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?
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
Can you find ways of joining cubes together so that 28 faces are
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
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?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Investigate the number of paths you can take from one vertex to
another in these 3D shapes. Is it possible to take an odd number
and an even number of paths to the same vertex?
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.
Can you create more models that follow these rules?
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
An activity making various patterns with 2 x 1 rectangular tiles.
How can you arrange these 10 matches in four piles so that when you
move one match from three of the piles into the fourth, you end up
with the same arrangement?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
Make new patterns from simple turning instructions. You can have a
go using pencil and paper or with a floor robot.
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?
This practical problem challenges you to create shapes and patterns
with two different types of triangle. You could even try
How many triangles can you make on the 3 by 3 pegboard?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
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.
In how many ways can you stack these rods, following the rules?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
What do these two triangles have in common? How are they related?
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!
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
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?
Investigate the different ways you could split up these rooms so
that you have double the number.
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
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.
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
Investigate this balance which is marked in halves. If you had a weight on the left-hand 7, where could you hang two weights on the right to make it balance?
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?
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?
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?
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?
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
Let's suppose that you are going to have a magazine which has 16
pages of A5 size. Can you find some different ways to make these
pages? Investigate the pattern for each if you number the pages.
48 is called an abundant number because it is less than the sum of
its factors (without itself). Can you find some more abundant