© 21st Century Math Projects
Project Title: Archaeological Dig
Standard Focus: Algebra & Functions
Time Range: 2-3 Days
Supplies: Basic Stuff
Topics of Focus:
-
Exponential Functions and Logarithms
-
Half-Life and Decay
Benchmarks:
Seeing Structure in
Expressions
A-SSE
1a. Interpret parts of an expression, such as terms, factors, and coefficients.
Creating Equations
A-CED
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Creating Equations
A-CED
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning
as in solving equations.
Interpreting Functions
F-IF
8b. Use the properties of exponents to interpret expressions for exponential
functions.
Building Functions
F-BF
5. (+) Understand the inverse relationship between exponents and logarithms and
use this relationship to solve problems involving logarithms and exponents.
Linear and Exponential
Models
F-LE
Linear and Exponential
Models
F-LE
1c. Recognize situations in which a quantity grows or decays by a constant
percent rate per unit interval relative to another.
4. For exponential models, express as a logarithm the solution to abct = d where
a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using
technology.
Procedures:
A.) Students will complete “Half Life”. In this assignment, students will use lab observations to calculate the
half-life of 5 mysterious elements. They’ll use this measurement to determine what the mystery element was.
B.) Students will complete “Carbon Dating”. In this assignment, students will determine whether five artifacts
are possibly authentic or fakes. Using the half-life of carbon-14, students will use the half-life formula to
calculate the approximate age of the artifacts and compare the dates to the dates of the ancient civilizations.
C.) Students will complete “Archaeological Dig”. In this assignment, students be given information about 7
authentic artifacts and must calculate the ages of the samples using carbon-14 dating and will be asked to
summarize their findings based on their research.
© 21st Century Math Projects
Half Life
Possible Isotopes
Isotope
Americium-241
Berkelium-247
Californium-251
Nickel-63
Plutonium-238
Radium-226
Silicon-32
Silver-108
Titanium-44
Uranium-232
Half-Life (years)
432.2
1,380
898
100.1
87.7
1,600
170
418
63
68.9
A critical concept in physics and chemistry, half-life is the
amount of time required for a quantity of a radioactive
isotope to fall to half of its current value. This concept has
proven very useful in modern medicine and archaelogy.
In this assignment, there are five mysterious isotopes that
have been kept in a lab for different time periods. Use the
information provided and the half-life formula to discover
what these mysterious isotopes actually are and calculate
when there will be 1 gram remaining.
Half Life
What is the half-life?
1
Radioactive isotope #1 has been kept for 1.5
years. Originally, there were 100 grams and
now there are 99.39 grams.
2
Radioactive isotope #2 has been kept for 2.75
years. Originally, there were 330 grams and
now there are 323.78 grams.
3
Radioactive isotope #3 has been kept for 1 year
and 3 months. Originally, there were 180 grams
and now there are 179.887 grams.
𝑡
1 ⁄ℎ
𝐴 = 𝐴0 (2)
Mystery
Element
When will
there be 1
gram left?
4 Radioactive isotope #4 has been kept for 3 and a
half years. Originally, there were 15 grams and
now there are 14.916 grams.
5
Radioactive isotope #5 has been kept for 6 years
and 9 months. Originally, there were 88 grams
and now there are 81.70 grams.
© 21st Century Math Projects
Carbon Dating
Invented in the 1940s by Willard Libby, carbon dating is a process by which
archaelogists measure the carbon-14 content in a fossil to determine its
approximate age. With a half-life of 5,730 years, carbon-14 is a byproduct
created through plants through photosynthesis. People and animals acquire carbon-14 by eating
plants and animals. Once an organism dies, the amount of carbon-14 in the organism begins to
decay. By measuring the amount of carbon-14 in a fossil and understanding how much carbon-14
lives in similar organisms, its approximate age can be estimated.
Ancient Egypt
3000 BC to 31 BC
Ancient Rome
753 BC to 476
Ancient China
2070 BC to 256 BC
Mayan Civilization
250 to 900
Medieval Europe
500 to 1509
Carbon-14 dating does have challenges and
opponents. One challenge is that scientists have to
consider how much carbon-14 was normal in the
time period because it is not exactly the same as it is
today and its difficult to know for sure. A second
challenge is that in this type of research accidentally
contaminating the fossil with new carbon-14 can
create significant problems with the results.
