Exploring and noticing

Can you find and prove the relationship between the area of a trapezium and the area of a triangle constructed within it?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a rhombus.

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

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

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

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?