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?
How many models can you find which obey these rules?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
These practical challenges are all about making a 'tray' and covering it with paper.
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.
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?
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?
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
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 a rectangle of paper and fold it in half, and half again, to
make four smaller rectangles. How many different ways can you fold
Can you order pictures of the development of a frog from frogspawn
and of a bean seed growing into a plant?
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.
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
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?
Can you create more models that follow these rules?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
This activity investigates how you might make squares and pentominoes from Polydron.
What are the next three numbers in this sequence? Can you explain
why are they called pyramid numbers?
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
This practical investigation invites you to make tessellating
shapes in a similar way to the artist Escher.
Can you fit the tangram pieces into the outline of this goat and giraffe?
Can you fit the tangram pieces into the outline of this plaque design?
Can you fit the tangram pieces into the outline of these rabbits?
Can you fit the tangram pieces into the outlines of the workmen?
Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?
Can you fit the tangram pieces into the outline of the telescope and microscope?
Make a cube out of straws and have a go at this practical
NRICH December 2006 advent calendar - a new tangram for each day in
the run-up to Christmas.
This problem invites you to build 3D shapes using two different
triangles. Can you make the shapes from the pictures?
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
What do these two triangles have in common? How are they related?
Exploring and predicting folding, cutting and punching holes and
Paint a stripe on a cardboard roll. Can you predict what will
happen when it is rolled across a sheet of paper?
Can you fit the tangram pieces into the outlines of the candle and sundial?
Can you fit the tangram pieces into the outline of Little Ming playing the board game?
Can you fit the tangram pieces into the outline of Little Fung at the table?
Can you fit the tangram pieces into the outline of this telephone?
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
Can you fit the tangram pieces into the outline of this junk?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?
Can you fit the tangram pieces into the outlines of these people?