What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
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
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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 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?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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. . . .
Can you discover whether this is a fair game?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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.
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
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?
Can you make a right-angled triangle on this peg-board by joining
up three points round the edge?
A metal puzzle which led to some mathematical questions.
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
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.
Six balls of various colours are randomly shaken into a trianglular
arrangement. What is the probability of having at least one red in
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. . . .
Use this animation to experiment with lotteries. Choose how many
balls to match, how many are in the carousel, and how many draws to
make at once.
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
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.
Can you beat the computer in the challenging strategy game?
Use Excel to explore multiplication of fractions.
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. . . .
The classic vector racing game brought to a screen near you.
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?
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.
Match the cards of the same value.
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
This set of resources for teachers offers interactive environments
to support work on loci at Key Stage 4.
Use an interactive Excel spreadsheet to investigate factors and
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 find all the 4-ball shuffles?
Can you give the coordinates of the vertices of the fifth point in
the patterm on this 3D grid?
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.
A simple spinner that is equally likely to land on Red or Black. Useful if tossing a coin, dropping it, and rummaging about on the floor have lost their appeal. Needs a modern browser; if IE then at. . . .
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?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
Can you beat Piggy in this simple dice game? Can you figure out
Piggy's strategy, and is there a better one?
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. . . .
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.
Use an Excel to investigate division. Explore the relationships
between the process elements using an interactive spreadsheet.
Square It game for an adult and child. Can you come up with a way of always winning this game?
Have you seen this way of doing multiplication ?