# Archimedes and Numerical Roots

##### Age 14 to 16

Challenge Level

### Why do this problem?

This problem offers students the opportunity to engage with and
make sense of a numerical method for finding roots.

### Possible approach

"How could I find the square root of three if I didn't have a
calculator?" Collect together students' suggestions - it is likely
that various methods of trial and improvement will be suggested, as
well as the observation that the value will be between 1 and
2.

"Trial and improvement takes time. Here is another numerical
method for finding roots." Introduce the algorithm for finding a
new approximation. Give students some time to experiment with the
method to get a feel for it and to observe how it converges to
$\sqrt{3}$.

"Can you adapt the method to find roots of other numbers? Can
you explain why it works?"

Again, give the students time to explore these two
questions.

Finally, bring the class together so they can share their
ideas and explanations.

### Key questions

What does the method $(\frac{(\frac{3}{n} + n)}{2})$
calculate, if $n$ is an approximation to $\sqrt{3}$?

How can you change the method to work out other square
roots?

How does the equation $n = (\frac{(\frac{3}{n} + n)}{2})$
help you to make sense of why the method works?

### Possible extension

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### Possible support