School went pretty quickly on Wednesday. I had planned my tutoring session,
slept well, and was in a good mood all day. No one hassled me in the hallways.
Since I’d started hanging out with the Razors, even my teachers keep
their distance.
As soon as my classes were over, I headed out to hitch a ride. But before
I’d walked more than a block and a half, I found myself swerving
towards of the post office. I hooked up with Mrs. Ellis and we drove off
without my being seen by any of the Razors. Or so I thought.
On the ride to Atkins Corner Mrs. Ellis described how Susie emptied her
piggy bank of pennies and spent an hour dividing them into groups, announcing, “This
is a half of the pennies; this is a fourth, and this is an eighth.” I
laughed and told Mrs. Ellis that I hadn’t shown Susie how to do that;
she’d figured it out on her own. Mrs. Ellis was thrilled. “You
know, Buzz,” she said leaning towards me as if she were telling me
in confidence, “when I saw Susie’s math grades, I was really
worried. I’d always thought she was reasonably intelligent.”
“
Oh, she’s smart enough,” I reassured her. “She just got
frustrated and gave up temporarily.”
“
She’s a very visual child, and I think rational numbers more abstract
than visual. I suppose she just couldn’t visualize fractions. Until
you came along.”
Mrs. Ellis pulled up in front of the library and thanked me again.
The tutoring session started well enough. Susie showed me what she’d
figured out with her pennies. “It happened because Mom watches
infomercials. That’s what she calls them, but they’re just
ads. So I was wishing she’d change the channel when this guy announced
that three in every five Americans will die before the age of eighty
because of poor calcium metabolism, whatever that is. And I started thinking
about my class. There are twenty-five of us. How many did the guy just
death-wish? I couldn’t figure it out in my head, so I got some
pennies out of my bank. I put out twenty-five on the bed. I figured that
fifteen of us would die if the guy on the stupid show was right. Then
I realized that I could do that in two ways.” She grinned at me,
her braces picking up a stray ray of sun. “Look!”
She opened a plastic sandwich container. Inside were at least two hundred
pennies. She counted out twenty-five and divided them into five equal
piles.
“
There are two ways to get three-fifths,” she announced. “You
can pull three of the piles or else you can take three pennies from each
pile.” Her grin got bigger. “Top that, Mrs. Pigface Barrows!” she
squealed, slapping the table and drawing an angry look from the librarian.
I shushed her and asked, “Your mom said you were doing better in
math, Susie. Isn’t Mrs. Barrows happy about that?”
Susie’s nose turned an angry red. “She doesn’t want
me to use pennies in class. She said she’d take them away if I
brought them out again. She’s a pig-faced meanie,” she said, “an
ugly, boring bag of a teacher, and she hates me. All she wants us to
do is put fractions on a diet.”
“
What?” I asked, feeling confused, but also tempted to laugh. “Put
fractions on a diet? What are you talking about, Susie?”
She shrugged. “Don’t ask me. It’s not my idea. Putting
fractions on a diet and finding common dominators.”
I wondered if Susie needed to get her ears checked. “Not common
dominators, Susie. Common deNOMinators. You need to find a common denominator
to add or subtract fractions.”
She stared at me blankly. “Why?”
I mentally tossed away my plans for the session and pulled out a page
of notebook paper. “Okay, what if I want to add half of the page
and a third of the page?”
She sneered. “What if you do?”
“
Show me how!” I ordered.
At first Susie tried to add and subtract fractions using her pennies,
but she became confused. “It doesn’t work,” she grumbled.
She glared at me, then at the paper. Then taking a deep breath, she folded
the paper in half and tore it. “There’s half,” she
said, setting one piece aside. “But how do I get a third from the
other piece?” She looked at me but I just shrugged.
After a few minutes, she said, “I don’t want a third of the
piece that’s left over because that would be a third of a half.” She
looked at me slyly. “I want two thirds of a half, don’t I?” She
grinned, folded the half sheet into thirds and tore off two of them. “There!” she
announced triumphantly, drawing another disapproving look from the librarian.
“
Good!” I said. “You showed me. Now put it into words.”
She scowled. “I took a half and I added two thirds of the other
half.” She nodded. “Yep, that’s what I did all right.”
“
Okay,” I conceded. “But now put it into math language.”
Her scowl deepened. “Well,” she said slowly, “I added
a half and two thirds of a half.”
