# Unusual Long Division - Square Roots Before Calculators

##### Age 14 to 16

Challenge Level

### Why do this problem?

This problem takes more able students into the realm of
'non-calculator' methods that lie beyond the arithmetic they became
familiar with when they were much younger. It usefully draws
attention to the need for validation in any algorithm whether
carried out electronically or 'by hand'.

### Possible approach

- Spend a little time looking at the validity of the standard
method for 'Long Division'. Discuss the historical need for
efficient algorithms before electronic calculators, when
computation was manual, and point out that calculators and
computers aren't 'magic' and there still has to be a valid
algorithm.

- Find some square roots of two-digit numbers to 2dp by trial and
improvement.

- Spend time understanding what this new method involves (maybe
use the audio link on the
Problem page ,while keeping the working still on view),
practise and then organise a time trial.

- Alternate between this method's algorithm and trial &
improvement, for the square roots of 30, 50, 60, 70, 80, and 90,
all to two decimal places. Record the calculation time for each one
and compare methods.

### Key questions

### Possible extension

Explain that a mathematician will always want to justify or
validate a procedure and leave that challenge with the group.

### Possible support

For less able students raising awareness that methods of
calculation need justifying can lead to a stronger and more
satisfying grasp of arithmetic procedures like 'long
multiplication' (traditional and alternative) and 'long
division'.