In this assignment, collectors have brought five items that they
believe are artifacts of ancient civilizations. Are they real or are
they fake? Using carbon dating, the half life formula and the date
ranges of the civilizations, decide if these items could be authentic.
Ancient Egypt
1.
Half-Life Formula
1
𝑡⁄
ℎ
𝐴 = 𝐴0 ( )
2
where A is the current amount
A0 is the initial amount
t is the time (in years)
and h is half-life (5,730 years)
A collector has a sample of a cloth that they think was from a mummy in Ancient Egypt.
After inspection it is estimated that the cloth would have had about 0.0000013 grams of
Carbon-14 when it was new. Today there was found to be 0.0000009 grams remaining.
Approximately how old is the cloth? Could it have been from Ancient Egypt?
Ancient Rome
2.
A collector owns a piece of a wooden shield that they believe was from Ancient Rome. It
was determined that it likely had about 0.004 grams of Carbon-14 when it was new. Today
there was found to be 0.0037 grams remaining. Approximately how old is the shield? Could
it have been from Ancient Rome?
© 21st Century Math Projects
Ancient China
3.
A collector purchased a set of pottery that they believe was from Ancient China. Based on the
composition of the pottery it is estimated to have had about 4.5 x 10-9 grams of Carbon-14 when
it was new. Today there was found to be 2.9 x 10-9 grams remaining. Approximately how old is
the pottery? Could it have been from Ancient China?
Mayan Civilization
4.
A collector found a bone near former Mayan territory. After careful research it has been
determined that the bone lost about 8% of its Carbon-14. Approximately how old is the
bone? Could it have been from the Mayan Civilization?
Medieval Europe
5.
A collector recently acquired a matching dress and a hat that they believe was from
Medieval Europe. After taking a sample it is estimated that the cloth had lost about 2.7%
of its Carbon-14. The hat had lost about 3.2% of its Carbon-14. Approximately how old are
the items? Could they have been a matching pair? Could it have been from Medieval
Europe?
© 21st Century Math Projects
Archaeological Dig
A series of successful archaelogical digs have taken place all around the world.
In your science lab, samples of interesting materials have been brought in with
the request that they are carbon-dated. With a half-life of 5,730 years, the
amount of carbon-14 that is present in the fossils will unlock an estimation of
an approximate age. For each of the samples use the half-life formula to determine an approximate
age of the material.
1
𝑡⁄
ℎ
Half-Life Formula 𝐴 = 𝐴0 ( )
2
where A is the current amount, A0 is the initial amount
t is the time (in years) and h is half-life (5,730 years)
Dead Sea Scrolls
Background: Found inside a cave near the
shore of the Dead Sea, the Dead Sea Scrolls
were discovered. The scrolls include texts of historical, religious and linguistic significance.
Samples pieces of the papyrus that they were written on were sent to the lab for carbon-14 dating.
Determine their age and an approximate date range that they were first written.
Sample 1
Sample 2
-6
Sample 1 was found to have 2.55 x 10 grams Sample 2 was found to still contain 77% of its
of carbon-14. It is estimated it originally would original carbon 14.
have contained 3.2 x 10-6.
Age:
Age:
Summarize your findings and an approximate date range of the scrolls.
© 21st Century Math Projects
Otzi the Ice Man
Background: In the Alps of Italy, a pair of German Hikers noticed a body lying in a
block of ice. Since the body and clothing was so well preserved, the hikers assumed he
had only recently died. A tissue sample of “Otzi the Ice Man” and a fiber of his clothing
have been shipped to a lab for analysis. How old was this “natural mummy”? Was he
recently deceased?
Tissue Sample
Fiber Sample
The tissue sample was found to have 0.0076 The fiber sample was found to still contain 51%
grams of carbon-14. It is estimated it originally of its original carbon 14.
would have contained 0.0145 grams of carbon-14.
