Fractions: Multiplication and Division

The operations that are easier than they look

Multiplying fractions is the least demanding thing in the whole topic: multiply the tops, multiply the bottoms, simplify. There is no common denominator to find and nothing to line up.

Division adds one move. Flip the second fraction, then multiply. After that it is the same operation you were already doing.

Which is why this sheet is often set too late. Children who are still struggling with adding fractions can usually multiply them fine, and doing so successfully is a useful thing to have happen partway through a difficult topic.

The thing worth teaching alongside it

Multiplication makes things smaller here, and that contradicts everything a child has learned about multiplication.

Half of a half is a quarter. Multiplying by 1/2 halves the number. Before anybody calculates anything, ask a class whether 3/4 × 2/3 will come out bigger or smaller than 3/4, and a good proportion of the room will say bigger. Getting that prediction wrong out loud, before the arithmetic, is worth more than the arithmetic.

Division does the opposite. 3/4 ÷ 1/2 is 3/2, which is larger than either. "How many halves fit into three quarters?" is the question that makes it make sense.

Setting it up

• Operation — multiplication only, division only, or a mixed sheet. Mixed is the harder one, because the first decision on every problem is which rule applies.
• Improper fractions — off keeps every fraction under 1. On allows tops bigger than bottoms, which is where multiplication stops always shrinking the number and the class's new intuition gets tested.
• Maximum denominator, 3 to 20.
• Problems, up to 30, plus the answer key, a title and Name and Date fields.

Every answer is fully simplified

Results are reduced before they are printed, so 8/12 appears as 2/3 on the key.

That is worth flagging to a class, because a pupil who has correctly worked out 8/12 and stopped there has the right value and the wrong form. Deciding in advance whether you are marking the value or the form saves an argument.

Cancelling before multiplying

Worth teaching from this sheet, since it is where it pays. In 4/9 × 3/8, the 4 and 8 share a factor and so do the 3 and 9, and cancelling first leaves 1/3 × 1/2 — small numbers, done in your head.

Multiply first and you land on 12/72, and now you have to spot that 12 is the highest common factor. Same answer. Considerably more work, and several more places to slip.

Print at 100% on A4 or US Letter.