Saturday, August 29, 2009

How Close is Close Enough?

For my 8th graders, homework for day 2 consisted of a worksheet where students determined which set(s) included given numbers. Pretty easy stuff. But I threw one of my favorite problems at 'em to see what they'd do with it.

What's the sum of 1/2 + 1/4 + 1/8...?

At first they're thinking, "not possible 'cause it goes on forever." I told them to try it anyway.

The two most popular answers were ".99..." and "1." But those who answered "1" were quick to admit that they just rounded off. We open the discussion and I was very pleased with how thoughtful and respectful everyone was. These kids were really interested in getting to the bottom of this. It was a great opportunity to demonstrate that often times drawing a picture will allow you to see things in a problem that you may not otherwise catch.

So we draw a square on the board and shade 1/2. Then we shade 1/4, then 1/8 and so on. They soon see that the square will eventually be full.

Me: "So is it 1 or is it just really close?"

"Really close. Because the square is never completely full. You always have half of the remaining area that is unshaded."

Good. So let's see how they handle this.

"Alright, what's 1/3 as a decimal?"


"Ok, and how about 2/3?"


"what's 1/3 + 2/3?"


"And what's .333...+ .666..."


"So does .999... = 1 or is it just really close?"

At this point they admit that it looks like it's equal but it just doesn't make sense. Time to to talk about what it means to be infinitely close to something. This is always a fascinating discussion.  We discussed the idea of a neighborhood and how if .999... does not equal 1, then there must be a number between them. 

"Give me the number and I'll shut up", I tell them. 

One kid says," How about .0 with a repetend, then a 1?"

But another student catches this, "If the zero goes forever, when do we add the 1?"

 It amazes me how these kids can grapple with the real "stuff" that is mathematics.  These same questions that got me hooked as I was taking my analysis classes in college are finding their way into the minds of 8th graders.  And you know what?  They get least as much as they possibly can. 

Man, I love this job!


Rick Regan said...

In grade school I wondered how something in the real world could be exactly 1/3 of an inch. I was hung up on the infinite 3s and how no ruler could measure it. Many years later I realized the answer had to do with our representation of numbers -- base 10 -- and not something inherent in numbers themselves. After all, 1/3 in base 3 is terminating - it's 0.1!

Darren Kuropatwa said...

Great lesson! What the world needs now is more math lessons like this one.

I always loved seeing the light of curiosity and thought appear in kids eyes when presented with a seeming paradox. Good on ya for bringing it out in your kids eyes.

David Cox said...

These were the types of questions that kept me coming to class while in college. It still bugs me that I never had any conversation like this in middle school or high school. How many fires could get sparked if we just allow for 15 minutes of a rabbit trail every now and then?

Thanks for the encouragement, Darren. BTW, I have taken the advice that many of you gave regarding my 8th grade class and have started taking a more analytical approach to the Geometry. The Exeter problems as well as some adaptations that Alison sent have been a great resource.

Alison Blank said...

This conversation comes up in every class I teach. The idea of the infintesimal just runs so deep through mathematics. I don't think I've ever been able to convince anyone that 0.999... = 1 though.

Darren Kuropatwa said...

@David That's great. When I started using a similar approach im my classes I saw more lightbulbs going on over kids heads that I had seen before. It also made teaching more interesting for me; I started to grok the connections between the seemingly disparate branches of mathematics.

@Alison Even when you used David's approach here?
Find the decimal representation of 4/9.
Find the decimal representation of 5/9.
Add the decimals, what do you get?
Add the fractions what do you get?
Same thing, no?

It really helps underscore that infinity is not a number; it's an idea. ;-)

David Cox said...

I had an interesting conversation with a colleague the other day about this. He approached the question from a philosophical standpoint. Are they actually =? Philosophically, I suppose we could say no. The numbers become ultimately equal (as per Newtons Lemma 1). He doesn't say that they ultimately become equal. Leave it to a philosopher, huh?

Darren Kuropatwa said...

Actually, I would say, yes, they are actually equal. 0.9999... out to infinity IS 1. It isn't 1 only if you stop somewhere, but if you stop somewhere then 0.9999... isn't an infinite decimal.

David Cox said...

I agree that they're equal. It's the infinite decimal that is the sticking point. His point of emphasis is "when will it reach 1 if it's infinite?" and my point of emphasis is "then give me a number between .999... and 1.". Still a fun conversation to have.