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?
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?
Sort the houses in my street into different groups. Can you do it in any other ways?
Explore the triangles that can be made with seven sticks of the
Can you find ways of joining cubes together so that 28 faces are
These pictures show squares split into halves. Can you find other ways?
Try continuing these patterns made from triangles. Can you create
your own repeating pattern?
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
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
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Can you create more models that follow these rules?
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.
Use your mouse to move the red and green parts of this disc. Can
you make images which show the turnings described?
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
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.
An activity making various patterns with 2 x 1 rectangular tiles.
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?
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?
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.
This practical problem challenges you to create shapes and patterns
with two different types of triangle. You could even try
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?
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
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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?
Make new patterns from simple turning instructions. You can have a
go using pencil and paper or with a floor robot.
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
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?
What do these two triangles have in common? How are they related?
How many models can you find which obey these rules?
Investigate the different ways you could split up these rooms so
that you have double the number.
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
In this investigation we are going to count the number of 1s, 2s,
3s etc in numbers. Can you predict what will happen?
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?
How could you put eight beanbags in the hoops so that there are
four in the blue hoop, five in the red and six in the yellow? Can
you find all the ways of doing this?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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!
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Follow the directions for circling numbers in the matrix. Add all
the circled numbers together. Note your answer. Try again with a
different starting number. What do you notice?
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?
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?