Conjecturing and generalising

A mother wants to share some money by giving each child in turn a lump sum plus a fraction of the remainder. How can she do this to share the money out equally?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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?

Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?