Keel the Constant-Keeper

They rarely find it by staring, so Keel gives them the boatwright's move: divide. Coins by baskets. Ten by two is five. Twenty by four is five. Thirty by six is five. "There," he says. "The steady five. Five coins for every basket, standing unmoving behind the whole shifting table. That is the constant of proportionality — the fixed multiplier that ties the two quantities together. Scholars write it k. Here, k is five."

The Advanced point, he insists, is what the constant buys you. "Once you own the k, you own the whole table and every row that isn't on it. Any number of baskets, times five, is the coins. Ninety baskets? Four hundred fifty coins — and you never had to extend the table. You do not memorise the pairs. You keep the one number and generate the pairs at will." He writes it as the scholars do: coins = 5 × baskets, or in the bare form, y = k x. "The little letter is just the steady number, dressed for travel."

He is careful to separate the constant from its cousins, because Advanced students confuse them. "The ratio of a single pair — ten coins to two baskets — is a comparison of that pair. The unit rate is that ratio reduced to per-one — five coins per basket. And the constant of proportionality is the claim that this same per-one number holds across every pair in the relationship. In a truly proportional relationship, the unit rate and the constant are the same number — five — and that is exactly why finding it once is enough."

He shows them the tell, too. "A relationship is proportional only if dividing output by input gives the same answer every time, and the line, drawn, passes clean through the corner where both are zero. Zero baskets, zero coins. If the line starts above the corner — if there is a fee before the first basket — then there is no single constant, and y = k x is the wrong tool. Check the corner. Check that the quotient repeats. Then you may trust the k."

A girl named Isla, quick but anxious, stayed after one afternoon, looking at the chalked five behind all those shifting numbers. Her trouble was not the division. It was that she had been trying to hold the whole table in her head — every basket, every coin — and it kept slipping.

Keel sat down beside her. "You are carrying the ship plank by plank," he said. "Carry the keel instead. Find the one number that stays, and let the rest hang off it. You do not need eleven facts. You need one, and the rule that turns it into the other ten." Isla divided one more row — thirty-five coins, seven baskets, five again — and felt the whole table collapse gratefully into a single number she could actually hold.