A common problem: decimal division

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 10¹
• 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.