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Solving An Equation

1. The Question

“Take a look at the paper in front of you,” they said.

I looked down and got my pencil out of my pocket. (In retrospect, I wonder if any of them noticed that instinctive reach for the pencil. There were generations of us raised with pencils as necessary appendages to our brains, something that I suspect is not true today.)

It was a simple linear equation with one unknown variable.

“Can you tell us how you would teach a freshman with a seventh grade understanding how to solve for the variable?”

I started with, “The first thing I would do is ask them if they understand what it means to say there is an unknown variable in this equation.”

I explained how in my experience for many students (not only those with only seventh grade understanding) this thing about letters in math is baffling. They just haven’t fully internalized what is going on with the letters when math is after all about numbers.

They seemed to like this, because they all wrote down some notes at that point.

My answer that followed was probably good. But of course it could have been better. Here is that better answer.

2. Understanding the Equation

Let’s give the student a name: Pat.

I sit down at a table with Pat and write the equation out on a piece of paper:

10a + 1 = 46.

“Here’s an equation,” I say. “Pat, do you understand what I mean when I call this an equation?” We talk.

“And look at this darned thing. This a. Do you understand what that a is? It’s called an unknown variable. Do you understand what that means?” Pat says no.

“Right,” I say. “You are not alone. It almost never gets explained well. And it’s important to understand this, believe it or not. So let’s talk about that before we go further.”

“Let’s think of a as a box. It has something inside it, but we don’t know what that something is, because the lid of the box is closed. Our purpose in life at this point is to discover what’s in the box. Is a holding a 5? Is a holding a 43? We don’t know. But we’re on a mission to figure that out. We’re on a mission to open the box (to discover what a holds) using … math.”

“By the way, a is just a name of the box,” I point out. Someone working the same problem, looking at the same equation, trying to figure out what’s in the box might write the problem using x.”

10x + 1 = 46.

“So don’t get hung up on the specific letter. It’s a name for the box which could be x or b or z or … Fred.”

With that in place — explaining the problem we’re trying to solve in terms of this black-box thing we’re calling a, I point back at the equation.

“So let’s come back to this darned equation on the paper, here,” I say. “What does this whole thing mean?”

I look at Pat. Of course math is taught in a way that virtually no one knows how to answer that question meaningfully. I tell Pat that. 

“Here’s what that equation means,” I say. “Someone has told us something about the box. They’ve given us a hint. The equation is the hint. It doesn’t tell us what’s in the box, but they promise us that the hint is a true statement about the box, something about a, something that we can rely on with 100% accuracy.”

I look at Pat.

“I know you don’t know why I’m telling you that.” I smile. “But do you hear what I’m saying. Our friend has told us that even though they won’t tell us what’s in the box, they promise that the following statement is true.”

10a + 1 = 46

I think Pat would nod at that point. And I would say, “Great! Let’s get to work!”

3. Matter and Anti-Matter

“So,” I say, kind of like we’re starting over. “What we know is this…” And I circle the equation dramatically.

“What we know with 100% certainty is that this equation is true. We know without a doubt that the left side of that equation equals the right side.”

“And here is the thing — the key to solving this problem. We already know enough about math to force this true statement that looks kinda useless into telling us what’s in the box.” 

So I circle the equation again a few times. I point out that this true statement isn’t doing us any good. Things are jumbled up. Sure, a is in there, but it’s mixed up with other numbers. It’s multiplied by 10. 1 is added to it. It’s all a mess.

A mess? I ask, rhetorically. So let’s clean up the mess!

I ask if Pat knows the opposite of +1. If not, I guide Pat into understanding that if you add -1 to +1 you get zero.

“It’s like matter and anti-matter!” I say. “You add -1 to +1 and they both evaporate into zero… into nothingness!”

“But remember, this true equation has two sides. Whatever we do to the left side, we must do to the right side to keep everything in balance.”

And I look at Pat. “Because if you don’t, you turn the equation (the only thing we know about a) into something bogus.”

4. Making Things Less Messy

“Ok. So we’ve got this messy +1 sitting there on the left side of the equation. Let’s add -1 to it. And to the other side of the equation so that things stay balanced.”

I underline the left side and write -1 below it. And I underline the right side (looking at Pat and emphasizing that we have to do it on both sides), and I write -1 under it.

Here’s what we get:

10a + 1 – 1 = 46 – 1

I get Pat to help me do the math. I use a red pencil to draw parentheses around the places where we’re doing our cleanup.

10a + (1 – 1) = (46 – 1)  

“Since we did the same thing to both sides, we know this is still a 100% true statement,” I say. 

I get Pat’s help simplifying just that red stuff, first on the left and then on the right. Here’s what we get.

10a + (0) = (45)

“And all we did was clean things up,” I say. “So this is also a 100% true statement.”

And in a dramatic flourish, I say “Poof! Matter/antimatter! Zero! Nothingness!” And I rewrite it like this.

10a = 45

“And this is also a 100% true statement, but look at it. Now we’re talkin’!” I say. “The mess looks … less messy, doesn’t it?”

5. Even Less Messy

Then I run through the matter/antimatter thing again with the 10. But I point out that 10 is multiplying a, so we can’t just add some antimatter to get rid of it. 

Here, I ask a leading question. “What do you get if you divide 10 by 10?” I’m expecting that a seventh grader can tell me that — that Pat can give me the answer I’m looking for.

“Right. 10 divided by 10 is 1. It’s a different way to clean things up. You’ll see. Let’s divide by 10.”

Wait! I look at Pat. If we do something to the left, we have to do it to the right, too. So I ask what we need to do to the right. (Right. Divide by 10.)

I underline the left and write a divide-by-10 annotation. I look over at Pat, “And over here?” I ask, pointing to the right side. “Right. Divide-by-10 on both sides.”

Here’s what we get:

10a / 10 = 45 / 10

I make the this-is-100%-true point yet again.

“Let’s rearrange this a bit,” I suggest.”We’re just rearranging, so this will still be 100% true.”

(10/10)a = (45/10)

With Pat’s help, we do the math.

(1)a = (4.5)

“And since we just cleaned things up, this is a 100% true statement. But look: Poof!” I say. A different kind of matter/antimatter.” And so we get this.

a = 4.5

“Is this statement still 100% true?” I ask. “Yes. As long as we kept things in balance.”

“So now you tell me: what is the value of a?”

We did it! We found what’s in the box!

6. Practice

At this point, I hope Pat understands the general idea. But in my opinion, general ideas like this need to be hammered in with reinforcement a few times with tiny (Tiny I say!) variations on the same problem.

So I would have Pat work thru some similar problems. To be honest, at this point it would work better if there were two students, so that they could work on these problems together. But Pat’s here alone.

“What if we rename the box?” I ask. “Show me how to solve this problem,” I say, writing down the same equation on a fresh piece of paper with a different name for the unknown variable:

10b + 1 = 46

“What if it’s 45 instead of 46?” I ask, writing down a different equation on a fresh piece of paper. “Show me how to solve this problem.”

10b + 1 = 45

“What if it’s 2 instead of 1?”

10c + 2 = 45 

“What if it’s 3 instead of 10?”

3d + 2 = 45 

And in that way, I hope to have taught a seventh grader how to solve that kind of linear equation.

For evidence of mastery, I write this equation down and say to Pat, “If you can take what we’ve talked about here and solve this equation, then you have completely mastered this. Try it. Come get me if you need help. I’ll be right over here.”

2x + 1 = -3x + 11