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?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
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?
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?
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?
Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.
How many triangles can you make on the 3 by 3 pegboard?
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 activity investigates how you might make squares and pentominoes from Polydron.
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.
Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
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?
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 it up?
How many models can you find which obey these rules?
Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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?
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.
An activity making various patterns with 2 x 1 rectangular tiles.
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
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?
This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.
Can you make the birds from the egg tangram?
Can you create more models that follow these rules?
Surprise your friends with this magic square trick.
How do you know if your set of dominoes is complete?
NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.
Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
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.
What do these two triangles have in common? How are they related?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
Can you fit the tangram pieces into the outline of this junk?
Here are some ideas to try in the classroom for using counters to investigate number patterns.
Can you fit the tangram pieces into the outline of this telephone?
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 Little Fung at the table?
Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?
Can you fit the tangram pieces into the outlines of the chairs?
Can you fit the tangram pieces into the outline of this shape. How would you describe it?
Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?
Can you fit the tangram pieces into the outline of the child walking home from school?
Can you fit the tangram pieces into the outlines of these clocks?
Can you fit the tangram pieces into the outline of Wai Ping, Wah Ming and Chi Wing?
Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?