Prise

One afternoon a young dockhand named Vell watched Prise clear a heap of mixed cargo and asked how she worked so much faster than anyone.

"I don't work faster. I look first," said Prise. "See these two loads? This one's six barrels. That's nine barrels. But look —" she tapped the pallets — "both are stacked on the same size sledge, three barrels to a sledge. So really it's two sledges here and three sledges there, and every sledge is the same. Pull the shared sledge out, and the whole pile makes sense." Vell frowned, then brightened. "So instead of six and nine as two separate messes, it's three, shared — and then two of them and three of them." "The shared three, pulled to the front," said Prise. That night she scratched inside her tool-chest lid: FIND WHAT THEY SHARE — the greatest part they hold in common — and pull it to the front. Six and nine both carry a three, so it's three of (two and three).

After that Prise saw shared parts in everything — two recipes both beginning with the same broth, three songs built on the same three chords, a row of houses all raised on one kind of stone. Underneath what looked separate there was so often one thing held in common, waiting to be pulled to the front.

A year later the EquationQuest academy, which wanted someone who could teach children to tidy an expression back down after it had been spread out, heard of a dockworker who "found what a heap shared and pulled it out in one pull." The master wrote to her. Prise, ready to trade river damp for a warm hall, accepted, and brought her worn pry-bar.

The board, Prise found, was a dock with the crates rubbed out and numbers chalked in — and numbers, like crates, so often shared a part. She wrote: 6x + 9.

"Two loads," she said. "6x here, 9 there. They look separate. Look at what they share." She circled the numbers. "6 is 3 × 2. 9 is 3 × 3. Both carry a 3 — and it's the biggest number they both carry, which is the one worth pulling; that's called the greatest common factor. The three is the shared strap." She set the chalk like a pry-bar against the expression. "So I prise the 3 out to the front and write what's left inside brackets: 6x + 9 becomes 3(2x + 3)."

She checked it the way she checked a crate had truly opened: "3 × 2x = 6x. 3 × 3 = 9. Yes — the same load, only tidier. And here's the part that matters: the value hasn't changed at all. 3(2x + 3) and 6x + 9 are the very same amount, wearing two coats — one spread out, one gathered up. Factoring doesn't make an expression smaller or bigger. It makes the hidden shared part visible, out front where you can use it."

"That's just the roller run backwards," said Vell's cousin, who had learned from Spread. "Spread rolls the three across. You prise it back out." "Exactly backwards," said Prise, pleased. "Spread spreads; I gather. Same seam, opposite pull. If you can undo what Spread does and land back where you began, you've factored true."