Only Addition and Only One (Rev 07/26/2007)

The buzzer in the hall sounded, ending the first class. The instructor put down the last pop quiz. "Congratulations. Every one of you made 100% on your quiz. I said you will learn math. I said you will not be afraid of math. I'm never wrong. Just ask me!" She grinned. "You've chosen partners. You have your assignment due for next class. Now go celebrate your victory. See you next week."

Jane and Alice picked up their belongings and headed out of the class. Outside the door Jane raised her arm with a clinched fist. "Well, Robin, we got through the first class. However. We're doomed. Let's get to the batmobile."

Alice said, "Nothing but good times ahead."

They arrived at Jane's car. Alice said, "Let's do what the instructor said and go celebrate.

Jane started the car, heading to the Sip and Suds Drivein.

"What time is it, Batman? That thing says 6:10."

Jane glanced at the car clock and her watch. "It's 9:05. That clock hasn't been right in a long time. I can't set it and drive, too. Robin fix clock. Batman drive."

Alice flicked down the cover and started holding the minute hand down, then punching it over and over again.

Jane stopped for the red light and idly glanced at Alice punching the minute hand. "Holy Toledo, Robin, I've got it!"

Alice sat back quickly. "What? Don't scare me like that. What do you have? The flu? An anxiety attack? What?"

The light changed, and Jane headed for the drivein.

Jane said jubilantly, "You're adding one over and over again! That's it. That's the answer to the first question. You only add one!"

"Oh? Oh! So I am! Adding one over and over again. But you made me go beyond the 10. Now I've go to go around again. Another almost 60. Oh! That's the answer to the second question."

"What? What's the answer to the second question?"

"At least for the one's thing, multiplication is adding one over and over again. See, going all the way around the clock is punching one 60 times. Or I can say 60 times 1 equals 60. I'll have to think about numbers greater than one later."

They arrived at the Sip and Suds, and their celebration drinks arrived. "Robin, we're almost half way there. The first question is 'you only add one' and the second question is 'multiply adds one multiple times.' So multiplication really is addition. What about the subtraction and division questions?"

"I have absolutely no idea. We've gotta get home. We've got, ugh, work tomorrow."

Jane started backing the car out of the parking space and stopped. They sat there.

"You're thinking too hard, Batman. That's dangerous. What now?"

"I'm thinking about backing up. I'm also thinking about how the instructor said we could use one or negative one."

"That's seriously heavy thinking. Can you finish backing up now so we can get out of here?"

Jane finished backing out and started forward. "Alice, pretend you're standing still."

"I can pretend that."

"Now take a step backward. Now take another step backward. Now take yet another step backward."

"I can pretend that, too. I've taken 3 steps back."

Jane paused for the red light and clapped three times and laughed. "See, taking a step backward is adding negative one. That's the same thing as subtracting one. That's the same thing as adding one, only in a negative direction. Subtraction really is addition.

"Taking 2 steps backward is adding a negative one twice. That's the same thing as subtracting two. Or adding 2 in a negative direction.

"Taking 3 steps backward is adding a negative one three times. That's the same as subtracting 3, and that's the same as subtracting 1 three times. And, tada, that's the same thing as dividing 3 by 3."

Alice frowned. "Try again, Jane. I'm not following this. How about doing division again using six divided by two? I already know the answer is three. So talk me through it."

Jane said, "OK, let's pretend you're on a path with stepping stones separated with lots of gravel. You're standing in the gravel with six stepping stones behind you. You take two steps backward so that now you're standing in the gravel with four stones behind you."

Alice nodded, "Yes, now we've subtracted two from six and have four left. So far addition rules!"

"Now take another two steps backward so that you have two stones behind you. We've subtracted another two from six and have a total of two left. Now take the last two steps backward so that you have no stones behind you. So in all we've subtracted two a total of three times. Or we can say we've added negative two a total of three times. Or we can say we've divided six by two and got an answer of three. Addition rules!"

"Holy Toledo, Batman, you've got it!"

Alice's expression changed. "What about the remainders? There aren't any in this case, but we do have remainders, like 7 divided by 3 equals 2 with a remainder of 1."

"Ummm. Robin, next time. We're only supposed to be thinking about one. Remainders aren't part of the questions. Here's your house. I refuse to think about remainders. Talk to you tomorrow."