mathematical origami - wikispaces€¦ · word origami? limping seagulls? jumping frogs? ......
TRANSCRIPT
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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Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY
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