math

due 2022/10/04

1.Read “count-like-an-egyptian”

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Submit a text-only one page (250 word) summary here (unless you are designated as a

discussant for this class).

2.Read about Egyptian fractions (“count-like-an-egyptian-p13-21”)

Fraction exercise #1

Find the problem to divide five by seven, on the right side of p.18 of the Egyptian Fractions

reading. Submit illustrations for each of the two solutions that are similar in spirit to the

author’s illustrations of earlier problems.

Fraction exercise #2: Exotic fraction ruler

Make a strip of paper of any length, like a ruler. One meter would be good, for instance.

Mark the point halfway along its length.

Mark 1/3 and 2/3 in a different color.

Mark 1/4, 2/4, 3/4 in a different color.

Mark all the fifths in a different color.

Mark all the sixths in a different color.

Mark all the sevenths in a different color.

Mark all the eighths in a different color.

Mark all the ninths in a different color.

Mark all the tenths in a different color.

If you do this precisely and pick nice colors, the result is a pleasing and intriguing object you

will want to keep.

Submit a photo.

Fraction Bonus Exercise (optional challenge)

A stick is divided by red marks into 7 equal segments and by green marks into 13 equal

segments. Then it is cut into 20 equal pieces.

Do any pieces contain no marks?

Do any pieces contain more than one mark? Why or why not?

1.Read “count-like-an-egyptian”
Submit a text-only one page (250 word) summary here (unless you are designated as a discussant for this class). 2
Fractions

The Sheltering Arms of the Desert

Egyptian Fractions and Decimals

Egyptian society was concurrent with a number of great
Mesopotamian empires; however, those empires rarely
lasted more than a couple of generations, being brought
down by barbarians or a warlord with a desire for an
empire of his own. Although it was a great civilization,
Mesopotamia suffered from constant turmoil and abrupt
change. Although equally as large, Egypt was more of a
nation than an empire. For almost three thousand years,
Egypt remained relatively stable. There were a few in-
termediate periods consisting of either internal strife or
foreign influence, but these were small compared to the
thousands of years of internal peace.

Egypt owed much of its stability to its unique geog-
raphy. The land simply kept the violent outsiders away.
In order to understand ancient Egypt, however, we must
realize that modern borders meant little in the ancient
world. Egypt was split into two parts: the Nile valley and
the northern delta. The valley is hundreds of miles long
but never more than about ten miles wide. A map of this
part of Egypt would look like a long string winding its
way through the desert. The western desert comprised the
harsh sands of the Sahara Desert. The eastern desert was
a stony, mountainous landscape. Neither side could sup-
port enough people to seriously threaten the Egyptians
who lived on the banks of the Nile, and both sides pre-
sented a sweltering, waterless barrier to outside invaders.

The Egyptians were also relatively protected in the
south. Most nations have trouble defending borders that
span up to a thousand miles long. In order to protect the
entire country, they would need to disperse their armies

Mediterranean

Marsh

Nubia

First Cataract

Desert

Ancient Egypt.

into small, vulnerable groups. The southern border of
Egypt was only a few miles wide because it consisted only
of the width of the Nile and its banks. It was easy to con-
centrate troops at this one point. Egyptians had little fear
of an armada sailing downriver from the south. The Nile
has a series of rapids, called cataracts, along its southern
portion. Large boats could not navigate these waters and
would have to be portaged over land to get around them.

14 Chapter 2

The Egyptians wisely built forts at these points to harass
any fleet that attempted this tactic.

The northern delta was also fairly safe. This triangular
region was also protected on the east and west by desert.
The delta did have some vulnerability due to its long bor-
der with the Mediterranean, but this was not as bad as
it might seem. During most of Egypts history, sea travel
was a risky business. Ship-building technology had not
progressed far enough at this time. Any king would be
taking a great risk by placing the army charged with pro-
tecting his kingdom in a series of not-so-seaworthy ships.
One storm could wipe out the bulk of his armed forces.

Even if an invading force managed to sail a fleet to the
delta, it would immediately encounter problems. Ships
that can survive in the sea are not very good on rivers and
would be at a serious disadvantage against Egypts river
fleet. If the force attempted to disembark and proceed on
foot, they would run into other problems. The delta con-
sists of many small islands surrounded by the branches of
the Nile. This area is swampy and not conducive to mov-
ing heavy military equipment. If an invading army con-
quered one of the islands, they would have to load all of
their equipment back in their boats and sail to another.
Holding an island would force them to leave men behind,
splitting their troops. In any case, the time such opera-
tions took would enable the Egyptians to mount a large-
scale response. Clearly, Egypt was safe until seafaring and
military technology advanced to the point of overcoming
these obstacles.

