Isn’t that why we start, traditionally, with proofs in Geometry? They can see the angles in question, so the abstraction is limited.

In introducing proof by Geometry deduction, I wonder how much use a “proof” of this sort is. Geometry proofs don’t use exhaustion, exclusion or induction, so it’s not a great lead in. (I prefer to get the strugglers to do geometry proofs first with some numbers, then repeat the process with variables/pronumerals to lower the mental barriers.)

I can see the proposed “proof” being much more relevant when approaching elementary algebra proofs — where we do actually use exhaustion, induction and exclusion.

So if you were leading a period with this, how much time do you spend on it? Is it a quick taster or a task to fill a period?

That depends on the convention. Many math teachers (like Michael Serra) insist that “one hundred and one” means 100.1, or something like that. I disagree, but you’ll meet many students who absolutely insist on that as well, because their teachers scolded them into it.

But that doesn’t matter, because 100, 101, … 999 all have “hundred”; 1000, 1001, … 999,999 all have “thousand”, and so on. Since “hundred”, “thousand”, and “*illion” all have “n”, it suffices to show that 1..100 works. If you’re using the million/million/billion/billiard convention, then “l” becomes the obvious link.

Hey, Chris: “A milliard” doesn’t have any of those letters, so your conjecture doesn’t work in some English naming conventions. ;)

Almost every number ends with “and one” or “and two” or “and three” etc. Right? So wouldn’t showing that 1,2,3,4,5,6,7,8 and 9 all fit this pattern suffice for almost every number?

The exception to this are numbers that end in zero, like “One million” or “one hundred forty-two thousand nine hundred and thirty”. Ignoring the fact for the moment that just about the entire number in my example has the same letters as the next one, wouldn’t you run into trouble when you get out to numbers that have not been named? There has to be a biggest number with a name doesn’t there? (Googolplex?) So what happens when you move beyond that number? If we’re talking “proof” wouldn’t you need to account for that? Am I being too demanding of the word Proof? (Not a rhetorical question.) :)

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?

Nice. Also added to the post.

What letter(s) of the alphabet do(es) not appear in the spelling of the first 999 whole numbers? Prove it.

That’s a beaut. Added to the post.

I know the argument. I disagree with it. “One hundred and one dollars” is perfectly understandable.

That also leads to the debate about whether “a hundred” or “one hundred and one” are “proper” names for 100 and 101, respectively. (I say they are, but a lot of math teachers, particularly in lower grades, apparently insist they’re not.)