Friday, September 14, 2012

Sometimes It's Good to Take a Detour

Probably one of the coolest things about teaching is when a student asks a really good question that lets you detour from your original plan to talk about something really super-awesome.

That happened in my class today.

We were discussing slope and going through a few example problems with the slope formula. I decided to show them one example that resulted in an undefined slope. I gave them the points (7, 3) and (7, 10), then we worked through the problem. We got to a point where we had 7/0 on the board and I asked the students what that meant. The consensus was that the slope was undefined because "we can't divide by zero."

Then, one of my students asked: "Mr. Brenneman, why can't we divide by zero?"

I stopped. I looked at him. I said, "I love that question! Let's put aside what we're doing and talk about this!"

I then launched into a brief explanation of proof by contradiction and asked them to put aside the laws of mathematics for one second. "Let's suppose that you can divide by zero," I said. "Let's consider what 0/0 would be equal to. What do you think?"

Many students chimed in with "0." Others chimed in with "1." I asked each side to back up their reasoning.

"Well, it would be zero because you're dividing zero by another number," one student said.

"I think it would be one, because 2/2 is 1, 4/4 is 1, so 0/0 would be 1," said another.

A few minds were blown when I told them they were both right.

Here's why:

Assuming we can divide by zero, the quotient of 0/0 yields two distinct yet equally valid results.

Suppose we choose a number a from all of the numbers in existence. We say that 0/a = 0 (the zero property of division) and a/a = 1 (a form of the multiplicative inverse property).

In this scenario, division by zero is allowable. (This is an important distinction, because normally the two properties I mentioned above specify that a must be nonzero.) So, 0/0 = 0 by the zero property. But, 0/0 = 1 by the multiplicative inverse property.

Thus, it is reasonable to conclude that 0/0 = 0 and 0/0 = 1.

In other words, 0 = 1.


The discussion can certainly stop here, because we have arrived at a conclusion that is mathematically absurd. Furthermore, this absurdity stems from the initial assumption that we can divide by zero; hence, we must conclude that we cannot divide by zero.

But I knew that ending our discussion at 0 = 1 wouldn't have been nearly quite as fun as proceeding with even more absurdity.

So, I asked the students, "what would 1 + 1 be equal to?"

Many said 2. Some said 1. They were all correct. I showed them why.

1 + 1 certainly equals 2. But, we've already established that 1 = 0, so we can also say that 1 + 1 = 1 + 0 = 1. Or, 1 + 1 = 0 + 0 = 0.

In other words, 0 = 1 = 2.

I extended it one more time by asking the students what 1 + 1 + 1 would equal. Some said 3, some said 2, some said 1. Again, they were all correct. Using similar reasoning as the "1 + 1" case, we concluded that 0 = 1 = 2 = 3.

At that point, the students came to realize that if we kept going, eventually we would conclude that all numbers would be equal to each other.

I told the students one of my favorite mathematically absurd things to say: "If Congress legalized division by zero, we could solve all of our economic problems. We wouldn't have a $15 trillion debt, because if we can divide by zero then 15 trillion would be equal to zero. We wouldn't owe anyone $15 trillion. Problem solved!"

My students seemed to love it. Sometimes it's fun to drop what we're doing and discuss something far more interesting when the opportunity arises.




3 comments:

  1. Love it. It's almost always worth it to go on a detour. There are many ways of approaching the concept of dividing by zero. It's a great opportunity to explore a concept like "proof by contradiction." In a different context, I think I've also detoured to talking about infinity, or limits... it depends on the specific way the question was worded I guess.

    Thanks for sharing!

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  2. My father, a gifted teacher would always tell me “Don’t let school interfere with your education”. I’m going to twist it a bit: “Don’t let the curriculum interfere with your teaching”. Wonderful post. Detours can be the best teaching moments. There is a lot of emerging research that shows math is best taught with practical applications. Students do better solving practical math problems at the grocery store than the identical problem on a math assignment, and a lot of it has to do with interest. Your students felt empowered when you shifted the day’s topic based on their question. They were as excited, I’m sure, to see how you would ad lib this one as they were to see you answer their question.

    Math anxious students, those who feel uncomfortable in the math class, could do with a healthy dose of empowerment. Showing the importance of consistency in rules we follow gives all a thought provoking reality check. Speaking of checks, I’ll send one for a million dollars for this informative post. Oops, that would be a check for 0 dollars wouldn’t it, or perhaps 15 million?

    Tim

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