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.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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 follow-up activity to Tiles in the Garden.
Explore one of these five pictures.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
This problem is intended to get children to look really hard at something they will see many times in the next few months.
How many tiles do we need to tile these patios?
I cut this square into two different shapes. What can you say about
the relationship between them?
An investigation that gives you the opportunity to make and justify
An activity making various patterns with 2 x 1 rectangular tiles.
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.
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?
What do these two triangles have in common? How are they related?
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?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Investigate the number of faces you can see when you arrange three cubes in different ways.
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?
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
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?
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.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
Can you find ways of joining cubes together so that 28 faces are
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?
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?
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
If the answer's 2010, what could the question be?
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?
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?
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
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?
Try continuing these patterns made from triangles. Can you create
your own repeating pattern?
Explore ways of colouring this set of triangles. Can you make
In this section from a calendar, put a square box around the 1st,
2nd, 8th and 9th. Add all the pairs of numbers. What do you notice
about the answers?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?