We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
A tilted square is a square with no horizontal sides. Can you
devise a general instruction for the construction of a square when
you are given just one of its sides?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
The number of plants in Mr McGregor's magic potting shed increases
overnight. He'd like to put the same number of plants in each of
his gardens, planting one garden each day. How can he do it?
Can you find a relationship between the number of dots on the
circle and the number of steps that will ensure that all points are
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Can you discover whether this is a fair game?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.
Can you find all the 4-ball shuffles?
To avoid losing think of another very well known game where the
patterns of play are similar.
Six balls of various colours are randomly shaken into a trianglular
arrangement. What is the probability of having at least one red in
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
A collection of resources to support work on Factors and Multiples at Secondary level.
Use the interactivity to listen to the bells ringing a pattern. Now
it's your turn! Play one of the bells yourself. How do you know
when it is your turn to ring?
A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .
Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses?
This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.
This is an interactivity in which you have to sort the steps in the
completion of the square into the correct order to prove the
formula for the solutions of quadratic equations.
Use the interactivity to play two of the bells in a pattern. How do
you know when it is your turn to ring, and how do you know which
bell to ring?
Show that for any triangle it is always possible to construct 3
touching circles with centres at the vertices. Is it possible to
construct touching circles centred at the vertices of any polygon?
You can move the 4 pieces of the jigsaw and fit them into both
outlines. Explain what has happened to the missing one unit of
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
Work out how to light up the single light. What's the rule?
What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?
7 balls are shaken in a container. You win if the two blue balls
touch. What is the probability of winning?
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?
Do you know how to find the area of a triangle? You can count the
squares. What happens if we turn the triangle on end? Press the
button and see. Try counting the number of units in the triangle
now. . . .
A game for 2 players. Can be played online. One player has 1 red
counter, the other has 4 blue. The red counter needs to reach the
other side, and the blue needs to trap the red.
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
Mr McGregor has a magic potting shed. Overnight, the number of
plants in it doubles. He'd like to put the same number of plants in
each of three gardens, planting one garden each day. Can he do it?
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
Find the vertices of a pentagon given the midpoints of its sides.
Here is a chance to play a version of the classic Countdown Game.
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
What is the relationship between the angle at the centre and the
angles at the circumference, for angles which stand on the same
arc? Can you prove it?
Learn how to use the Shuffles interactivity by running through these tutorial demonstrations.
Can you make a right-angled triangle on this peg-board by joining
up three points round the edge?
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
Can you beat the computer in the challenging strategy game?