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.
An investigation that gives you the opportunity to make and justify
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Explore one of these five pictures.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
Have a go at this 3D extension to the Pebbles problem.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
I cut this square into two different shapes. What can you say about
the relationship between them?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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
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?
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?
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?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
How can you arrange these 10 matches in four piles so that when you
move one match from three of the piles into the fourth, you end up
with the same arrangement?
An activity making various patterns with 2 x 1 rectangular tiles.
What happens if you join every second point on this circle? How
about every third point? Try with different steps and see if you
can predict what will happen.
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
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?
What do these two triangles have in common? How are they related?
A follow-up activity to Tiles in the Garden.
Investigate the number of faces you can see when you arrange three cubes in different ways.
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?
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!
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?
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
How many tiles do we need to tile these patios?
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
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 how this pattern of squares continues. You could
measure lengths, areas and angles.
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
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.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
A description of some experiments in which you can make discoveries about triangles.