What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
Billy's class had a robot called Fred who could draw with chalk
held underneath him. What shapes did the pupils make Fred draw?
What shape has Harry drawn on this clock face? Can you find its
area? What is the largest number of square tiles that could cover
How can you make an angle of 60 degrees by folding a sheet of paper
Draw three straight lines to separate these shapes into four groups
- each group must contain one of each shape.
A group activity using visualisation of squares and triangles.
Four rods, two of length a and two of length b, are linked to form
a kite. The linkage is moveable so that the angles change. What is
the maximum area of the kite?
A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.
Find a way to cut a 4 by 4 square into only two pieces, then rejoin the two pieces to make an L shape 6 units high.
Can you cut a regular hexagon into two pieces to make a
parallelogram? Try cutting it into three pieces to make a rhombus!
Investigate how the four L-shapes fit together to make an enlarged
L-shape. You could explore this idea with other shapes too.
This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!
These are pictures of the sea defences at New Brighton. Can you
work out what a basic shape might be in both images of the sea wall
and work out a way they might fit together?
An extension of noughts and crosses in which the grid is enlarged
and the length of the winning line can to altered to 3, 4 or 5.
How many different triangles can you make on a circular pegboard that has nine pegs?
Exchange the positions of the two sets of counters in the least possible number of moves
A game for two players on a large squared space.
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.
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?
It is possible to dissect any square into smaller squares. What is
the minimum number of squares a 13 by 13 square can be dissected
These points all mark the vertices (corners) of ten hidden squares.
Can you find the 10 hidden squares?
How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
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?
Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?
What is the greatest number of squares you can make by overlapping
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?
In this problem, we have created a pattern from smaller and smaller
squares. If we carried on the pattern forever, what proportion of
the image would be coloured blue?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
Imagine a 4 by 4 by 4 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will not have holes drilled through them?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?
A game has a special dice with a colour spot on each face. These
three pictures show different views of the same dice. What colour
is opposite blue?
Anne completes a circuit around a circular track in 40 seconds.
Brenda runs in the opposite direction and meets Anne every 15
seconds. How long does it take Brenda to run around the track?
Can you fit the tangram pieces into the outlines of the watering can and man in a boat?
Which of these dice are right-handed and which are left-handed?
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
Can you fit the tangram pieces into the outlines of the workmen?
Here are four tiles. They can be arranged in a 2 by 2 square so
that this large square has a green edge. If the tiles are moved
around, we can make a 2 by 2 square with a blue edge... Now try. . . .
Can you fit the tangram pieces into the outlines of the candle and sundial?
This article for teachers describes how modelling number properties
involving multiplication using an array of objects not only allows
children to represent their thinking with concrete materials,. . . .
Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?
Can you maximise the area available to a grazing goat?
You have 27 small cubes, 3 each of nine colours. Use the small
cubes to make a 3 by 3 by 3 cube so that each face of the bigger
cube contains one of every colour.
ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.
Seven small rectangular pictures have one inch wide frames. The
frames are removed and the pictures are fitted together like a
jigsaw to make a rectangle of length 12 inches. Find the dimensions
of. . . .
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP
: PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED.
What is the area of the triangle PQR?