1.Read
the following text about the Origins of Mathematics.
Tallies and Tablets - The Origins of Mathematics
By Colleen
Messina
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1 Have you
ever wondered what a teraflop is? No, it is not a clumsy, prehistoric fish. A
teraflop is a unit that measures the speed of computer calculations. One
teraflop is 1 trillion calculations per second, and today there are computers
that can sustain speeds of 35.86 teraflops! These incredible electronic
calculations originated with the idea of numbers and counting. Most of us take
math for granted, but numbers and counting have taken thousands of years to
develop. So how did it all begin?
2 Tens
of thousands of years ago, our ancestors found their food by hunting for meat
and gathering wild plants. Survival was a constant struggle. Little did they
realize that some mathematics could vastly improve their lives. For example, if
they knew when certain berries were ripe, they could save themselves a lot of
wandering time by only going to the berry thickets at precisely the right
moment. The hunters and gatherers of ancient times needed something constant in
their environment to help them track time.
3 The
most constant thing in their world was the sky since the landscape changed
through the seasons of the year. Early peoples observed the geometry in nature,
the cycles of the seasons, and the splendor of the Milky Way. Our ancestors
noticed the moon's pattern of becoming full, then slender, then full again in a
recurring thirty-day cycle. This cycle gave them a key to solving the dilemma
of tracking time. With this knowledge, they observed that sour, green berries
took approximately a full cycle of the moon to ripen, so they began to cut
notches in a tree or a stick to keep track of the days of the lunar cycle.
Harvesting the berries and other food became much more efficient with this new
system.
4 The
idea of keeping track of the lunar cycle sounds simple, but it was a momentous
event in the evolution of mathematics. Our ancestors were keeping a tally,
for the first time, and they probably began to use this form of counting in
other areas of their lives. The earliest known tallies were carvings in bones
dated approximately 15,000 years ago, which were discovered in the area now
known as the Middle East. Putting pebbles or
shells in a pile was another way of keeping a tally. Keeping track of items by
using simple marks or objects was still a long way off from the invention of
numerals, but it was a big step forward.
5 Another
way that early people kept track of things was by using "body
counting." Different parts of the body represented different amounts of
things. For many thousands of years people counted using their ten fingers, and
some tribes took this idea even further. The Paiela tribe, who lived in the
highlands of Papua New Guinea, counted by pointing to different parts of their bodies
to represent different numbers. For example, their little fingers represented
the number "one." Other fingers, wrists, elbows, shoulders, ears and
eyes all represented different numbers up to twenty. Body counting worked fine
as long as there was no need for large numbers.
6 When
our ancestors became farmers, they needed to keep track of larger amounts of
things. Farming probably started when the hunters and gatherers visited a
campsite where they had lived during the previous season and noticed grain
growing from seeds they had accidentally dropped on the ground. They learned to
save seeds and sow crops rather than gather wild plants. They also learned to
keep sheep, goats, and cows in pens and slaughter them rather than hunt for
wild animals. Life became easier, and villages formed since no one had to
wander around to survive. A better system of counting evolved because the men
who became shepherds had to count their animals, and the men who became farmers
had to keep track of their harvest.
2. Answer the
following questions.
1.
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What is
a teraflop?
A unit of measurement for computer calculations
The missing link in man's evolution
A clumsy, prehistoric fish
An archaeological artifact
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2.
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What
does the word, "lunar," mean in paragraph 3?
Moon
Crazy
Musical
Sun
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3.
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Which of
the following is not a fact from the article?
Tallies began about 15,000 years ago
A lunar cycle is 6 months long
A written system of numbers was developed about 5000 years ago
(3000 B.C.)
Sometimes pebbles were used to keep a tally
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4.
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How long
is a lunar cycle?
30 days
3 days
Thousands of years
The article does not say
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5.
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Where
were the earliest known devices for keeping tallies found?
Africa
The Middle East
North America
South America
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6.
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Where
did the Sumerians live?
South America
Africa
Iraq
Iran
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7.
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What is
missing in the Sumerian system of numbers?
1
0
60
10
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8.
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What is
a quipas?
An ancient joke
An ancient numeral
A knot in a cord used for counting
An ancient game
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3.
Read the following text about the Age of Discovery.
