© 21St Century Math Projects

- Half-Life and Decay

Benchmarks: Seeing Structure in A-SSE 1a. Interpret parts of an expression, such as terms, factors, and coefficients. Expressions 2. Create equations in two or more variables to represent relationships between Creating Equations A-CED quantities; graph equations on coordinate axes with labels and scales. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning Creating Equations A-CED as in solving equations. 8b. Use the properties of exponents to interpret expressions for exponential Interpreting Functions F-IF functions. 5. (+) Understand the inverse relationship between exponents and logarithms and Building Functions F-BF use this relationship to solve problems involving logarithms and exponents. Linear and Exponential 1c. Recognize situations in which a quantity grows or decays by a constant F-LE Models percent rate per unit interval relative to another. 4. For exponential models, express as a logarithm the solution to abct = d where Linear and Exponential F-LE a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using Models 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.

Possible Isotopes A critical concept in physics and chemistry, half-life is the Isotope Half-Life (years) amount of time required for a quantity of a radioactive Americium-241 432.2 isotope to fall to half of its current value. This concept has Berkelium-247 1,380 proven very useful in modern medicine and archaelogy. Californium-251 898 Nickel-63 100.1 In this assignment, there are five mysterious isotopes that Plutonium-238 87.7 have been kept in a lab for different time periods. Use the Radium-226 1,600 information provided and the half-life formula to discover Silicon-32 170 what these mysterious isotopes actually are and calculate Silver-108 418 when there will be 1 gram remaining. Titanium-44 63 Uranium-232 68.9

Half Life When will 푡 Mystery What is the half-life? 1 ⁄ℎ there be 1 퐴 = 퐴 ( ) Element 0 2 gram left?

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.

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.

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 Carbon-14 dating does have challenges and opponents. One challenge is that scientists have to Ancient Rome 753 BC to 476 consider how much carbon-14 was normal in the Ancient China 2070 BC to 256 BC time period because it is not exactly the same as it is today and its difficult to know for sure. A second Mayan Civilization 250 to 900 challenge is that in this type of research accidentally contaminating the fossil with new carbon-14 can Medieval Europe 500 to 1509 create significant problems with the results.

In this assignment, collectors have brought five items that they Half-Life Formula believe are artifacts of ancient civilizations. Are they real or are 푡 1 ⁄ℎ they fake? Using carbon dating, the half life formula and the date 퐴 = 퐴 ( ) 0 2 ranges of the civilizations, decide if these items could be authentic. where A is the current amount A0 is the initial amount t is the time (in years) Ancient Egypt and h is half-life (5,730 years)

1. 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?

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?

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. 푡 1 ⁄ℎ Half-Life Formula 퐴 = 퐴 ( ) 0 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 Sample 1 was found to have 2.55 x 10-6 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.

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.

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) The torch sample was found to have 8.42 x 10-15 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?

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Half Life When will 푡 Mystery What is the half-life? 1 ⁄ℎ there be 1 퐴 = 퐴 ( ) Element 0 2 gram left?

1 Radioactive isotope #1 has been kept for 1.5 years. Originally, there were 100 grams and 1127.95 now there are 99.39 grams. 170 Silicon more years

2 Radioactive isotope #2 has been kept for 2.75 years. Originally, there were 330 grams and 834.7 more now there are 323.78 grams. 100.1 Nickel years

3 Radioactive isotope #3 has been kept for 1 year and 3 months. Originally, there were 180 grams 10337.5 and now there are 179.887 grams. 1380 Berkelium more years

4 Radioactive isotope #4 has been kept for 3 and a half years. Originally, there were 15 grams and 1685.2 now there are 14.916 grams. 432.2 Americium more years

5 Radioactive isotope #5 has been kept for 6 years and 9 months. Originally, there were 88 grams 400.2 more and now there are 81.70 grams. 63 Titanium years

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.

1. 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.

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

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.

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

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.

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.

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.