Gather and Stretch

Gather sat down with a heavy flump and explained, because he believed in being honest with a kid.

"Here's my whole secret," he said. "Adding fractions is just counting — when the pieces are the same size. Two fifths plus one fifth is three fifths, because every piece is a fifth. I just count them. One, two, three. Done." He tipped his basket so Pip could see the matching pieces from an earlier problem, all the same neat size.

"But a third is a big piece, and a fourth is a smaller piece," he went on, holding one of each. "If I count 'one piece, two pieces,' I'm lying, because the pieces aren't equal. Two different sizes don't make 'two' of anything. It's like saying one apple plus one grape is two apples. It just isn't true."

Pip looked at the mismatched ribbons. "So you can never add a third and a fourth?"

"I can't," Gather said. "Not the way they are. Somebody has to make these pieces the same size first. And making things the same size, Pip" — he brightened, and turned toward the door — "well, that's not my craft. But I know exactly whose it is. STRETCH!"

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

Before he was a teacher, Stretch had been a glass-blower's apprentice, and his whole craft had been making things a perfect, even size — rolling hot glass along the marble slab until it was exactly the same width from end to end. Now he did the same thing with fractions.

"Mismatched pieces," he said, eyeing the third and the fourth. "My favorite. Watch." He laid both ribbons on his slab. "I can't change how much ribbon there is — that would be cheating. But I can cut each one into more, smaller, equal pieces, so they finally match each other."

He rolled the slab. "A third — if I slice it into four equal slivers each, it becomes four-twelfths. Same amount of ribbon, just shown in smaller pieces." He rolled again. "And the fourth — if I slice it into three equal slivers each, it becomes three-twelfths. Again, same amount, smaller pieces." He sat back, pleased. "Now look. Four-twelfths and three-twelfths. Both measured in twelfths. The pieces finally match. I haven't added a thing — I've just made them speak the same size."

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

"Twelfths, both of them," said Stretch. "That's all a common size is. A size both fractions can agree to be."

Gather peered over Stretch's shoulder and his whole face lit up.

"Oh, now I can work," he said. He scooped up the four-twelfths and the three-twelfths — all the same size now, every sliver a twelfth — and dropped them into his basket. He gave it his familiar little shake.

"Four pieces, then three more pieces. All twelfths. All matching." He held up the combined ribbon, beaming. "Seven-twelfths. One-third plus one-fourth is seven-twelfths."

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

"Never could," said Gather happily. "I bring things together. But I can only ever bring together things that are the same. Stretch makes them the same. He stretches; I gather. That's the whole dance." Stretch gave a small, proud nod from behind his slab. "Half the answer is making the pieces match," Gather said, "and half is bringing them in. We each only ever do one half."