Proper, Improper and Mixed Fractions

The Story

Two pizzas arrived, each cut into eight slices. Between them the class ate eleven slices.

Ms. Rao asked how much pizza had been eaten. Anaya said eleven eighths. Kabir said one and three eighths.

"Eleven eighths sounds like more," said Kabir. "Eleven is bigger than one."

"Draw it," said Ms. Rao.

Anaya drew two circles, each cut into eight. She shaded eight in the first — the whole thing — then three in the second. She counted the shaded pieces: eleven.

Kabir looked at the same picture. One whole circle, and three eighths of another. One and three eighths.

"It's the same drawing," he said.

"It is the same pizza," said Ms. Rao. "Eleven eighths counts the SLICES. One and three eighths counts the WHOLE PIZZAS and then the leftover slices. Neither is more. They are two ways of reading one picture."

She showed them how to move between the two. "To go from eleven eighths to a mixed number, ask how many whole pizzas eleven slices makes. Divide: eleven divided by eight is one, remainder three. The one is the whole pizza. The three is the slices left. The eight never changes, because the slices never got any bigger."

Kabir tried the other direction with 5 2/3. "Five wholes, each cut into three, is fifteen thirds. Plus the two extra thirds is seventeen. So 17/3."

"Multiply the whole by the denominator, add the numerator, keep the denominator," said Ms. Rao. "You have just worked out why that rule says what it does."

Anaya was still thinking about the argument. "So 4/4 is a whole pizza. Is that proper or improper?"

"Improper. Proper means the top is SMALLER than the bottom — less than one whole. Four quarters is exactly one, so it does not qualify."

"Improper sounds like it's badly behaved."

"It is just the old name for top-heavy," said Ms. Rao. "There is nothing wrong with it. Sometimes it is by far the easier form to calculate with."