177
Some Experiences with Egyptian Fractions
Jarmila Novotná, Marie Kubínová
Projects of the type “We Calculate as the Ancient …” are based on a
genetic parallel. The use of the idea of a genetic parallel led to a
significant reinforcement of pupils’ intrinsic motivation and also to
pupils’ interest in techniques of calculation, types of numbers,
mathematical symbols etc. used by our predecessors. The project also
proved itself to be an excellent, non-standard tool for checking the level
of knowledge necessary for calculations with fractions.
In this case, “We Calculate as the Ancient Egyptians Did”, helped to:
• Create a ‘natural’ need to raise fractions to higher terms and to
reduce them in situations which later led to the creation of an
algorithm for adding and subtracting fractions. This approach was
quite different from the ‘normal’ school situation.
• Create interesting non-standard conditions for diagnosis of the level
of understanding the development of algorithms for calculations with
fractions.
• To show again that an unusual approach may mean a challenge for
discovering new properties of numbers which can be a real pupil’s
masterpiece, and an intellectual challenge.
Pupils entering lower secondary level come with separate models of
fractions (one half, one quarter, one third, three quarters, two thirds,
sometimes even one fifth and one tenth). They have learnt most of them
outside school when solving everyday problems (a half of a cake, a
quarter of loaf of bread, a third of the houses in our street, he came three
quarters of an hour late, one tenth of a litre). When solving simple
problems pupils usually use fractions as operators and they work with
the object image of the fraction. When deducing rules for adding and
subtracting fractions, the phase of “naïve” mathematics ends and the
phase of formal mathematics begins. Pupils will fail here even if given a
powerful tool – the algorithm for adding fractions. It is because this
algorithm does not enable the majority of pupils to make the transition
from understanding fractions as quantities, (as parts of a whole) which is
178
visual and relies on physical experience to fractions as operators and as
rational numbers which is conceptual. Pupils do not get this insight and
consequently they cannot solve more complicated problems. Pupils have
to be taught to abstract from the object image of the part-whole and they
have to be led to the discovery that fractions can be added directly if they
understand the principle of equivalence of rational numbers.
The traditional approach of the primary school has been by
• the use of suitable modelling of situations (predominantly
geometrically) with the use of various models as each of them
reflects the different life experience of pupils and is the introduction
to the application of fractions in various areas.
• leading pupils towards understanding that the basis of the whole
process of adding is reduction or raising fractions to higher terms
with a common denominator. The fractions that pupils are asked to
add are of different quality – of different denominator. Therefore it is
necessary to express them through the same quality – the same
denominator. But it must not be forgotten that it is not a drill of
methods of searching for the common denominator in the beginning,
but the realization of the need to find a common denominator.
When adding fractions, the task of preparing the transition from the
conscious process, which is based mostly on work with geometrical
models but is time consuming, to an automatic process, which is based
on using algorithms and which, if grasped correctly makes work much
faster, cannot be ruhed.
Our effort to base the exposition of concepts like “reduce or raise
fractions to higher terms by a given number” on work with unit fractions
was caused by the need to eliminate “artificial” situations which are used
in textbooks of mathematics for the introduction of the concepts, and
also by our previous experience of using unit fractions when attempting
to make knowledge about fractions more secure.
Calculations with unit fractions captured the attention of several
pupils in our classes to such an extent that they carried on in their work
without any direct link to school work. Several interesting pupils’ works
were created. These later influenced other teachers to change the order of
presenting the subject matter and their teaching methods in some of their
179
mathematics lessons. The teaching process showed that the projects “We
Calculate as ........” present strong elements of motivation for the pupils.
Pupils, freely and actively participated in solving the problems within the
projects and many of them also set themselves their own goals which
surpassed the level expected. An poster of rational numbers with
denominators 2 – 14 was thus created and it remained on the noticeboard for more than two months and was an inspiration for several pupils
to further work with fundamental fractions.
Evaluation
Pupils of the class were generally good when solving standard tasks in
the subject matter “Fractions”. With the exception of two pupils, they
acquired algorithms for calculating with fractions well and they were
able to apply them when solving a wide range of problems. The situation
in another class of the same year group was similar. Therefore we were
looking for “problem” settings in which our pupils could utilize their
earlier knowledge and in which we could at the same time check that
algorithms for calculations with fractions have been acquired on such a
level that pupils would be able to use them competently in non-standard
situations. In the end we decided to use the setting of unit fractions.
