This interactivity allows you to sort letters of the alphabet into two groups according to different properties.
Published October 2001,January 2004,February 2011.
We live in a universe of patterns.
So begins Ian Stewart's book Nature's Numbers, a fieldtrip
that takes the reader sightseeing in the mathematical universe that
is the world around us. It is interesting that Stewart sees fit to
take his adult readers on that intriguing and important journey
into pattern as he explores what mathematics is for and what it is
about. Interesting because many teachers query why investigating
pattern has such a central role in the mathematics' curriculum for
younger explorers. In asking why pattern, what is it studied for
and what is it about, we are actually asking about the role and
purpose of mathematics itself.
Mathematics is the study of patterns. Studying pattern is an
opportunity to observe, hypothesise, experiment, discover and
create. By understanding regularities based on the data we gather
we can predict what comes next, estimate if the same pattern will
occur when variables are altered and begin to extend the pattern.
Practical activities that allow us to construct knowledge for
ourselves with all of the ingredients for a meaningful, thought
provoking and mentally and physically engaging mathematics
Study of pattern integrates both the strands of mathematics and
a variety of curricular areas. We can use and extend skills and
knowledge of number, measurement, geometry, data collection and
statistics, probability and algebraic thinking. It allows us to
bring together mathematics with music, visual art and craft,
vocabulary building, creative writing and verbal communication,
social studies, science and environmental studies, talent and
technology. What better way to build a rich, developmentally
appropriate curriculum for youngsters?
If we ask the youngest of learners, "What are patterns?",
calling on their prior experiences and knowledge, they often use
fabric, wallpaper, wrapping paper or building bricks to indicate
their understanding. Somewhere is the idea of repeating over and
over again. But, what are some of the pattern 'pieces' we can
choose from so that we don't feel as though we are repeating the
same instructional material over and over and over again?
Although used so frequently in early classrooms, we don't often
connect the pattern of repeated sound and words in story, song and
rhyme and relate it to maths. Is it a simple A, B pattern? An A, B,
A, B pattern or more complex, perhaps with three or more elements.
What symbols can be used to represent the pattern - can it be
clapped, clicked, counted, drawn? Is the pattern generalisable -
can we find other tunes, poems that have the same rhythm or beat?
What will come next, what happens if we switch two of the elements,
will A, A, B, B sound the same as B, B, A, A?
No fieldtrip into the local environment is complete without
looking for the patterns that brighten and enliven our world. What
patterns are present, are they from the natural or manmade world?
What are the elements of pattern? What shapes do we see, how are
colours used, do the patterns have meaning or purpose, how many
varieties are there, how do they differ, what elements are the
same? Looking at the overlapping tiles or wooden shingles on a roof
we see a wide variety of patterns.
We see the same type of pattern in the leaves of certain plants,
on the scales on fish and snakes and fir tree cones. They are not
just pretty they are protection. Animals having overlapping body
structures are flexible; they can curl up. We might not be able to
see an armadillo first hand, but meal worms (known also as pill
bugs and roly-polys) can be found easily in most damp outdoor
environments. Looked at through a hand lens, the pattern of the
overlapping structure can be seen.
A trick with overlapping playing cards demonstrates one of
nature's fundamental laws. Overlap one card in a line on the next
and flip the fist card laid down. All of the cards move and turn.
Change to the pattern in one element of a system causes change and
disruption to all elements of the system.
Typically we consider the elements of shape and colour, often
colour simply emphasises shape. Tartans are created through the
same basic element, repeated lines, intersecting. The variety is
immense as the thickness and colour of lines alternate, as we see
in books detailing Scottish clans and their tartans. We could be
even more creative in designing our own tartans and clan history
than the Celts were.
Some lines are not all that they seem. Zebras' stripes
have pattern, but how can we describe it? The stripes are the
fingerprints of the animal, each has its own unique pattern. They
are an example of how the purpose of pattern comes into its own in
the natural world. In the animal kingdom, like colour, they are
signal friend or foe. The stripes of a zebra act to distort what
the predators see. We could investigate the effect of different
shapes in helping animals hide and keep safe.
Stripes, lines or bands of colour, can cause optical illusions.
Here is an opportunity to investigate the causes after-images on
the retina. Pattern and colour is seen to change as the speed of
the motion increases. Colour wheels or discs made of lines, circles
or coloured sectors can be stuck (masking tape works very well) on
the end of a hand-held electric beater and observed as the speed
increases. The results are fascinating.
Colours are nature's alarm clocks waking us up to the different
and changing pattern in the seasons. Being able to read the pattern
of the seasons was man's first way of measuring passing time.
Recording daily changes in temperature, amount of sunshine,
rainfall etc. provides children with an opportunity to develop ways
to record, organise and retrieve, as well as interpret, use and
display real data. Data, information that can be used to solve
problems, is full of pattern and the purpose of analysing data is
to enable people to predict and plan. We can do plenty of that with
climate, weather and other seasonal data.
Motifs on buildings and in fabrics, in the trimmings and
decorations that brighten our lives, often illustrate repeated
patterns. We are surrounded by translational or rotational symmetry
patterns. Look around: bottles in a rack, wallpaper and fabrics,
borders and ribbons, fruit cut in half, letters of the alphabet and
houses in the street. Mathematical vocabulary, sorting and
classifying skills are developed and extended as attributes and
properties of geometric shapes in these patterns are recognised and
Measurement, design and technology can be brought in as
accumulated knowledge of pattern is applied. A motif can be
designed, cut into polystyrene, lino or potato and printed on to
large sheets of paper.