The Age of Discovery - Gravity and Gauss
By Colleen
Messina
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1 By the
seventeenth century, mathematics had come a long way from the tallies and abacuses
of the ancient world. Mathematicians had finally adopted the new Arabic
numbers, as well as the symbols for addition, subtraction, multiplication, and
division. Logarithms made difficult problems much easier, and calculus opened
up new possibilities in science. Mathematicians applied these new tools in
exciting ways ranging from world exploration to astronomy. Ships crisscrossed
the oceans to new places, and telescopes scanned the skies and discovered the
elliptical orbits of planets. The understanding of gravity revolutionized
military science. It was truly an age of discovery.
2 The
discovery of gravity especially changed how people viewed the world. Up until
the 16th century, people thought that heavy objects fell faster. A feisty
Italian named Galileo Galilei had another idea. In 1585, he climbed to the top
of the leaning Tower
of Pisa, made sure no one
was down below, and dropped two objects. One object was heavy and the other was
light, but both reached the ground at the same time. Galileo proved that
objects fall at the same rate and accelerate as they fall. Eventually, military
engineers understood that a cannonball shoots out in a straight path, but the
force of gravity makes the cannonball fall downward in a curve called a
parabola. The engineers could then fortify their strongholds in the right
places, and artillerymen could shoot their cannons more accurately. Galileo's
experiment revolutionized military science.
3 Galileo
also did experiments with pendulums that helped clockmakers design accurate
clocks. Seamen needed accurate time-keeping devices to navigate during long
journeys. The weight-driven clocks of the previous centuries were not accurate
enough; now seamen needed to measure minutes and seconds, so the new clocks
were invaluable. Navigators then accurately plotted the daily positions of
their ships on maps that had vertical and horizontal lines of latitude and
longitude. When they connected the dots on these grids, they saw an accurate
record of the ship's journey.
4 Rene
Descartes, a great French mathematician and philosopher, also liked grids. He
had a big nose and a sheath of black hair that came down to his eyebrows. He
always stayed in bed until late in the morning and said that that was the only
way to get ready to do mathematics! Descartes tied geometry and algebra
together by writing equations for a geometric shape, like a parabola, on a
graph. His analytic geometry became the foundation of the higher mathematics of
today, and some people call him the first modern mathematician. The Cartesian
coordinate system is named after Descartes.
5 Another
mathematician who laid the foundation for higher math was a number-crunching
prodigy. In 1779, three-year-old Carl Friedrich Gauss watched his father add up
the payroll for a crew of bricklayers and pointed out a mistake his father made
in the calculations! When Gauss was 14, a wealthy Duke noticed his incredible
abilities and was so impressed that he sponsored Gauss's entire education. This
patronage was well deserved, as Gauss dominated mathematics of the nineteenth
century.
6 Gauss
first became famous when an Italian astronomer discovered an asteroid in 1801.
Joseph Piazzi accidentally found a minor planet and then lost sight of it in
the bright sky near the sun. This new planet, called Ceres, caused a great rush
of excitement all over Europe. When it
disappeared, astronomers were upset because they didn't know how to find the
new planet again. Gauss used the tables of logarithms he had memorized to
predict where Ceres would reappear. The tiny planet showed up on the other side
of the sun just where Gauss said it would! Gauss received many honors from
scientific societies because of this triumph.
4. Answer the following questions.
1.
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Which
scientist performed an experiment from the top of the Tower of Pisa?
Galileo
Newton
Descartes
Gauss
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2.
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What
field was affected by Galileo's experiments with gravity?
Magnetism
Counting machines
Electricity
Military science
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3.
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Check
which discoveries Galileo made from the Tower of Pisa
experiment.
Objects accelerate as they fall.
Objects fall at the same rate regardless of weight.
A scientist should warn people before dropping objects from
great heights.
The Tower leaned too much to make the experiment useful.
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4.
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What
kind of geometry did Descartes develop?
Lateral
Analytic
Longitudinal
Topographic
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5.
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Why did
Carl Gauss first become famous?
He located a lost asteroid.
A wealthy duke financed his education.
He loved complicated equations.
He was brilliant at an early age.
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6.
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What is
unique about a complex number?
It involves negative numbers.
It involves the square root of minus one.
It is very large.
It is used in complicated equations.
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7.
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What was
John Napier's counting machine called?
Napier's bones
Napier's multiplier
Napier's calculator
Napier's abacus
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8.
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Who was
the first computer programmer?
Albert Einstein
John Napier
Charles Babbage
Ada Lovelace
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