The Beginnings
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Module 1
The Beginnings
What is this all about?
This module will give an insight on the beginnings of
Mathematics which are important in the succeeding lessons.
It discusses with the beginnings to with conceptions of
number and form take place and how early abstractions
develop. Also, this module will discuss the primitive counting
in the ancient civilization.
At the end of the lesson, you will come up with a
poster in order to assess your understanding on the
application of your knowledge.
What you should know?
At the end of the lesson, you will be able to:
 identify the happenings during the old stone age, new
stone age and bronze age
 perform the Quipu knots
 solve the primitive countings
 create a poster
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Get Started
Answer the following:
1.
2.
3.
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Are you ready?
Below is a list of knowledge that maybe you have already encountered
in your previous mathematics subjects.
Check the appropriate box
corresponding to your agreement to the statement using the given scale
below.
5-Advanced
4- proficient
3 –approaching proficiency
Statements
2- developing
5
4
1-Beginning
3
2
1
I can identify the happenings in the beginning of
Mathematics:
a. old stone age
b. new stone age
c. bronze age
I can perform the Quipu knots.
I can solve the primitive countings
a. Quipu knots
b. Mayan Numeration System
I know how to use the following apps:
a. PowerPoint presentation
b. Google Classroom (Moodle)
c. Zoom
d. Microsoft word
Others, pls specify
________________________
________________________
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Module 1
Read and Learn
The Beginnings
History of mathematics is an area of
study that explores on the persons, events,
and places that bear notably on the growth of
mathematical ideas. This narration ranges
from thousands of years before Christ to the
beginnings of mathematics as a science in the
15th Century and spilling over into the new
millennium. The beginning of Mathematics
started during the prehistoric times where
people kept written records.
Photo retrieved from https://www.shorthistory.org/prehistory/language-and-spiritual-culture-in-old-stoneage/
Dirk J. Struik believed that the first conceptions of numbers were during the Old Stone
Age, also called as Paleolithieum or Paleolithic Age. This period is considered the
longest part of prehistoric times. Throughout this period, people lived in caves. They
made weapons like daggers and spears for hunting and fishing. They also developed a
language in order to communicate with each other.
There was a little progress occurred from
hunting and fishing to agriculture. With
this, the passive attitude of man turned
to active one. This transition is called the
New Stone Age, also called as
Neolithieum or Neolithic Age. The
great happening in the history of
mankind occurred ten thousand years
ago. The ice sheet which covered Europe
and Asia began to melt and it made room
for forests and deserts. Fishermen and
hunters were replaced by primitive farmers. Farmers remained in one place as long as
the soil stayed fertile. They developed pottery, carpentry, and weaving. Inventions ere
made like the potter’s wheel ad wagon wheel. These remarkable inventions occurred
only within the local areas. As compared with the paleolithic times, technical
improvement was enormously accelerated.
Photo retrieved from
The Beginnings
https://www.slideshare.net/gremarmar/neolithic-age-21830345
Page 5
The
discovery
of
the
arts
of
manufacturing and smelting was during
the Bronze Age. The stone was replaced
by bronze as preferred material for
making tools and weapons. This led
to improvements in agriculture and
brought with it changes in the way
people live.
There is further formation of languages.
Also, there was already some simple
numerical terms. Many Australian,
American, and African tribes were in this
stage in their first contact with white
men.
is the mathematics that is persistent in particular cultural
groups. It is derived from ethnos (within a cultural background), mathema
(explaining and understanding in order to transcend, managing and coping with reality
in order to survive and thrive), and tics (techniques such as counting, ordering, sorting,
measuring, weighing, ciphering, classifying, inferring, and modeling).
Ethnomathematics
1.1 Numerical Terms
The ancient numerical terms were qualitative rather than quantitative. The ancient
qualitative origin of numerical conceptions still be detected in certain languages such as
Greek or Celtic. The number concept were first formed by addition: 3 adding 2 and 1, 4
adding 2 and 2, 5 by adding 2 and 3.
Here is an example from some Australian tribes:
Murray River: 1 = enes, 2 = petcheval, 3 = petcheval-enes, 4 = petcheval petcheval
Kamilaroi: 1 = mal, 2 = bulan, 3 = guliba, 4 = bulan bulan, 5 = buls guliba,
6 = guliba guliba
The development of the commerce and of the crafts stimulated the concept of
numbers. Numbers were arranged and bundled into larger units, usually by the use of
the fingers of the hand or both hands, a natural procedure in trading. This led to
numeration first with five, later with ten as a base, completed by addition and
sometimes by subtraction. So twelve was conceived as 10 + 2, or 9 as 10 – 1.
Sometimes 20, the number of fingers and toes, was selected as a base.
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1.2 Primitive Counting
Three or four thousand years ago, in ancient Egypt and Babylonia, there
already existed a significant body of knowledge that we should describe as
mathematics.
