In how many ways can you fit all three pieces together to make
shapes with line symmetry?
Consider a watch face which has identical hands and identical marks
for the hours. It is opposite to a mirror. When is the time as read
direct and in the mirror exactly the same between 6 and 7?
How many different symmetrical shapes can you make by shading triangles or squares?
Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
A triangle ABC resting on a horizontal line is "rolled" along the
line. Describe the paths of each of the vertices and the
relationships between them and the original triangle.
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly. . . .
A train leaves on time. After it has gone 8 miles (at 33mph) the
driver looks at his watch and sees that the hour hand is exactly
over the minute hand. When did the train leave the station?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Two motorboats travelling up and down a lake at constant speeds
leave opposite ends A and B at the same instant, passing each
other, for the first time 600 metres from A, and on their return,
400. . . .
The triangle ABC is equilateral. The arc AB has centre C, the arc
BC has centre A and the arc CA has centre B. Explain how and why
this shape can roll along between two parallel tracks.
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
To avoid losing think of another very well known game where the
patterns of play are similar.
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4
respectively so that opposite faces add to 7? If you make standard
dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
Mathematics is the study of patterns. Studying pattern is an
opportunity to observe, hypothesise, experiment, discover and
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.
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
Blue Flibbins are so jealous of their red partners that they will
not leave them on their own with any other bue Flibbin. What is the
quickest way of getting the five pairs of Flibbins safely to. . . .
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
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. . . .
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
How efficiently can you pack together disks?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Jo made a cube from some smaller cubes, painted some of the faces
of the large cube, and then took it apart again. 45 small cubes had
no paint on them at all. How many small cubes did Jo use?
Have a go at this 3D extension to the Pebbles problem.
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
An introduction to bond angle geometry.
When dice land edge-up, we usually roll again. But what if we
A package contains a set of resources designed to develop pupils'
mathematical thinking. This package places a particular emphasis on
“visualising” and is designed to meet the needs. . . .
Can you mark 4 points on a flat surface so that there are only two
different distances between them?
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. . . .
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
An irregular tetrahedron has two opposite sides the same length a
and the line joining their midpoints is perpendicular to these two
edges and is of length b. What is the volume of the tetrahedron?
Four rods are hinged at their ends to form a convex quadrilateral.
Investigate the different shapes that the quadrilateral can take.
Be patient this problem may be slow to load.
Here are four tiles. They can be arranged in a 2 by 2 square so
that this large square has a green edge. If the tiles are moved
around, we can make a 2 by 2 square with a blue edge... Now try. . . .
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.
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?