Conjecturing and generalising

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

There are nasty versions of this dice game but we'll start with the nice ones...

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

Can all unit fractions be written as the sum of two unit fractions?

Investigate what happens to the equation of different lines when you translate them. Try to predict what will happen. Explain your findings.

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

Investigate what happens to the equations of different lines when you reflect them in one of the axes. Try to predict what will happen. Explain your findings.

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

Can you decide whether two lines are perpendicular or not? Can you do this without drawing them?

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.