Age:
Age:
Summarize your findings and an approximate date range for when Otzi the Ice Man lived.
Stonehenge
Background: One of the most famous sites in the world, Stonehenge,
stands in Wiltshire England. Researchers believe the first step in its
construction was assembling the bluestones. They believe one of the
final steps was laying the top slabs. Determine an approximate age for
the monument and an approximate length of time that it took to construct it.
Bluestone Sample
Top Slab Sample
The bluestone sample was found to have 1.79 x The top slab sample was found to have lost 36.5%
10-9 grams of carbon-14. It is estimated it of its original carbon 14.
originally would have contained 3.32 x 10-9.
Age:
Age:
Summarize your findings and an approximate date range of the construction of Stonehenge.
© 21st Century Math Projects
Chauvet Cave
Background: In the south of France, archaeologists uncovered the most
well-preserved cave paintings in the world. The paintings include
hundreds of animals from what is believed to be two distinct time periods.
Samples have been taken from torch marks and the paintings for dating.
Torch Sample (Early Period)
Paint Sample (Later Period)
-15
The torch sample was found to have 8.42 x 10
The paint sample was found to have lost 97.35%
grams of carbon-14. It is estimated it originally of its original carbon 14.
would have contained 5.81 x 10-13.
Age:
Age:
Summarize your findings and an approximate dates the Chauvet Cave paintings. Were they from
two different periods?
Wrangel Island Mammoths
Background: Located in the Arctic Ocean between Russia and the United
States, bones of the extinct woolly mammoth were found on Wrangel Island
and the nearby St. Paul Island. It is estimated that mammoths were extinct in
the mainland 10,000 years ago. Based on these fossils, scientists would like to
determine if they lived longer.
St. Paul Bone Sample
Wrangel Island Bone Sample
The St. Paul mammoth bone sample was found The Wrangel mammoth bone sample was found
to have 4.15 x 10-3 grams of carbon-14. It is to have 61.7% of its original carbon 14.
estimated it originally would have contained 9.0
x 10-3.
Age:
Age:
Summarize your findings and an approximate date range of the mammoth populations. Did
mammoths exist longer than previously accepted?
© 21st Century Math Projects
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Half Life
Possible Isotopes
Isotope
Americium-241
Berkelium-247
Californium-251
Nickel-63
Plutonium-238
Radium-226
Silicon-32
Silver-108
Titanium-44
Uranium-232
Half-Life (years)
432.2
1,380
898
100.1
87.7
1,600
170
418
63
68.9
A critical concept in physics and chemistry, half-life is the
amount of time required for a quantity of a radioactive
isotope to fall to half of its current value. This concept has
proven very useful in modern medicine and archaelogy.
In this assignment, there are five mysterious isotopes that
have been kept in a lab for different time periods. Use the
information provided and the half-life formula to discover
what these mysterious isotopes actually are and calculate
when there will be 1 gram remaining.
Half Life
What is the half-life?
1
𝑡
1 ⁄ℎ
𝐴 = 𝐴0 (2)
Mystery
Element
When will
there be 1
gram left?
Radioactive isotope #1 has been kept for 1.5
years. Originally, there were 100 grams and
now there are 99.39 grams.
170
Silicon
1127.95
more years
Radioactive isotope #2 has been kept for 2.75
years. Originally, there were 330 grams and
now there are 323.78 grams.
100.1
Nickel
834.7 more
years
Radioactive isotope #3 has been kept for 1 year
and 3 months. Originally, there were 180 grams
and now there are 179.887 grams.
1380
Berkelium
10337.5
more years
4 Radioactive isotope #4 has been kept for 3 and a
half years. Originally, there were 15 grams and
now there are 14.916 grams.
432.2
Americium
1685.2
more years
63
Titanium
400.2 more
years
2
3
5
Radioactive isotope #5 has been kept for 6 years
and 9 months. Originally, there were 88 grams
and now there are 81.70 grams.
