This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .
This article looks at levels of geometric thinking and the types of
activities required to develop this thinking.
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.
Imagine you have six different colours of paint. You paint a cube
using a different colour for each of the six faces. How many
different cubes can be painted using the same set of six colours?
Try this interactive strategy game for 2
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
I found these clocks in the Arts Centre at the University of
Warwick intriguing - do they really need four clocks and what times
would be ambiguous with only two or three of them?
This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands?
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
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
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
Imagine you are suspending a cube from one vertex (corner) and
allowing it to hang freely. Now imagine you are lowering it into
water until it is exactly half submerged. What shape does the
surface. . . .
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
A rectangular field has two posts with a ring on top of each post.
There are two quarrelsome goats and plenty of ropes which you can
tie to their collars. How can you secure them so they can't. . . .
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
How many different ways can I lay 10 paving slabs, each 2 foot by 1
foot, to make a path 2 foot wide and 10 foot long from my back door
into my garden, without cutting any of the paving slabs?
Find all the ways to cut out a 'net' of six squares that can be
folded into a cube.
What can you see? What do you notice? What questions can you ask?
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .
Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?
Can you maximise the area available to a grazing goat?
You can move the 4 pieces of the jigsaw and fit them into both
outlines. Explain what has happened to the missing one unit of
In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
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
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. . . .
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?
Make a cube out of straws and have a go at this practical
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
These points all mark the vertices (corners) of ten hidden squares.
Can you find the 10 hidden squares?
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,. . . .
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
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.
The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .
Draw a pentagon with all the diagonals. This is called a pentagram.
How many diagonals are there? How many diagonals are there in a
hexagram, heptagram, ... Does any pattern occur when looking at. . . .
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
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 bus route has a total duration of 40 minutes. Every 10 minutes,
two buses set out, one from each end. How many buses will one bus
meet on its way from one end to the other end?
Here's a simple way to make a Tangram without any measuring or
Can you fit the tangram pieces into the outline of this goat and giraffe?