# Teacher Notes

## Student Activity Kit

### Materials Included In Kit

4-sided dice, 20
6-sided dice, 50
8-sided dice, 20
10-sided dice, 20
12-sided dice, 20
20-sided dice, 20
Plastic coins, 100

Cardboard boxes, 15 (optional)

### Disposal

No disposal is necessary.

### Lab Hints

• This activity can easily be completed in a typical 50-minute class period. The Prelab Questions may be assigned as homework to help students prepare for class. It may also be helpful to demonstrate the half-life of coins for the students before they do this activity (see the Background section). One hundred plastic coins have been included in the kit. The cardboard boxes provide a convenient containment for the thrown dice. Alternately, cups can be used to shake the dice and then quickly inverted to roll the dice.
• This activity is also an excellent take-home exercise for students, and may be used as a make-up assignment for students who miss other Radiation” experiments.
• Explain to students that rolling 10 dice ten times is equivalent to rolling a set of 100 dice just once. After the first set of 100 dice has been rolled, students must calculate how many times they must roll their dice in the next round and in each subsequent round. If, for example, 79 dice remain after the first round, then the 10 dice would be rolled 7 times and then 9 dice would be rolled once.
• Assign different groups of students different decay numbers.” For an interesting post-lab discussion, assign a few groups two decay numbers or even three decay numbers rather than just one. (If the assigned decay numbers are 1 and 6, discard all of the dice that land on either of these numbers.) Compare the shape of the decay curve for two or three decay numbers versus that for just one decay number. All of the decay curves will have the same general shape, but the rate of decay will be greater—and the half-life will be smaller—as the number of decay numbers increases. The decay curve obtained using three decay numbers for the 6-sided dice should be the same as the decay curve obtained for coins (see the Background section).
• The rate of decay will decrease, and the half-life will increase, as the number of sides on the dice increases (assuming only one number is chosen for the decay number”). See Table 3 for theoretical half-life values (number of dice rolls) for multifaceted dice as a function of the number of decay numbers assigned.
{12599_Hints_Table_3_Theoretical Half-life Values for the Decay of Multifaceted Dice}

### Teacher Tips

• In addition to simulating radioactive decay and half-life calculations, rolling dice can also be used to simulate the build-up and decay of daughter products in a radioactive decay series. See the Further Extensions section.
• This activity is an excellent opportunity for students to develop spreadsheet skills. The data can be compiled in a Microsoft™ Excel™ spreadsheet, decay curves created from the data, and cell equations created to calculate the various half-lives.
• Radioactive decay is a first-order process—the rate of decay depends on the initial number of atoms or nuclei in the sample (No). After any time t has elapsed, the number of nuclei remaining in the sample may be calculated using Equation 1, where k is the rate constant for radioactive decay. The half-life is the time needed for one-half of the nuclei in a sample to decay. Thus, t½ is equal to the time elapsed when N = 0.5 No. Equation 1 can be rearranged to give the equation for the half-life (Equation 2):
{12599_Tips_Equation_1}

ln(0.5 No/No) = ln(0.5) = –k t½

{12599_Tips_Equation_2}

### Further Extensions

To model a two-step decay series for a radioisotope, start with 20 four-sided dice (parent isotopes). Select a decay number and roll the dice. Record the number of dice that decayed and the number of dice remaining. After round one, remove the four-sided dice that decayed and replace them with six-sided dice. (The six-sided dice represent the newly produced daughter isotopes, which are also radioactive.) Record the number of six-sided dice as well as the number of four-sided dice that remain for round two. (There should be a total of 20 dice in the collection.)

Select a decay number for the six-sided daughter” isotope and roll all 20 dice again. Count and record the number of both four- and six-sided dice that decayed. At the end of the round, replace the four-sided dice that decayed with more six-sided dice. Replace all six-sided dice that decayed with buttons—the buttons represent non-radioactive atoms produced via the decay of the daughter isotopes. Record the number of four-sided and six-sided dice in the sample, as well as the number of buttons. Remember that the total number of “objects” in the sample should be 20 (constant) throughout the simulation. Roll the dice again and repeat the entire process until all of the dice (radioactive isotopes) have decayed—about 30 rounds. A graph of the number of parent isotopes (four-sided dice), daughter isotopes (six-sided dice), and non-radioactive products (buttons) versus the number of dice rolls (rounds) is shown in Figure 7.

{12599_Discussion_Figure_7}

### Science & Engineering Practices

Developing and using models
Planning and carrying out investigations
Analyzing and interpreting data
Using mathematics and computational thinking
Obtaining, evaluation, and communicating information

### Disciplinary Core Ideas

MS-PS1.A: Structure and Properties of Matter
HS-PS1.A: Structure and Properties of Matter
HS-PS1.C: Nuclear Processes

### Crosscutting Concepts

Patterns
Energy and matter
Scale, proportion, and quantity

### Performance Expectations

MS-PS1-1: Develop models to describe the atomic composition of simple molecules and extended structures.
MS-PS1-2: Analyze and interpret data on the properties of substances before and after the substances interact to determine if a chemical reaction has occurred.
MS-PS1-4: Develop a model that predicts and describes changes in particle motion, temperature, and state of a pure substance when thermal energy is added or removed.
HS-PS1-1: Use the periodic table as a model to predict the relative properties of elements based on the patterns of electrons in the outermost energy level of atoms.
HS-PS1-2: Construct and revise an explanation for the outcome of a simple chemical reaction based on the outermost electron states of atoms, trends in the periodic table, and knowledge of the patterns of chemical properties.
HS-PS1-3: Plan and conduct an investigation to gather evidence to compare the structure of substances at the bulk scale to infer the strength of electrical forces between particles.
HS-PS1-4: Develop a model to illustrate that the release or absorption of energy from a chemical reaction system depends upon the changes in total bond energy.

