Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
In this article for teachers, Bernard gives an example of taking an
initial activity and getting questions going that lead to other
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?
Complete these two jigsaws then put one on top of the other. What
happens when you add the 'touching' numbers? What happens when you
change the position of the jigsaws?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
Investigate what happens when you add house numbers along a street
in different 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?
Investigate the different ways you could split up these rooms so
that you have double the number.
How many models can you find which obey these rules?
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?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
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?
"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?
Sort the houses in my street into different groups. Can you do it in any other ways?
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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 create more models that follow these rules?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Here are many ideas for you to investigate - all linked with the
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
When Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?
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?
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.
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.
Explore ways of colouring this set of triangles. Can you make
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?
48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?
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?
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?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
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?
If the answer's 2010, what could the question be?
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
Here is your chance to investigate the number 28 using shapes,
cubes ... in fact anything at all.
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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.
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
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?
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?
What do these two triangles have in common? How are they related?