Take a rectangle of paper and fold it in half, and half again, to
make four smaller rectangles. How many different ways can you fold
If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?
This problem focuses on Dienes' Logiblocs. What is the same and
what is different about these pairs of shapes? Can you describe the
shapes in the picture?
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
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?
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
An activity making various patterns with 2 x 1 rectangular tiles.
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
These practical challenges are all about making a 'tray' and covering it with paper.
In how many ways can you fit two of these yellow triangles
together? Can you predict the number of ways two blue triangles can
be fitted together?
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?
This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?
Our 2008 Advent Calendar has a 'Making Maths' activity for every
day in the run-up to Christmas.
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?
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
Can you make the birds from the egg tangram?
Here is a version of the game 'Happy Families' for you to make and
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
The Man is much smaller than us. Can you use the picture of him
next to a mug to estimate his height and how much tea he drinks?
In this challenge, you will work in a group to investigate circular
fences enclosing trees that are planted in square or triangular
NRICH December 2006 advent calendar - a new tangram for each day in
the run-up to Christmas.
How many triangles can you make on the 3 by 3 pegboard?
These pictures show squares split into halves. Can you find other ways?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
How many models can you find which obey these rules?
Can you create more models that follow these rules?
Explore the triangles that can be made with seven sticks of the
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?
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?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
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.
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.
Ideas for practical ways of representing data such as Venn and
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
Make a cube out of straws and have a go at this practical
Have a go at drawing these stars which use six points drawn around
a circle. Perhaps you can create your own designs?
Can you fit the tangram pieces into the outline of this telephone?
Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?
Can you fit the tangram pieces into the outline of this junk?
We can cut a small triangle off the corner of a square and then fit
the two pieces together. Can you work out how these shapes are made
from the two pieces?
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
Kaia is sure that her father has worn a particular tie twice a week
in at least five of the last ten weeks, but her father disagrees.
Who do you think is right?
This practical problem challenges you to create shapes and patterns
with two different types of triangle. You could even try
Follow the diagrams to make this patchwork piece, based on an
octagon in a square.
Can you see which tile is the odd one out in this design? Using the
basic tile, can you make a repeating pattern to decorate our wall?
Exploring and predicting folding, cutting and punching holes and