### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### Doodles

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

### Russian Cubes

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

# Archimedes and Numerical Roots

### 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?