© 21st Century Math Projects
Carbon Dating
Invented in the 1940s by Willard Libby, carbon dating is a process by which
archaelogists measure the carbon-14 content in a fossil to determine its
approximate age. With a half-life of 5,730 years, carbon-14 is a byproduct
created through plants through photosynthesis. People and animals acquire carbon-14 by eating
plants and animals. Once an organism dies, the amount of carbon-14 in the organism begins to
decay. By measuring the amount of carbon-14 in a fossil and understanding how much carbon-14
live similar organisms, its approximate age can be estimated.
Ancient Egypt
3000 BC to 31 BC
Ancient Rome
753 BC to 476
Ancient China
2070 BC to 256 BC
Mayan Civilization
250 to 900
Medieval Europe
500 to 1509
Carbon-14 dating does have challenges and
opponents. One challenge is that scientists have to
consider how much carbon-14 was normal in the
time period because it is not exactly the same as it is
today and its difficult to know for sure. A second
challenge is that in this type of research accidentally
contaminating the fossil with new carbon-14 can
create significant problems with the results.
In this assignment, collectors have brought five items that they
believe are artifacts of ancient civilizations. Are they real or are
they fake? Using carbon dating, the half life formula and the date
ranges of the civilizations, decide if these items could be authentic.
Ancient Egypt
1.
Half-Life Formula
1
𝑡⁄
ℎ
𝐴 = 𝐴0 ( )
2
where A is the current amount
A0 is the initial amount
t is the time (in years)
and h is half-life (5,730 years)
A collector has a sample of a cloth that they think was from a mummy in Ancient Egypt.
After inspection it is estimated that the cloth would have had about 0.0000013 grams of
Carbon-14 when it was new. Today their was found to be 0.0000009 grams remaining.
Approximately how old is cloth? Could it have been from Ancient Egypt?
It is estimated that the cloth is 3039.8 years old. Thus it would have been around 1000 BC which
does fall under the Ancient Egypt time frame so it is possible.
Ancient Rome
2.
A collector owns a piece of a wooden shield that they belief was from Ancient Rome. It was
determined that it likely had about 0.004 grams of Carbon-14 when it was new. Today their
was found to be 0.0037 grams remaining. Approximately how old is shield? Could it have
been from Ancient Rome?
It is estimated that the shield is 644.48 years old. Thus it would be from around 1350-1400 which
is not ancient Rome.
© 21st Century Math Projects
Ancient China
3.
A collector purchased a set of pottery that they believe was from Ancient China. Based on the
composition of the pottery it is estimated to have had about 4.5 x 10-9 grams of Carbon-14 when
it was new. Today their was found to be 2.9 x 10-9 grams remaining. Approximately how old is
pottery? Could it have been from Ancient China?
It is estimated that the shield is 3,632.1 years old. Thus it would be from around 1642 BC which is
in the possible range for Ancient China.
Mayan Civilization
4.
A collector found a bone near former Mayan territory. After careful research it has been
determined that the bone lost about 8% of its Carbon-14. Approximately how old is bone?
Could it have been from the Mayan Civilization?
It is estimated that the bone was 689 years old. This would mean the bone was from around 1350
or so. This is out of the range.
Medieval Europe
5.
A collector recently acquired a matching dress and a hat that they believe was from
Medieval Europe. After taking a sample it is estimated that the cloth had lost about 2.7%
of its Carbon-14. The hat had lost about 3.2% of its Carbon-14. Approximately how old are
the items? Could they have been a matching pair? Could it have been from Medieval
Europe?
It is estimated that the dress is 226.3 years old. This would been it would have been from the late
1700s. This is out of the range so it cannot be from Medieval Europe. The hat would have
been 268.9 years old so it is also not from Medieval Europe. With an approximate 52 year
difference it is not likely they were from the same time period.
© 21st Century Math Projects
Archaeological Dig
A series of successful archaelogical digs have taken place all around the world.
In your science lab, samples of interesting materials have been brought in with
the request that they are carbon-dated. With a half-life of 5,730 years, the
amount of carbon-14 that is present in the fossils will unlock an estimation of
an approximate age. For each of the samples use the half-life formula to determine an approximate
age of the material.