Strontium-90 is a radioactive isotope with a half-life of 29 years. Assume that 10,000 atoms of Sr-90 are generated in a nuclear reaction and then stored.

1. Complete the following table to determine how much time will have elapsed when 1,250 atoms of Sr-90 remain in the storage facility.

87 years will have elapsed when 1250 atoms remain.

2. Graph the data for the radioactive decay of Sr-90 and draw a continuous, smooth-fit curve through the data points.
3. Describe the shape of the radioactive decay curve shown.

The rate of radioactive decay is described mathematically as an exponential decay curve. The number of Sr-90 atoms remaining decreases exponentially as the number of years increases.

4. What is the probability that a specific Sr-90 atom will have decayed after (a) 29 years and (b) 58 years?
1. There is a 50% chance that a specific Sr-90 atom will have decayed after 29 years.
2. After 58 years (two half-lives), the probability that a specific Sr-90 atom will have decayed increases to 75%.

### Sample Data

{12599_Data_Table_5}

1The number of dice remaining will equal the number of initial dice for the next round.
2Record the number of dice that decay from each roll of 10 dice here. After the required number of rolls has been completed to match the initial number of dice for each round, add up the total number of dice that decayed and record the result.
(2 decay numbers)

{12599_Data_Table_6}

1The number of dice remaining will equal the number of initial dice for the next round.
2Record the number of dice that decay from each roll of 10 dice here. After the required number of rolls has been completed to match the initial number of dice for each round, add up the total number of dice that decayed and record the result.

1. Graph the results obtained for the “radioactive decay” of dice. Note: Include a point on the graph for “100” as the number of dice “remaining” after zero rolls of the dice (Round zero). (In this example, six-sided dice were used.)
2. Determine the half-life: Choose two points on the y-axis, where the first point is about twice as large as the second point (e.g., 80 dice and 40 dice). How many rounds are needed for one-half of the dice to decay?

Compare the results for Round 3 (65 dice) and round 6 (34 dice). Three rounds are required for approximately one-half of the dice to decay. The estimated half-life is three rounds.

3. Verify the half-life value by choosing another set of two points on the y-axis. Is the half-life a “constant” for the decay of the dice?

Compare the results for Round 5 (45 dice) and round 8 (22 dice)—three rounds are required for approximately one-half of the dice to decay. The half-life appears to have a constant value (t½ = 3 rounds). However, as the number of dice being rolled decreases, the half-life is no longer constant. Radioactive decay is a random process. Large numbers of dice are needed to simulate a random process.

4. Compare the half-life value with that obtained by another group using the same-sided dice but a different decay number. Does the half-life depend on the decay number that was assigned? Explain, based on probability.

The half-life for six-sided dice does not depend on the “decay number that was selected. There is an equal probability (1/6 or 16.7%) of rolling any decay number.

5. Using the value of the half-life determined in Questions 2 and 3, predict how many dice should have remained after 15 rounds. Compare this with the number of dice that actually remained (see the data table). What factors might account for any difference between the predicted and actual number of dice remaining after 15 rounds?

If one half-life is equal to three rounds, then 15 rounds equal five half-lives. The number of dice remaining after 5 half-lives should be three (each arrow in the following sequence represents one half-life).

The actual number of dice remaining after 15 rounds was six! The difference between the predicted and the actual number of dice remaining after 15 rounds is due to the fact that the process is no longer statistically random when the total number of dice is very small.

6. Using the concept of half-life, predict the number of rounds that would be needed to reduce the number of dice from 10,000 to 625 using 6-sided dice and one “decay number.

Each arrow in the following sequence represents one half-life (3 rounds). Four half-lives (12 rounds) would be needed to reduce the number of dice from 10,000 to 625.

7. Does the “decay curve” for dice have the same general shape as the decay curve for Sr-90 (see the Prelab Questions)? Would the shape of the decay curve be different if you had started with 1000 dice instead of 100?

The “decay curve” for dice has the same general shape as the exponential decay curve for Sr-90. The shape of the curve does not depend on the initial number of dice.

# Student Pages

### Introduction

Radioactive decay is a spontaneous and completely random process. There is no way to predict how long it will take a specific atom of a radioactive isotope to disintegrate and produce a new atom. The probability, however, that a specific atom will decay after a certain period of time can be simulated by studying other random processes, such as a coin toss or a "roll of the dice."