According to one of the Egyptian theologies, Ptah
formed the world out of chaos, which still surrounded
it. Egypt was the center of this world, and as you moved
away from Egypt, the closer you came to this disorder.
The violent and barbaric ways of the outsiders probably
seemed natural to the Egyptians world view because the
foreigners bordered the primal chaos. Egyptians had little
respect for and wanted little to do with non-Egyptians. In
many ways, this view was justified despite its seemingly
racist overtones. Imagine how a pharaoh whose position
was based on a thousand-year-old tradition would view
a Mesopotamian king who recently secured his position
through violence and plunder. Egypt had little contact

with the outside world and had little desire to change that
situation. As a result, Egypt developed in its own way and
did things radically differently from the rest of the world.
This is not to say that there was no foreign influence on
the Egyptians, but they had the ability to reject or accept
ideas as they saw fit, and they were predisposed to reject
them. Nowhere is this more apparent than in the way they
dealt with fractional values.

The Mesopotamians had a system remarkably close to
our decimal system except that they used base sixty. When
we want to express a half, we write 0.5 in decimal form
because 5 is half of 10. When the Mesopotamians wanted
to write one-half, they wrote something like 0.30, since 30
is half of 60. Actually, they didnt have zeros or a point
symbol, so they would just write a symbol that repre-
sented 30. In any case, this is why today half an hour is 30
minutes. Were still writing the decimal value of time in
ancient Mesopotamian. We essentially use the Mesopota-
mian system today in part because they were not isolated
from the rest of the world of which we are a part.

The Egyptian method seems decidedly strange to the
modern mind. They only used parts, like a fourth or a
tenth. The hieroglyphic representation of a fourth was
simply the number four under the symbol of a mouth.
The mouth symbol makes an r sound in the Egyptian
language. Its possible that they added er to the end of a
number, just as we add th to the end of a number like ten
to form a part, like a tenth.

One-fourth written in hieroglyphs.

So a fifth could be written OAAAAA and a thirteenth as OSAAA.
We will use the modern shorthand of placing a bar over
the number, so f will be used to represent an Egyptian
fourth. Today these numbers are called unit fractions. A
fourth is the fraction . The word unit refers to the 1

fractions 15

in the numerator of the fraction. This is a misnomer be-
cause Egyptian fractions are not fractions in the modern
sense. There is no 1 in the numerator because there is no
numerator.

When modern mathematicians are confronted with
Egyptian fractions, they often shake their heads in dis-
belief. Remember that Egyptian notation could express
fractions only where we would put a 1 in the numera-
tor. There simply was no way to write 25. When Egyptians
wanted to express this value, they had to write d ag. This is
25, since 13 + 115 is 25. Who would possibly want to express
non-integer values as the sum of fractions? The answer is
simple. We would, and we do it every day.

Every math student eventually learns the approxima-
tion 3.14 for . What does the representation 3.14 mean?
If youre sufficiently familiar with the base system, you
should recognize that this is the sum of three parts: a unit,
a tenths, and a hundredths part. So when we write 3.14 we
really mean this:

3.14 = 3 + 110 + 4100

But this can be simplified to the following:

3.14 = 3 + 110 + 4100 = 3 + 110 + 125 = 3 + a + g

Just as we ignore the plus signs between the parts of our
decimal representations, so did the ancient Egyptians.
They might represent the quantity 3.14 as 3 a g or in
hieroglyphs as AAA IS PSSAAAAA. I think part of the problem mod-
ern mathematicians have in appreciating Egyptian math-
ematics is the phrase we use to describe their numbers,
unit fractions. We compare their system to our system
of fractions instead of to our decimal system, with which
it has far more in common.

To truly comprehend the Egyptian system of fractions,
we must deeply understand the properties of our deci-
mal system that make it so effective. Imagine that you
won the lottery. Someone calls you up on the phone and
tells you that you won 28 million and . . . . At this point
your phone drops the signal. The caller was going to say
$28,732,593, but they only got off two of the eight digits.

