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
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?
These pictures were made by starting with a square, finding the
half-way point on each side and joining those points up. You could
investigate your own starting shape.
Investigate these hexagons drawn from different sized equilateral
Investigate and explain the patterns that you see from recording
just the units digits of numbers in the times tables.
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?
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?
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?
In this investigation we are going to count the number of 1s, 2s,
3s etc in numbers. Can you predict what will happen?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Here are many ideas for you to investigate - all linked with the
Investigate the different ways these aliens count in this
challenge. You could start by thinking about how each of them would
write our number 7.
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
Bernard Bagnall describes how to get more out of some favourite
What happens when you add the digits of a number then multiply the
result by 2 and you keep doing this? You could try for different
numbers and different rules.
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.
Investigate the numbers that come up on a die as you roll it in the
direction of north, south, east and west, without going over the
path it's already made.
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
Arrange your fences to make the largest rectangular space you can.
Try with four fences, then five, then six etc.
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?
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?
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?
Explore ways of colouring this set of triangles. Can you make
This problem is intended to get children to look really hard at something they will see many times in the next few months.
Investigate what happens when you add house numbers along a street
in different ways.
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?
Why does the tower look a different size in each of these pictures?
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?
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
How many different sets of numbers with at least four members can
you find in the numbers in this box?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
What is the largest cuboid you can wrap in an A3 sheet of paper?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
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?
48 is called an abundant number because it is less than the sum of
its factors (without itself). Can you find some more abundant
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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?
Make new patterns from simple turning instructions. You can have a
go using pencil and paper or with a floor robot.
Investigate the number of faces you can see when you arrange three cubes in different ways.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?