In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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?
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!
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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?
These pictures show squares split into halves. Can you find other ways?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
Have a go at this 3D extension to the Pebbles problem.
It starts quite simple but great opportunities for number discoveries and patterns!
There are to be 6 homes built on a new development site. They could
be semi-detached, detached or terraced houses. How many different
combinations of these can you find?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
What happens if you join every second point on this circle? How
about every third point? Try with different steps and see if you
can predict what will happen.
An activity making various patterns with 2 x 1 rectangular tiles.
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?
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?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
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 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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
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.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
Investigate the different ways you could split up these rooms so
that you have double the number.
Sort the houses in my street into different groups. Can you do it in any other ways?
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?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
Explore one of these five pictures.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
How many models can you find which obey these rules?
Can you create more models that follow these rules?
We think this 3x3 version of the game is often harder than the 5x5 version. Do you agree? If so, why do you think that might be?
This challenge extends the Plants investigation so now four or more children are involved.
This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
In how many ways can you stack these rods, following the rules?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
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?
An investigation that gives you the opportunity to make and justify
These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?
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?
In this article for teachers, Bernard gives an example of taking an
initial activity and getting questions going that lead to other