Yet only hearing 28 million, you have a very good ap-
proximation of what youve won. Technically, the number
is the sum

The sum of the last six terms is dwarfed by the first two
numbers, so we dont really need to add all the terms to
get a good idea of the total value.

The same holds true for decimals. Consider the follow-
ing approximation for the square root of two, 1.414214. To
an untrained eye, this expression is overly complicated.
Its the sum of an integer and six different fractions, each
with a different denominator. It consists of parts mea-
sured in millionths, parts too small to intuitively grasp. Yet
when we look at it, we immediately see a number close
to one and a half, or a little more than 1.4. The beauty
of the decimal system is that it gives as little or as much
information as you need. I can view the representation
of the approximation of the root as a little more than 1.4
or a little more than 1.41 or a little more than 1.414. With
each phrase Im forced to tolerate more complexity in ex-
change for more accuracy. The best part is that the choice
is ours. The rapidly declining significance of the place val-
ues gives us power to adjust our number interpretations
to our needs. It gives us a quick approximation together
with an accurate estimation.

The Egyptian system does exactly the same thing.
Their representation of 3.141, 3 a g a , can be quickly
assessed as a little more than 3 110 . If they need more ac-
curacy, they can include more digits. For contrast, lets
compare this system to our fractional system. Try to
guess the rough value of 45861310. Can you come up with

20,000,000

8,000,000

700,000

30,000

2,000

500

+ 3

16 Chapter 2

an estimate? Even if you can, do you have any idea how
close your approximation is? However, in Egyptian this
number is 3 ada , which is obviously a number less than
a thousandth away from 3 . Clearly the Egyptian system
has more in common with our place-value system. Just
like our decimal system, theres an easy balance between
accuracy and estimation.

One common observation of the Egyptian system of
fractions is that they never would write the same fraction
twice within one number. So you never see a number like
7 g g. This so-called rule is more likely the misinterpreta-
tion of a more general rule of thumb. Think of our deci-
mal representation of this number:

7 g g = 7 + 15 + 15 = 7 + 0.2 + 0.2 = 7.4

Our placement system refuses to accept repeated dig-
its, which is perhaps the reason we take this for granted. If
we tried to force 7 g g into our decimal system we might
get something like this: 7.2.2. We would then interpret this
as a number close to 7.2, but it isnt. Its actually 7.4, and
7.2 is a bad approximation. Similarly the Egyptians would
not tolerate g g. Its not close to a fifth. Its twice as big as
a fifth. To be off by 100% in an approximation is terrible.
The Egyptians would write this fraction as d ag. The frac-
tion is close to a third, not a fifth. Its a third and a little bit
more. Its important that ag is significantly smaller than d
because it is a refinement of an approximation. So they
know its basically a third, and if they need more accuracy,
they can add the fifteenth.

The rule applies to more than just equal fractions. The
Egyptians would never write g h, since a sixth is too close
to a fifth to be a refinement. This number is not close to
a fifthit is almost twice as much. So they would instead
write d d . We can easily check to see that these represent
the same value as follows:

g + h = 15 + 16 = 630 + 530 = 1130

d + d = 13 + 130 = 1030 + 130 = 1130

As a result, Egyptian fractions are fairly easy to read.
But isnt it difficult to work with fractions that have radi-
cally different denominators? We will see that it is true
only if you stick to the modern method, but not if you
calculate like an ancient Egyptian.

The Best Thing since Sliced Bread

An Intuitive Model for Egyptian Fractions

One of the most common duties of an ancient Egyptian
scribe was to pay the workers. This was not always easy
because they often received shares for pay and not a set
salary. A scribe would receive some amount of food and
have to divvy it out fairly according to each workers share
value. Not every job had equal shares, just as today not
every job has an equal salary. For the purpose of this sec-
tion, however, well assume everyone has a share value of 1.

Assume that seven workers get a loaf of bread to share.
This isnt very realistic because they were usually paid in
grain, but lets accept it for arguments sake. The scribe
realizes that one loaf divided between seven men means
that each gets a seventh, which the scribe would record as
j in his records.

So far everything is working out well. However, the
next day the workers get four loaves of bread. The scribe
could cut each loaf of bread into seven pieces and give
each worker four, but it seems like too much cutting and
the workers wont appreciate all the small pieces. The
scribe gets an idea and decides to cut each loaf in half.
Now he has eight pieces, enough to give each worker one
and he has one left over.

