Exploring and noticing

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?

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

Starting with two basic vector steps, which destinations can you reach on a vector walk?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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.

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

Which curve is which, and how would you plan a route to pass between them?

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.

In this twist on the well-known Countdown numbers game, use your knowledge of Powers and Roots to make a target.