I cut this square into two different shapes. What can you say about
the relationship between them?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
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?
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
Explore one of these five pictures.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
The ancient Egyptians were said to make right-angled triangles
using a rope with twelve equal sections divided by knots. What
other triangles could you make if you had a rope like this?
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?
How many triangles can you make on the 3 by 3 pegboard?
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?
What do these two triangles have in common? How are they related?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
An activity making various patterns with 2 x 1 rectangular tiles.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
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?
Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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
An investigation that gives you the opportunity to make and justify
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?
A follow-up activity to Tiles in the Garden.
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.
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?
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
How many tiles do we need to tile these patios?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Investigate the number of faces you can see when you arrange three cubes in different ways.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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
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.
What is the largest cuboid you can wrap in an A3 sheet of paper?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
In how many ways can you stack these rods, following the rules?
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?
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!
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?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
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?
Why does the tower look a different size in each of these pictures?
A challenging activity focusing on finding all possible ways of stacking rods.