Rank

When Rank was small, comparing fractions made his head hurt.

His family ran the village relay races, and every race ended in arguments: whose runner had covered more of the track? One had run three-fifths of the way. Another had run two-thirds. The kids would shout numbers at each other and nobody could tell who'd won.

Little Rank tried to settle it by converting everything — cutting the track into tiny matching pieces, fifteenths, counting them all up. It worked, but it was slow, and by the time he finished, everyone had wandered off.

His grandfather, who judged the races, took him aside. "You don't always need the whole calculation," he said. "Look. Is three-fifths more than half the track, or less?"

Rank thought. Half of five-fifths would be two-and-a-half fifths. Three-fifths was past that. "More than half."

"And two-thirds?"

Half of three would be one-and-a-half thirds. Two-thirds was past that, too. "Also more than half." Rank frowned. "That doesn't settle it."

"No," his grandfather agreed, "but now you only have two close ones to think hard about, instead of a whole jumble. You've already thrown out the easy cases. Half did most of the work for you."

Rank stared at the track. Half. The middle marker. Suddenly comparing didn't feel like a wall of math anymore. It felt like sorting.

Rank came to the academy with a deck of fraction cards he could shuffle and sort faster than anyone.

Slice, the old tortoise, fanned out four cards: *2/5, 4/4, 1/6, 7/8*. "Order them," he said. "Out loud. Tell me how."

Rank didn't reach for a pencil. "Four-fourths is a whole one — that's the biggest, it's a complete thing. One-sixth is a tiny sliver, way under half — smallest. Two-fifths is under half too, but bigger than that little sliver. And seven-eighths is almost a whole, just under four-fourths." He laid them out left to right. "One-sixth, two-fifths, seven-eighths, four-fourths."

"And if two are both close to half?" Slice asked.

"Then I think harder," Rank said. "Same top number? The one with the smaller bottom is bigger — fewer, fatter pieces. Same bottom number? The one with the bigger top is bigger — more pieces. And if I'm really stuck, I'll do it Stretch's way and give them a common bottom. But most of the time, reasoning gets me there first."

Slice looked at the tidy row of cards. He had taught comparison as always convert first for sixty years. He had never watched a student sort by thinking out loud. "You may stay," he said.

In his corner, Rank met a worried vole named Wisp clutching two cards.

"I have to figure out which is bigger," Wisp said, "three-fourths or two-thirds, and I always get them mixed up."

"Middle marker first," Rank said. "Are they over half or under half?"

Wisp thought. "Three-fourths... half of four is two, three is past two, so over half. Two-thirds... half of three is one-and-a-half, two is past that, so also over half." Wisp sagged. "They're both over half. Now I'm stuck again."

"Good — you've earned the harder look," Rank said warmly. "Now think about how far past a whole they each fall short. Three-fourths is missing just one-fourth to be a whole. Two-thirds is missing one-third to be a whole." He held up two fingers a little apart. "Which gap is bigger — a fourth or a third?"

Wisp pictured the pieces. "A third is bigger than a fourth. So two-thirds is missing a bigger chunk."

"So which one is closer to a whole?"

"Three-fourths! It's only missing a little fourth. So three-fourths is bigger!" Wisp nearly dropped the cards. "I figured it out by thinking about what was missing."

"That's the magpie's trick," Rank grinned. "Sometimes the easiest way to see which is bigger is to see which one has less missing."