Times

When Times was small, she got the surprise herself, and it bothered her for a week.

She was helping her uncle in his greenhouse, where seedlings grew under colored cloth. "Take half of that tray," he said, pointing to a tray that was already only a third full of sprouts.

Little Times scooped up half of the third-full tray and carried it over. Then she stopped, holding the small handful, confused.

"This is way less than half a tray," she said. "But you said half. Half is a lot."

Her uncle knelt down. "Half of what, though? Half of a whole tray would be big. But you took half of a third. You took part of a part. A part of a small thing is a smaller thing still."

Times stared at the little handful of sprouts. Half of a whole would have been huge. But half of a third was tiny. The word "multiply" had always meant more to her. Now here it was, quietly meaning less.

"Part of a part," she whispered. It made her dizzy and delighted at the same time.

That afternoon she went home and laid two strips of paper across each other, just to watch the overlap shrink. She never got tired of it.

Times arrived at the academy carrying her two glass strips wrapped in a cloth.

Slice, the old tortoise, set her a problem. "What is two-thirds of three-quarters?" he asked. "No tricks. Show me."

Times laid down her first strip to cover three-quarters of the white floor patch. Then she turned her second strip sideways and laid it across to cover two-thirds of the patch the other way.

Where they overlapped, a block of doubled color appeared. She counted the little squares the two strips had cut the patch into. "The whole patch is cut into twelfths now — four columns, three rows. And the overlap covers six of those twelfths." She looked up. "Six-twelfths. Which is the same as one-half."

Slice raised an eyebrow. He had taught multiplication with rules — top times top, bottom times bottom — for sixty years. He had never seen a student lay the answer out in light before.

"The rule still works," Times added quickly. "Two times three is six, four times three is twelve, six-twelfths. But I like to see it. The overlap doesn't lie."

"You may stay," Slice said.

In her sunny corner, Times met a grumpy beetle named Cricket who slammed a worksheet down.

"Multiplying fractions is broken," Cricket announced. "I did one-half times one-fourth and got one-eighth. One-eighth is smaller than both of them. Multiplication is supposed to make things bigger. The math is wrong."

"The math is fine," Times said gently. "It's the word that's tricking you. Watch." She laid down a strip covering one-fourth of the patch, then crossed it with a strip covering one-half the other way. The overlap was a tiny rectangle. "One-half of one-fourth. You're not stacking two things up. You're taking a part of a part. A part is always smaller than the whole thing you took it from."

Cricket squinted at the little overlap. "So 'times' here means 'of.'"

"That's the secret," Times said. "When you multiply by a fraction, 'times' means 'of.' Half times a fourth is half of a fourth. And half of something small is smaller still."

Then she did something that made Cricket's antennae twitch. She picked up the strips and swapped them — laid the fourth where the half had been, and the half where the fourth had been. The overlap rectangle was exactly the same size.

"Half of a fourth, or a fourth of a half," Times said. "Same answer. One-eighth, either way. The order doesn't change the overlap."

Cricket stared. "Two different sentences. One little purple square."

"One little purple square," Times agreed, delighted.