I assigned the following problem last week in my class:
Giuseppe likes to count on the fingers of his left hand, but in a peculiar way. He starts by calling the thumb 1, the first finger 2, the middle finger 3, the ring finger 4, and the pinkie 5, and then he reverses direction, so the ring finger is 6, the middle finger is 7, the first finger is 8, the thumb is 9, and then he reverses again so that the first finger is 10, the middle finger is 11, and so on.
One day his parents surprise him by saying that if he can tell them some time that day what finger the number 1,234,567 would be, he can have a new sports car. Giuseppe can only count so fast, so what should he do?
Here’s how it went down:
Some of my students realized instantly that counting up to 1,234,567 on their fingers wouldn’t be a very effective use of their time (or Giuseppe’s!). Others counted up to about 180 until they “abandoned ship” on the brute force method.
They were working in self-selected groups of 3, so each group had its own flavor. My tired group kinda stared at each other after they hit 180 on their fingers. My wired group almost immediately decided to figure out where all the tens hit—maybe they could figure out where 1,234,560 would hit. My careful group started writing out numbers on the fingers of Giuseppe’s hand just to see.
The first major “insight” was from the careful group. They realized that Giuseppe’s pointer and ring fingers would be landed on by even numbers only. So, 1,234,567 couldn’t possibly land on either of those. They moved on to look at the other fingers—if certain numbers landed in a certain way. The wired group realized that multiples of ten would alternate between the pointer and ring in a specific pattern. They then spent a loooong time tracing the 10’s to 100’s and debating the pattern that multiples of 100 would make.
By now, the tired group had tried counting up to different numbers (45, 60, 67, 167) on their fingers. They guessed a finger that 1,234,567 would fall on because that’s where 67 fell, but they couldn’t justify why that should remain constant over 1,234,500 more counts. I encouraged them to try to find more information that they could use to justify their guess.
The careful group was already naming fingers at this point. One student had the insight that one finger was 8x+1, another was 8x+2, 8x+3, 8x+4 and 8x+ 5 for the different multiples of 8. The rest of the group “bought” the naming scheme, but didn’t know what to do with it. I asked them on which finger would a multiple of 8 fall? They could tell me, but they didn’t know where to go. I asked a couple leading questions about where would 64 fall? 65? 99? They debated a lot because they didn’t know which direction they would land on the multiple of 8 from. Would they go pointer-thumb or pointer ring?
Back to my wired group. They had an elaborate system for where each 10, 100, and 1000 would land. So, they were now building up to 1,234,500. By now, though, someone had realized that every 1000, the system repeats. 1001 lands back on the thumb, and counts up the same way you would starting from 1. Thus, they had simplified the problem to, where does 567 land? Luckily, they knew where 500 landed, so it was just a matter of figuring out which direction to move after 500 to count up to 67! They were in the home stretch, and they knew it!
My tired group had figured out different characteristics of each finger—the 7’s land on these fingers, only even’s land on these fingers, and 9’s land here. But they weren’t giving air-tight arguments. I argued the case for a different finger based on their observations, and they looked down at the floor. Then one of them said, “it just seems like it’s going to be the middle.” So, a great intuition!
The good morale was completely disappearing in the careful group. They didn’t’ know what their patterns meant or how to use them. It seems that my wired group had stolen all that good energy, because they were just bouncing with excitement about their answer. We took a break, then came back together as a class. I asked each group to explain their approach and findings. Tired volunteered their characteristics of each finger. Careful gave their algebraic classification, which we were able to use to solve the problem during whole-class discussion in a very elegant and exciting way! Finally, wired gave their scheme, while tired counted on their fingers to make sure 100 and 200 did switch fingers!