Conjecturing and generalising

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

Ben's class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

In how many different ways can you break up a stick of seven interlocking cubes? Now try with a stick of eight cubes and a stick of six cubes. What do you notice?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Think of a number, square it and subtract your starting number. Is the number you're left with odd or even? How do the images help to explain this?

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

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.