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
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?
How many tiles do we need to tile these patios?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
I cut this square into two different shapes. What can you say about
the relationship between them?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
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.
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
How many triangles can you make on the 3 by 3 pegboard?
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.
Bernard Bagnall describes how to get more out of some favourite
In this investigation we are going to count the number of 1s, 2s,
3s etc in numbers. Can you predict what will happen?
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
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 the answer's 2010, what could the question be?
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?
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Here are many ideas for you to investigate - all linked with the
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.
When Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?
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?
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
"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
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
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?
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
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.
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?
What do these two triangles have in common? How are they related?
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?
What is the largest cuboid you can wrap in an A3 sheet of paper?
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?
Investigate the number of faces you can see when you arrange three cubes in different ways.
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?
An activity making various patterns with 2 x 1 rectangular tiles.