If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
A right-angled isosceles triangle is rotated about the centre point
of a square. What can you say about the area of the part of the
square covered by the triangle as it rotates?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
A metal puzzle which led to some mathematical questions.
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 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?
This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.
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.
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?
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
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
Cellular is an animation that helps you make geometric sequences composed of square cells.
Square It game for an adult and child. Can you come up with a way of always winning this game?
There are thirteen axes of rotational symmetry of a unit cube. Describe them all. What is the average length of the parts of the axes of symmetry which lie inside the cube?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
To avoid losing think of another very well known game where the
patterns of play are similar.
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?
A counter is placed in the bottom right hand corner of a grid. You
toss a coin and move the star according to the following rules: ...
What is the probability that you end up in the top left-hand. . . .
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
Can you give the coordinates of the vertices of the fifth point in
the patterm on this 3D grid?
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 make a right-angled triangle on this peg-board by joining
up three points round the edge?
Match the cards of the same value.
Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.
Find all the ways of placing the numbers 1 to 9 on a W shape, with
3 numbers on each leg, so that each set of 3 numbers has the same
in how many ways can you place the numbers 1, 2, 3 … 9 in the
nine regions of the Olympic Emblem (5 overlapping circles) so that
the amount in each ring is the same?
This set of resources for teachers offers interactive environments
to support work on loci at Key Stage 4.
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?
How good are you at finding the formula for a number pattern ?
An environment that enables you to investigate tessellations of
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Discover a handy way to describe reorderings and solve our anagram
in the process.
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
A java applet that takes you through the steps needed to solve a
Diophantine equation of the form Px+Qy=1 using Euclid's algorithm.
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 put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?
An environment that simulates a protractor carrying a right- angled
triangle of unit hypotenuse.
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
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.
Have you seen this way of doing multiplication ?
Can you beat Piggy in this simple dice game? Can you figure out
Piggy's strategy, and is there a better one?
Use Excel to practise adding and subtracting fractions.
Find the vertices of a pentagon given the midpoints of its sides.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Can you find all the 4-ball shuffles?
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.
Use Excel to explore multiplication of fractions.