with whole numbers, and now you're bringing fractions to the party. Exactly.

Speaker 1:

It feels like the rules just changed in the middle of the game.

Speaker 2:

And I get it, but this lesson plan does a nice job of reminding teachers to emphasize that fractions, they're just numbers too.

Speaker 1:

Okay. That makes sense. So how do they drive that point home?

Speaker 2:

Well, remember those visual aids we talked about?

Speaker 1:

Oh, you mean our secret weapon for conquering perfect squares?

Speaker 2:

Exactly. They're back in action here. Like, imagine you're trying to explain what 1 half squared means. You could draw a square divided into 2 equal parts, and then

Speaker 1:

Sure. That squaring a half is basically finding the area of one of those smaller squares. Right?

Speaker 2:

Boom. You got it. Visuals make all the difference.

Speaker 1:

It's like that old saying, a picture is worth a 1,000 equations.

Speaker 2:

Something like that. But speaking of things that can trip students up, we can't forget about our old pal, the negative sign.

Speaker 1:

Oh, those sneaky little minus signs. Yeah. Yeah. What kind of trouble do they stir up in the world of perfect squares?

Speaker 2:

Well, you know how a negative number times a negative number equals a positive number.

Speaker 1:

Yeah. It's like 2 wrongs make a right, at least in the math universe.

Speaker 2:

Exactly. But students, sometimes they forget that little rule when they're dealing with perfect squares.

Speaker 1:

I can see how that would happen.

Speaker 2:

Right. Like, a student might remember that 4 times 4 is 16, but when they see negative 4 squared, their brains short circuit.

Speaker 1:

So how do we prevent those short circuits?

Speaker 2:

Repetition, emphasis, and, of course, real world examples, like, imagine we're talking about temperature. If it's negative 5 degrees outside and it gets twice as cold, well, you're essentially squaring a negative number right there.

Speaker 1:

Oh, I like that. It's like those perfect squares are hiding in plain sight.

Speaker 2:

Exactly. You just gotta know where to look.

Speaker 1:

This has been such a fascinating deep dive. It's amazing to see how much thought and intentionality goes into crafting an effective lesson plan like this.

Speaker 2:

Right. It's like who knew there was so much to unpack about perfect squares?

Speaker 1:

Seriously. But you know what I really appreciate about this lesson plan?

Speaker 2:

What's that?

Speaker 1:

It's not just about teaching math. It's about teaching students how to think critically, how to problem solve, how to learn. Those are skills that will serve them well no matter where life takes them.

Speaker 2:

100%. It's like they're not just building a foundation in algebra. They're building a foundation for lifelong learning.

Speaker 1:

Well said. A huge thank you to the authors of Illustrative Math for giving us so much to think about.

Speaker 2:

Absolutely. And to you, dear listeners, thank you for joining us on this deep dive into the fascinating world of perfect squares.

Speaker 1:

Until next time. Stay curious.

Speaker 2:

See you next time.