One loaf of bread cut up to feed seven workers.

The loaves cut in half make one piece for each of the
seven workers with one left over.

Being an honest scribe, he doesnt keep the one re-
maining piece for himself but cuts it into seven pieces,

fractions 17

one for each worker. The size of the smaller pieces is a
seventh of a half, which is a fourteenth, since 7 2 is 14.

When the eighth piece is cut into seven, the smaller pieces
can be handed out to the workers.

So each worker gets a half loaf and a fourteenth of a loaf,
which the scribe records as af.

Note that weve just performed our first Egyptian divi-
sion involving fractions. We now know that 4 7 = af.
Although this was not their typical method of division,
it does show how natural the Egyptian system is. Its also
possible that their method of fractions was formed from
similar considerations.

Lets try another division.

Example: Divide 3 by 16 using sliced bread.

Before we can begin, we need at least 16 slices so each
worker can get one. If we were to cut each loaf into four
pieces, we would get only 12 slices. Mathematically, this
tells us that 3 16 is smaller than f, so we cant write the
division as f plus some other fractions. If we cut each loaf
into 6 slices, we get 18 slices, which is more than enough

for the 16 workers. Well perform the first slice and mark
off the 16 slices that will be distributed.

We now know that each worker gets h of a loaf with
2 slices left over. If the scribe now cuts the remaining 2
slices into eight pieces, he gets 16 smaller slices, which is
one for each worker. The size of these smaller slices is fk
since we cut an eighth of a sixth and 8 6 is 48. This makes
316 equal to h fk. The final answer is as follows:

Solution: h fk

Three loaves to be divided between 16 workers. After cutting
them in sixths, each worker gets one, leaving two slices.

The remaining two slices get cut into eighths and
distributed to the 16 workers.

Practice: Divide 2 by 5 using sliced bread. Make your
first cuts as thirds.

Answer: d ag

Practice: Divide 4 by 18 using sliced bread. Make your
first cuts as fifths.

Answer: g fg

Note that some divisions cant be done in two slices.
Consider the following problem:

Example: Divide 4 by 5 using sliced bread.

Four loaves cut in half with each of five workers getting one slice.
Three halves are left over.

18 Chapter 2

If we cut the loaves in two, well get eight slices. We
hand out five of them to each of the five workers, leaving
three halves.

We can treat these three remaining halves just as
we would three whole loaves. In order to divide three
loaves into five, we can cut each in half. Note that these

are already halves so half of a half is a fourth. Now the
three halves become six pieces of size f, of which we can
give one to each worker leaving one of the fourths.

We can now cut the remaining fourth into five pieces,
giving each worker a piece of size . This makes 45 equal
to f .

Answer 1: f

The above example shows us that divisions might re-
quire three or more fractions in the solution. This should
not come as a surprise since Egyptian fractions are closer
to our decimals than to our fractions. You should real-
ize that just as decimals may require a lot of digits, like
8.77928347723, so Egyptian math may require many
fractions.

Note that in the above problem we did not need to cut
all of the original four pieces in half. By cutting the first

The three remaining halves can be cut in half. Five of the six
pieces are distributed between the workers.

three pieces in half we would get six halves, enough to
give one to each of the five workers.

We can now cut both the half and the remaining whole
into five pieces each. So now each worker gets a half, a
fifth of a half, and a fifth, or equivalently a g.

The remaining quarter is cut in five pieces and given to the workers.

This time only three of the loaves are cut in half.

Answer 2: g a

Note that we got two different answers for the same
problem. This means that Egyptian fractions can be writ-
ten in more than one way. Specifically we see for the
above problem that f is the same as g a .

Practice: Divide 5 by 7 in two ways using sliced bread.

Answer: h f ahk or j af

You might argue that having more than one way to
write a number is extremely awkward, but as I mentioned
in the introduction, we see difficulties in alien systems far
more easily than we notice them in our own. In our num-
ber system all of the following are exactly the same.

1.75=175%=74=134=6336

At least the Egyptian system is consistent with the way
it portrays numbers even though they are not exactly the
same.

