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 the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
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
Here are many ideas for you to investigate - all linked with the
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
How many tiles do we need to tile these patios?
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
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?
Bernard Bagnall describes how to get more out of some favourite
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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.
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.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
Which times on a digital clock have a line of symmetry? Which look
the same upside-down? You might like to try this investigation and
Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
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 is the largest cuboid you can wrap in an A3 sheet of paper?
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?
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?
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?
This problem is intended to get children to look really hard at something they will see many times in the next few months.
Why does the tower look a different size in each of these pictures?
When Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?
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?
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
How many triangles can you make on the 3 by 3 pegboard?
An investigation that gives you the opportunity to make and justify
Here is your chance to investigate the number 28 using shapes,
cubes ... in fact anything at all.
This article for teachers suggests ideas for activities built around 10 and 2010.
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?
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Investigate these hexagons drawn from different sized equilateral
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?
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
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
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?
Investigate the number of faces you can see when you arrange three cubes in different ways.