Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
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 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.
Can you find ways of joining cubes together so that 28 faces are
This practical problem challenges you to create shapes and patterns
with two different types of triangle. You could even try
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.
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?
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
What do these two triangles have in common? How are they related?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
How many models can you find which obey these rules?
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
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.
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?
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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?
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
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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.
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 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?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
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?
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?
We need to wrap up this cube-shaped present, remembering that we
can have no overlaps. What shapes can you find to use?
In how many ways can you stack these rods, following the rules?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
This challenge is to design different step arrangements, which must
go along a distance of 6 on the steps and must end up at 6 high.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
In this investigation, you must try to make houses using cubes. If
the base must not spill over 4 squares and you have 7 cubes which
stand for 7 rooms, what different designs can you come up with?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Explore one of these five pictures.
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Investigate the different ways you could split up these rooms so
that you have double the number.