 Lesson plan and activities for teaching U-Pb radiometric dating
By
Britta Bookhagen, Noah McLean, Frank Dudas, Sam Bowring & Jennifer Hamon

 Grade Level 10-12 Time required: This lesson plan is designed for a 90 min period, but could easily be divided into two separate periods of 45 min each Content objective: Students will learn how scientists use radioactive decay to determine the ages of rocks. They will be introduced to the concepts of half-life and decay rate, and learn to use the decay equation, which is fundamental to geochronology.

Interested teachers can now download the entire lesson plan as one pdf here.

Synopsis of this lesson plan (click for link or scroll down):

Overview of the activities | Principal concept | Associated concept | Context | Objectives | Required materials | Instructions | Pre-knowledge

Overview of the activities:
There are three interrelated parts in this exercise and each builds upon the acquired knowledge from the previous one. Students may work in groups (preferably 3 students per group).

• Part 1: The Concept of Half-life
Part 1 introduces the fundamental concept of radioactive decay and the definition of half-life through worksheets and graphs, and connects the determination of a date with the ratio of daughter to parent atoms. The student workload involves completion of a worksheet and a page of short-answer questions.

• Part 2: Ratio measurement: a hands-on age determination exercise
Isotopic measurements and geochronological age determinations are based on using isotopic ratios - not absolute numbers of atoms. This part is designed so that the students understand this approach. They determine ratios using colored discs to model parent and daughter isotopes; they then calculate a date for their sample and that of another group's, using the graphs from part 1.

• Part 3: Ratio measurement: a hands-on isotope dilution exercise
This exercise demonstrates that it is possible to determine the total number of atoms present in a sample without counting them all. Students add a known number of red ("tracer") jellybeans to a large unknown number of jellybeans of various colors (excluding red). Students then take several small samples of the large number of mixed jrelly beans, count the ratio of red beans to other colors in each small sample, then average these ratios. Using these small-sample averages as a precise measure of the large unknown sample, they can calculate the total number of jellybeans in the original mixture. These averages can also be used to determine an age, again using the graphs from part 1.

Principal Concept: Dates and Rates
Radiometric dating is a method used to determine how old rocks and minerals are. The dates can then be used to determine when a specific geologic event, like meteorite impact or volcanic eruption, happened. In addition, dates allow us to estimate rates (event/time) of processes such as climatic change, sediment accumulation, mountain building, and evolutionary change. No Dates=No Rates

Associated Concept: Measurement and Interpretation
This exercise explores the relationship between physics, chemistry, and geology, including making and interpreting scientific measurements. There is the possibility for the exercise to be extended to discuss mathematical formulation of the measured relationships as well as the estimation of uncertainties within the measurements.

Context:

This exercise is intended to be compatible with any geoscience curriculum - to illustrate the methods and applications of radiometric dating to understanding important events in earth history and how we calibrate the geologic time scale. If needed the exercise can be modified and used to illustrate and compare the fundamental geologic concepts of absolute dating (e.g., U/Pb, radiocarbon dating, dendrochronology, etc.) and relative dating (e.g., superposition, cross-cutting relationships, index fossils and biostratigraphy, etc).
Important concepts from physics and chemistry (atomic structure, isotopes, radioactivity) as well as geosciences (planetary development, plate tectonics, climate change) and biology (paleontology, evolution) on which this exercise depends will be utilized and reinforced. The exercise gives students insight into the multidisciplinary nature of geology and how scientists work together to determine and apply accurate and precise radiometric dates.

