Lesson 7

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 \(\frac12\) in 4 hours, the decay factor for one hour, say \(b\), must be such that \(b \boldcdot b \boldcdot b \boldcdot b = \frac12\) or \(b^4 = \frac12\).

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 \(t\) 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 \(\frac12\) 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 \(\frac12\) 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).

Give students a brief overview of radioactive dating. Explain that a radioactive element is an element that emits rays or particles of energy when its atoms break down. Some radioactive elements break down at an exponential rate, and their decay rate is measured in terms of half-lives. Carbon-14 is one of those radioactive elements whose decay rate is known: its half-life is 5,730 years. Because of this, scientists commonly use the amount of carbon-14 in ancient artifacts to estimate their age or date of origin.

Point out that the amount of carbon-14 found in most artifacts is incredibly small. Although carbon is one of the most abundant elements, the most common form of carbon is carbon-12 (which has the same number of protons but a different number of neutrons than carbon-14). For every one part of carbon-14, there are about 1,000,000,000,000 (1 trillion) parts of carbon-12!

If students struggle figuring out how much carbon-14 remains after 1,910 years, consider asking them how 1,910 compares to 5,730. (It is one third.)

Some students may mistake the decay factor after 1,910 years as \(\frac13 \boldcdot \frac12\) or \(\frac16\) because 1,910 is a third of 5,730, and then calculate the amount of carbon-14 left as \(3 \boldcdot \frac 16\) or 0.5 picogram. Ask them if this amount makes sense given the amount of carbon-14 after 5,730 years (which is 1.5 picograms). Remind them of earlier activities in which we learned that if, for example, the hourly growth factor of an exponential function is 16, the factor for a quarter hour is \(\sqrt[4]{16}\), not \(\frac {16}{4}\).

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 \(\left(\frac12\right)^\frac13\) or \(\sqrt [3]{\frac12}\) because the function at hand is exponential and 1,910 years is \(\frac13\) 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 \(\left(\frac12\right)^4\) or \(\frac{1}{16}\) 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. 

In case students are unfamiliar with papyrus, explain that it is a material that was used as a writing surface in ancient times, similar to how paper is used today. Papyrus is made out of the papyrus plant and once it is harvested and turned into a type of paper, the amount of carbon-14 in the papyrus starts decreasing.

As in the previous activity, students may mistakenly think that after about \(\frac{1}{2}\) of a half-life, only about \(\frac{1}{4}\) of the carbon-14 remains. Ask them if more or less than \(\frac{1}{2}\) of the carbon-14 remains after 2,600 years. (Less since this is less than a half-life.)

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, \(\frac{1}{2}\) half-life)
• “By what factor does the amount of carbon-14 decrease in one half-life? What about in half of a half-life?” (\(\frac{1}{2}\) and \(\frac{1}{\sqrt{2}}\))

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 only used 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?” (\(\frac{1}{2^9}\) or \(\frac{1}{512}\))