Times

When Times was small, the surprise found her first, and it gnawed at her for a solid week before it let go.

She had been helping her uncle in his greenhouse, a glass-walled place where seedlings grew under sheets of coloured cloth that turned the light to syrup. He pointed her toward a wooden tray that was already only about a third full of sprouts. "Take half of that tray," he told her, "and carry it across to the bench."

Little Times scooped up half of the third-full tray, carried the small green handful across, and then stopped dead, holding it, genuinely confused. "This is hardly anything," she said. "It's nowhere near half a tray. But you said half. Half is meant to be a lot."

Her uncle knelt down to her level. "Half of what, though — that's the question you skipped. Half of a whole tray would be a generous heap. But you didn't take half of a whole. You took half of a third. You took a part of a part. And a part of a small thing can only ever be a smaller thing still."

Times stared down at the little clutch of sprouts in her hands. Half of a whole would have filled both arms. Half of a third barely filled one palm. The word multiply had meant more to her for as long as she could remember — it was practically the definition. Yet here it sat in her hands, quietly, undeniably, meaning less.

"Part of a part," she whispered, and the phrase made her feel dizzy and thrilled at the same time, the way a true idea sometimes does. That afternoon she went home and laid two paper strips across each other on the kitchen floor, just to watch the overlap shrink, and she found she never quite got tired of doing it.

Years later, Times arrived at the academy with her two glass strips wrapped carefully in a square of cloth and very little else to her name.

Slice, the old tortoise who ran the place, looked her over and set her a problem without preamble. "What is two-thirds of three-quarters?" he asked. "No memorised tricks. Show me the thing itself."

Times did not reach for a pencil. She knelt, unrolled her cloth, and laid the first strip across the white floor patch until it covered three-quarters of it. Then she turned the second strip sideways and laid it across the first, the other way, until it covered two-thirds of the patch in its own direction.

Where the two strips crossed, a block of doubled colour appeared, and the crossing lines had quietly diced the whole patch into a neat grid. "The strips have cut the patch into twelfths now," she said, counting under her breath — "four columns one way, three rows the other, twelve little squares in all. And the overlap, the part both strips agree on, covers six of those twelfths." She looked up. "Six-twelfths. Which is only another name for one-half."

Slice raised a single, ancient eyebrow. He had taught multiplication by the rule — top times top, bottom times bottom — for the better part of sixty years, and the rule was perfectly good. But he had never once watched a student lay the answer out in light before, and see it arrive on its own.

"The rule still holds, of course," Times added quickly, as if worried he might think she scorned it. "Two times three is six, four times three is twelve, six-twelfths. I'm not throwing it away. I simply prefer to see the thing happen. The overlap doesn't lie, and it doesn't ask me to take anything on faith."

"You may stay," Slice said.

In her sunny corner, some weeks on, Times met a grumpy beetle named Cricket, who marched in and slapped a worksheet down on the floor as though it had personally insulted him.

"Multiplying fractions is broken," Cricket announced, "and I can prove it. I did one-half times one-fourth and the answer came out as one-eighth. One-eighth is smaller than both of them. Multiplication is supposed to make things bigger — everyone knows that — so the whole method must be wrong."

"The method is fine," Times said gently. "It's the word that's been tricking you, and honestly it's an easy trap to fall into. Watch." She laid down a strip covering one-fourth of the patch, then crossed it with a strip covering one-half the other way, and the overlap settled into a tiny, modest rectangle. "One-half of one-fourth. You aren't stacking two things on top of each other to build something taller. You're reaching into a piece and taking a part of it — and a part is always smaller than the whole you reached into."

Cricket squinted at the little overlap for a long moment. "So 'times,' here, really means 'of.'"

"That is the entire secret," Times said. "When you multiply by a fraction, times means of. Half times a fourth is just half of a fourth — and half of something already small is bound to be smaller still."

Then she did a thing that made Cricket's antennae twitch. She picked up the two strips and swapped them — laid the fourth where the half had been, and the half where the fourth had been, crossing them again. The overlapping rectangle was exactly, precisely the same size as before.

"Half of a fourth, or a fourth of a half," Times said. "Two different sentences, two different orders — and the very same answer. One-eighth, whichever way round you say it. The order you cross the strips in never changes the overlap they make."

Cricket stared, his complaint quietly evaporating. "Two whole different sentences," he said slowly, "and one little purple square."

"One little purple square," Times agreed, thoroughly pleased with him.