Mathematics is an activity that has been present from the earliest days of human
experience. It is commonly accepted that mathematics originated with the practical
problems of counting and recording numbers. Anthropologists tell us that there has
hardly been a culture, however primitive.
Numerical records were kept by means of bundling, strokes on a stick, knots on a
string, pebbles or shells arranged in heaps of fives. The earliest and most immediate
technique for visibly expressing the idea of number is tallying. The term tally comes
from French verb tailler, “to cut”, like the English word tailor; the root is seen in the
Latin taliare, meaning “to cut”. It also interesting to note that the English word write
can be traced to Anglo-Saxon writan, “to scratch,” or “to notch.”
Count were maintained by making scratches on stones, by cutting notches in
wooden sticks or pieces of bone, or by tying knots in strings of different color or
lengths. It is likely that grouping by pairs came first, soon abandoned in favor of groups
of 5, 10, or 20.
A.
Tally Sticks
Tally sticks, a notched stick, have been used since the beginning of counting.
Their use has been universal. Tallying is the most basic form of keeping a local record
about the larger world. Tally sticks can and were used by all peoples that have a ready
piece of wood and something sharp. But it was not limited to primitive peoples. The
acceptance of tally sticks as promissory notes or bills of exchange reached all levels of
development in the British Exchequer tallies (12th century onwards). It took an act of
parliament in 1846 to abolish the practice.
The double tally stick was used by the
Bank of England. If someone lent the Bank
money, the amount was cut on a stick and the
stick was then cut in half - with the grain of the
wood. The piece retained by the Bank was called
the foil, and the other half was called the stock.
It was the receipt issued by the Bank. The holder
of said became a “stockholder” and owned “bank
stock”. When the holder would return the stock
was carefully checked against the foil; if they
agreed, the owner would be paid the correct
amount in kind or currency. A written certificate
that was presented for remittance and checked
against its security later became a “check”.
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Photo retrieved from https://www.math.tamu.edu/~dallen/hollywood/counting/counting.htm
Note how the tally marks match up exactly on the two split sticks. With forgery almost
impossibility, records kept using double tally sticks were very, very secure. It is no
wonder they persisted so long.
B.
Notches as Tally Mark
Bone artifacts bearing incised markings seem to indicate that the people of the Old
Stone Age had devised a system of tallying by groups as early as 30,000 B.C. The
most impressive example is a shinbone from a young wolf, found in Czechoslovakia in
1937; about 7 inches long, the bone is engraved with 55 deeply cut notches, more or
less equal in length, arranged in group of five.
For many years such notched bones were interpreted as hunting tallies and the
incisions were thought to represent kills.
A more recent theory, however, is that the first recordings of ancient people were
concerned with reckoning time. The markings on bones discovered in French cave sites
in the late 1880s are grouped in sequences of recurring numbers that agree with the
number of days included in successive phases of moon.
Another example of an incised bone was unearthed at Ishango along the shores of
Lake Edward, one of the headwater sources of the Nile. The best archeological and the
geological evidence dates the site to 17, 500 B.C. or some 12, 000 years before the first
settled agrarian communities appeared in the Nile valley.
row A
row B
row C
Photo retrieved from https://www.researchgate.net/figure/A-drawing-of-two-sides-of-theIshango-bone-showing-the-grouped-notches_fig1_334988609
It contains groups of notches arranged in three definite rows; the odd, unbalanced
composition does not seem to be decorative. In row A, the notches occur in eight
groups, following order: 3, 6, 4, 8, 10, 5, 5, 7. In row B, the groups are composed of
11, 13, 17 and 19 notches. The last row has four groups consisting of 11, 21, 19,
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and 19 individual notches. The pattern does not necessarily indicate a familiarity of
prime numbers.
Because 11+13+17+19 = 60, and 11+21+19+9 = 60.
It might be argued that the Ishango bone are related to lunar count.
C.
The Peruvian Quipus: Knots as Numbers
In the New Stone Age, the number string is best illustrated by the knotted cords,
called quipus, of the Incas of Peru. They were originally a South American Indian
Tribe, or a collection of kindred tribes, living in the central Andean mountainous
highlands.
Photo retrieved from https://blog.goway.com/globetrotting/2015/08/the-incas-perus-rich-cultural-heritage
The Incas became renowned for their engineering skills, constructing stone
temples and public building of a great size.
The quipus were important in the Inca
Empire, because apart from these knots no
system of writing was developed there.The
quipu was made of a thick main cord or
crossbar to which were attached finer cords
of different lengths and colors, ordinarily the
cords hung down like the strands of a mop.
To appreciate the quipu fully, we should
notice the numerical values represented the
tied knots.
