Backwards Math

aloft, sideways & backwards
Photo by Complicated.

Princess Kitten is recovering from her cold and getting some energy back. She came to me and said wistfully, “I wish I could do backwards math.”

I looked up from my keyboard. “Backwards math? What do you mean?”

“Umm. It’s kinda hard to explain, but I can show you.”

Negative Number Addition

A brief scramble through the piles on my desk produced a single, blank index card. She found a pencil and wrote:

-6 + -2 =

“I think I know what it is,” she said.

I started to worry. Kitten is a perfectionist who hates to have to change an answer. I said, “Negative numbers are tricky. Maybe you should just tell me what you think it is, before you write anything down.”

But she also hates to admit uncertainty. She didn’t say anything. After several seconds of thought, she wrote:

-6 + -2 = -8

I was impressed. “You figured it out. That’s exactly right!”

“Do you want to know how I got it?” She turned the card over and wrote out her proof, which she still couldn’t put into words. The squiggle means, “…is like the following, only backwards…”

backwards math 1

“You thought that through very well,” I said.

“I’m going to try some bigger numbers.” She laughed and added, “I mean smaller ones.”

Subtraction with Negative Numbers

She started to write a new problem, while I went back to my typing. After a few minutes, I glanced over to see how she was doing. She was on her third equation:

-6 - -4 =

Oh, no! Pre-algebra students always struggle with that sort of thing, and Kitten is only in 3rd grade. I was sure we were headed for another traumatic meltdown.

“Be careful,” I warned. “That’s a super hard problem. Do you want a hint?”

She ignored me. In her intense concentration, I am not sure that she even heard. She wrote her answer:

-6 - -4 = -2

“You got it! Wow, that was a tough one, and you figured it out just right.”

She smiled proudly. “This is more fun than my math book.”

No argument there. In her math workbook, she has been doing 3- and 4-digit subtraction with borrowing. Just about anything is more fun than that.

On to Multiplication

Kitten worked a couple more problems. Then she said, “But what I really want to do is timesing.”

Timesing? In addition to her workbook, Kitten has been doing daily practice pages to master her multiplication facts. But with negative numbers? Now I started to panic. “You want to multiply the negative numbers?”

She nodded.

“Okay, but before you try it, let me show you something,” I said. I turned her card over, so as not to ruin her list of equations. I wrote:

3 \times -2 =

I started to lecture. “Remember, you can think of the times symbols as ‘of’ —”

She rolled her eyes at me. “I know, Mom. That’s how I always think of it.”

Sigh. My kids never have liked lectures. I handed back her pencil. “Okay, then I’ll bet you can figure this problem out.”

She did.

“Now, here is the big, important problem,” I said. “If you get this one, then you can multiply negative numbers. Think about the problem you just solved, and see if you can decide what this would be:

backwards math 2

She thought for awhile, but she wasn’t satisfied with her result. With a frown, she asked, “Would it be -6?”

“That’s a nice try, but remember that 3 of the -2′s make -6.” At a time like this, I am allowed to sneak in a mini-lecture. “If 3 of the -2′s make -6, … [pause] … and -3 is the opposite of 3, … [pause] … then the opposite of three -2′s …”

“It’s SIX?”

“That’s right!” I laughed. “It turns into positive 6. Isn’t that funny?”

Princess Kitten’s Masterpiece

Kitten took her pencil back and worked a little bit longer before losing interest. When she was finished, I told her to autograph the card, and we saved it to show Dad how good she was at backwards math.

backwards math 3


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17 comments on “Backwards Math

  1. I just have to say that this is one of the cutest and adorablest posts I’ve read in a while. The scans really drive home the cuteness factor. I think Princess Kitten is on her way to win the Field’s Medal in a year, at the rate she’s going.

    But seriously, what a wonderful day that must have been for you (and her!).

  2. Thank you, samjshah and Gary. It was fun in retrospect, but she really did have me worried at the time. I could just see a teary breakdown that would set back her recovery from that persistent cold…

    Reverse polish notation? Not me! Her Dad’s an engineer. If he wants her to learn that, he can teach it on his own.

  3. Wow! I wish my 7th graders had the patience and willingness just to sit down and THINK about problems like that… it’s like pulling teeth making them operate with integers!

  4. Pingback: Princess Kitten

  5. My daughter called negative numbers “backwards” because they count in the opposite direction from positive numbers: the smaller they are (the farther down below zero), the bigger they look.

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