Solving with the Reciprocal
by Liana Krajnak

The sheep on the computer game I was playing descended in rows of three, each threatening to cross a fence at the bottom of the screen, at which time the game would end. A math problem was written on each of the sheep’s backs. To stop the sheep’s descent, you had to solve the problem. The game was called Math Nightmare.

"Dad and I are going to pick you up early on Friday, Liana," my mother informed me suddenly. "I have a doctor’s appointment."

"And Dad’s going with you? Why?" It was strange enough to be picked up early, but it was outrageous to be picked up early by both parents. But it still wasn’t outrageous enough to make me turn from the screen to look at her.

"I wanted him to. It’s a special appointment."

"Why’s it special?" On the screen, a sheep labeled 12 x 13 appeared on the screen and began to descend quickly.

"It just is, okay?" It wasn’t okay, but I could tell she was a little upset.

Quick, what is twelve times thirteen? I quizzed myself. Um…twelve times twelve is 144. Plus twelve?

I keyed in 156. The sheep disappeared. I’d solved the problem.

Several more sheep appeared and disappeared before my mom asked me if I really wanted to know more about her appointment.

"Okay. Sure," I shrugged. It wasn’t much to me.

"Really, really?"

"Mm-hmm," a sheep labeled 165 ÷ 11 had appeared. How was I supposed to know that?

My mom pulled up a chair to sit beside me. I turned to look at her, giving up on my game. When she took my hands, and I began to worry about what she was going to say.

"Sweetie, Mommy found—I found…" I looked at her expectantly; she looked at our hands. "I found a lump in my breast."

Miss Dyer, the elementary school guidance counselor, a few weeks before this news: "Now, I want you girls to watch this and take it seriously. No giggling."

All the girls in fourth, fifth and sixth grades sat together on the floor, talking and giggling, crowded into the school’s tiny library. To our chagrin, we were all forced to watch the video Miss Dyer was putting into the VCR.

A woman on the screen began talking about "how the body changes", and from the feet up, she described what was supposedly happening to us. I listened to her for lack of anything better to do.

Eventually, she started talking about breasts. Mostly she just repeated herself, but there were a few sentences that somehow caught my ear:

"Women in their forties and above sometimes find lumps in their breasts. This is the first sign of breast cancer, which is a very bad disease."

The screen cut to pictures of bald, weak-looking women in hospital beds. The narrator’s voice described the effects of breast cancer and the treatments. Also, she announced that any girl with cancer in her family was two and a half times more likely to get breast cancer than anyone else. I shut my eyes and tried to ignore it. I was horrified at the thought of anyone I knew getting breast cancer.

Oh, well, I thought, No one in my family has a lump, do they? Nope. Couldn’t possibly. That only happened to other people. It really couldn’t, could it? Who would it be?

No one, that’s who.

So why do I have this sinking feeling?

"Oh, honey, it’s okay, it’ll be okay," I had started to cry, and was hugging my mother. "I’m going to be fine."

Well, I knew that.

My mom was going to be okay. I had no doubt of it. There were treatments. There was chemotherapy. There were lots of doctors. She would live and be just like before. I didn’t see any reason why she wouldn’t.

No, my mom had misunderstood. I wasn’t crying for her. I was much more selfish than that—I was crying for me. I was crying for my new risk factor. I was crying for the fact that I was probably going to get it. I was crying because, though Mommy couldn’t die, I could.

But at the same time, I knew I was being selfish. I knew my mom was more likely to die than I was, and I should be sad for her. I should worry about her.

I had two attitudes, and I couldn’t accept either of them.

So I compiled them in my mind. Mom wasn’t going to die. Neither was I. I probably wasn’t going to get it. Mom probably wasn’t going to die from it. I was okay. She was okay. The world was okay and I could go back to playing Math Nightmare.

I didn’t cry again for a month, and when I did, it wasn’t because of my mom’s tumor, which had, by now, been officially diagnosed as cancerous.

Instead, I was crying because I couldn’t divide fractions.

My dad had explained to me the night before: "You switch the top and bottom of the second number and multiply. Multiply by the reciprocal. Easy."

"What? Why?" I was confused.

"You just do it. I don’t know why. You just do."

"But why?"

"They’re reciprocal, and dividing numbers is the same thing as multiplying by the reciprocal," he looked at me. He looked exasperated.

"I still don’t get it."

Mrs. Hess, my fifth grade teacher, announced that she was giving an oral pop quiz. She’d give us one question to answer, and if we were able to answer it, we would receive 100%. The questions were over multiplication and division of fractions. I prayed for a multiplication problem.

She gave me the first division problem, and asked me what to do in order to solve it. My mind went blank. What was I supposed to do again?

"I…I—switch the first number and then cross multiply the two numbers."

"No! Where did you get that? Where are you this morning, Liana?" She seemed amused.

"I mean, not the two numbers. Switch the answer and then…" Tears were welling in my eyes. What was happening? "I—I don’t know!"

I let out a loud, wracking sob, and ran to the bathroom. I kept crying long after I’d calmed myself from the fraction division problem.

Mrs. King, a teacher I’d never had for class at any time, managed to coax me out of the bathroom. We walked behind the school, through the playground, and beyond. For several long minutes she said nothing, just let me cry, but she finally asked, "Why don’t you tell me why you started crying?"

"I can’t divide fractions. I didn’t know how to do a fraction division problem," I sobbed.

"But why were you crying?" she was incredulous. "That’s not all, is it, Liana?"

It was. I felt fine, except for that. I said nothing.

"I think the fraction problem was just the straw that broke the camel’s back, huh?"

I didn’t know what she meant, or how that expression applied to me. I still said nothing.

"Didn’t you just find out your mom has breast cancer?" her large eyes studied me. "Aren’t you upset about that?"

"Yes," I replied cautiously—mostly because I thought this was the answer she wanted.

"I want you to talk about it," her tone was firm, as if she were going to force it out of me.

"I really don’t want to, Mrs. King," I told her, sobbing weakly. I didn’t feel upset. I couldn’t think of anything to tell her.

"Well, maybe I’ll talk, and then you’ll feel like talking to me about it."

"Okay." Go ahead, I thought. Talk away, I thought.

She described her sister, or her mother, or perhaps an aunt who’d had breast cancer and had to get a "double mastectomy". What was a mastectomy anyway? Was that the surgery Mom kept talking about?

I wasn’t really listening, anyway. I was thinking about fractions, trying to solve Mrs. Hess’s problem.

Okay, what do you have to do? Think, Liana! Don’t you have to flip the second number and then just multiply the tops and bottoms? And oh! You switch the second number because any division’s the same thing as multiplying by the reciprocal.

—

That’s it. That’s what you do. You just do it. You just deal with it. You just solve it with the reciprocal.

—

I stopped crying later, when my friends cheered my up.

My mom is now a five-year breast cancer survivor.

I’m okay with my risk factor.

And I can now divide fractions with ease.