1
𝑡⁄
ℎ
Half-Life Formula 𝐴 = 𝐴0 ( )
2
where A is the current amount, A0 is the initial amount
t is the time (in years) and h is half-life (5,730 years)
Dead Sea Scrolls
Background: Found inside a cave near the
shore of the Dead Sea, the Dead Sea Scrolls
were discovered. The scrolls include texts of historical, religious and linguistic significance.
Samples pieces of the papyrus that they were written on were sent to the lab for carbon-14 dating.
Determine their age and an approximate date range that they were first written.
Sample 1
Sample 2
-6
Sample 1 was found to have 2.55 x 10 grams Sample 2 was found to still contain 77% of its
of carbon-14. It is estimated it originally would original carbon 14.
have contained 3.2 x 10-6.
Age: 1877 years
Age: 2160 years.
Summarize of your findings and an approximate date range of the scrolls.
They would have be approximately written between 146BC to 137AD
© 21st Century Math Projects
Otzi the Ice Man
Background: In the Alps of Italy, a pair of German Hikers noticed a body lying in a
block of ice. Since the body and clothing was so well preserved, the hikers assumed he
had only recently died. A tissue sample of “Otzi the Ice Man” and a fiber of his clothing
have been shipped to a lab for analysis. How old was this “natural mummy”? Was he
recently deceased?
Tissue Sample
Fiber Sample
The tissue sample was found to have 0.0076 The fiber sample was found to still contain 51%
grams of carbon-14. It is estimated it originally of its original carbon 14.
would have contained 0.0145 grams of carbon-14.
Age: 5340 years
Age: 5566.3 years.
Summarize of your findings and an approximate date range for when Otzi the Ice Man lived.
He would have approximately lived between 3552 BC and 3326 BC.
Stonehenge
Background: One of the most famous sites in the world, Stonehenge
stands in Wiltshire England. Researchers believe the first step in its
construction was assembling the bluestones. They believe one of the
final steps was laying the top slabs. Determine an approximate age for
the monument and an approximate length of time that it took to construct it.
Bluestone Sample
Top Slab Sample
The bluestone sample was found to have 1.79 x The top slab sample was found to have lost 36.5%
10-9 grams of carbon-14. It is estimated it of its original carbon 14.
originally would have contained 3.32 x 10-9.
Age: 5107 years
Age: 3754 years
Summarize of your findings and an approximate date range of the construction of Stonehenge.
They would have been built approximately between 3093 BC and 1740 BC. This would be it took
approximately 1353 years to complete its construction.
© 21st Century Math Projects
Chauvet Cave
Background: In the south of France, archaeologists uncovered the most
well-preserved cave paintings in the world. The paintings include
hundreds of animals from what is believed to be two distinct time periods.
Samples have been taken from torch marks and the paintings for dating.
Torch Sample (Early Period)
Paint Sample (Later Period)
-15
The torch sample was found to have 8.42 x 10
The paint sample was found to have lost 97.35%
grams of carbon-14. It is estimated it originally of its original carbon 14.
would have contained 5.81 x 10-13.
Age: 35002 years
Age: 30,013 years
Summarize of your findings and an approximate dates the Chauvet Cave paintings. Were they from
two different periods?
They would have been created approximately between 32,988 BC and 27,999 BC which would
support two different periods.
Wrangel Island Mammoths
Background: Located in the Arctic Ocean between Russia and the United
States, bones of the extinct woolly mammoth were found on Wrangel Island
and the nearby St. Paul Island. It is estimated that mammoths were extinct in
the mainland 10,000 years ago. Based on these fossils, scientists would like to
determine if they lived longer.
St. Paul Bone Sample
Wrangel Island Bone Sample
The St. Paul mammoth bone sample was found The Wrangel mammoth bone sample was found
to have 4.15 x 10-3 grams of carbon-14. It is to have 61.7% of its original carbon 14.
estimated it originally would have contained 9.0
x 10-3.
Age: 6399 years
Age: 3992 years
Summarize of your findings and an approximate date range of the mammoth populations. Did
mammoths exist longer than previously accepted?
They would have been existed until approximately between 4385 BC on St. Paul and 1978 BC on
Wrangel. This means they lived much longer than first thought.
© 21st Century Math Projects