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 it…at least as much as they possibly can.
Man, I love this job!