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
How many tiles do we need to tile these patios?
I cut this square into two different shapes. What can you say about
the relationship between them?
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 do these two triangles have in common? How are they related?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
What is the largest cuboid you can wrap in an A3 sheet of paper?
Investigate the number of faces you can see when you arrange three cubes in different ways.
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Explore ways of colouring this set of triangles. Can you make
An activity making various patterns with 2 x 1 rectangular tiles.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
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?
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?
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?
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
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.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Investigate the different ways these aliens count in this
challenge. You could start by thinking about how each of them would
write our number 7.
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?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Here are many ideas for you to investigate - all linked with the
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 we are going to count the number of 1s, 2s,
3s etc in numbers. Can you predict what will happen?
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?
This problem is intended to get children to look really hard at something they will see many times in the next few months.
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Why does the tower look a different size in each of these pictures?
When Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?
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.
Bernard Bagnall describes how to get more out of some favourite
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?
A follow-up activity to Tiles in the Garden.
Here is your chance to investigate the number 28 using shapes,
cubes ... in fact anything at all.
If the answer's 2010, what could the question be?
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?
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios 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
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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 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?
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?
These pictures show squares split into halves. Can you find other ways?
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.
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.