Stretch

Forge's workshop made the best vials in three provinces. The secret was the stretching.

Stretch — whose given name was Hadrian, though everyone called him Stretch by the time he was sixteen — had been apprenticed to Forge at twelve. His first year was spent sweeping the workshop. His second year was spent feeding the furnace. His third year, finally, was spent at the marble slab.

The marble slab was the centerpiece of the workshop. It was a long flat piece of polished marble, about six feet long and two feet wide, set on a sturdy iron frame. The slab was used for stretching molten glass. A glass-blower would take a small ball of molten glass from the furnace, attach it to a long iron rod, and roll it back and forth across the marble slab — patiently, evenly — until every part of the glass had the same diameter.

This was the operation Stretch eventually understood as finding a common denominator.

The molten glass, when it left the furnace, was not uniform. It was thick in some places and thin in others. It was, in fraction terms, like fractions with different denominators — not directly comparable until brought to a common scale. To make the glass into a usable tube, you had to roll it. The rolling forced every part of the glass to the same diameter. Once every part was the same diameter, the tube was uniform. You could cut it into vials. Each vial was the same width as every other.

Stretch learned this, slowly, over years. By the time he was eighteen he could roll a six-foot tube of one-thumb diameter that was uniform within a hair's-width across its entire length. He was, by workshop standards, very good.

He stayed at the workshop until he was twenty-six. Then his father died, and Stretch inherited his father's small farm, and he left the workshop to manage the farm. He stayed on the farm for two years. He hated it. (He had no aptitude for sheep and the farm had been a sheep-farm.) He sold the farm. He returned to Anneal.

But Forge had retired. The workshop had a new master who did not need an old apprentice. Stretch, who was now twenty-eight and unsure what to do with himself, sat on the harbour wall and thought.

What he thought, eventually, was that the rolling-the-glass-to-uniform-diameter operation was an arithmetic operation. He had spent his apprenticeship doing it physically. But the same operation — bring all the parts to a common scale — was the same operation needed to add or compare fractions with different denominators.

The connection was clear, once he saw it.

He went to the FractionForge academy. He told the academy master what he had been thinking. The academy master listened patiently. The academy master said: "You are correct. Bringing fractions to a common denominator is the same operation as bringing the molten glass to a uniform diameter. We need teachers who can explain this physically. Would you teach?"

Stretch said yes.

That was twelve years ago. He has been teaching common denominators ever since.

In his classroom, he begins every first-day lesson the same way. He brings, from Anneal (he goes back twice a year to visit the new workshop's owner and the new owner has been kind enough to supply him with a teaching prop), a small piece of stretched glass tube. He places it on the desk. He holds up a small caliper. He measures the tube at one end, in the middle, and at the other end. He says: "This tube is one-thumb diameter at every point. To make it that way, the glass-blower rolled the molten glass on a marble slab. The rolling brought every part to a common diameter."

The children — always — examine the tube.

Then he writes on the board: 1/3 + 1/4. He says: "These two fractions have different denominators. Three and four. They cannot be added directly. We need to bring them to a common denominator. The smallest common denominator is twelve. 1/3 scaled to twelfths is 4/12. 1/4 scaled to twelfths is 3/12. Now we can add them: 4/12 + 3/12 = 7/12. They have been stretched to a common scale."