Creating and manipulating expressions and formulae

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?

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value. Can you find that x, y pair?

How good are you at finding the formula for a number pattern ?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Can you match up these equivalent algebraic expressions?

The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?