## 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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