I cut this square into two different shapes. What can you say about
the relationship between them?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
An investigation that gives you the opportunity to make and justify
Investigate all the different squares you can make on this 5 by 5
grid by making your starting side go from the bottom left hand
point. Can you find out the areas of all these squares?
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.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
If I use 12 green tiles to represent my lawn, how many different
ways could I arrange them? How many border tiles would I need each
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
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.
When newspaper pages get separated at home we have to try to sort
them out and get things in the correct order. How many ways can we
arrange these pages so that the numbering may be different?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
Explore one of these five pictures.
What do these two triangles have in common? How are they related?
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?
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?
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
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?
Have a go at this 3D extension to the Pebbles problem.
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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!
If you have three circular objects, you could arrange them so that
they are separate, touching, overlapping or inside each other. Can
you investigate all the different possibilities?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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?
How many tiles do we need to tile these patios?
The challenge here is to find as many routes as you can for a fence
to go so that this town is divided up into two halves, each with 8
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
Ana and Ross looked in a trunk in the attic. They found old cloaks
and gowns, hats and masks. How many possible costumes could they
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
This article for teachers suggests ideas for activities built around 10 and 2010.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Investigate the number of faces you can see when you arrange three cubes 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?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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?
What is the largest cuboid you can wrap in an A3 sheet of paper?
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
An activity making various patterns with 2 x 1 rectangular tiles.
Investigate what happens when you add house numbers along a street
in different ways.
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?