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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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.
Have a go at this 3D extension to the Pebbles problem.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
An investigation that gives you the opportunity to make and justify
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
Investigate and explain the patterns that you see from recording
just the units digits of numbers in the times tables.
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?
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
Can you find ways of joining cubes together so that 28 faces are
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.
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.
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.
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?
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.
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 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?
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?
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?
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?
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
A challenging activity focusing on finding all possible ways of stacking rods.
An activity making various patterns with 2 x 1 rectangular tiles.
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
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 how many ways can you stack these rods, following the rules?
How many tiles do we need to tile these patios?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
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?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
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?
Can you create more models that follow these rules?
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?
Explore one of these five pictures.
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
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?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.