ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
What is the total number of squares that can be made on a 5 by 5 geoboard?
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?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
It would be nice to have a strategy for disentangling any tangled ropes...
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Can you tangle yourself up and reach any fraction?
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
A game for 2 players
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
Can you use the diagram to prove the AM-GM inequality?
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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 loser is the player who takes the last counter.
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?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?
The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.
Can all unit fractions be written as the sum of two unit fractions?
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.
What's the largest volume of box you can make from a square of paper?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
A collection of games on the NIM theme
Charlie has moved between countries and the average income of both has increased. How can this be so?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Can you find the values at the vertices when you know the values on the edges?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?