The ancient Egyptians were said to make right-angled triangles
using a rope with twelve equal sections divided by knots. What
other triangles could you make if you had a rope like this?
I cut this square into two different shapes. What can you say about
the relationship between them?
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.
Using different numbers of sticks, how many different triangles are
you able to make? Can you make any rules about the numbers of
sticks that make the most triangles?
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?
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?
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
While we were sorting some papers we found 3 strange sheets which
seemed to come from small books but there were page numbers at the
foot of each page. Did the pages come from the same book?
How many triangles can you make on the 3 by 3 pegboard?
What do these two triangles have in common? How are they related?
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?
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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.
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.
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Investigate the number of faces you can see when you arrange three cubes in different ways.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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
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?
An activity making various patterns with 2 x 1 rectangular tiles.
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
Why does the tower look a different size in each of these pictures?
What is the largest cuboid you can wrap in an A3 sheet of paper?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
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!
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?
In how many ways can you stack these rods, following the rules?
A follow-up activity to Tiles in the Garden.
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.
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?
Investigate the different ways you could split up these rooms so
that you have double the number.
How many tiles do we need to tile these patios?
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?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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?
How many different shaped boxes can you design for 36 sweets in one
layer? Can you arrange the sweets so that no sweets of the same
colour are next to each other in any direction?
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?
What happens when you add the digits of a number then multiply the
result by 2 and you keep doing this? You could try for different
numbers and different rules.
Make new patterns from simple turning instructions. You can have a
go using pencil and paper or with a floor robot.
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
How many models can you find which obey these rules?
When Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?
Can you create more models that follow these rules?