There are **112** NRICH Mathematical resources connected to **Creating and manipulating expressions and formulae**, you may find related items under Algebraic expressions, equations and formulae.

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 make sense of these three proofs of Pythagoras' Theorem?

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

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 ?

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

If a sum invested gains 10% each year how long before it has doubled its value?

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?

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

Can you find the area of a parallelogram defined by two vectors?

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

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers.

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?

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

Can you prove that twice the sum of two squares always gives the sum of two squares?

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.

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

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

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

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

Five equations... five unknowns... can you solve the system?

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.

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

By proving these particular identities, prove the existence of general cases.

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

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?

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

An algebra task which depends on members of the group noticing the needs of others and responding.

A task which depends on members of the group noticing the needs of others and responding.

Fancy learning a bit more about rates of reaction, but don't know where to look? Come inside and find out more...