Addition and Subtraction of Unlike Fractions

The Story

The question was simple enough: 1/2 + 1/3.

Anaya added the tops: one and one is two. Then the bottoms: two and three is five. 2/5.

She wrote it down and then stopped, because something felt wrong.

"Draw all three," said Ms. Rao.

Anaya drew a bar and shaded half. She drew another and shaded a third. Then she drew a third bar and shaded two fifths.

The two fifths was visibly smaller than the half.

"I started with a half and added something, and got less than a half."

"Because you changed what the pieces WERE while you were counting them," said Ms. Rao. "The bottom number is not a quantity. It is the NAME of the piece. Halves, thirds, fifths. When you added 2 and 3 to get 5, you invented a new size of piece that nobody had."

She drew the two bars again, and cut both into sixths. "A half is three sixths. A third is two sixths. Now they are the same kind of piece — sixths — and I can count them."

Three sixths plus two sixths. Five sixths.

Anaya checked against her drawing. Five sixths was bigger than a half. It looked right.

"Why sixths?" asked Kabir.

"Because six is the smallest number that both 2 and 3 divide into. You have met it before under another name — the LCM."

Kabir sat up. "The bells."

"The same idea doing a different job. You need a size of piece that both halves and thirds can be cut into evenly, and the LCM is the smallest one that works."

Later Anaya tried 4 2/3 − 2 5/6 and got tangled trying to take five sixths from two thirds.

"Turn both into improper fractions first," said Ms. Rao. "14/3 and 17/6. Over sixths that is 28/6 and 17/6. Now it is just 28 minus 17."

Eleven sixths. One and five sixths. No borrowing needed anywhere.