How many triangles can you make on the 3 by 3 pegboard?
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?
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?
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 is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
The ancient Egyptians were said to make right-angled triangles
using a rope with twelve equal sections divided by knots. What
other triangles could you make if you had a rope like this?
Try continuing these patterns made from triangles. Can you create
your own repeating pattern?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
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?
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?
Investigate the different ways you could split up these rooms so
that you have double the number.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
We need to wrap up this cube-shaped present, remembering that we
can have no overlaps. What shapes can you find to use?
This challenge is to design different step arrangements, which must
go along a distance of 6 on the steps and must end up at 6 high.
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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?
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?
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 make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
An activity making various patterns with 2 x 1 rectangular tiles.
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.
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
This challenge extends the Plants investigation so now four or more children are involved.
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 challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
I cut this square into two different shapes. What can you say about
the relationship between them?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
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?
An investigation that gives you the opportunity to make and justify
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!
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Suppose there is a train with 24 carriages which are going to be
put together to make up some new trains. Can you find all the ways
that this can be done?
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.
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.
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?
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
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
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?