Prove that 3 times the sum of 3 squares is the sum of 4 squares. Rather easier, can you prove that twice the sum of two squares always gives the sum of two squares?

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

Solve the system of equations: ab = 1 bc = 2 cd = 3 de = 4 ea = 6

Rotate a copy of the trapezium about the centre of the longest side of the blue triangle to make a square. Find the area of the square and then derive a formula for the area of the trapezium.

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?

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.

Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.

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 ?

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.

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

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

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

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

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

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.

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

Can you hit the target functions using a set of input functions and a little calculus and algebra?

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 a rule which relates triangular numbers to square numbers?

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

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.

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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

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?

Can you find the value of this function involving algebraic fractions for x=2000?

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

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

Several graphs of the sort occurring commonly in biology are given. How many processes can you map to each graph?

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

Relate these algebraic expressions to geometrical diagrams.

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

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.

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

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

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