“
That would be a pretty complicated thing to write down in numbers, wouldn’t
it?” I asked. “Finding a common denominator is a way of simplifying
it,” I said.
She stared at me. “It wasn’t simple when Pigface put it on
the board.”
“
Well, we don’t have a board,” I said. “How could you
make a single fraction that would describe the paper?”
She wrinkled up her nose. “What do you mean?”
“
Can you make halves or thirds out of it?”
She shook her head.
“
What did you make when you tore the half into thirds?” I prompted.
She thought for a moment. “Sixths,” she said.
So when you added one half and one third using the sixths,” I started.
She grunted. “One third and one half is five sixths,” she
said.
“
And what’s the common denominator?”
She shrugged. “Haven’t got a clue.”
“
What’s a denominator?”
She squinted at the wall behind my head. “The bottom number of
a fraction?”
“
Right,” I said, “but what does it mean?”
She nodded. “The number of pieces all together that I made out
of the paper. It’s six. The common denominator is six.”
I grinned at her. “Got it!”
She grinned back. “Why was it so hard when Mrs. Barrows put it
on the board? She took three-fourths and one-sixth and multiplied six
times four and put it on the bottom. Then she multiplied the tops by
something, but I forget what and then she put the fractions on a diet
and got something else.”
I almost laughed. But at the same time I was a little worried about how
Susie seemed to translate what she heard into weight loss.
“
Susie, she reduced the fraction, she didn’t put it on a diet. What
is it with you and losing weight? You’re skinny as a rail.”
She stared at me as if I were ignorant. “Everybody has to lose
weight,” she said, “sooner or later.” I wondered if
I should mention this conversation to Mrs. Ellis. Did girls Susie’s
age get eating disorders?
I sighed. “Okay, Mrs. Barrows made the problem a lot harder than
she had to by multiplying four by six. But I guess she wanted to show
you how to reduce fractions.”
“
That’s what I said,” Susie snapped at me. “She put
them on a diet.”
I shook my head. “No, reducing fractions has to do with simplifying,
not dieting, Susie. It’s hard to work with big numbers, so if possible,
you make them smaller.”
She inhaled loudly. “All right,” she said. “Give me
another one.”
It only took her a couple of minutes this time.
“
Three quarters and one sixth,” she intoned. “The common dominator
is twelve and the answer is nine-twelfths and two-twelfths. That makes
eleven of them.”
Then she looked at me with a very bright smile. “I understand it
now, Buzz. Thanks a lot.”
“
You’re welcome,” I replied, smiling at her. Her face got
so red that I could hardly see her freckles. Then she looked up at the
ceiling as if she were reading a message up there.
“
So, if Mrs. Barrows won’t let you use pennies, will she let you
fold and tear paper?” I inquired.
She shook her head. “I don’t think so. She says we’re
too old to do stuff like that. We should be able to figure everything
out with pencil and paper.”
My future as a math tutor did not look good. I looked at her seriously.
“
What?” she demanded.
“
Susie, have you ever learned to visualize?” Her expression told
me no. “We have a choice,” I explained. “I can show
you how to find common denominators and how to reduce fractions on paper,
and you can memorize the steps. Or else we can practice seeing the pennies
or pieces of paper in your mind.” I took a deep breath. “If
Mrs. Barrows is going to be picky about your using props, maybe you can
learn to imagine the props and what you do with them. It’s probably
harder than just memorizing a paper and pencil procedure,” I added.
She nodded. “Yeah, but maybe that way I’ll understand what
I’m doing.” She smiled at me. “Besides, I wouldn’t
totally have to see it in my head. I could draw on the back of my test
paper.”
So Susie and I spent the rest of the session trying to develop her visualization
skills and apply them to math problems. It wasn’t easy, but she
was surprisingly good at it.
“
Mrs. Barrows is coming to talk to us tomorrow,” I reminded her.
She made a face. “Yeah, Mom said. Someone else is coming too,” she
said. “That old lady that used to teach math, but now she’s
the town meditator.”
I laughed. “Mediator, not meditator.”
Susie shrugged. “Whatever!”
“
We’ll be fine,” I said, trying to sound reassuring. “Just
don’t mention putting fractions on a diet.”
She glared at me.
“
Just kidding Susie, but honestly, you don’t need to worry about
your weight. You’re just right the way you are.”
She blushed even more brightly than before and looked up at the ceiling
again. It was time for both of us to go home.
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