Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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.
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Can you find the area of a parallelogram defined by two vectors?
Think of a number, add one, double it, take away 3, add the number
you first thought of, add 7, divide by 3 and take away the number
you first thought of. You should now be left with 2. How do I. . . .
Explore the effect of reflecting in two parallel mirror lines.
Take a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the
Can you tangle yourself up and reach any fraction?
Find the vertices of a pentagon given the midpoints of its sides.
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 would be nice to have a strategy for disentangling any tangled
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?
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.
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. . . .
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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?
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.
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
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.
Can you find the values at the vertices when you know the values on
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. . . .
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Charlie has moved between countries and the average income of both
has increased. How can this be so?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
In how many ways can you arrange three dice side by side on a
surface so that the sum of the numbers on each of the four faces
(top, bottom, front and back) is equal?
Explore the effect of reflecting in two intersecting mirror lines.
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
It starts quite simple but great opportunities for number discoveries and patterns!
Can you describe this route to infinity? Where will the arrows take you next?
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.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?