What images spring to mind when you hear the

word origami? Limping seagulls? Jumping frogs?

Water lilies? Fighter aircraft? One piece of paper

can take on manifold forms. Yet people seldom

associate origami, the ancient art of paper folding,

with the rather austere requirements of the national

curriculum. In fact ‘origami’ is an ideal vehicle for

tackling the shape and space element of the maths

curriculum.

Many resources which are commonly used for

shape and space work have to be specially bought.

One, however, is readily available – A size paper. By

folding A size paper you can magically manufacture

a variety of 2D and 3D mathematical shapes. In

addition you can acquire a party piece capable of

entertaining a railway carriage of passengers

throughout a journey from Fort William to

Penzance on a day when the snow is of the wrong

type and the leaves aren’t that good either.

The magical properties of A size paper are due

to a system which is rational and mathematical.

Each A size rectangular sheet is made by folding in

half the size numerically below it e.g. folding A3 in

half creates A4 etc. (Figs 1a and 1b). By doing this,

two similar rectangles are produced and if the

process is continued, a never ending family of

similar shapes (Fig 2). This is made possible

because the sides of any A size paper are in the ratio

1: �2 (1.4142...).

Why not try some folding yourself, have fun and

see how easy it all is. You’ll be amazed at the range

of maths you encounter, the amount of discussion

generated and the use of mathematical vocabulary.

Here are a few activities for you to have a go at. All

you need to do is fold equilateral triangles (Fig 3).

Remember to always start with A size paper and the

well known saying ‘PRACTICE MAKES PERFECT’!

• Explore the properties of an equilateraltriangle

Investigate the angles.

How many lines of symmetry does it have?

What about the order of rotational symmetry?

Unfold the equilateral triangle and work out

(using known angle facts) the sizes of the angles

made by the fold lines (Fig 4).

Make polygon characters and families with

different sized equilateral triangles and name them

e.g. Ellie and Ewan Equilateral (Fig 5).

MATHEMATICS TEACHING 176 / SEPTEMBER 2001 23

trains and folds and planes

MATHEMATICAL ORIGAMILiz Meenan

ffig 1a

ffig 1b

ffig 2

Fig 3: folding an

equilateral triangle

ffig 5

ffig 4

Further information andsuggestions on paperfolding (and other suchactivities includingtiling, printing weavingand reflecting) areincluded in the Shape,space and measures(maths from design)video (£14.99),teachers’ guide (£3.95)and activity book(£6.95). Theseresources are for 7- to12-year-olds andavailable from4Learning, PO Box 444,London SW1P 2WD.

unfold➔

MT176-2 5/12/02 2:51 pm Page 23

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MATHEMATICS TEACHING 176 / SEPTEMBER 200124

How are the tiles

made? Describe the

patterns on them.

At Christmas these

‘tiles’ could make a

‘forest’ of mathe-

matical equilateral

Christmas trees.

How many spirals can you see being

formed here?

•Make and investigate tiling patterns using‘tiles’ folded from coloured A4, A5, A6 . . .equilateral triangles (Fig 6).

•Use the equilateral triangles to makeother polygons e.g. with two equilateral

triangles you can make a rhombus.Investigate the polygons you can make with

three equilateral triangles, with four, with fiveand so on. Then use ‘families’ of similar

equilateral triangles to make colourful polygonpatterns including, large equilateral triangles,

rhombi, trapeziums, regular hexagons, 6-pointed starsand of course the Star of David (Fig 8).

• Make 3D shapes with the equilateraltriangular ‘tiles’ taped together –including a square based pyramid anda pentagonal pyramid (Fig 7).

ffig 6

ffig 7

This shape can be used to illustrate the

equivalence of halves, thirds and sixths.

Can you see how the number of triangles in this

shape generate the counting sequence, the odd,

the triangle and the square number sequences?

How many different shapes can you see in this

triangle?

At Christmas this design can be used as an

advent calendar or large Christmas tree.

MT176-2 5/12/02 2:51 pm Page 24

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MATHEMATICS TEACHING 176 / SEPTEMBER 2001 25

• Make a washing number line.

• Make quite a few congruent colouredequilateral triangles. Join them in pairsto make rhombi. Use therhombi to make 2Dillustrations of 3Dshapes (Fig 9).

Can you see the 3D shape made out of cubes in

this illustration?

How many cubes would be needed to turn this

shape into a large cube?

• Make ‘triangular limping seagulls’ by firstfolding equilateral triangles (Figs 10a, b, c, d).Then follow the instructions and slot theseorigami units together using the ‘wings’ tomake a regular tetrahedron, a regularoctahedron and a regular icosahedon.

How many shapes can you see

in this star?

ffig 9

Start with an equilateral triangle

Fold and unfold along the 3

lines of symmetry

Fold and unfold each

corner to the centres of

their opposite sides.

Cut from the corners along to

the creases made by the

second set of folds.

Choose whether to have ‘left’

or ‘right’ flaps, then fold the

unwanted flaps down and

interlock them to complete

the triangular seagull.

ffig 10a

ffig 10b

ffig 10c

ffig 10d

• And now for the party trick!

Fold an equilateral triangle (instructions

on page 23)

The equilateral triangle is divided in

2 parts by a fold line.

Fold the small triangle away.

An isosceles trapezium

Turnover.

ffig 8 Star of David

MT176-2 5/12/02 2:52 pm Page 25

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MATHEMATICS TEACHING 176 / SEPTEMBER 200126

Fold in turn, each of the two outer

triangles over the middle triangle.

Congruent rhombuses

Fold up the 3 outer triangles

To secure punch holes in the top 3 corners &

tie with ribbon or use tape. Great as a

Christmas bauble or a package for a small

present.

A net of a tetrahedron

Unfold

This is great as a package for a present.

Note: If you make 20 of these truncated tetrahe-

drons and glue them all together, you make a

unique icosahedron. Peep inside this magical solid

and you will see another smaller icosahedron.

Unfold all the folds and you

have one sheet of A4. One word

of warning; – care must be taken

not to get too enthralled lest you miss Penzance and

find yourselves carried back Northwards.

Now you’ve got the hang of it, why not explore

how to fold other triangles, other polygons and

other polyhedra?

A regular tetrahedron

Unfold tetrahedron and find

centre of triangle by pinch

folding sides together.

Fold corners to centre.

A regular hexagon

Refold the six folds.

Tuck one of the smaller

triangles into one of

the others and fold the

last one on top.

A truncated tetrahedron

Unfold the truncated tetrahedron. Reverse fold each

of the 3 small folds. Fold each trapezium into the

centre one at a time. (Secure it by tucking in flap *.)

Note: A bigger Star of David could have been formed

by making two larger equilateral

triangles and sticking them

together (Fig 8).

A Star of David

*

Pull the Star apart

slightly and a star dish

is formed

Liz Meenan is a 4Learning Education Officer.

The photographs for thisarticle were taken byColin Ross

MT176-2 5/12/02 2:52 pm Page 26

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