Improper Fractions: A Mathematical Trauma
Nearing the end of Miquon Blue today, my youngest daughter encountered fractions greater than one. She collapsed on the floor of my bedroom in tears.
The worksheet started innocently enough:
Those Frustrating Fractions
I wonder if that was a typo, because the next several questions followed a pattern that begged for a beginning:
“Mom, I’m stuck!” she complained.
“Hmm. Those do look tricky. Could you do this one if it was 1/8?” I pointed at the 2/8 question, rewording it with our familiar translation. “Remember that the multiplication symbol means of. What would 1/8 of 8 be?”
She still looked puzzled. When she decides something is too difficult, she starts to build a mental brick wall, and it can take awhile to break it back down.
Back to the Basics
I pointed to the picture at the top of the page and reminded her how to make a fraction.
“If we took this bar and cut it into eight pieces, how big would one of the pieces be? What is 1/8 of 8?”
She thought for a moment. A few bricks tumbled. “One.”
“Good! And if 1/8 of 8 is one, then what would two of the pieces be? What is 2/8 of 8?”
The wall came down, letting in the light. She bowed her head over the worksheet, shielding it with her arm so I couldn’t watch her fill in the boxes.
It’s Too Much!
She worked happily until…
“I can’t do it!”
“What do you mean?”
She pointed at the fraction 9/8. “That’s impossible!”
“Well, if 1/8 is one, then how much would it be if you had nine of the pieces? You already figured out that 8/8 of 8 is eight, so lets put one more of the little pieces here.” I sketched a small square at the end of the rectangle labeled 8. “If we have one more piece, that would make 9/8, right?”
“It’s more than eight,” she said.
“That’s right. It’s more than eight. How much is it?”
“I can’t do more than eight. It’s too much!”
It was too much numerically, since the fractions were originally made (in her mind) by cutting a single bar into pieces. How could there be more pieces than the bar she had started with? It was too much emotionally, as well. She threw herself onto a pile of blankets and hid her face in the folds of cloth.
What’s in a Name?
“It’s okay to have fractions that are bigger than a whole thing, you know.” I tried my most encouraging voice. “If we had grilled cheese sandwiches for lunch, you could have three halves of a sandwich, couldn’t you?”
Whimper. Sniffle. No reply.
“You don’t like these fraction, do you? Somebody else who didn’t like this sort of fractions gave them their name. He called them improper fractions. Do you know what improper means?”
She looked up. “Not right?”
“You’ve got it! Fractions bigger than one just don’t seem right, do they? So he called them improper fractions, and that’s what we still call them today.”
A weak smile.
I had mercy on her — after all, she is only in second grade. I told her she could skip all the fractions that were greater than one whole thing. She will have to learn how to work with improper fractions in a few years, but for now, I gave her my permission to ignore them. She happily scribbled over the questions and put her workbook away.
Still, she’s a bright girl, and somewhat stubborn. She is rarely content to leave a topic unconquered. The idea of improper fractions must have continued to simmer in her brain, because when I went back later to check the worksheet, I found that she had erased her scribbles and answered every question.
And they were all correct.