Wrapping gifts in your own paper makes it special. Why not wear
your maths lesson? Children could cut their motif from pop-up
sponge, which then swells in water, with fabric paint they can
personalise a plain coloured tee shirt.
The questions to be asked are: how does the motif itself have
pattern; how will the motif be used in a design; will it be
repeated across the material; will the design be symmetrical? There
is more than one form of symmetry and this is an ideal opportunity
to discover the different types of pattern created by different
symmetries. Do the shapes tessellate?
No study of pattern would be complete without regard to
tessellation. Finding shapes that fit together without leaving gaps
between them has long been a preferred way for people to add visual
decoration to their built environment.
Mosaic and stained glass windows are the usual way to explore
tessellations with young children, and the hexagon is the shape of
As well as examining soccer balls and honeycombs, there are
other hexagonal patterns to be explored. Try to arrange a set of
the same sized coins so that they fit as tightly together as
possible. Six of them will always cluster around the central one to
form a shape similar to a honeycomb. What happens with different
here is no doubt that the circle has a unique beauty. Concentric
circles, those endless series of ripples and rings are fascinating.
Investigating the pattern created when one or more stones are
dropped into a pool allows children to dip into concepts of
gravity, force, resistance, motion and surface tension. No nature
fieldtrip is complete without investigating tree rings.
How do local species compare with a Giant Sequoia from Nevada
which, through its rings, was found to be 3500 years old? What
important history unfolded during those lifetimes? Why are some
rings thicker than others, how can we know about the climate during
the years the tree was alive?
Children are never too young to learn the value of tools in
mathematics, especially for geometric constructions. Little hands
may require some help and guidance with safety compasses, but the
fascinating patterns they can create are worth the effort. Having
children experiment to find out how, with a little help from their
friends, they can turn a piece of string and pencil into a compass
may produce a low-tech instrument but it is an exercise in lateral
Where else do we see concentric circles? On old records, compact
disc surfaces? Well, yes to the first but no to the second. A
strong magnifying lens will help the careful observer see that in
fact the CD is made of millions of tiny dots arranged in tight
Spirals start at a central point and coil around. They are
easily seen on nautilus shells and ammonite fossils, in springs and
the threads of screws and the tight coils of tendrils on climbing
plants. Is there a purpose to this natural and common manmade
shape? If we look at and think about spiral staircase we begin to
get a clue. They take up very little space and in some structures
are very strong - springs are tight and tough!
Spirographs, whether the wheels of varying sizes or the newer
battery operated pen style, allow youngsters to create neat and
regular spiral shapes. Again, string, pencil and a friend to gently
pull and shorten the string length, act together to make an
admirable tool. If we pull out those magnifying lens again we will
discover that spiders' webs also spiral out from the centre. Not
what we expect!
What we probably expect to find in a web is a radial pattern,
that is to say, one in which straight lines radiate from a central
point. If not in a spider's web where do we see radial patters?
Where roads meet at a roundabout, on a dart board which is a
combination of concentric circles and radial sections, in cactus
spines where they meet the barrel and in flowers like the waterlily
and the gerber daisy. In these flowers the purpose of such an
arrangement is to attract insects to their centres. The Sea Anemone
is a radial animal, and so is a Starfish, its five arms allow it to
move in any direction.
When children make a paper flower or snowflake by folding a
circle into eighths or sixths from the centre point and then
cutting patterns into the folded sides and edge (circumference),
they are part of a long tradition. Almost 500 years ago, Kepler
wrote a book called the 6 Cornered Snowflake. He concluded upon
examining the structure and pattern of snowflakes that matter is
composed of identical units or atoms.
Logo programs can be written to simulate Von Koch's snowflake.
The edges are equilateral triangles. The pattern is one of ever
decreasing size as each new generation is added. The straight lines
take on an illusion of curves, in the same way as curved stitching
does. Information and pictures are easily attainable on the
Very young learners can use gummed paper triangles of different
sizes to build a snowflake. By examining the pattern of growth they
are able to estimate the number of triangles each generation
Similar Logo programs can be written to show other branching
patterns wherein sections get progressively smaller. The same type
of branching that is seen in deer antlers, fern leaves, blood
vessels and TV aerials as well as trees. The results of these
programmes amaze young learners and give them early exposure to the
concept of fractals!
All of these patterns and we have hardly mentioned the simplest
of all pattern, numeric. Even the natural world is loaded with
numeric pattern: from the regular 28 days lunar cycle, the annual
cycle of 365 and a quarter days to the number of legs on animals.
Legs? Yes, humans have 2, cows have 4, bees have 6, spiders have 8.
Why even flowers' petals are not exempt from the power of pattern.
Lilies have 3, buttercups have 5, delphiniums have 8, marigolds
have 13, asters 21 and daisies 34. It took 800 years to explain why
this series of numbers that Leonardo of Pisa (Fibonacci) identified
is rampant in nature, just like the rabbits in the problem he
devised. We haven't even got to the golden ratio, Pascal's or Omar
Khayam's triangles, multiples and ...
As Ian Stewart noted, all pattern can be expressed by number. We
can recreate nature's shapes by finding and plotting co-ordinates.
Did we mention co-ordinates? No, the problem is not lack of
material to choose from in studying pattern, the problem is lack of