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?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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 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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
These pictures show squares split into halves. Can you find other ways?
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?
Here are many ideas for you to investigate - all linked with the
An activity making various patterns with 2 x 1 rectangular tiles.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
How many tiles do we need to tile these patios?
A follow-up activity to Tiles in the Garden.
Explore one of these five pictures.
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?
What do these two triangles have in common? How are they related?
An investigation that gives you the opportunity to make and justify
I cut this square into two different shapes. What can you say about
the relationship between them?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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.
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
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?
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
Investigate the number of faces you can see when you arrange three cubes in different ways.
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?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
What is the largest cuboid you can wrap in an A3 sheet of paper?
Place this "worm" on the 100 square and find the total of the four
squares it covers. Keeping its head in the same place, what other
totals can you make?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
This problem is intended to get children to look really hard at something they will see many times in the next few months.
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?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
Investigate and explain the patterns that you see from recording
just the units digits of numbers in the times tables.
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
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?
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 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?
In this investigation we are going to count the number of 1s, 2s,
3s etc in numbers. Can you predict what will happen?
Explore ways of colouring this set of triangles. Can you make
Try continuing these patterns made from triangles. Can you create
your own repeating pattern?
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?