Gather and Stretch

He sat down with a heavy flump, because he believed a child deserved the honest version.

"Here's the whole secret nobody tells you," he said. "Adding fractions is just counting — but only when every piece is the same size. Two fifths plus one fifth is three fifths because each piece is a fifth; I line them up and count. One, two, three. There's no cleverness in it at all."

He held up the two mismatched ribbons, one clearly larger than the other. "But a third is a big piece and a fourth is a smaller piece. If I count 'one, two' across these, I'm lying — because two things of different sizes don't make 'two' of anything. It's the apple-and-grape trick. One apple plus one grape isn't two apples; it's just two things you can't honestly total."

Pip frowned. "So a third and a fourth can never be added?"

"I can't add them — not as they stand," Gather said. "Someone has to make the pieces the same size first. And making things the same size, I'm afraid, is nowhere in my repertoire." Then he brightened, and turned toward the door with the relief of a man who knows exactly whom to call. "But I know precisely whose craft it is. STRETCH!"

Stretch arrived carrying a long, smooth marble slab under one arm, as he always did.

Before he taught, he had apprenticed to a glass-blower, and his entire trade had been evenness — rolling molten glass along the marble until it was exactly the same width end to end. He did the same, now, with fractions.

"Mismatched pieces," he said, surveying the third and the fourth with quiet appetite. "My favorite kind. Watch — and notice what I don't do. I don't change how much ribbon there is; that would be cheating. I only cut each piece into more, smaller, equal slivers, until the two finally agree."

He laid them on the slab and rolled. "Slice the third into four, and it becomes four-twelfths — the identical amount of ribbon, merely shown in finer pieces. Slice the fourth into three, and it becomes three-twelfths — again, the same amount, finer pieces." He sat back. "Now look at them. Four-twelfths and three-twelfths. Both measured in twelfths. I've added nothing whatsoever. I've only made them speak the same size."

Pip stared. "They're the same kind of piece now."

"Twelfths, both," said Stretch. "That is all a common size ever is — a size both fractions can be persuaded to agree on."

Gather leaned over the slab, and his whole face lit like a lamp.

"Oh — now I can work." He scooped up the four-twelfths and the three-twelfths, every sliver identical, and tipped them into his basket with obvious relief. The old familiar shake. "Four pieces, then three more, all of them twelfths, all of them matching. Seven-twelfths. One-third plus one-fourth is seven-twelfths."

Pip's mouth fell open, and then, slowly, understanding arrived. "You needed each other," she said. "Stretch couldn't add them up — that's your work. But you couldn't add them at all until Stretch made them match. Neither of you could finish alone."

"Never could, never will," Gather said, content. "I bring things together — but only ever things that are already the same. Stretch makes them the same; I gather them in. He stretches, I gather; that's the whole dance. Half the answer is making the pieces match, and half is bringing them home, and neither half is the whole."