A point P is selected anywhere inside an equilateral triangle. What
can you say about the sum of the perpendicular distances from P to
the sides of the triangle? Can you prove your conjecture?
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. . . .
To avoid losing think of another very well known game where the
patterns of play are similar.
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.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Can you discover whether this is a fair game?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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.
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
Can you beat the computer in the challenging strategy game?
Cellular is an animation that helps you make geometric sequences
composed of square cells.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
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
Find the vertices of a pentagon given the midpoints of its sides.
You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.
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.
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. . . .
A metal puzzle which led to some mathematical questions.
Use Excel to explore multiplication of fractions.
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?
This set of resources for teachers offers interactive environments
to support work on loci at Key Stage 4.
There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being
visible at any one time. Is it possible to reorganise these cubes
so that by dipping the large cube into a pot of paint three times
you. . . .
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.
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?
Discover a handy way to describe reorderings and solve our anagram
in the process.
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
How good are you at finding the formula for a number pattern ?
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
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.
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?
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
Can you find all the 4-ball shuffles?
P is a point on the circumference of a circle radius r which rolls,
without slipping, inside a circle of radius 2r. What is the locus
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. . . .
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Have you seen this way of doing multiplication ?
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Place a red counter in the top left corner of a 4x4 array, which is
covered by 14 other smaller counters, leaving a gap in the bottom
right hand corner (HOME). What is the smallest number of moves. . . .
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?
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?
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.
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.
An environment that enables you to investigate tessellations of