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
Let's suppose that you are going to have a magazine which has 16
pages of A5 size. Can you find some different ways to make these
pages? Investigate the pattern for each if you number the pages.
There are ten children in Becky's group. Can you find a set of
numbers for each of them? Are there any other sets?
Investigate and explain the patterns that you see from recording
just the units digits of numbers in the times tables.
Can you design a new shape for the twenty-eight squares and arrange
the numbers in a logical way? What patterns do you notice?
EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.
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.
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?
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?
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.
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.
How many different sets of numbers with at least four members can
you find in the numbers in this box?
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?
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
Make new patterns from simple turning instructions. You can have a
go using pencil and paper or with a floor robot.
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.
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
Investigate the different ways you could split up these rooms so
that you have double the number.
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?
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?
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?
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?
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.
How many tiles do we need to tile these patios?
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
Here are many ideas for you to investigate - all linked with the
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
When Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?
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?
Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
Explore Alex's number plumber. What questions would you like to
ask? Don't forget to keep visiting NRICH projects site for the
latest developments and questions.
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!
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Here is your chance to investigate the number 28 using shapes,
cubes ... in fact anything at all.
48 is called an abundant number because it is less than the sum of
its factors (without itself). Can you find some more abundant
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 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 challenge encourages you to explore dividing a three-digit number by a single-digit number.
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?
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?
Investigate what happens when you add house numbers along a street
in different ways.
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?
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?
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?
Explore ways of colouring this set of triangles. Can you make
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why