Here is a very valuable conjecture:
The spelling of every whole number shares at least one letter with the spelling of the next whole number.
Which is to say that:
- “one” and “two” both share an “o”
- “two” and “three” both share a “t”
Could that possibly be true for every whole number?
If I were starting a course on geometry or a unit on proof or an activity on deductive logic, I would introduce this conjecture very early in the process. Let me explain what I find so very valuable about this conjecture.
Deduction is hard. It’s an abstract mental act that adults find difficult. (See: the van Hiele’s and their levels.) Too often we rush students to that abstract act, rushing them past the lower van Hiele levels, and we ask them to argue deductively about objects that, to them, are also abstract.
I suspect that, to many students, those proof prompts read something like this:
Given that the base bangles are twice the tonnage of the circumwhoozle and the diagonalized matrox is invertible, prove that all altimeters cross the equation at Quito, Ecuador.
The word “prove” is weird. And, unfortunately, so is every other word in the sentence.
So I cherish opportunities to help students argue deductively with concrete objects, which is what we’re working with here, with the spelling of whole numbers. This conjecture also gives students several different angles on the proof act.
You can ask students to find a counterexample, for example, a useful strategy when first interrogating a conjecture.
Once students have tried several different numbers they may satisfy themselves that the conjecture is true. This is one of the naive proof schemes Harel & Sowder observed in the students they studied. When this proof scheme surfaces in conjectures about geometric shapes, it’s challenging to summon up one new shape after another to challenge the student’s proof by example. It’s trivial, by comparison, to summon up one new number after another and ask the student to check her hypothesis again.
At a certain point in this process, likely after you give several numbers in the millions, your students may transform in two ways:
- They’ll get tired of trying example after example. “Proof by examples means you have to try all the examples,” you can say, giving you both a moment to reflect on the need for a more rigorous proof scheme, like deductive reasoning.
- They’ll notice that every number in the millions shares an “n” with every other number in the millions. And same for the billions. And same for the trillions. And … same for the hundreds. And so on.
And suddenly we’re on our way to a proof by exhaustion, which is much more rigorous than a proof by example. Nice.
This conjecture also leaves ample room for you and your students to pose follow-up conjectures. Like, “Does it work for all integers, or just whole numbers?”
I saw the conjecture and saw its value immediately. This is a very valuable kind of conjecture, I thought. But I don’t have many of them. Do you have another you can trade?
[via Futility Closet.]
BTW. You’re worse off in at least one way now than before you knew the conjecture was true. Now, when you ask your students, “Could that possibly be true?” you’re going to have to pretend.
What I like about game strategies is you go from “what seems to work,” to “will this always work,” to “here’s why this does/doesn’t work” pretty seamlessly.
Is it true for any other language with an alphabet? It fails for German (5/6 using umlauts, 7/8 otherwise), French (2/3), and Spanish (7/8).
What letter(s) of the alphabet do(es) not appear in the spelling of the first 999 whole numbers? Prove it.
List all the factors of every number from 1-100. What do you notice? Which numbers have an even number of factors? An odd number? What do you notice about numbers with an odd number of factors? Can you prove which numbers beyond 100 will have an odd number of factors?