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?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
Six balls of various colours are randomly shaken into a trianglular arrangement. What is the probability of having at least one red in the corner?
Can you make a right-angled triangle on this peg-board by joining up three points round the edge?
7 balls are shaken in a container. You win if the two blue balls touch. What is the probability of winning?
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?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
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.
Interactive game. Set your own level of challenge, practise your table skills and beat your previous best score.
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?
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 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.
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.
Can you explain the strategy for winning this game with any target?
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 hit?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
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...
An animation that helps you understand the game of Nim.
Practise your diamond mining skills and your x,y coordination in this homage to Pacman.
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.
Can you locate the lost giraffe? Input coordinates to help you search and find the giraffe in the fewest guesses.
Can you find all the 4-ball shuffles?
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.
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 area.
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.
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.
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?
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.
Can you spot the similarities between this game and other games you know? The aim is to choose 3 numbers that total 15.
Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?
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?
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.
Learn how to use the Shuffles interactivity by running through these tutorial demonstrations.
Carry out some time trials and gather some data to help you decide on the best training regime for your rowing crew.
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?
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.
A game for 1 person to play on screen. Practise your number bonds whilst improving your memory
Can you find triangles on a 9-point circle? Can you work out 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.
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 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.
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.