Rich example

Things you might have done

Without using a calculator, find a reasonable estimate of: $1 \div \sqrt{2}$

There are several approaches that could have been used, some of which are discussed below.

What can your thinking about the original problem tell you about other divisions of the form $1 \div \sqrt{n}$?

• Can you select $n$ such that you can write down the value of the division calculation more easily?

• Can you select $n$ such that your answer to the original problem is useful in finding the value of the division?

If we think about the different divisions we would get if we took $n$ to be a value between $1$ and $9$ we see there will be some simple calculations when $n = 1, 4$ and $9$, i.e. when $n$ is a square number. We have left $n = 3, 5, 6, 7$ and $8$. We could estimate these as we did with $\sqrt{2}$ above, but if we look at $\sqrt{8}$, we might recognise that it can be written as $\sqrt{4\times 2}$ or $2\sqrt{2}$, and then we could use our original answer to help us.

Without using a calculator, find a reasonable estimate of: $\sqrt{50} \div 10$

Using square numbers, we know $\sqrt{49} = 7$, so $\sqrt{50} \approx 7.1.$ We could improve this estimate, but already we can see that $\sqrt{50} \div 10 \approx 0.71.$

Compare this to your original problem: $1 \div \sqrt{2}$.

$\sqrt{50} \div 10$ was a simple calculation to do, as the square root was relatively easy to approximate, and the divisor was a whole number, i.e. this fraction is already in a rationalised form.
You might have also noticed that your answer is very similar, or the same, as your answer for $1 \div \sqrt{2}.$ Can you see why this is? What happens if you simplify $\frac{\sqrt{50}}{10}$?