In this activity, students use fractional exponents to answer questions about amounts of radioactive isotopes in old artifacts. Unlike in previous activities in which the input values of the exponential functions were straightforward, more reasoning is needed here to determine the input values to use. Students are not expected to answer rigorous questions about finding the ages of artifacts at this point, as doing so requires finding unknown exponents. That work will occur later, when students are equipped with the knowledge of logarithms.

To answer the last question about the amount of carbon-14 in the fossil today, students may:

• Find the exact (or estimated) number of years since 20,000 BCE, substitute it for in the equation of the function, and calculate the output.
• Make a new table counting by intervals of 5,730 in the years column and multiplying the value in the mass column by for each additional row until the years column is about 22,000. (This would include using a spreadsheet tool.)
• Estimate the number of half-lives between 20,000 BCE and today, and multiply the amount by that many times.

Monitor for students who use each strategy to share their reasoning during the discussion.

Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

If students mistake the decay factor after 1,910 years as (or ) because 1,910 is a third of 5,730, and then calculate the amount of carbon-14 left as , or 0.5, picogram, consider asking:

• “Can you explain how you found how much carbon-14 remains after 1,910 years.”

• “Recall the Warm-up in which we learned that if the four-hour growth factor of an exponential function is, for example, , the hourly growth factor is . How does the four-hour growth factor connect to the hourly growth factor?”

Begin the discussion by inviting students to share how they completed the table, focusing on the second row (1,910 years). Make sure students recall that finding the amount of carbon-14 after 1,910 years means multiplying the original amount (3 picograms) by a factor of , or , because the function at hand is exponential and 1,910 years is of 5,730 years.

Next, select previously identified students to share their strategies for the last question, having as many strategies as possible represented. If no one brings up the fact that the present time is less than four half-lives from 20,000 BCE, which means more than , or , of the carbon-14 remains, point this out. Connect this to the idea of equal factors over equal intervals—the intervals being 5,730 years in this case.

In this activity, students continue to reason about half-lives and exponential functions to solve problems. Both questions provide possible reasoning for a calculation and then prompt students to construct an argument to justify their position (MP3).

The context continues to be the decay of carbon-14, and students investigate two different situations. In each case, the amount of time that has passed since the carbon-14 began to decay is not a multiple of the half-life. This means that students need to make estimates, either in terms of a fraction of a half-life or a whole number of half-lives. 

Focus the discussion on the estimates students made and the properties of exponential functions that they used. Here are some questions for discussion:

• “About how old is the papyrus? About how many half-lives of carbon-14 is that?” (2,600 years, half-life)
• “By what factor does the amount of carbon-14 decrease in one half-life? What about in half of a half-life?” ( and )

One thing to note to students is that carbon-14 dating, while improving, is not an exact calculation but rather a good approximation for dating items from long ago. Consider telling students that carbon-14 dating is used only for objects that are less than about 50,000 years old. Ask,

• “About how many half-lives is that?” (9)
• “About what fraction of the carbon-14 would be left in a 50,000-year-old artifact?” ( or )