So I showed them a number trick I learned recently. (Was it at AMATYC? Was it online? Or was it somewhere else? I haven't a clue.)

**The Trick**

If you don't want to do a subtraction, say one with lots of borrowing, you can add instead.

Let's try an example:

Lots of borrowing needed here, and two borrows in a row can get confusing. So instead we'll do this...

- Find the "9's complement"
- Add
- Subtract one from the highest-value digit (leftmost position) of the result, and add that one in the ones place.

Now we add, and do a little fussing...

**Why it Works**

The real fun, with Helen on the plane and with my students, was showing why it works. (One student wanted to know why they hadn't been taught this before. I said I thought it was too much of a trick, and you might forget just how to do it, and math shouldn't be that way. But maybe it would improve some people's subtraction enough to be worth it for them.)

Let x-y represent the subtraction problem. Let's start with supposing y is 3 digits (or less), and see if we can find a way to generalize, which is harder to write down. Find y complement, which I'll call y

_{c}, by finding 999-y. If y

_{c}=999-y, then y=999-y

_{c}. And

x-y

= x- (999-y

_{c})= x-999+y

_{c}= x+y

_{c}-999= x+y

_{c}- (1000-1)= x+y

Done._{c}-1000+1.I like that! And, more importantly, so did my students. I used to think number tricks were silly, but the ones I did at the start of this semester, along with this one, have convinced me to change my mind about equations versus expressions.

I used to feel like equation solving was the heart of algebra, but the algebraic explanations of number tricks start with an expression representing the numbers, and then involve simplifying until something comes out which really does explain why the trick works. Simplifying expressions suddenly seems much cooler than it used to. (I used to also think you couldn't check your answer when you simplified an expression, but all you need to do is put in a random number to the original and your answer, and make sure they match. Try 2 or 10 for an easy number, since 0 and 1 aren't good choices for catching mistakes.)

That IS a fun trick!!

ReplyDeleteI sell it to high school freshmen as "no-carry" subtraction, but frame the arithmetic slightly differently.

ReplyDeleteStart with a little story. I've got 15 dollars more than Teacher Y. If we both get an extra 2 dollars, how much more do I have?

Then, maybe:

562 - 97

= (562 + 3) - (97 + 3)

= 565 - 100

Vertical format:

562 + 3

-97 + 3

--------

and we see how the 3s don't matter.

457 - 179 =

(457 + 21) - (179 + 21)

Works.

Jonathan