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 faces you can see when you arrange three cubes in different ways.
Can you create more models that follow these rules?
What do these two triangles have in common? How are they related?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
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?
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.
Sort the houses in my street into different groups. Can you do it in any other ways?
Try continuing these patterns made from triangles. Can you create
your own repeating pattern?
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.
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
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?
Bernard Bagnall describes how to get more out of some favourite
We need to wrap up this cube-shaped present, remembering that we
can have no overlaps. What shapes can you find to use?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Investigate all the different squares you can make on this 5 by 5
grid by making your starting side go from the bottom left hand
point. Can you find out the areas of all these squares?
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.
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
An investigation that gives you the opportunity to make and justify
Explore one of these five pictures.
How many models can you find which obey these rules?
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Why does the tower look a different size in each of these pictures?
A follow-up activity to Tiles in the Garden.
This problem is intended to get children to look really hard at something they will see many times in the next few months.
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Have a go at this 3D extension to the Pebbles problem.
Make new patterns from simple turning instructions. You can have a
go using pencil and paper or with a floor robot.
Use your mouse to move the red and green parts of this disc. Can
you make images which show the turnings described?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
In this article for teachers, Bernard gives an example of taking an
initial activity and getting questions going that lead to other
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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?
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.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
This practical problem challenges you to create shapes and patterns
with two different types of triangle. You could even try
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?