Photo retrieved from https://www.atlasobscura.com/articles/khipus-inca-empire-harvard-universitycolonialism
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Just three types of knots were
used, a figure-eight knot standing
for 1, a long knot denoting one of
the values through 2 to 9,
depending on the number of twists
n the knot, and a single knot
indicating 1.
The figure-eight and long knot
appear only in the lowest (units)
position on a cord, while clusters
of single knots can appear in the
other space positions.
Example 1. Recalling that ascending positions carry place value for
successive powers of ten, let us suppose that a particular cord contains the
following, in order: a long knot with four twists, two single knots, an empty
space, seven clustered single knots, and one single knot.
Solution:
4+(2∙10)+(0∙
)+(7∙
)+ (1∙
) = 4 + 20 + 0 + 7,000 +10,000
=17,024
Example 2.
A long knot with eight twists, three single knots, six clustered single
knots, an empty space, and two single knot.
Solution:
8+(3∙10)+(6∙
D.
)+(0∙
)+ (2∙
Mayan Numeration System
) = 8 + 30 + 600 + 0 + 20,000
= 20,638
Another New Stone culture that used a place numeration system was that
of the ancient Maya. The people occupied a broad expanse of territory
embracing the southern Mexico and parts of what is today Guatemala, El
Salvador, and Honduras.
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Photo retrieved
civilization/
from
https://www.ancienthistorylists.com/maya-history/top-10-inventions-of-mayan-
The Mayan civilization lasted existed
over 2000 years, with the time of its greatest
flowering being the period 300-900 A.D. The
Mayan calendar year was composed of 365
days divided into 18 months of 20 days each,
with a residual period of 5 days.
Photo retrieved from
http://www.webexhibits.org/calendars/calendar-mayan.html
Numbers were expressed
symbolically in two forms. To
indicate the numbers 1 to 19.
The common people recorded the
same
numbers
with
the
combinations bars and dots, where a
short horizontal bar represented
5 and a dot 1. A particular feature
was a stylized shell that served as a
symbol for zero.
Photo retrieved from https://en.wikipedia.org/wiki/Maya_numerals
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Example 3.
Solution:
Example 4.
Solution:
The symbols representing numbers larger than 19 were arranged in a
vertical column with those in each position, moving upward, multiplied by successive
powers of 20; that is, by 1, 20, 400, 8000, 160,000, and so on. Because the Mayan
numeration system was developed primarily for calendar reckoning, there was a minor
variations when carrying out such calculations. The symbol in the third column was
multiplied by 18 ∙ 20 rather than 20 ∙ 20, the idea being 360 was better approximation
to the length of the year than was 400. That is, the multiples are 1, 20, 360, 720,
144,000, and so on.
Example 5.
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Solution:
Example 6.
Solution:
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Module 1
PRACTICE EXERCISE
I. Perform the indicated statements. Show your solution.
1. A long knot with eight twists, three single knots, an empty space,
nine clustered single knots, and five single knot
2. Seven clustered single knots, six single knots, a long knot with five
twists, and nine single knot
II. Write the symbols of Mayan counting system.
1. 19, 0, 3, 5
2. 7, 14, 7, 17, 0
3. 12, 5, 4, 15, 18
4. 16, 19, 19, 11, 10
5. 13, 3, 0, 9, 6
III. Solve using the Mayan counting system. Show your solution.
1. 36
2. 78
3. 95
4. 124
5. 790
IV. True or False
1. 10 = coil of rope
2. 100, 000 = frog
3. 1 = single stroke
4. 100 = God
5. 10, 000 = finger
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Module 1
CONNECT TO THE WORLD
You connect…
. . .with the class
 Submission of practice exercises to Moodle.
Solve at least 2 items in the practice exercise daily.
Write your solutions in your notebook.
Take a photo of your solutions and submit it every Saturday.



 Active participation in an open discussion using Facebook messenger.
Post your queries, comments and even solutions you want to
share to the class in our Facebook Messenger group chat.

. . .with the world


https://www.youtube.com/watch?v=cy-8lPVKLIo
https://www.youtube.com/watch?v=OmJ-4B-mS-Y


Watch the videos to further understand the history of
mathematics.
Share your experience to the class by posting your thought about
it in our group chat.
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Module 1
MAIN TASK
Do this…
Task Prompt:
1. This is an individual activity.
2. You are given two weeks for the preparation of your postermaking.
3. Create a poster illustrating your skills on the following:
a. Math in Old Stone Age, New Stone Age, and Bronze
Age
b. Notches as Tally Mark
c. The Peruvian Quipus: Knots as Numbers
d. Mayan Numeration System
4. You will use ¼ illustration board and may use different
z
coloring materials.
5. When it is done, you may now take a photo and upload in
our Messenger group chat.
6. Your poster-making will be rated according to the criteria
shown in the rubric.
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How you are rated?
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Look back . . . share your thoughts
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