I was talking with one of our 5th graders, S, last week about the difference between showing a few examples of numerical computations and developing a way to know something was true no matter what numbers were chosen. I hadn’t started our conversation thinking about introducing proof. Once we turned in that direction, I anticipated scaffolding him in a completely different direction, but S went his own way and reinforced for me the importance of listening and giving students the encouragement and room to build their own reasoning.
SETUP: S had been telling me that he “knew” the product of an even number with any other number would always be even, while the product of any two odds was always odd. He demonstrated this by showing lots of particular products, but I asked him if he was sure that it was still true if I were to pick some numbers he hadn’t used yet. He was.
Then I asked him how many numbers were possible to use. He promptly replied “infinite” at which point he finally started to see the difficulty with demonstrating that every product worked. “We don’t have enough time” to do all that, he said. Finally, I had maneuvered him to perhaps his first ever realization for the need for proof.
ANTICIPATION: But S knew nothing of formal algebra. From my experiences with younger students sans algebra, I thought I would eventually need to help him translate his numerical problem into a geometric one. But this story is about S’s reasoning, not mine.
INSIGHT: I asked S how he would handle any numbers I asked him to multiply to prove his claims, even if I gave him some ridiculously large ones. “It’s really not as hard as that,” S told me. He quickly scribbled
on his paper and covered up all but the one’s digit. “You see,” he said, “all that matters is the units. You can make the number as big as you want and I just need to look at the last digit.” Without using this language, S was venturing into an even-odd proof via modular arithmetic.
With some more thought, he reasoned that he would focus on just the units digit through repeated multiples and see what happened.
FIFTH GRADE PROOF: S’s math class is currently working through a multiplication unit in our 5th grade Bridges curriculum, so he was already in the mindset of multiples. Since he said only the units digit mattered, he decided he could start with any even number and look at all of its multiples. That is, he could keep adding the number to itself and see what happened. As shown below, he first chose 32 and found the next four multiples, 64, 96, 128, and 160. After that, S said the very next number in the list would end in a 2 and the loop would start all over again.
He stopped talking for several seconds, and then he smiled. “I don’t have to look at every multiple of 32. Any multiple will end up somewhere in my cycle and I’ve already shown that every number in this cycle is even. Every multiple of 32 must be even!” It was a pretty powerful moment. Since he only needed to see the last digit, and any number ending in 2 would just add 2s to the units, this cycle now represented every number ending in 2 in the universe. The last line above was S’s use of 1002 to show that the same cycling happened for another “2 number.”
DIFFERENT KINDS OF CYCLES: So could he use this for all multiples of even numbers? His next try was an “8 number.”
After five multiples of 18, he achieved the same cycling. Even cooler, he noticed that the cycle for “8 numbers” was the 2 number” cycle backwards.
Also note that after S completed his 2s and 8s lists, he used only single digit seed numbers as the bigger starting numbers only complicated his examples. He was on a roll now.
I asked him how the “4 number” cycle was related. He noticed that the 4s used every other number in the “2 number” cycle. It was like skip counting, he said. Another lightbulb went off.
“And that’s because 4 is twice 2, so I just take every 2nd multiple in the first cycle!” He quickly scratched out a “6 number” example.
This, too, cycled, but more importantly, because 6 is thrice 2, he said that was why this list used every 3rd number in the “2 number” cycle. In that way, every even number multiple list was the same as the “2 number” list, you just skip-counted by different steps on your way through the list.
When I asked how he could get all the numbers in such a short list when he was counting by 3s, S said it wasn’t a problem at all. Since it cycled, whenever you got to the end of a list, just go back to the beginning and keep counting. We didn’t touch it last week, but he had opened the door to modular arithmetic.
I won’t show them here, but his “0 number” list always ended in 0s. “This one isn’t very interesting,” he said. I smiled.
ODDS: It took a little more thought to start his odd number proof, because every other multiple was even. After he recognized these as even numbers, S decided to list every other multiple as shown with his “1 number” and “3 number” lists.
As with the evens, the odd number lists could all be seen as skip-counted versions of each other. Also, the 1s and 9s were written backwards from each other, and so were the 3s and 7s. “5 number” lists were declared to be as boring as “0 numbers”. Not only did the odds ultimately end up cycling essentially the same as the evens, but they had the same sort of underlying relationships.
CONCLUSION: At this point, S declared that since he had shown every possible case for evens and odds, then he had shown that any multiple of an even number was always even, and any odd multiple of an odd number was odd. And he knew this because no matter how far down the list he went, eventually any multiple had to end up someplace in his cycles. At that point I reminded S of his earlier claim that there was an infinite number of even and odd numbers. When he realized that he had just shown a case-by-case reason for more numbers than he could ever demonstrate by hand, he sat back in his chair, exclaiming, “Whoa! That’s cool!”
It’s not a formal mathematical proof, and when S learns some algebra, he’ll be able to accomplish his cases far more efficiently, but this was an unexpectedly nice and perfectly legitimate numerical proof of even and odd multiples for an elementary student.