I cut this square into two different shapes. What can you say about
the relationship between them?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Arrange your fences to make the largest rectangular space you can.
Try with four fences, then five, then six etc.
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?
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
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
A follow-up activity to Tiles in the Garden.
Explore one of these five pictures.
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 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?
The ancient Egyptians were said to make right-angled triangles
using a rope with twelve equal sections divided by knots. What
other triangles could you make if you had a rope like this?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Place the 16 different combinations of cup/saucer in this 4 by 4
arrangement so that no row or column contains more than one cup or
saucer of the same colour.
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?
How many triangles can you make on the 3 by 3 pegboard?
An investigation that gives you the opportunity to make and justify
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?
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?
Investigate the different shaped bracelets you could make from 18
different spherical beads. How do they compare if you use 24 beads?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
What do these two triangles have in common? How are they related?
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?
Investigate the number of faces you can see when you arrange three cubes in different ways.
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?
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
An activity making various patterns with 2 x 1 rectangular tiles.
How many tiles do we need to tile these patios?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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?
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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
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
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 many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
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?
Investigate what happens when you add house numbers along a street
in different ways.
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?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
A description of some experiments in which you can make discoveries about triangles.
48 is called an abundant number because it is less than the sum of
its factors (without itself). Can you find some more abundant
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
What is the largest cuboid you can wrap in an A3 sheet of paper?
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?
Have a go at this 3D extension to the Pebbles problem.
This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.