Fractions as numbers — Mathematics, 11–13 years
See a fraction as a number with a place on the number line, not only as a piece of a cake.
A fraction has a place
A fraction tells how many equal steps we have taken when one whole is split into equal parts. In 3/4, the 4 says the size of each step and the 3 says how many steps we have, so 3/4 belongs between 0 and 1 on the number line.
Why we need fractions
Whole numbers are not enough for measuring lengths, sharing food or timing a race. People needed a way to record amounts such as half a metre or three quarters of an hour, without pretending that the amount was a whole number.
Adding unlike fractions
To add 1/3 and 1/4, first use equal-sized parts. Twelve works because 12 is divisible by both 3 and 4: 1/3 = 4/12 and 1/4 = 3/12. Now add the pieces: 4/12 + 3/12 = 7/12.
Do not add the bottoms
A common first attempt is 1/3 + 1/4 = 2/7, because adding the top and bottom looks neat and symmetrical. But the bottoms describe different-sized pieces, so they must first become the same; only then can the top numbers count pieces together.
Fractions outside school
Recipes use fractions to scale ingredients, and rulers mark fractions of a centimetre or inch. Music also uses fractions: a half note lasts twice as long as a quarter note, so the fraction describes a part of a measured whole.
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