These pictures were made by starting with a square, finding the
half-way point on each side and joining those points up. You could
investigate your own starting shape.
What do these two triangles have in common? How are they related?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
The challenge here is to find as many routes as you can for a fence
to go so that this town is divided up into two halves, each with 8
Ana and Ross looked in a trunk in the attic. They found old cloaks
and gowns, hats and masks. How many possible costumes could they
What is the largest number of circles we can fit into the frame
without them overlapping? How do you know? What will happen if you
try the other shapes?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
When newspaper pages get separated at home we have to try to sort
them out and get things in the correct order. How many ways can we
arrange these pages so that the numbering may be different?
I cut this square into two different shapes. What can you say about
the relationship between them?
Place the 16 different combinations of cup/saucer in this 4 by 4
arrangement so that no row or column contains more than one cup or
saucer of the same colour.
This article for teachers suggests ideas for activities built around 10 and 2010.
You cannot choose a selection of ice cream flavours that includes
totally what someone has already chosen. Have a go and find all the
different ways in which seven children can have ice cream.
If you have three circular objects, you could arrange them so that
they are separate, touching, overlapping or inside each other. Can
you investigate all the different possibilities?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
What happens to the area of a square if you double the length of
the sides? Try the same thing with rectangles, diamonds and other
shapes. How do the four smaller ones fit into the larger one?
Start with four numbers at the corners of a square and put the
total of two corners in the middle of that side. Keep going... Can
you estimate what the size of the last four numbers will be?
Let's say you can only use two different lengths - 2 units and 4
units. Using just these 2 lengths as the edges how many different
cuboids can you make?
What is the largest cuboid you can wrap in an A3 sheet of paper?
Investigate the different ways you could split up these rooms so
that you have double the number.
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?
We went to the cinema and decided to buy some bags of popcorn so we
asked about the prices. Investigate how much popcorn each bag holds
so find out which we might have bought.
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
The red ring is inside the blue ring in this picture. Can you
rearrange the rings in different ways? Perhaps you can overlap them
or put one outside another?
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
An activity making various patterns with 2 x 1 rectangular tiles.
In my local town there are three supermarkets which each has a
special deal on some products. If you bought all your shopping in
one shop, where would be the cheapest?
I like to walk along the cracks of the paving stones, but not the
outside edge of the path itself. How many different routes can you
find for me to take?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
This activity asks you to collect information about the birds you
see in the garden. Are there patterns in the data or do the birds
seem to visit randomly?
Take 5 cubes of one colour and 2 of another colour. How many
different ways can you join them if the 5 must touch the table and
the 2 must not touch the table?
Sort the houses in my street into different groups. Can you do it in any other ways?
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
Try continuing these patterns made from triangles. Can you create
your own repeating pattern?
Explore ways of colouring this set of triangles. Can you make
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
Can you find ways of joining cubes together so that 28 faces are
How many models can you find which obey these rules?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Can you create more models that follow these rules?
In this investigation we are going to count the number of 1s, 2s,
3s etc in numbers. Can you predict what will happen?
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?
Why does the tower look a different size in each of these pictures?
This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?
In how many ways can you stack these rods, following the rules?
How many tiles do we need to tile these patios?