Pivot the Rule-Switcher

The driver would call out a town. "Northgate!" Or "Easton!" Sometimes "Southport!" Once in a while, "The capital!" Or even, "I don't know. My lady just said to follow this road." If that happened, Pivot walked. He went to the lady in the coach. He asked her himself.

Then Pivot would tell them. "Take the north fork." Or "Take the east fork." Or "Take the south fork." If they needed the capital, it was trickier. "You want the central road," he'd say. "It's the middle one. Between the north and east forks. Look for the lion-and-star signs. That's the capital's crest."

The coach would proceed. The next coach would arrive.

Pivot did this work every day. Eleven hours a day. Six days a week. For twelve long years. He counted every coach. More than two hundred thousand! He never sent one the wrong way. The road office praised him often. He even won awards. Twice he got the "Reliable Service" award. It was a quiet award. But it meant a lot.

Pivot learned something important. It made him a great teacher later on. His job had a pattern. It was like a game with rules. The rules changed. They changed based on what the coachman said.

If a coach said "Northgate," Pivot's rule was "take the north fork." If a coach said "Easton," Pivot's rule was "take the east fork." If a coach said "Southport," Pivot's rule was "take the south fork." If a coach said "the capital," Pivot's rule was "take the central road; follow the lion-and-star signs."

Different words from the driver. Different rules from Pivot. The job was always the same. It was about directing coaches. But the answer changed. It changed based on where they wanted to go.

Pivot didn't know the fancy math word for it. Not yet. But what he did was called a piecewise function.

Pivot was thirty-one. The road office sent him help. A young man named Cobble arrived. Cobble was new. He just finished school. He knew a lot about math. On his second day, Pivot was busy. He told a driver to take the south fork. Cobble spoke up. "Sir," he said, "that's a piecewise function."

Pivot said: "A what?"

Cobble explained. "In math, it's a special kind of function. It has different rules. The rules change for different numbers. Like, if 'x' is small, you use one rule. If 'x' is big, you use another. It's still the same big function. But it has different parts. You do this every day, sir. You hear where the coach wants to go. That's the 'input.' Then you pick the right rule. You tell them which road to take."

Pivot thought about it. He thought for a few days. Then he told Cobble, "That's a good word. Piecewise. I'll keep that in mind."

Pivot turned thirty-three. He wrote a letter. It went to the FunctionForge academy. His letter said: "I've been a junction-master for twelve years. I've sent two hundred thousand coaches the right way. I just learned my job is a piecewise function. I want to teach this. I think the road office can find someone new for Threefork. And honestly, I'm tired of standing all day."

The head of the academy invited him. "Come teach!" they said. Pivot said yes. He left his job at the junction. The road office sent a new person. Later, Cobble took over Threefork.

Pivot has been teaching piecewise functions at the academy for eleven years.

In his classroom, Pivot always starts the same way. On the first day, he stands up front. "Imagine I'm at a road fork," he says. "Three coaches drive up. The first one wants to go to Northgate. What do I tell it?"

The children — always — say take the north fork.

Pivot says: "The second coach is bound for Easton."

The children say take the east fork.

Pivot says: "The third coach is bound for Southport."

The children say take the south fork.