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

### 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 ((5 works just as well))
• 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.