Gather

When Gather was a kid, he learned this the hard way.

He grew up helping at his family's button stall in the market. People brought scraps of cloth and his job was to add up how much they had. One afternoon a tailor dumped a pile on the counter: some pieces were cut into fifths, some into quarters, all jumbled together.

"How much cloth altogether?" the tailor asked.

Little Gather started counting pieces. One, two, three, four, five, six. "Six pieces!" he said proudly.

The tailor frowned. "Six pieces of what, though? These three are fifths. These three are quarters. They're not the same size. You can't just say six."

Gather looked at the pile. The fifths were skinny. The quarters were wider. He had counted them like they were all the same — but they weren't. Six pieces, sure. But six pieces of nothing in particular.

His ears went hot. He had been adding things up his whole life, and he'd been doing it wrong — or at least, half-right. You could count pieces. But only if the pieces were the same size.

"How do I fix it?" he asked.

The tailor smiled. "You make them match first. Then you count."

Gather came to the academy years later, basket on his back, determined to teach the one thing that had tripped him up.

Slice, the old tortoise who ran the place, set a test on the table: two-sixths of a chocolate bar and three-sixths of a chocolate bar. "Add them," he said.

Gather dropped both into his basket and gave it a shake. "Five-sixths," he said at once. "Both in sixths. Pieces match. Two plus three is five. Five-sixths."

Then Slice slid over a trickier pair: one-half of a bar, and one-third of a bar. "And these?"

Gather did not shake his basket. He stopped. He set the two pieces side by side and looked at them carefully. The half was big. The third was smaller. They didn't match.

"I can't add these yet," he said. "Not honestly. The pieces are different sizes. If I just say 'two pieces,' I'd be lying about how much there is." He looked up. "First I have to make them the same. I'd cut both into sixths. The half becomes three-sixths. The third becomes two-sixths. Now they match. Now I can gather them. Five-sixths."

Slice nodded slowly. "Most people shake the basket first and think later," he said. "You think first. You may stay."

In his corner of the academy, surrounded by baskets, Gather met a frustrated otter named Wren.

"I hate adding fractions," Wren announced, dropping into a chair. "I did one-fourth plus one-half and I got two-sixths. My teacher said it's wrong. But I added the tops AND the bottoms, which is what you do with regular numbers!"

"Ah," Gather said kindly. "That's the trap. Let's gather them for real." He took out a fourth-piece and a half-piece and set them in the basket. "Look. These don't match. The half is bigger. If we just smush the numbers together, the basket doesn't know what size the pieces are."

He took the half-piece and showed Wren how it could be cut into two equal fourths. "Watch. One half is the same as two-fourths. Now both pieces are fourths. They match." He dropped them in. "One-fourth plus two-fourths. Three pieces, all fourths. Three-fourths."

Wren stared. "So the bottom number stays the same once they match?"

"The bottom number tells you the size of the pieces," Gather said. "Once the sizes match, you stop touching the bottom. You just gather the tops. Three-fourths."

Wren laughed, a little stunned. "I was adding the sizes together. That's like saying two baskets plus two baskets equals four... basket-baskets."

"Exactly," Gather grinned. "Keep the size. Gather the count."