With Older Pupils
When working with fundamental fractions it became evident that pupils
have acquired algorithms for calculations with fractions well. The
evidence is the fact that the pupils were able to use the algorithm for
adding, as one of them said, even “from the other side”. What he meant
is that he had decomposed unit fractions into the sum of two suitable
fractions.
The pupils were interested in fundamental fractions and five out of
six groups even carried on with their work at home. One pupil (Hanka)
gathered results of work of all groups and she organized them neatly into
a table. This table was used for further work.
We became interested in the following questions:
•
Why is it possible that different fractions can be recorded in the
same way with the use of fundamental fractions?
180
•
•
•
How comes that the same fraction can be recorded in different ways
with the use of fundamental fractions?
How many fundamental fractions do we need to record a particular
fraction?
How could fractions be sorted out with regard to their record
through fundamental fractions?
When answering this, the following ideas emerged:
•
Some fractions (for example those with denominator five, seven,
eleven, thirteen, …) show the same characteristics as primes?
• Can we record a fraction through a limitless number of fundamental
fractions?
• Would it be possible to describe the search for fundamental fractions
in a systematic way?
It can therefore be said that pupils in this particular class created for
themselves space for the introduction of some mathematical concepts
(transfer of characteristics of primes to other objects, rational number as
a class of decomposition, the sum of infinite number series, the concept
of variable), which many pupils will not perhaps encounter in full
perspective in their future studies. Nevertheless, thinking about them
may, according to our own experience, considerably contribute to the
better cultivation of their mathematical thinking.
With respect to diagnosis of the level of acquisition of calculations
with fractions, it was interesting to observe how different groups of
pupils chose various strategies when looking for unit fractions. The most
common strategy was the one based on subtraction of fractions. This
strategy was used by three groups.
When looking for decomposition, one group used the process of
raising fractions to higher terms (see the decomposition of 2/15 below).
The other two groups did not work in a systematic way. They combined
various estimations and calculations, when working with fractions with
small denominators they often used their fraction wall to help them find
fundamental fractions. Nevertheless, all pupils, while working on the
project, acquired algorithms for calculations with fractions at such a high
181
level that calculating with fractions in standard situations was no longer
a problem for them.
Our basic goal is exposition and enabling pupils to develop secure
algorithms for numerical calculations with fractions. As we have stated
above, it is equally as important that this project implicitly allows, on
various levels, the introduction of a number of important mathematical
concepts and the cultivation of better mathematical thinking in our
pupils.
Egyptian Fractions - Plans for lessons from a Czech classroom
Learning objectives
Operating with Egyptian fractions is a suitable tool for developing ideas
about unit fractions in a context which has historical and cultural
connections. Pupils can solve simple problems and discuss the ways of
approaching various types of problems dealing with the Egyptian way of
calculating.
Programme
¾ Exploration and discussion:
Introductory task:
You are sure to remember that many years ago Egyptians wrote numbers
in a different form than we do. Even when counting with them they used
different methods than we do now.
a) On a prepared notice board show the method used by the Egyptians
for multiplying two natural numbers. They called it duplication. Try to
describe how the Egyptians divided numbers by using their knowledge
of multiplication methods. Show some examples. Help the pupils to
write their numbers as in ancient Egypt.
Note:
The ancient method of calculation by doubling and halving is very
easy to do and can be adapted for use in a number of contexts.
182
Simple mental and oral exercises to start with may be making the
sequence of ‘doubles’ starting with easy numbers like 2 or 3:
2, 4, 8, 16, 32, 64, ... etc.
3, 6, 12, 24, 48, 96, ... etc.
Children are often fascinated by this and try to see how far they can go
with the sequence.
Alternative ways of multiplying numbers can be developed and
practised with simple situations at first:
257 x 13 is 25 x (10 + 3) or, 270 + (27 x 3)
To multiply 27 x 3 put
double it
and add
so we get
27
1
54
2
87
3
270 + 87 = 357
Another hint to remember is that to multiply by 5, we multiply by 10
and halve the result:
23 x 15 becomes
halve it
so the result is
230
115
345
10
5
15
Since they became fluent in this technique, Egyptian Scribes could
handle quite complicated calculations in fractions as well as whole
numbers.
b) Division of a pair of natural numbers which have a common factor
give a whole number result. However, it becomes clear that it is not
possible to divide any two numbers without obtaining a remainder.
Remainders of division can be expressed in the form of fractions. The
ancient Egyptians used only ‘unit’ (or fundamental) fractions of the form
1/n, where n is a natural number.
Try to express some fractions less than 1 with the denominators from
2 to 9 as a sum of different unit fractions of this form.
