Multiply and Divide Fractions
Lessons in this chapter
About Multiply and Divide Fractions
Fractions and their Multiplication and Division completes fraction arithmetic, and both operations behave in ways that contradict whole-number intuition.
Multiply Fractions is mechanically the easiest fraction operation: multiply the numerators, multiply the denominators, done. No common denominator is needed, which surprises kids who have just worked hard for one. The idea underneath is taking a part of a part, so half of a third is a sixth.
Divide Fractions introduces the invert-and-multiply rule, and it is worth knowing why rather than just that. Dividing by a half asks how many halves fit into the amount, and since two halves fit into each whole, the answer doubles. That is why dividing by a fraction less than one makes the answer bigger.
Solve Applied Problems on Fraction Operations puts both into context, where the difficulty is choosing the operation. Three quarters of a recipe is multiplication; splitting three quarters of a pizza between two is division, and the phrasing is all that distinguishes them.
All three lessons open as a 5th grade math worksheet in the page. Multiply Fractions, Divide Fractions and Solve Applied Problems on Fraction Operations each set math problems for 5th graders to work one at a time, scored as you answer them. They are free to try, with no login needed for your first 3.
Frequently asked questions
Multiply Fractions is mechanically the easiest fraction operation, with no common denominator needed, and the idea underneath is taking a part of a part. Divide Fractions introduces invert and multiply, and it is worth knowing why: dividing by a half asks how many halves fit, which is why the answer grows. Solve Applied Problems on Fraction Operations then puts both into context, where choosing the operation is the difficulty. The chapter sits in Number and Operations with Fractions (5.NF), though the Common Core keeps 5th grade division to unit fractions and whole numbers and leaves the general case to its 6th grade Number System domain (6.NS).
Multiply the numerators and multiply the denominators. No common denominator is needed, which makes it easier than adding.
Because you are taking a part of a part. Half of a third is a sixth, smaller than both. It only feels wrong because whole-number multiplication always makes things bigger.
Because dividing by a fraction asks how many of it fit into the amount. Dividing by a half doubles the answer because two halves fit into each whole, and flipping and multiplying does exactly that.
Because the thing you are dividing by is smaller than one, so more of them fit. 3 ÷ one half is 6, since there are six halves in three.
Taking a fraction of something is multiplication. Splitting something into fractional parts, or asking how many fit, is division. The phrasing is the only signal.