Lessons · Teaching · adding and subtracting fractions

You cannot add pieces of different sizes

Adding fractions needs the same denominator, because a third and a quarter are different-sized pieces and there is no name for their total until both are cut to a common size. Find a common denominator, rewrite both, add the numerators, and leave the denominator alone.

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What it is for

Adding the denominators is the single most common fraction error in the world, and it has an answer that ends it: one half plus one half would be two quarters, which is one half, so half a cake plus half a cake is half a cake. Said once, with a picture, it does not come back.

How to think about it

Find the lowest common multiple of the two denominators. Rewrite each fraction with that denominator by multiplying top and bottom by whatever the bottom needs. Add or subtract the numerators only. Simplify at the end, and convert to a mixed number if asked.

Worked example

1/3 + 1/4: the denominators 3 and 4 have lowest common multiple 12

Twelfths are small enough that both thirds and quarters fit exactly.

1/3 = 4/12 and 1/4 = 3/12

Each was multiplied top and bottom, so neither changed value.

4/12 + 3/12 = 7/12, and the denominator does not move

Seven pieces of the same size. The size itself did not change.

2/3 - 1/2 = 4/6 - 3/6 = 1/6

Subtraction is the same move with a different sign.

Your turn

Name what stays unchanged when two fractions with the same denominator are added.

Add the numerators and leave the  alone

The trap

Adding both tops and both bottoms. It gives an answer that is always between the two fractions rather than above both, which is a check worth teaching: one third plus one quarter must be more than one third, and 2/7 is not.

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