Pivot Pia

"Oh. Then... one-fifth of the larger one. Because one-fifth of a huge pizza is like two-fifths of the first pizza. And two-fifths is bigger than one-quarter."

"Right."

"Is that right? Let me think."

"Take your time," Pia said. "The question changed when I changed the pizzas. So your answer should change, too. Most kids stick with their first answer. They don't like to change their minds. But you re-thought it. That's what I teach."

"That's the skill you teach?"

"Yep. It's a small skill. But you'll see it everywhere."

Pivot Pia had a small, tricky smile. Maya noticed it never went away.

Over the next few weeks, Maya learned Pia's first skill. It was simple. Notice when you're about to answer the wrong question.

This happened a lot more than Maya thought.

One day, Pia gave her a problem. "There are 20 marbles in a jar. 8 are red. The rest are blue. What's the ratio of red to blue?"

Maya typed 8 to 20.

Pia said, gently, "That's red to total. The question asked for red to BLUE. Read it again."

Maya read it again. Pia was right. Maya typed a new answer: 8 to 12.

"There you go," Pia said. "You answered a different question the first time. That happens all the time. The best way to catch it is to read the question. Then, before you answer, say it again in your head. Make sure your answer matches the question."

Maya thought about this.

"That seems easy."

"It is easy," Pia said. "And almost nobody does it."

"Why not?"

"Your brain is fast," Pia explained. "Super fast. It skims the words and sees a pattern. It sees 'red marbles, 8, total 20.' Then it starts solving. Vroom! But it solved before it finished reading. It missed the word BLUE. The trick is to catch your brain. Slow it down before it gives the wrong answer."

"That sounds hard, actually."

"It's hard at first," Pia said. "But it's the most useful trick for math. It catches tons of wrong answers. Just read the question twice. If you do that, you'll get about half as many problems wrong. I'm not making that up. I read it somewhere."

Pivot Pia had another trick, too. It was the one she was named for.

The trick was pivoting.

Some problems look really hard. But they have an easier question hiding inside. Pia's job was to find that easier question.

For example: "If you walk at 4 miles per hour, how many minutes does it take to walk a quarter mile?"

That sounded complicated. It had decimals and hours and minutes.

But a pivot could help. An easier question was hiding inside. "If it takes an hour to walk 4 miles, how long to walk one mile? Fifteen minutes. So how long to walk a quarter of a mile? A quarter of fifteen minutes. A little less than four minutes."

Same answer. Different question. The second one was way easier.

"That's pivoting," Pia explained. "You take the question and flip it around. You find an easier way to ask the same thing. Sometimes the flip is small. Sometimes it's huge. When you find a flip that makes the problem super easy? That's the best feeling in the world."

Maya started looking for pivots on her own.

She found them in weird places. A problem about sharing cookies: "How do 4 friends share 13 cookies?" That pivoted to a different question. "If each friend gets 3 cookies, how many are left over?" It was easier to think about, but it gave the same answer. Three cookies each, with one left over.

A problem about a recipe: "If a recipe for 6 people uses 2 cups of rice, how much for 9 people?" That pivoted, too. "How much rice per person? One-third of a cup. Okay, so for 9 people? Three cups."

A problem about percents: "What's 15% of 80?" That pivoted into two smaller questions. "What's 10% of 80? That's 8. What's 5% of 80? That's 4. So 15% is 8 plus 4. It's 12."

The pivots were always the same trick. Find an easier question with the same answer. Then answer THAT one.

"That's the secret," Pia said when Maya told her this. "Most hard problems have an easy twin. The trick is finding it. That's what real mathematicians do all day. People think they just crunch numbers like a calculator. Nope. They pivot. They turn hard questions into easy ones. Then the answer just pops out."

Pia paused.

"Also," she added. "Would you rather have ten dollars now, or one penny that doubles every day for thirty days?"

Maya thought about it.

She said, "The penny that doubles. I read about that one."

"Smart kid. How much does it become?"