Multiplying fractions
Five copies of three fourths, or half of a third of a banner. What does it mean to multiply with fractions, and why do the tops and bottoms multiply?
Where you are 4 inside
What this area is about
Multiplying by a whole number counts copies, and multiplying by a fraction takes that fraction of an amount: the word “of” means times. Both readings lead to one rule, for whole numbers a, c and nonzero b, d:
How these fit together
Copies. A fraction times a whole number takes several copies of one fraction, added up, and says which two whole numbers the product sits between.
A part of a part. A fraction times a fraction cuts a rectangle both ways to find a fraction of a fraction, which is where the rule comes from.
Using the rule. Multiplying mixed numbers splits a mixed number into its parts, or rewrites it as an improper fraction, before multiplying. Multiplying as scaling judges a product without computing it: for a positive number N, a factor less than 1 gives less than N, a factor greater than 1 gives more.
A path through it
A fraction times a whole number, then a fraction times a fraction. Mixed numbers and scaling both come after that, in either order.
This area leads straight to a fraction of a whole number and dividing fractions, and later to volume with fraction edge lengths and probabilities on tree diagrams.