*I'm a big advocate of error logs: notebooks in which students analyse their mistakes. I recommend a three-column approach: in the first, write the question, in the second, what went wrong, and in the last, how to do it correctly. Oddly, that's the format for this post, too.*

### The question

Decimal division: something like 14.4 ÷ 1.2

### What went wrong

Got 1.2 instead of 12.

### How to do it right

**Approach 1: Estimation.** 14 ÷ 1 is 14, so an answer of 1.2 is way off – 12 seems more reasonable.

**Approach 2: Decimal fractions.** (A little bit of commentary here: at this point, students normally groan and say "I can't do fractions" or similar. It would be rude to point out that they clearly can't yet do decimal division, either.)

- Treat the sum as $\frac{14.4}{1.2}$
- Decide that the bottom is ugly.
- "Cancel up": multiply top and bottom by 10
^{1}
- You now have $\frac{144}{12}$, which is obviously 12.

**Approach 3: Fractional fractions.** Even a naive approach to dividing fractions is simple here:

$14.4 \div 1.2 = \frac{144}{10} \div \frac{12}{10}$

$\frac{144}{10} \div \frac{12}{10} = \frac{1440}{120} = \frac{144}{12} = 12$.

There are probably a dozen other ways to approach this. What are your favourites?

* Edited 2017-05-16 to add a category.

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008.
He lives with an espresso pot and nothing to prove.

## Stephen Cavadino

My favourite approach is the one you have used here as “Approach 3”. But I would always advocate using estimation as a sense check after completion.