All types of mathematical problems serve a useful purpose in
mathematics teaching, but different types of problem will achieve
different learning objectives. In generalmore open-ended problems
have. . . .
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?
There are three tables in a room with blocks of chocolate on each.
Where would be the best place for each child in the class to sit if
they came in one at a time?
Numbers arranged in a square but some exceptional spatial awareness probably needed.
In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.
In this article for teachers, Bernard gives an example of taking an
initial activity and getting questions going that lead to other
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
How will you decide which way of flipping over and/or turning the grid will give you the highest total?
Explore one of these five pictures.
It starts quite simple but great opportunities for number discoveries and patterns!
A follow-up activity to Tiles in the Garden.
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?
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?
Sort the houses in my street into different groups. Can you do it in any other ways?
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!
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
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?
Here is your chance to investigate the number 28 using shapes,
cubes ... in fact anything at all.
Can you find ways of joining cubes together so that 28 faces are
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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?
48 is called an abundant number because it is less than the sum of
its factors (without itself). Can you find some more abundant
What do these two triangles have in common? How are they related?
What is the largest cuboid you can wrap in an A3 sheet of paper?
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?
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
Try continuing these patterns made from triangles. Can you create
your own repeating pattern?
Explore ways of colouring this set of triangles. Can you make
An activity making various patterns with 2 x 1 rectangular tiles.
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?
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
The red ring is inside the blue ring in this picture. Can you
rearrange the rings in different ways? Perhaps you can overlap them
or put one outside another?
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 what happens when you add house numbers along a street
in different ways.
In this investigation, you are challenged to make mobile phone
numbers which are easy to remember. What happens if you make a
sequence adding 2 each time?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
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?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
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.
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?
Why does the tower look a different size in each of these pictures?
This challenge extends the Plants investigation so now four or more children are involved.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Can you create more models that follow these rules?
How many models can you find which obey these rules?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?