Objectives:

• Students will:
• work in learning groups and learn the value of team work;
• complete a table from given initial values;
• plot these values in labeled graphs (decay curves and ratio of remaining daughter isotopes) and interpret the graphs;
• work with and compare isotope ratios (parent/daughter ratios at a given age do not change with sample size); and
• measure the age of a sample (with a model) using parent/daughter ratios.
Required Materials for this exercise

• prepared worksheets (note: 2 slightly different types)
Worksheet 1 A
Worksheet 1 B
• graph paper
• discs with different colors on each side (e.g., black on one side, white on the other)--if only one color is available, they can to be glued together or marked/labeled on one side
• jellybeans, 3 or 4 different colors (at least five 9 oz bags. =1500 beans is ideal.)
• bowls to contain and mix jelly beans
• calculators
• white board for sketching results (or a computer with PowerPoint and projection capability)
Estimated cost for materials (jelly beans, bingo chips) sum to ~ \$20; most materials can be re-used.

Instructions

Part 1: Understanding Half-life (20-25 min)

Introduce the topic of radioactive decay. To initiate a discussion you may ask the following questions in the beginning part of your lecture session. What are some example of radiometric dating? What physical principle is being used?
You will realize that many students may have heard of one or two methods of absolute dating, such as radiocarbon dating or potassium-argon dating; the general theme to develop is that the same principle - radioactive decay - is used in all forms of radiometric dating. Introduce the concept of half-life, linking atomic abundance with time. Depending on your students, you may find it useful to distribute a copy of the included glossary of important background terms. Good ways to discuss radioactive decay and decay constants are to explain exponential growth/decay.
For instance, one could use the example of an hour glass, where sand grains (atoms) move from one location to another at a constant rate. If we know the rate we can calculate elapsed time by counting grains and dividing by the rate they fell. In contrast, with radioactive decay the number of decays changes with time, following an exponential curve. The number of daughter atoms that are generated by radioactive decay isn't constant, but instead depends on the number of parent atoms around. Since the number of parent isotopes declines as they decay, the number of daughter atoms generated during through time decreases. Put another way, each half life generates only half as many new daughter atoms as the half-life before. We can't divide the number of new atoms by a constant rate that they're created; instead we use the radioactive decay equation, which is exponential.

Worksheet 1 A
Worksheet 1 B
Teachers' Solutions

Notes:
• There are two different worksheets, starting with different numbers of parent atoms. Different groups should not know that other groups have worksheets with a different number of parent atoms, because the ultimate point is that the daughter/parent ratio, as a function of half-life, is independent of the number of initial parent atoms. For the worksheets appended to this exercise, one starts with 128 parent-isotope atoms, and one with 96.
• Depending on your students, you might want to go through the first columns of the table to make sure they have the right idea before students start filling in the worksheet. If possible, this should be done by helping the groups separately while they work on their tables, so you can avoid telling them that different groups have different initial numbers of parent atoms.
• Point out that the worksheet asks for the ratio of daughter- to parent-isotope atoms and remind the students that the ratio is asking for the daughter atoms to be in the numerator. This is important for plotting the "data" graph.
• As a helpful tip, and a way of encouraging students to check their work, remind your students that the total number of atoms in each column (the total of parent- and daughter- atoms) always stays the same: matter is conserved!
After finishing the table, students should graph their results.
One plot should show the change in number of parent atoms over time, one the change in number of daughter atoms over time, and the third the change of the daughter to parent atom ratio over time.
Challenge the students to think before they plot the first two graphs (Talking Point: Are the number of parent or daughter atoms decreasing or increasing with time, and how should the graphs look to represent this?). Remark that the x-axis in all three graphs represents time.
Briefly discuss the first two graphs. The parent-isotope graph shows the initial rapid decrease in the number of parent-isotopes with time, and the slower decrease as time passes. The daughter-isotope graph shows an initial rapid increase, then a slower increase in the number of daughter-isotopes with time.
The third graph shows the increase of the daughter/parent isotope ratio with time.

This graph of the "daughter/parent vs. time" should be projected.
Either prepare two overlays with the ratio vs. time grid on it ahead of time, or have two groups of fast students (from groups with different numbers of initial parent-isotopes) plot their data on the overlays. Alternatively, you could prepare to show the plots using PowerPoint. In either case, be sure to emphasize that the ratio graphs are identical even though the groups had different numbers of parent isotope atoms to start with! To clarify this, you can also write a table on the board where students plot their different starting numbers and the ratios (they will easily see that the ratio is the same).

