Year 11+ Exploring and noticing

The items in the shopping basket add and multiply to give the same amount. What could their prices be?

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

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Quadratic graphs are very familiar, but what patterns can you explore with cubics?

Can you find the area of a parallelogram defined by two vectors?

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

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?

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

What is the same and what is different about these circle questions? What connections can you make?