### Concepts

• Half-life
• First-order rate
• Probability

### Background

Radioactive nuclei disintegrate via different processes and at different rates. The amount of time required for different radioactive nuclei to decompose varies widely, from seconds or minutes for very unstable nuclei to a billion years or more for long-lived radioactive nuclei. Polonium-218, for example, emits alpha particles and decays very quickly—within minutes. Uranium-238 also decays via alpha-particle production, but the decay takes place over billions of years! The relative rate of decay of different radioactive isotopes is most conveniently described by comparing their half-lives. The half-life (t½) of a radioactive isotope (called a radioisotope) is the amount of time needed for one-half of the atoms in a sample to decay. Every radioisotope has a characteristic half-life which is independent of the total number of atoms in the sample. Thus, the half-life of polonium-218 is about three minutes while the half-life of uranium-238 is more than 4 billion years. Regardless of the total number of atoms in a sample of polonium-218, one-half of the atoms will always “disappear” (decompose to produce other atoms) within three minutes.

To better understand half-lives, consider the following example. Iodine-131 is an artificial radioisotope of iodine that is produced in nuclear reactors for use in medical research and in nuclear medicine. It has a half-life of eight days. If 32 grams of iodine-131 are originally produced in a nuclear reactor, after eight days only 16 grams of iodine-131 will remain. After two half-lives, or 16 days, only 8 grams will be left, and after three half-lives (24 days) only 4 grams will be left. Every eight days, the amount of iodine-131 that remains will decrease by 50%.

The process of radioactive decay may be modeled by studying a random process such as a coin toss. Imagine placing 100 coins heads-up in a box to start, shaking the box, and then discarding the coins that land tails or “decay.” Since the probability of a specific coin “decaying” (landing tails) is 50%, we would predict that only 50 coins will remain in the box after the first coin toss. Repeating the coin toss with 50 coins remaining in the box should result in an additional 25 coins “decaying” in the second round, and so on (see Table 1).

A simulated “radioactive decay curve” obtained by graphing the data (see Figure 1) shows that the “half-life” of coins is equal to “one coin toss.” The number of coins remaining in the box decreases by 50% after each coin toss.

### Experiment Overview

The purpose of this activity is to simulate radioactive decay by studying the probability of a random process—rolling dice. The radioactive decay of dice will be studied by rolling 10 dice ten times in Round 1 and recording the number of dice that display a specific “decay number, for example, all dice that read six. (Rolling 10 dice ten times is equivalent to rolling 100 dice once.) The total number of dice that “decayed (landed on six) during Round 1 will then be counted and subtracted from the total number of dice rolled. This is the number of dice remaining that will be rolled in Round 2. This process will be repeated until no dice remain. The “half-life of dice will be determined by graphing the number of dice remaining after each round.

### Materials

Cardboard box (optional)
Dice, multi-sided, 10

### Prelab Questions

Strontium-90 is a radioactive isotope with a half-life of 29 years. Assume that 10,000 atoms of Sr-90 are generated in a nuclear reaction and then stored.

1. Complete the following table to determine how much time will have elapsed when 1250 atoms of Sr-90 remain in the storage facility.
{12599_PreLab_Table_2}
2. Graph the data for the radioactive decay of Sr-90 and draw a continuous, “smooth-fit” curve through the data points.
{12599_PreLab_Figure_2}
3. Describe the shape of the “radioactive decay curve” shown.
4. What is the probability that a specific Sr-90 atom will have decayed after (a) 29 years and (b) 58 years?

### Procedure

1. Obtain an assigned set of “multi-sided” dice (e.g., 4-sided, 6-sided, 8-sided, 10-sided, 12-sided, 20-sided dice). The teacher will assign a “decay number” to each group. Record the number of sides on the dice and the assigned decay number in the data table.
2. Roll all 10 dice in a box or on a table top. Hint: Roll the dice so they all have an opportunity to have a “random” roll and all of the dice land flat. Re-roll any dice that do not land flat.
3. Assume that any dice landing on the assigned decay number have “decayed.” Record the number of dice that decayed in the Data column of the data table.
4. Repeat steps 2 and 3 until the 10 dice have been rolled 10 times (for a combined total of 100 dice rolls). After each roll, record the number of dice with the assigned decay number in the Data column of the data table.
5. Add together the number of dice that decayed from the Data column. Enter this number—the total number of dice that decayed in the first round—in the “Number of dice that decayed” column of the data table.
6. Subtract the number of dice that decayed from the initial number of dice (100 for the first round) to determine the number of dice remaining. Record the “Number of dice remaining” in the data table and enter this number as the initial number of dice for the second round.
7. Repeat steps 2–6. Roll the dice as many times as needed to match the number of dice (initial) for Round 2. Record all data in the data table. For example, if 79 dice remain after the first round, then the 10 dice would be rolled seven times, and then nine dice would be rolled once so that the total number of dice rolled in Round 2 is equal to 79 [(10 x 7) + 9 = 79].
8. Repeat step 7 until no dice remain or until 20 rounds of “radioactive decay” have been completed.

### Student Worksheet PDF

12599_Student1.pdf

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