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
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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.
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?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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?
This resources contains a series of interactivities designed to support work on transformations at Key Stage 4.
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
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 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?
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 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. . . .
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?
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.
Can you make a right-angled triangle on this peg-board by joining up three points round the edge?
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
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?
Can you explain the strategy for winning this game with any target?
Can you discover whether this is a fair game?
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.
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.
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.
Can you give the coordinates of the vertices of the fifth point in the patterm on this 3D grid?
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 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.
The interactive diagram has two labelled points, A and B. It is designed to be used with the problem "Cushion Ball"
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?
This resource contains a range of problems and interactivities on the theme of coordinates in two and three dimensions.
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 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?
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?
7 balls are shaken in a container. You win if the two blue balls touch. What is the probability of winning?
An Excel spreadsheet with an investigation.
Use Excel to practise adding and subtracting fractions.
This set of resources for teachers offers interactive environments to support work on loci at Key Stage 4.
Use an Excel to investigate division. Explore the relationships between the process elements using an interactive spreadsheet.
This resource contains interactive problems to support work on number sequences at Key Stage 4.
An animation that helps you understand the game of Nim.
Use Excel to explore multiplication of fractions.
Use an interactive Excel spreadsheet to investigate factors and multiples.
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.
Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?