What is the sum of the infinite series 1 + 1/2 + 1/4 + 1/8 ...
On reflection, I decided this would make a nice connection with converting repeating decimals back to fractions. I had actually tried this 2 years ago and it went okay. Most kids can convert fractions to decimals but can only handle non repeating decimals in the other direction. But in the intervening time I had lost the worksheet I used back then. This time, I wanted to risk it and just work on the whiteboard, have the kids go off and experiment and come back and discuss what they found.
Planned Questions1. What is .999999... equal to and why?
2. How can we represent .99999.... as a series of fractions.
3. Warm up with some easier ones.
S = 1 + 1/2 +1/4 ....
S = 2/1
S = 1 + 1/3 + 1/9 ....
S = 3/2
S = 1 + 1/4 + 1/16
S = 4/3
4. Find the pattern and then come up with the general case:
S = 1 + 1/n + 1/n^2 .....
S = n/n-1
5. Ok let's go back to decimals
S = 1/10 + 1/100 + 1/1000 just like above. Can you use the same technique?
How about if the digits differ
S = 12/100 + 12/10000 + 12/100000
1 = 2 / 3 - 1 vs 2 = 2 / 3 -2 as continued fractions.
I actually started by having everyone talk about the Julia Robinson festival. A couple kids mentioned the final flatland talk and this was of sufficient interest that I ended up spontaneously repeating a huge section of it for those who weren't there. My retelling was accurate except I didn't have any klein bottle pictures on hand other than one on my phone. This ended up taking at least 10 minutes and I would repeat and make a day of it based on how it well it was received.
Basically you have a town in a 2 dimensional world and the inhabitants assume they live in an infinite plane but have never explored it. Then finally one tries it out and discovers if he goes north and leaves a trail he arrives back in the town from the south side etc. Given the behavior when the inhabitants go N, E and then NE you conjecture the existence of a sphere, torus and then klein bottle.
As I result I ended up skipping my planned kenken warm up. We made it through about question 5 from above but by this time I had exhausted the focus of the group, it was getting harder to keep everyone on task. So I made the executive decision to pull out the kenken puzzles after all and "cool off". Fortunately, that pulled everything together again. My take away from this is:
- Kids were aware that .9999 = 1 but the explanation was a bit fuzzy (no numbers between 9 and one) but I didn't have enough time to circle back at the end and show why this must be the case.
- This was still too much material, I need to break it up with something "lighter" if I try again. I think I want either a visual interlude (color in one of these infinite series?) or to gameify the middle somehow.
I went with this puzzle from The Guardian: