Hint Hertha and Stretch Sage

She walked back to her table. She wrote slowly:

I need 2/3 of 3/4 of a cup.

She turned that over in her head. Two-thirds of three-quarters. It wasn't an addition problem. It wasn't a subtraction problem. It was a taking-part-of-a-part problem.

She had done these before. With whole numbers. Two-thirds of fifteen. You divide by three, then multiply by two. Fifteen divided by three is five. Five times two is ten. So two-thirds of fifteen was ten.

The same recipe must work for fractions.

Three-quarters divided by three is one-quarter. One-quarter times two is two-quarters. Two-quarters is one-half.

The baker uses half a cup of flour.

Maya checked it. Half a cup is two-thirds of three-quarters of a cup. She drew a quick picture. A glass with three-quarters of a cup. Mark off two-thirds of that height. Yes. The mark landed exactly at one-half.

She laughed quietly. She wrote 1/2 cup in the answer box. She put down the pencil.

The pencil felt much lighter than it had ten minutes ago.

Maya was packing up to go home when she heard a low rumble of a voice behind her.

"What's that on the worksheet?"

She turned around. Stretch Sage was standing in the aisle, holding his sketchbook under one arm. He had come over from the alcove. He must have been watching.

"It's a fraction problem," Maya said. "I figured it out."

"May I see?"

She handed him the worksheet.

Stretch read the problem. He read Maya's work. He nodded slowly. He turned his crooked smile on her.

"This is good," he said. "Hertha got you to see it as taking a part of a part. That's the real meaning. Most kids never get that meaning. They just memorize multiply across the top, multiply across the bottom. You actually saw it."

"Thank you."

"What would happen," Stretch said, tapping the page, "if the baker wanted to make seven-eighths of a batch instead of two-thirds?"

Maya thought.

"Then I'd need seven-eighths of three-quarters."

"And how would you do that?"

"Three-quarters divided by eight is three-thirty-seconds. Times seven is twenty-one thirty-seconds. I'd use twenty-one thirty-seconds of a cup."

"Whatever-thirty-seconds is small," Stretch said. "Is that right?"

"It's less than a cup. The original was three-quarters of a cup. Seven-eighths is almost a whole batch. So the flour should be almost three-quarters of a cup. Twenty-one thirty-seconds is..."

She did the math.

"Twenty-four thirty-seconds is three-quarters. Twenty-one is three less than twenty-four. So it's three thirty-seconds less than three-quarters. That's a little less than a teaspoon less than three-quarters. That sounds right."

"That sounds right to me too," Stretch said. He nodded approvingly.

He flipped the worksheet over.

"Now, what would happen," he said, "if the recipe called for 2/3 cup of flour instead of three-quarters, and the baker wanted 3/4 of a batch instead of two-thirds?"

Maya squinted. "That's... the same numbers. Just swapped."

"Is it the same answer?"

She paused. Three-quarters of two-thirds. Two-thirds of three-quarters. She had figured both of those.

Three-quarters of two-thirds. Two-thirds divided by four is two-twelfths. Times three is six-twelfths. Which is one-half.

The same answer.

Half a cup.

Maya's eyes went wide.

"It's the same?"

"It's the same."

"But that doesn't make sense. The recipe is different. The batch size is different."

"And yet," Stretch said.

Maya stared at her worksheet.

"The order doesn't matter," she said slowly. "Two-thirds of three-quarters is the same as three-quarters of two-thirds. The flour needed is the same."

"Why might that be?" Stretch asked.

Maya thought hard. The answer was hovering somewhere just out of reach.

"Because... we're taking the same total amount of stuff each time? Two-thirds and three-quarters multiplied together is the same as three-quarters and two-thirds multiplied together. Multiplication doesn't care about order. Two times three is the same as three times two. So when we multiply fractions to get a part of a part, it doesn't matter which is the part and which is the whole."

"That's a deep observation," Stretch said. "That's actually a property of multiplication. Mathematicians call it commutativity. It holds for whole numbers. It holds for fractions. It holds for almost every kind of number you'll meet. Multiplication doesn't care about order. Addition doesn't either. Subtraction does. Division does. You can tell a lot about an operation by asking whether it cares about order."

Maya looked at the worksheet again. The original problem felt very, very small now. It had grown into something much wider.

She had started the afternoon stuck on one fraction-baking problem.

She had ended it with a property of multiplication.