Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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
Explore the effect of reflecting in two parallel mirror lines.
Find the vertices of a pentagon given the midpoints of its sides.
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
Can you find the area of a parallelogram defined by two vectors?
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
List any 3 numbers. It is always possible to find a subset of
adjacent numbers that add up to a multiple of 3. Can you explain
why and prove it?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Can you describe this route to infinity? Where will the arrows take you next?
Explore the effect of reflecting in two intersecting mirror lines.
What is the volume of the solid formed by rotating this right
angled triangle about the hypotenuse?
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.
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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?
A game for 2 players
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.
Can you find sets of sloping lines that enclose a square?
Find some examples of pairs of numbers such that their sum is a
factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and
16 is a factor of 48.
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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. . . .
A collection of games on the NIM theme
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.
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. . . .
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Can you use the diagram to prove the AM-GM inequality?
It would be nice to have a strategy for disentangling any tangled
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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Can all unit fractions be written as the sum of two unit fractions?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
It starts quite simple but great opportunities for number discoveries and patterns!
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Can you find the values at the vertices when you know the values on
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 tangle yourself up and reach any fraction?
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?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...