If you have particularly sharp students, they may ask about the mathematical formulation of the daughter/parent isotope relationship, and you can extend the discussion of the ratio vs. time graph to develop the exponential decay equation, D = P0eΛt. (D is the number of daughter atoms; P0 is the number of parent atoms; Λ is the decay constant; t is time). This relationship is general and applies to all radioactive decay systems. For different systems, the value of Λ changes, making different decay systems applicable to somewhat different ranges of geological time.

Part 2: Ratio measurement and age determination (20-25 min)
In this exercise, students "create" dates using the poker chips as a model for parent and daughter isotopes. Explain that students are going to work with the decay of the parent isotope 235U to the daughter isotope 207Pb. One side of each poker or bingo chip represents the parent-isotope atom (e.g., white for 235U), while the other side represents the daughter-isotope atom (e.g., black for 207Pb). By flipping a poker chip from white to black, the students are modeling radioactive decay.
Each student group should get 20 to 30 poker chips with a different color on each side. Have the students start with all the chips parent-side up, then create isotope ratios by flipping the chips over. Each group can pick an arbitrary ratio, and simulate radioactive decay to achieve their ratio. This mimics the state scientists find today when they measure a U/Pb ratio in a sample. Alternatively, you can select the ratios you want the different groups have. Given these ratios, each group should then figure out how to determine the age of their "sample". If they need help, give them a hint that they should use the plot with the ratio of daughter/parent vs. time from Part 1. This is a simple exercise in reading graphs. They get an estimated age by finding the ratio on the y-axis, and then tracking parallel to the x-axis until they hit the curve they plotted in Exercise 1.

Depending on time, the groups can exchange ratios and repeat the age-determination process for several daughter/parent values.

If time allows, the exercise can be used to stimulate a discussion of how sensitive the age determination is to the daughter/parent ratio. Early in the decay history, there is only a little of the daughter isotope, and small changes in the D/P ratio result in large changes in calculated age. Conversely, late in the decay history, there is only a little of the parent left, and relatively large changes in the ratio produce relatively small changes in the calculated age. This discussion can be extended to include the concept of uncertainty in ratio measurement, and the effect of this uncertainty on the calculated age.

Part 3: Ratio measurement and isotope dilution (20-25 min)
Isotope dilution: Explanation for teacher
In geochronology, one must develop a bookkeeping method for parent and daughter isotopes. Every step in the process of separating and measuring parent and daughter isotopes is not perfectly efficient, and we cannot assume that we end up with every atom we started with. To make sure we account for this, we use a clever technique called isotope dilution. The technique is conceptually simple: we add a precisely determined number of tracer isotope atoms to our sample when we begin our work. Often these tracer isotopes are not present in the sample, and in some cases do not occur in nature. Once the tracer is added, the ratio of the naturally occurring isotopes to the added tracer isotope remains the same?no matter how much of the mixture we lose along the way.

Let students model measurement of isotope abundances using jelly beans.

In this part, students learn that measuring isotopes is all about measuring ratios rather than absolute abundances of uranium and lead isotopes. This exercise models a procedure called isotope dilution, in which a known amount of a tracer isotope is added to a sample, "diluting" the naturally occurring isotopes, before the isotopic ratios in the mixture are measured. From the ratio of tracer atoms to naturally occurring atoms, the number of naturally occurring atoms in the sample can be determined.

Place a bowl with a large number of well-mixed, multi-colored jellybeans in front of the class. Ask the students how they could measure (not guess!) the number of jellybeans without counting them all. If they imagine that the jellybeans represent billions of atoms of various isotopes of uranium and lead, it becomes obvious that it would be impossible to count them all.

