Conjecturing and generalising

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

What is the largest number which, when divided into these five numbers in turn, leaves the same remainder each time?

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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.

A red square and a blue square overlap. Is the area of the overlap always the same?

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.