In this article for teachers, Bernard gives an example of taking an
initial activity and getting questions going that lead to other
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?
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?
Which times on a digital clock have a line of symmetry? Which look
the same upside-down? You might like to try this investigation and
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
Well now, what would happen if we lost all the nines in our number
system? Have a go at writing the numbers out in this way and have a
look at the multiplications table.
Bernard Bagnall describes how to get more out of some favourite
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
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?
Have a go at this 3D extension to the Pebbles problem.
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.
If the answer's 2010, what could the question be?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
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!
Investigate what happens when you add house numbers along a street
in different ways.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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?
An investigation that gives you the opportunity to make and justify
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?
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 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 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.
In this section from a calendar, put a square box around the 1st,
2nd, 8th and 9th. Add all the pairs of numbers. What do you notice
about the answers?
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.
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?
This problem is intended to get children to look really hard at something they will see many times in the next few months.
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?
While we were sorting some papers we found 3 strange sheets which
seemed to come from small books but there were page numbers at the
foot of each page. Did the pages come from the same book?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
Can you design a new shape for the twenty-eight squares and arrange
the numbers in a logical way? What patterns do you notice?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
Start with four numbers at the corners of a square and put the
total of two corners in the middle of that side. Keep going... Can
you estimate what the size of the last four numbers will be?
It starts quite simple but great opportunities for number discoveries and patterns!
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?
Investigate this balance which is marked in halves. If you had a weight on the left-hand 7, where could you hang two weights on the right to make it balance?
Why does the tower look a different size in each of these pictures?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
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?
A follow-up activity to Tiles in the Garden.
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. . . .
Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
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?
What is the largest cuboid you can wrap in an A3 sheet of paper?
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?
I cut this square into two different shapes. What can you say about
the relationship between them?