If I use 12 green tiles to represent my lawn, how many different
ways could I arrange them? How many border tiles would I need each
What is the largest cuboid you can wrap in an A3 sheet of paper?
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?
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
What do these two triangles have in common? How are they related?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
This activity asks you to collect information about the birds you
see in the garden. Are there patterns in the data or do the birds
seem to visit randomly?
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?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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?
Here are many ideas for you to investigate - all linked with the
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Investigate the numbers that come up on a die as you roll it in the
direction of north, south, east and west, without going over the
path it's already made.
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 your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
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?
Investigate the different ways you could split up these rooms so
that you have double the number.
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
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 find ways of joining cubes together so that 28 faces are
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?
How many models can you find which obey these rules?
Can you create more models that follow these rules?
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
Why does the tower look a different size in each of these pictures?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
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?
Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .
In how many ways can you stack these rods, following the rules?
How many tiles do we need to tile these patios?
These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.
What statements can you make about the car that passes the school gates at 11am on Monday? How will you come up with statements and test your ideas?
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?
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
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?
If the answer's 2010, what could the question be?
Have a go at this 3D extension to the Pebbles problem.
What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.