## Fractions are Hard!

### 1.4 Even division is confusing

There are actually two different ways to think about division. Consider, for example, $$20 \div 5$$.

DIVISION AS GROUPING

The question $$20 \div 5$$ can be interpreted as: How many groups of five can you find among twenty?

We see four groups of five.

We have $$20 \div 5 = 4$$.

DIVISION AS SHARING

The question $$20 \div 5$$ can be interpreted as: I have 20 pies to share equally among 5 people. How many pies per person does this give?

We see that each person gets four pies.

We have $$20 \div 5 = 4$$.

These two approaches are philosophically different yet give the same final answer of 4 in each case. Is it obvious that they should?

Thinking Challenge: Explain why counting groups of 12 among 3900 objects must yield the same numerical answer as the number of objects per person in distributing 3900 objects equally among 12 people. Can you do it?

ON DIVIDING FRACTIONS (OPTIONAL, AS THIS IS HARD)

Some curricula do have young folk visualize the division of fractions by returning to parts-of-a-whole thinking on the number line. Can you see the answers to these five examples?

Example 1: Evaluate $$2 \div \dfrac{1}{3}$$.

Answer: Thinking of Division as Grouping this is asking for how many thirds we see in a picture of two.

We see six thirds in this picture: $$2 \div \dfrac{1}{3} = 6$$.

Example 2: Evaluate $$3 \div \dfrac{2}{3}$$.

Answer: Thinking of Division as Grouping again we are looking for groups of two-thirds in a picture of three.

Thinking parts-of-a-whole we see four-and-half two-thirds in three:  $$3 \div \dfrac{2}{3} = 4\frac{1}{2}$$.

Example 3: Evaluate $$2 \div 1\dfrac{3}{4}$$.

Answer: We are looking for groups of $$1\frac{3}{4}$$ in a picture of two.

Another one-seventh of the blue line brings us up to two. So we see one and one-seventh copies of it in two: $$2 \div 1\dfrac{3}{4}= 1\frac{1}{7}$$.

Example 4: Evaluate $$\dfrac{2}{3} \div \dfrac{1}{2}$$.

Answer: We are looking for groups of a half in a picture of two-thirds.

Do you see $$\dfrac{2}{3} \div \dfrac{1}{2} = 1\frac{1}{3}$$? (Another third of the blue line brings us up to two-thirds.)

Example 5: Evaluate $$\dfrac{1}{2} \div 3$$.

Answer: This time use Division by Sharing (why not?) and divide a half into three equal parts.

We see that $$\dfrac{1}{2} \div 3 = \dfrac{1}{6}$$. (EXTRA: Can you see this by using division by grouping too? How many threes fit inside a half?)

UPSHOT: Can you now explain why dividing two fractions is the same as multiplying by the reciprocal? It is not clear how this tricky visual work – which has us go back to parts-of-a-whole thinking and mixes approaches to division – aids us here. It seems more confusing than enlightening.

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