This Warm-up reinforces the idea that finding the growth or decay factor for a fractional interval of input involves taking a root of (rather than dividing) the factor for a whole-number interval. Given a description of how an exponential function changes over some hours, students explain why the growth or decay factor for one hour is a certain number. In doing so, students practice reasoning quantitatively and abstractly (MP2).

Invite students to share their explanations. If no students reason about the hourly factor in terms of what it must equal when raised to a particular exponent (6 for the first function and 4 for the second), as shown in the Student Responses, bring this up.

For example, for the second question, point out that if the substance decays by in 4 hours, the decay factor for one hour, say , must be such that , or .

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).

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.