Challenge them to think about how measuring ratios could help in measuring the total number of jellybeans. Develop the idea that, by adding a small, known number of red jellybeans (representing a tracer isotope) to the whole bowl, mixing them together, and then measuring the ratios of the unknown beans to the red beans in a small sample, we can make a "measurement" of the total number of jellybeans. Have one group count the red beans before they are mixed with the rest (or tell them the number of the red beans), then mix them together.

Each group should then take a sample of beans (about full cup) and determine the ratios for green to red, yellow to red and so on. Compile the ratios from each group of students in a table on the board (green to red, yellow to red and so on), and have them calculate the average for each ratio across groups. From the mean value for each ratio, they can determine the number of each color of beans (total amount of red beans multiplied by the ratio), and thus measure the total number of beans. It will be obvious that this measurement is an estimate, and that the range of ratios between different groups can be used to estimate the uncertainty of the measurement. After students have calculated these numbers, tell them this is the principal scientists use to measure isotopes in a mass spectrometer: the mass spectrometer determines the ratio, not the total abundance of atoms. As the number of ratio determinations increases, the uncertainty of the average ratio decreases; for most measurements, the number of ratio determinations can be adjusted to achieve a pre-determined level of precision.

If there is time left, you can have the students count the different colors and see how close their estimates were, or you can tell the students the actual number of beans if you've counted in advance.

Notes:
• In practice, the initial bowl of jellybeans should contain beans of 3 or 4 colors only ? it is hard to keep track of a larger number of colors. The initial mixture of jellybeans should not include any jellybeans of the same color as the "tracer" jellybeans. It is best if the total number of jellybeans in the initial mixture is exactly known (i.e., the teacher or a teaching assistant should count the jellybeans of each color separately; this is part of set-up for this exercise), so that the uncertainty of the "measurements" compared to the "real" can be calculated.
• There is a temptation to eat some of the jellybeans. Because they are or have been handled by several people, this is unhygienic: DO NOT LET STUDENTS EAT THE JELLYBEANS! But the problem of eating jellybeans can be used to illustrate a fundamental principle of dating: if the sample that is being dated has lost any atoms (eaten jellybeans), or gained any atoms (somebody added jellybeans to the mix while the teacher wasn't looking!), the "measurement" of the isotope ratio will be incorrect (compared to the value that the teacher expected). Samples for dating have to be closed systems with respect to the parent and daughter elements.
• This exercise works poorly if the total number of jellybeans is less than about 1500, if the jellybeans are not well-mixed, and if the individual groups have samples of beans smaller than about 200. In each cases the issue is that we are dealing with the statistics of small numbers: a single red jellybean added to or removed from a small sample will produce a large change in the "measured" ratio. Averaging the "measured" ratios over 5 or more student groups helps to reduce the uncertainty. In reality, the mass spectrometer, under computer control, makes 100 or more ratio measurements of thousands to millions of atoms per ratio, so that the statistical uncertainty of the measurement is minimized.
• Perceptive students might ask how this method could work if the original bowl of jellybeans already contained some of the red beans we used as a tracer. In practice, this can be a concern: in some cases, the tracer isotope is also a naturally occurring isotope. There are two approaches to this problem. Isotope ratios in the natural sample, without the tracer, can be measured directly; this measurement doesn't give any abundance data, but when combined with a second measurement on a sample with the tracer, the difference in the ratios of "spiked" and "unspiked" samples yields the measurement of the naturally occurring abundances. A second approach is to take identical samples of the naturally occurring material, and to add known but different amounts of the tracer to the samples. By plotting appropriate ratios of the tracer isotope to naturally occurring, non-tracer isotopes, the abundance of the naturally occurring tracer isotope can be calculated: this is similar to the isochron method of dating, and is common practice in analytical chemistry (the method of standard additions).
• Always keep in mind that radiometric dating can generally be used for a variety of rock types. In geology it is often used for igneous rocks, as this method determines the amount of time that has passed since a mineral or rock formed .

Pre-Knowledge: Chemistry and Physics Vocabularies

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