It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Six balls of various colours are randomly shaken into a trianglular
arrangement. What is the probability of having at least one red in
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
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?
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
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
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.
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 reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
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.
7 balls are shaken in a container. You win if the two blue balls
touch. What is the probability of winning?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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. . . .
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.
Can you spot the similarities between this game and other games you know? The aim is to choose 3 numbers that total 15.
Can you locate the lost giraffe? Input coordinates to help you
search and find the giraffe in the fewest guesses.
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.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
Meg and Mo still need to hang their marbles so that they balance,
but this time the constraints are different. Use the interactivity
to experiment and find out what they need to do.
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 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.
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. . . .
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Interactive game. Set your own level of challenge, practise your table skills and beat your previous best score.
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 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?
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
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?
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. . . .
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?
Learn how to use the Shuffles interactivity by running through these tutorial demonstrations.
Practise your diamond mining skills and your x,y coordination in this homage to Pacman.
Two engines, at opposite ends of a single track railway line, set
off towards one another just as a fly, sitting on the front of one
of the engines, sets off flying along the railway line...
Carry out some time trials and gather some data to help you decide
on the best training regime for your rowing crew.
Mo has left, but Meg is still experimenting. Use the interactivity
to help you find out how she can alter her pouch of marbles and
still keep the two pouches balanced.
Imagine picking up a bow and some arrows and attempting to hit the
target a few times. Can you work out the settings for the sight
that give you the best chance of gaining a high score?
A game for 1 person to play on screen. Practise your number bonds
whilst improving your memory
Meg and Mo need to hang their marbles so that they balance. Use the
interactivity to experiment and find out what they need to do.
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
An animation that helps you understand the game of Nim.
How many different triangles can you make which consist of the
centre point and two of the points on the edge? Can you work out
each of their angles?
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.
Can you find all the 4-ball shuffles?
Two circles of equal radius touch at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P?
The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?