408. Borrowing Across a Zero, Part One

I’ve always enjoyed teaching children to do what I’ve called “borrowing across a zero.” It’s what you do when you solve a problem like 302 – 158. I’ve always told children a story, using the digits involved as characters. I know this approach bothers some mathematicians, who like to keep mathematics pure, and I know the term “borrowing” isn’t used as much as it used to be, but my story has helped some children, and I’d like to tell it to you. When I tell it, I use different voices for the different digits, but you do what you want.
Once upon a time, there were two ones. They were very happy in the ones place, and thought that’s where they’d always be. The ones place was small, but ones are small anyway, and don’t need much space.
But one day, one of the ones, whose name was Adeen, said to the other, named Uno, “I think there’s eight ones downstairs. And I think we’re supposed to subtract them.” Adeen and Uno were used to subtracting, but so far, they’d only subtracted zero, one, and sometimes two (they didn’t like subtracting two, because when they did, they had nothing left).
‘We can’t subtract eight!” cried Uno. “We’re only two! And don’t tell me we’re going to go to the tens place to borrow ten. That place always makes me nervous.”
“There isn’t any other way to do it,” answered Adeen. “The tens have always been nice about letting us have one ten. And the job has to be done. We can’t have those eight ones living downstairs. The place is too small for all ten of us. Let’s go. I’ll do the talking.”
Uno knew that Adeen was right, and agreed to go along, but did not agree to like it. So they went to the tens place and Adeen knocked on the door. There was a long silence. Adeen knocked again. More silence. “There’s nobody home,” said Adeen.
That delighted Uno. “Let’s come back tomorrow. They’ve probably gone fishing or something.”
“No,” said Adeen. “I know what we’re supposed to do. We’re supposed to go to the hundreds place.” And Adeen pointed to the huge building next door.
“We don’t need a hundred,” said Uno. “We just need ten.”
“Right,” replied Adeen, “but every hundred has ten tens.”
“Yes, but they’re not going to break up a set just for us.” Uno was already nervous, and the thought of having to go to the hundreds place was not helping. Still, on they went.
At the hundreds place, Adeen knocked on the door. The sound of the knocked echoed, and both Uno and Adeen trembled. Soon, they could hear footsteps. Loud, heavy footsteps.
To find out what happened next, read my next article, “Borrowing Across a Zero, Part Two.”