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

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

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.

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?

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

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?

Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

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

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

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

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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

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

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?

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

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?”

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 ?

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?

Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .

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?

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

Can you find a rule which connects consecutive triangular numbers?

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

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

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...

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .

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

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

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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

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?

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.

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

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. . . .

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

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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?

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?

What is the total number of squares that can be made on a 5 by 5 geoboard?

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

Show that all pentagonal numbers are one third of a triangular number.