Why two fifths is wrong

Before you read anything

$\frac{1}{2} + \frac{1}{3}$. Write down what you think it is, honestly, before you read another word.

If you wrote $\frac{2}{5}$, you are in very good company. Keep it where you can see it.

Look where it lands

A half, a third, and the two fifths that adding the tops and bottoms claims. Two fifths is shorter than the half it started from.

Two fifths is less than a half. You added something to a half and ended up with less than a half, which cannot be right whatever the rule says.

Why the rule is not a rule

Halves and thirds are different sizes. Adding the bottoms says "one piece and one piece is two pieces" — true, but two pieces of what? There is no such thing until the pieces match.

Otim

asked what a "fifth" had to do with halves and thirds

Otim is asking the right question. Nothing in $\frac{1}{2}$ or $\frac{1}{3}$ is a fifth. The five came from adding two numbers that were never counts of anything.

Make them match

Both cut into sixths. The shading does not move — three sixths and two sixths — and now they can be counted: five sixths.

Cut both into sixths, which module 9 says changes nothing, and module 10 taught you how to choose. Now they are like slices, and adding is counting again.

$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$. Bigger than a half, as it must be.

Check yourself

Six to try

The cutting is done for you in each one — the common slice size is in the question. All you have to do is say how many of them there are.

1

$\frac{1}{2} - \frac{1}{3}$. Cut both into $6$ths: $\frac{\square}{6}$. What goes in the box?

2

$\frac{1}{3} + \frac{2}{4}$. Cut both into $12$ths: $\frac{\square}{12}$. What goes in the box?

3

$\frac{2}{3} + \frac{2}{4}$. Cut both into $12$ths: $\frac{\square}{12}$. What goes in the box?

4

$\frac{1}{3} + \frac{4}{5}$. Cut both into $15$ths: $\frac{\square}{15}$. What goes in the box?

5

$\frac{2}{4} + \frac{1}{5}$. Cut both into $20$ths: $\frac{\square}{20}$. What goes in the box?

6

$\frac{3}{4} - \frac{1}{5}$. Cut both into $20$ths: $\frac{\square}{20}$. What goes in the box?

Make them match, then count · page 1 of 2 Print it

Make them match, then count · page 2 of 2 Print it

Show that you have it

The mistake, named

The first asks what is actually wrong with adding the bottoms. Answer it in a sentence before you look at the second.

1

Otim works out $\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$. What is wrong with that?

2

$\frac{2}{4} + \frac{2}{5}$. Cut both into $20$ths: $\frac{\square}{20}$. What goes in the box?

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