Filling the Void: The fraction 3

Once each year the star Sirius would disappear behind the
sun. Toward the end of this period, the Egyptians would
scan the skies at dawn, searching for its return. The first
day of its reappearance marked the beginning of a new
year. This sign from the gods foretold the coming of the
inundationsoon the Nile would flood, ending the har-
vest season. All of Egypt would then be covered by water
except for the settlements on the banks and the higher
islands of the delta. There wasnt much for the Egyptians

The half and the whole loaf are each cut into fifths.
Each worker gets two slices, one small and one large.

fractions 19

to do except wait for the flood to recede, and yet this was
the most crucial time in the Egyptian year.

The flood carried with it the two most valuable re-
sources in Egypt, water and topsoil. Obviously water is as
good as gold in a land surrounded by desert. The Egyp-
tians built barriers to trap the flood waters and use them
for irrigation in the drier seasons. Water was so precious
that restricting its flow onto your neighbors farm was an
offense punishable by damnation during your souls final
judgment in the Hall of Osiris.

The flood waters also brought the most fertile topsoil
found in the Mediterranean. During its four-thousand-
mile journey, the waters of the Nile picked up nutrients
and then scattered them throughout Egypt, enabling a
huge civilization to flourish in the desert. The color of this
rich soil lent the very name used by the Egyptians to de-
scribe their home, the Black Land.

The size of the flood varied from year to year. Some
farmland was replenished annually. Other areas received
the life-bringing waters only in high-flood years. The
scribes of Egypt assessed the value of farmland based on
how likely it was to be inundated. Too many years of low
waters could result in famine for a culture that was overly
dependent on the bounty of the Nile. However, if the
flood was too high, their homes on the banks would have
been threatened. The Egyptians needed to know the ex-
tent of a years flood, so they invented the nilometer. This
measuring device is essentially a stone stairway that de-
scended to the Nile. As the waters rose, individual stairs
would be covered in water, and marks on the nilometer
would give the depth of the water. Hence the nilometer
was essentially a giant ruler built into the Nile.

Reading a ruler tells us a lot about the way decimals
are used to make approximations. Consider measuring a
pencil with a ruler. In the diagram below, the ruler shows
us that the pencil is somewhere between 8 and 9 inches.
If we approximate the length as 8 inches, well be off by
at most 1 inch.

If we want more accuracy, we could look at the smaller
ruler marks indicating tenths of an inch between 8 and
9. Below we can see that the tip of the pencil falls be-
tween the sixth and seventh tick mark between 8 and 9.
So the pencil length is between 8.6 and 8.7 inches. If we

approximate the pencil length as 8.6, we will be off by at
most 0.1 inches.

There are a couple of things we should note. The first
is that the distance between the rulers marks determines
the accuracy of the measurement. In fact, this distance is
exactly the maximum error. The latter can be calculated
by subtraction. The above pencil point falls between the
8.6 and 8.7 mark, and hence the maximum error is 8.7
8.6 = 0.1. We should also note that in a decimal measure-
ment system, the distance between marks is uniform. This
makes the error the same no matter where on the ruler
our measurement occurs. This is not true for Egyptian
fractions.

If we made a ruler marked with Egyptian fractions,
it would look like the following diagram. In the middle
would be the mark for since its half way. Similarly,
one-third of the way over, we would find d, and so on.
The marks would become more tightly packed for the
smaller measurements. They get so close that eventually
we would have to stop marking the ruler so they didnt
overlap.

The pencil being measured below is 316 of a foot long.
The ruler shows it being longer than a sixth of a foot but

The length is estimated by the marks the point falls between.

The pencil is between 8.6 and 8.7 inches.

The marks are close below :2, and hence the
measurement is more accurate.

20 Chapter 2

less than a fifth. If we convert 316 into Egyptian fractions,
we get h fk. The Egyptian number verifies that it is a little,
specifically a forty-eighth, more than a sixth.

Someone measuring the pencil with the Egyptian ruler
can tell that its more than a sixth but cant exactly tell
by how much. Since the marks are not uniformly distrib-
uted, its not as easy to know the largest possible error
as it would be with a more conventional ruler. However,
we can calculate the error by finding the distance be-
tween the h and g marks using subtraction. Since I havent
taught you yet how Egyptians subtract fractions, well use
modern methods.

g h = 15 16

The common denominator is 30 and we can get both to be
30 by multiplying the 15 by 6 and the 16 by 5 giving

1 65 6 1 56 5 = 630 530 = 130 = d

Note that the error of a measurement between g and h
is d and 30 is 5 6. Its not difficult to show using algebra
that this is always true provided the Egyptian fractions
are adjacent numbers. Hence the error measurement be-
tween a and aa would be at most aa . A mathematician
would phrase this as the error of an Egyptian fraction is
roughly the square of the smallest term.

