Number Tricks Show the Power of Algebra: Subtract by Adding

I gave a mastery test last Tuesday, and my first class had worked hard to get ready for it. I wanted to show them something fun the next day, instead of diving into our next unit: roots, completing the square, and the quadratic formula. I really enjoy the buildup of all that, but it takes some serious work on their part. I wanted to start with something more .. light-hearted.

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...

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

To find the nines complement, subtract each digit from 9, and write the result in place of the digit.

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_c-1000+1. 

Done.

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.)