These practical challenges are all about making a 'tray' and covering it with paper.
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
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?
This activity investigates how you might make squares and pentominoes from Polydron.
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?
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.
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?
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.
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 triangles can you make on the 3 by 3 pegboard?
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
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?
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?
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
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?
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?
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?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?
In this town, houses are built with one room for each person. There
are some families of seven people living in the town. In how many
different ways can they build their houses?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
These pictures show squares split into halves. Can you find other ways?
If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?
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?
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
How do you know if your set of dominoes is complete?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?
What are the next three numbers in this sequence? Can you explain
why are they called pyramid numbers?
Here is a version of the game 'Happy Families' for you to make and
What do these two triangles have in common? How are they related?
Explore the triangles that can be made with seven sticks of the
Can you make the birds from the egg tangram?
NRICH December 2006 advent calendar - a new tangram for each day in
the run-up to Christmas.
Cut a square of paper into three pieces as shown. Now,can you use
the 3 pieces to make a large triangle, a parallelogram and the
Can you make five differently sized squares from the tangram
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Can you put these shapes in order of size? Start with the smallest.
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?
This project challenges you to work out the number of cubes hidden under a cloth. What questions would you like to ask?