We actually didnt even need to subtract the two val-
ues; we could just have easily noticed that the two lines on
the ruler are really close. Theyre all really close on the
left side of the ruler, and hence all measurement on this
half would be fairly accurate. However, there seems to be
a problem with the right side. The following pencil is 56 of
a foot long. The best our ruler can do now is to estimate it
as more than . In fact we can write 56 as d in Egyptian
fractions. Unfortunately, a half is a bad approximation for
56. If we treat the as the approximation and the d as the

error term, the error is 66 23% of the estimate, which is too
big to be considered accurate.

This seems to fly in the face of our interpretation of
Egyptian fractions as being a system of arbitrarily good
approximations. However, the Egyptians solved this
problem, and they did it in the most obvious way. Think-
ing in terms of rulers, the difficulty arises because there
are no marks on the right-hand side of the ruler. So the
Egyptians included an extra a mark or two.

By far, the more common of the two fractions added to
their ruler is 23, symbolized by the following hieroglyph.
We will use the modern transcription, , for this symbol.

The choice of symbol is truly inspired. It looks like the
mouth and is used to denote a fraction over the number
1 and a half. This is in fact what 23 is. In modern terms
we get

1 12 =(33)= 13/2 = 23 =

Well look at this relation in more detail later. Right
now lets see the impact it has on approximations in mea-
surement. Heres our previous pencil measurement using
a ruler with the mark added.

Objects that extend into the right half of the ruler are subject to bad
approximations. The error is the distance from the :2 to the pencil tip.

The hieroglyphic 2/3.

The distance between the :3 mark and the pencil tip is smaller than that
of the :2 mark. Hence the error in measurement is reduced.

Now the ruler reads 23 and a little more, and the 23 is a
much better approximation than the 12 we had previously.

fractions 21

The Egyptians would express the length of the pencil as
h rather than d. If we treat the length as roughly

with an error of h, we get a 25% error, which is much bet-
ter than the 66 23% error we got with d.

You might argue that the gap on the ruler to the right
of the is still a bit large. If you do, youll find that a fairly
small group of ancient Egyptians agree with you. Its ex-
tremely rare, but there are instances of a special symbol
for being used. Most Egyptians seem to have felt that
the 23 symbol was sufficient, and we will restrict ourselves

to , since the mathematical rules involving it are clearly
spelled out by ancient texts.

When I first learned of the symbol, I was uncomfort-
able. Mathematicians like consistency and order. This sym-
bol, being unique, bucked the rules and had to be treated
with special operations. However, as I became more pro-
ficient with Egyptian mathematics, I began to understand
that this gave their mathematical system added flexibility.
In order to begin to appreciate the versatility of Egyptian
math, well need to learn its basic operations. 2
Fractions

The Sheltering Arms of the Desert

Egyptian Fractions and Decimals

Mediterranean

Marsh

Nubia

First Cataract

Desert

Ancient Egypt.

14 Chapter 2

The Egyptians wisely built forts at these points to harass
any fleet that attempted this tactic.

religion

discuss the views of the commentators regarding these verses. Then offer your own reflections on these verses such as what you found interesting, how this topic relates to our own society,

Paper 3

The following verses with the commentaries deal with how the Quran modifies the concept of
generosity from how it was understood in pre-Islamic Arabia. For this paper, discuss the views
of the commentators regarding these verses. Then offer your own reflections on these verses
such as what you found interesting, how this topic relates to our own society, etc.

Directions and requirements:

Essays must be entered by 3:00 PM on Tuesday.
No late assignment will be accepted, and no assignment will be accepted through any

method other than D2L. I will not respond to an email with an attached assignment.
In each essay, you must do the following:

a. Put the critical ideas into your own words. Do not copy and paste the text unless it
is impossible to rephrase it in your own words.

b. Use the chapter: verse number when referencing the verses (dont copy the text of
the verse), and the page number(s) when referencing the commentaries.

c. Entries should not be less than two pages in length
d. Your grade is based primarily on your effort, including content and mechanics.
e. Use a 12-point font and avoid unnecessary blank spaces in your essay.

Here are the assigned verses: 2:261-264; 17:29-30; 2:195; 9:75-77; 64:16-18

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