Numbers arranged in a square but some exceptional spatial awareness probably needed.
How will you decide which way of flipping over and/or turning the grid will give you the highest total?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
A challenging activity focusing on finding all possible ways of stacking rods.
Take 5 cubes of one colour and 2 of another colour. How many
different ways can you join them if the 5 must touch the table and
the 2 must not touch the table?
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?
Let's say you can only use two different lengths - 2 units and 4
units. Using just these 2 lengths as the edges how many different
cuboids can you make?
We went to the cinema and decided to buy some bags of popcorn so we
asked about the prices. Investigate how much popcorn each bag holds
so find out which we might have bought.
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
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
This challenge is to design different step arrangements, which must
go along a distance of 6 on the steps and must end up at 6 high.
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
An activity making various patterns with 2 x 1 rectangular tiles.
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
How many triangles can you make on the 3 by 3 pegboard?
Investigate the number of paths you can take from one vertex to
another in these 3D shapes. Is it possible to take an odd number
and an even number of paths to the same vertex?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Can you find ways of joining cubes together so that 28 faces are
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.
This practical problem challenges you to create shapes and patterns
with two different types of triangle. You could even try
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?
Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.
Can you create more models that follow these rules?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
In how many ways can you stack these rods, following the rules?
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?
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
Let's suppose that you are going to have a magazine which has 16
pages of A5 size. Can you find some different ways to make these
pages? Investigate the pattern for each if you number the pages.
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
This article for teachers suggests ideas for activities built around 10 and 2010.
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?
Make new patterns from simple turning instructions. You can have a
go using pencil and paper or with a floor robot.
How many different shaped boxes can you design for 36 sweets in one
layer? Can you arrange the sweets so that no sweets of the same
colour are next to each other in any direction?
I like to walk along the cracks of the paving stones, but not the
outside edge of the path itself. How many different routes can you
find for me to take?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Arrange your fences to make the largest rectangular space you can.
Try with four fences, then five, then six etc.
What happens if you join every second point on this circle? How
about every third point? Try with different steps and see if you
can predict what will happen.
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
Suppose we allow ourselves to use three numbers less than 10 and
multiply them together. How many different products can you find?
How do you know you've got them all?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
How many models can you find which obey these rules?
Complete these two jigsaws then put one on top of the other. What
happens when you add the 'touching' numbers? What happens when you
change the position of the jigsaws?
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.
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
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.
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?
An investigation that gives you the opportunity to make and justify