A follow-up activity to Tiles in the Garden.
Explore one of these five pictures.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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?
It starts quite simple but great opportunities for number discoveries and patterns!
The letters of the word ABACUS have been arranged in the shape of a
triangle. How many different ways can you find to read the word
ABACUS from this triangular pattern?
Arrange your fences to make the largest rectangular space you can.
Try with four fences, then five, then six etc.
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.
Investigate the number of faces you can see when you arrange three cubes in different ways.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
This challenge extends the Plants investigation so now four or more children are involved.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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.
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
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.
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 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?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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?
An investigation that gives you the opportunity to make and justify
We think this 3x3 version of the game is often harder than the 5x5 version. Do you agree? If so, why do you think that might be?
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.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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 description of some experiments in which you can make discoveries about triangles.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
What is the largest cuboid you can wrap in an A3 sheet of paper?
A challenging activity focusing on finding all possible ways of stacking rods.
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?
What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?
An activity making various patterns with 2 x 1 rectangular tiles.
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
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
What do these two triangles have in common? How are they related?
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 tiles do we need to tile these patios?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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 find ways of joining cubes together so that 28 faces are
I cut this square into two different shapes. What can you say about
the relationship between them?
48 is called an abundant number because it is less than the sum of
its factors (without itself). Can you find some more abundant
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?
In this article for teachers, Bernard gives an example of taking an
initial activity and getting questions going that lead to other
All types of mathematical problems serve a useful purpose in
mathematics teaching, but different types of problem will achieve
different learning objectives. In generalmore open-ended problems
have. . . .