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.
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
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 is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Here are many ideas for you to investigate - all linked with the
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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?
This problem is intended to get children to look really hard at something they will see many times in the next few months.
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?
A follow-up activity to Tiles in the Garden.
How many tiles do we need to tile these patios?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Explore one of these five pictures.
I cut this square into two different shapes. What can you say about
the relationship between them?
Can you find ways of joining cubes together so that 28 faces are
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?
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
An investigation that gives you the opportunity to make and justify
These pictures show squares split into halves. Can you find other ways?
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"?
What do these two triangles have in common? How are they related?
Have a go at this 3D extension to the Pebbles problem.
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.
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.
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
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?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
Here is your chance to investigate the number 28 using shapes,
cubes ... in fact anything at all.
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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?
If the answer's 2010, what could the question be?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
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?
Let's suppose that you are going to have a magazine which has 16
pages of A5 size. Can you find some different ways to make these
pages? Investigate the pattern for each if you number the pages.
Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possibilities that could come up?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.