Note:
Easy combinations are
3/4 = 1/2 + 1/4 and 4/9 = 1/3 + 1/9 and 3/5 = 1/3 + 1/5 + 1/15
183
The convention was to place the fractions in order of increasing
denominator. However, we find that this procedure sometimes gives us
two equally valid representations of the same rational number:
2/15 = 1/10 + 1/30 but also 2/15 = 1/15 + 1/20 + 1/60
Clearly, there are many possibilities here.
Obtain examples of students’ solutions to problems of this kind and
analyse their attempts.
¾ Ideas for the classroom:
Our experience from students’ work
• Some students' initial enthusiasm gradually diminished. They found
that writing even very simple calculations in the non-positional
system was very tedious. Therefore many of them simplified the
task. They used ancient Egyptian methods but wrote Arabic digits
and used the decimal number system.
• Multiplication by duplication was used by all participants, but only
ten of 28 students used this method for division.
• When working with unit fractions students were significantly
influenced by the geometric interpretation (the part-whole
representation) of fractions as taught in earlier mathematics lessons1.
• When solving the task students themselves grasped the importance of
using the appropriate mathematical language for conveying the
results to others.
• Students got acquainted with methods of multiplication and division
of natural numbers different from these generally used in Czech
schools. Thus, they had the opportunity to compare them.
• Solving this problem led to interesting discussion about the
possibility of expressing every fraction in the desired form. One boy
offered the solution in the form of as infinite series.
For example, one method of conversion can lead to an infinite
sequence. From the example above we can obtain:
2/15 = 1/15 + 1/15 or 1/15 + 1/30 + 1/30 or 1/15 + 1/30 + 1/60 + 1/60 etc.
1
See Pitta-Panzani, D. and Gray, E. in ELTMAPS: Theory, Principles and
Research.
184
¾ Possible ideas for follow-up:
•
•
•
Negative numbers (China)
Time, angles (Sumerians)
Imperial measures (England)
¾ Samples of suitable tasks
Problems from a Czech textbook (Novotná et al., 1998)
Old Egyptians knew only the so called fundamental fractions. These are
fractions with 1 in the numerator and a natural number in the
denominator, e.g. 1/2, 1/3, 1/6, 1/7, 1/12. All other fractions were
expressed in the form of fundamental fractions. They cared to have all
fractions in the expression different.
a) Express the fractions (i) 3/4, (ii) 2/5, (iii) 17/23, (iv) 43/15 in the form
of fundamental fractions.
(i) 3/4 = 2/4 + 1/4 = 1/2 + 1/4 (the fraction is expressed as a sum of
fractions with the same denominator in such a way that we can simplify
at least in one of them)
(ii) 2/5 = 1/5 + 1/5 = 1/5 + 2/10 = 1/5 + 1/10 + 1/10 =
= 1/5 + 1/10 + 5/60 + 1/60 = 1/5 + 1/10 + 1/12 + 1/60
(When it is not possible to use the previous strategy, we choose one of
the fractions and multiply the numerator and denominator by the same
number so that we can write as a sum of fractions; we continue this
procedures until the strategy (i) can be applied.)
(iii) 17/23 = 1/23 + 16/23 = 1/23 + 32/46 = 1/23 + 23/46 + 9/46 =
= 1/23 + 1/2 + 27/138 = 1/23 + 1/2 + 2/138 + 25/138 =
= 1/23 + 1/2 + 1/69 + 23/138 + 2/138 =
= 1/23 + 1/2 + 1/69 + 1/6 + 2/138 =
= 1/23 + 1/2 + 1/69 + 1/6 + 4/276 =
= 1/23 + 1/2 + 1/69 + 1/6 + 3/276 + 1/276 =
= 1/23 + 1/2 + 1/69 + 1/6 + 1/92 + 1/276
185
(iv) 43/15 = 2 + 13/15 = 2 + 26/30 = 2 + 11/30 =
= 2 + 15/30 + 1/3 + 1/30.
c) Betka found another way how of solving the problem a). Example:
2/5 = 6/15 = 1/15 + 5/15 = 1/15 + 1/3
Express 17/23 in Betka’s way and show that it holds:
17/23 = 1/2 + 1/6 + 1/23 + 1/46 + 1/138.
Compare Betka’s expression of 2/5 and 17/23 with the expressions in the
problem a).
Reference
Novotná, J. – Kubínová, M. – Sýkora, V. (1998). Matematika s Betkou 3.
Praha: Scientia. (Czech textbooks for Basic schools.)