In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
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?
These pictures show squares split into halves. Can you find other ways?
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?
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.
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
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?
Can you find ways of joining cubes together so that 28 faces are
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
This practical problem challenges you to create shapes and patterns
with two different types of triangle. You could even try
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
Use your mouse to move the red and green parts of this disc. Can
you make images which show the turnings described?
We need to wrap up this cube-shaped present, remembering that we
can have no overlaps. What shapes can you find to use?
An activity making various patterns with 2 x 1 rectangular tiles.
Sort the houses in my street into different groups. Can you do it in any other ways?
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?
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.
Try continuing these patterns made from triangles. Can you create
your own repeating pattern?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
In this article for teachers, Bernard gives an example of taking an
initial activity and getting questions going that lead to other
What do these two triangles have in common? How are they related?
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?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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?
How many models can you find which obey these rules?
Investigate the number of faces you can see when you arrange three cubes in different ways.
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
Have a go at this 3D extension to the Pebbles problem.
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Explore the triangles that can be made with seven sticks of the
Make new patterns from simple turning instructions. You can have a
go using pencil and paper or with a floor robot.
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 challenging activity focusing on finding all possible ways of stacking rods.
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!
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?
In how many ways can you stack these rods, following the rules?
Numbers arranged in a square but some exceptional spatial awareness probably needed.
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?
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?
Suppose we allow ourselves to use three numbers less than 10 and
multiply them together. How many different products can you find?
How do you know you've got them all?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Investigate the different ways you could split up these rooms so
that you have double the number.
It starts quite simple but great opportunities for number discoveries and patterns!
A follow-up activity to Tiles in the Garden.
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?