Published February 2011.

All too often we ignore the creative aspects of mathematics in our classroom teaching, and focus on routine exercises and the repetition of procedures. In doing so we are unlikely to spark enthusiastic responses to the subject from the children. Not many children say that maths is their favourite lesson and yet it could be so easily.

On the NRICH website there are many ideas for making connections between mathematics and other subjects, as well as links to other valuable resources on the Internet. Mathematics is often described as the study of pattern, which might help to explain its strong connections with art, leading to some of the most creative aspects of mathematics. Let us consider tessellations as just one aspect of pattern. Let's start by looking at tessellating equilateral triangles and then go on to look at various other kinds of triangles and then quadrilaterals.

Equilateral triangles have three sides the same length and three angles the same. Can you make them fit together to cover a sheet of paper without any gaps between them? This is called "tessellating".

What about triangles with two equal sides? These are isosceles triangles. Can you tessellate all isosceles triangles? How do you know?

Now try with right angled triangles. These have one right angle or 90 degree angle.

Some triangles have sides that are all different. Can you tessellate these?

Could you tessellate any triangle?

What about squares? Rectangles? Kites? Any other polygons with four sides?

You could try polygons with more than four sides. What do you find?

There is a range of resources that you can use to explore such ideas, such as Polydron or ATM mats -playing around with shapes that you can move is satisfying and productive but some children become frustrated that they don't have a record of their results. You could use a digital camera to record (and make a diplay) or ask the children to transfer their ideas onto paper, perhaps using cardboard templates.

Whenever I think of tessellations I always think of Escher, an artist who has certainly explored this area thoroughly. It is indeed questionable whether he was more of an artist or more of a mathematician. Exploring how his tessellations link to underlying repetitions of quadrilaterals with "bits" cut out and added on gives children a creative way into developing their own more elaborate tessellating patterns. It might be worth avoiding the more sinister patterns though! Websites presenting Escher's work abound and can easily be found through search engines. The official website is a good place to start: www.mcescher.com.

One living mathematician who has done a lot of work in this
area is Roger Penrose. He has explored tiling patterns that cover
the whole plain like a tessellation but are not regular in the way
in which the pattern repeats itself. His designs are stunning in
their beauty and underlying simplicity, and we have developed one
of them into an interactive activity called "Building
Stars ".

This design uses two quadrilateral shapes -a kite and a dart
-and they are repeated over and over again but do not fit together
in a regular way. As well as exploring this pattern interactively
on the website you could also use the templates to create your own
kites and darts either cut out in sticky paper or in sponge so that
you could print a pattern using the shapes.

One of the key features of the kites and darts that are used
in this pattern is the ratio between the lengths of the sides,
which is the Golden Ratio. Investigating the importance of this
number in art and architecture as well as more mundane things like
the shape of A4 sheets of paper is yet another fruitful avenue of
mathematical study and once again it links to many different
curriculum areas.

The mention of architecture takes us into Islamic art, of
which decorative patterns are a central feature owing to the
prohibition of representation of natural forms in Islam. As is
often the case with constraints of one kind or another, this
restriction has led Islamic artists to become masters of abstract
geometrical patterns and tilings. One frequently used tiling is a
stars and crosses design which can be constructed using a ruler and
a pair of compasses.

The template, below, can also be found here
in the form of an interactive game to play with the pattern.
Constructing the pattern oneself is quite a challenge even for the
highest attainers.

Another kind of pattern that is found in art and that has
mathematical significance is the spiral. In fact, there are many
different spirals and they arise out of different contexts.
Archimedes, the Greek mathematician, has one named after him which
can be constructed using a cotton reel, strip of paper and a pencil
and looks like the pattern below: it is the path followed by an
object travelling away from a given point at constant
velocity.

Of course there are also spirals all around us - from screws
to the pattern of segments in a pineapple, or the way leaves are
spiralled round the stem of a plant. Did you even make twizzlers as
a child? Draw a spiral on each side of a circular piece of card,
make two holes either side of the centre and thread a length of
string through them. Tie it to make a loop, keeping the card in the
middle, twist both ends and then pull. The twizzler will work and
you can experiment to see what visual effects can be made. Make
links to colour mixing too, by colouring the spirals in different
colours.

What other connections can we make with mathematics across the
curriculum? One approach to teaching maths that I and many others
find useful is to contextualise it within a story, and there are
many stories that we can tell that have mathematics in them. One of
my favourites is a story about "Teddy
Town " where different bears of different colours live in
different streets. It offers a context for an interesting problem
about shape, space and combinations.

In Teddy Town, teddies are either red, yellow or blue and they
live in red, yellow or blue houses. There are nine teddies - three
red, three yellow, three blue - and nine houses - three red, three
yellow and three blue.

What are the nine different combinations of teddies and
houses?

Here is a map showing Teddy Town:

The streets are very special. If you walk along a street from
east to west, or west to east, all the houses are a different
pattern and the teddies living in the houses are a different
pattern too. The same is true if you walk along the streets in a
north-south or south-north direction.

In other words, looking at the map grid, each row and column
must have different patterned houses and different patterned
teddies.

Can you arrange the nine different combinations you've found
on the map grid?

There are plenty of other stories that you may be familiar
with that have mathematics in them. For example, Eric Carle's
picture books such as The Very
Hungry Caterpillar and The
Bad Tempered Ladybird , which can help to introduce young
children to counting and the concept of time on the clock. For
older children, Hans Ensenberger's The Number Devil is a delightful
story that describes a young boy dreaming about numbers which he is
introduced to by the Number Devil. It includes all sorts of
different numbers from Fibonacci numbers to triangle numbers and
prime numbers, with a chapter on each. Wouldn't it be refreshing to
have a shared reading of a mathematics book for a change?

A couple of years ago I was involved in a project in Lewisham
in which I worked with an expert Theatre in Education team called
Make Believe Arts
to develop some mathematical stories that had maths built into them
as part of the story itself. The children acted out the stories and
in each case a particular mathematical problem had to be resolved
in order for the story to progress. It was tremendous fun and
proved a memorable experience for the children. A traditional story
that is similar is an Indian tale called Sissa's
Reward :

According to an old Indian myth, Sissa ben Dahir was a
courtier for a king. Sissa worked very hard and invented a game
which was played on a board, similar to chess. The king decided to
reward Sissa for his dedication and asked what he would like. Sissa
thought carefully and then said, "I would like one grain of rice to
be put on the first square of my board, two on the second square,
four on the third square, eight on the fourth and so on." The king
thought this was a silly request, but little did he know...

Left is a chess board with the first few squares filled with
grains of rice as Sissa asked for.

- How many grains of rice would there be on the eighth square?
- How many grains would you need altogether in order to fill up to the 15th square?
- Estimate how many grains you would need in total to fill the entire board in this way. Explain your thinking.
- Perhaps Sissa was cleverer than the king thought!

I have already mentioned quite a few mathematicians of
historical significance and the connections between maths and
history are very strong and well worth investigating. There is the
history of mathematics itself, which is fascinating in its own
right. It is hard to imagine how the Romans did arithmetic when one
looks at their number system, and the development of our own
conventions are fascinating. How did they manage without a symbol
for zero for so long? Delving back into ancient history, the
Egyptians were brilliant at geometry and their methods helped them
to construct the Pyramids, but what about fractions? They had an
interesting method which only allowed them to express fractions as
unit fractions with one as the numerator. Expressing fractions as
the sum of unit fractions is tricky but is an interesting
mathematical problem:

This problem can be found on the website here
.

Did you know that the Egyptians wrote all their fractions
using what we call unit fractions? A unit fraction has 1 as its
numerator (top number). Here are some examples:

$1/5 \quad 1/3 \quad 1/2$

They expressed all fractions as the sum of unit fractions, but they weren't allowed to repeat the same unit fraction in the addition. So we couldn't write:

$3/8=1/8+1/8+1/8$

because we've used $1/8$ three times. However, this would be fine:

$3/8=1/4+1/8$

How could the Egyptians write $3/4$? Are there any other ways?

What is $2/3$ written as the sum of unit fractions? Again, investigate different ways of doing this.

Find some more fractions (say three or four) which you can write as the sum of unit fractions.

What about the great mathematicians? I have already mentioned
Archimedes but there are so many others: Pythagoras, Escher, Euler
and of course Fibonacci. St Andrew's University in Scotland has a
website dedicated to the history of mathematics and it is possible
to find a different mathematician for every day of the year there.
There are a number of sources of information on the web about
Fibonacci but one of the best must be that developed by an
enthusiast,
Ron Knott . He offers enough ideas to fill your maths lessons
for weeks as well as your history, science and art lessons! There
are lesson ideas, challenging questions and links to other websites
galore. Here is a sample:

Fibonacci's Rabbits

The original problem that Fibonacci investigated (in the year
1202) was about how fast rabbits could breed in ideal
circumstances.

Suppose a newly-born pair of rabbits, one male, one female,
are put in a field. Rabbits are able to mate at the age of one
month so that at the end of its second month a female can produce
another pair of rabbits. Suppose that our rabbits never die and
that the female always produces one new pair (one male, one female)
every month from the second month on. The puzzle that Fibonacci
posed was:

How many pairs will there be in one year?

- At the end of the first month, they mate, but there is still only one pair.
- At the end of the second month the female produces a new pair, so now there are two pairs of rabbits in the field.
- At the end of the third month, the original female produces a second pair, making three pairs in all in the field.
- At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making five pairs.

The number of pairs of rabbits in the field at the start of
each month is $1, 1, 2, 3, 5, 8, 13, 21, 34\ldots$

Can you see how the series is formed and how it
continues?

The Fibonacci connection brings me more or less to the end of
this article, although I have only scratched the surface of the
possibilities for linking maths with other curriculum areas in
exciting and challenging ways. I haven't even mentioned geography,
with maps, bearings, contours just as starters, or physics, where
we could explore speed, time, rates of change, measurement and a
host of other things. I will leave those for another time. The
creative possibilities are endless and are a fruitful field for
exploring.

Carle, Eric (1982) The
Bad-Tempered Ladybird . Picture Puffins, London. ISBN
0140503986

Carle, Eric (1970) The Very
Hungry Caterpillar . Hamish Hamilton, London. ISBN
024101798X Enzensberger, Hans Magnus (2000) The Number Devil: A Mathematical
Adventure . Henry Holt & Company. ISBN 0805062998

This article first appeared in
Maths Coordinator's File issue 19, published by pfp
publishing.(Although the File is no longer produced, some articles
may be available through Optimus Publishing http://www.teachingexpertise.com/-821.)