3/15/01
Jerry White jwhit216@uwsp.edu

I. Project Objectives
The primary objective of this project is to introduce secondary students to an experiment that uses the laws of probability to model radioactive decay. The experiment is intended to strengthen the students’ knowledge of the various concepts of probability which they have been exposed to in the classroom. Although the experiment is simply ‘modeling’ an actual event, it is designed such that the students will gain deeper insight into the laws of probability. Taking classroom theory into the lab and physically applying it to an everyday occurrence (radioactive decay) will accomplish this.
The secondary students will use the laws of probability to model the prevalence of radioactive decay in everyday life. The premise upon which the project is based is one in which the rate of change in the number of objects during an event is directly proportional to the number of objects existent before the event. That is, the value of n i
depends on the value of n i1
.
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The natural occurrence of radioactive decay has close ties to the concept of conditional probability. This fact will be explained to the students; in addition, it will become increasingly more apparent as the students perform the experiment and complete an assigned worksheet.
A secondary objective of the project is to modify the experiment in such a way that middle school students will be able to benefit from the experience, albeit on a much less complex level. We will briefly discuss the event of flipping a coin in the air and the probability (or ‘likelihood’) associated with the coin landing in either a ‘heads-up’ or a ‘tailsup’ position. Explanation that P (heads) = P (tails) = 1/2 for one flip and, that results from
1 The project will be done on a non-technical level. The students will need no prior knowledge of recursion relations, nuclear physics, or exponential decay functions.
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successive flips are independent, will be provided and experimented with. In regard to the term “independent,” the middle school students will come to understand that the outcome from a flip of one penny does not affect the outcome from a flip of another penny. In this respect, they will come to understand that the ‘events are not dependent’ upon each another.
Assumptions
This project is based on the assumption that the secondary students have been exposed to the basic theories of probability. More specifically, it is assumed that they have completed
Chapter 4 of the M350 text and are familiar with the following concepts: general addition and multiplication rules, conditional probability, and tree diagrams. It is assumed that they understand the word ‘probability’ as it refers to long-term relative frequency; also, they will have a clear understanding of disjoint and independent events.
With regard to the middle school students, the assumption is that they are familiar with the term probability as “the likelihood that an event occurs.” For instance, in the event of flipping a coin, they will understand that there are two possible outcomes and that both are equally likely. In addition, they have been introduced to the fact that, in the long run (after many flips of the coin), the same number of heads as tails will have occurred.
I. Project Materials
The materials needed to perform the experiment consist of five rolls of pennies and five shoeboxes (assuming a class size of approximately 20 to 25 students; we will divide up into groups of four to five). For the secondary students, additional materials required to complete the project will include a written explanation of the experiment and its procedure, along with an Activity Worksheet. The worksheet will include tables for data entry, a plot, and questions to answer as homework.
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For the middle school students, the pennies and the shoeboxes will suffice for their less sophisticated version of the activity.
I will provide all the materials needed to perform both activities with the secondary and middle school students; the cost should not exceed $2.50.
Sections III. Through V. of this report are devoted to the secondary students’ activity, while Section VI. illustrates modifications to the activity that will make it suitable for middle school students to participate in.
III. Motivation for the Activity
Radioactivity / Isotopes / Radioactive Decay
It is assumed that the students performing this activity have no prior knowledge of nuclear physics. This assumption has previously been discussed. Nevertheless, some basic information on the subject will be prove to be invaluable in the performance of the experiment and, moreover, painless to acquire.
The term ‘radioactivity’ describes the property of an isotope 2
to emit energy in the form of radiation. If an isotope is emitting radiation energy, this means that it is transforming into less energetic isotopes or elements. In essence, it is ‘decaying;’ hence the term
‘radioactive decay.’
One useful example of radioactive decay is the concept of carbon dating, whereby the age of ancient objects (on the order of several thousand years old) can be accurately determined. Conversely, a harmful example of radioactive decay is one of radon gas accumulation in basements. This accumulation occurs when Uranium-230, a naturally occurring radioactive isotope found in the bedrock and granite underground, radioactively
2 The term ‘isotope’ refers to any of two or more forms of the same element. For example, U-235, U-238, and
U-239 are all isotopes of Uranium.
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decays into the harmful, gaseous isotope Radon-218, which is then able to permeate through basement walls.
Now, one very important property inherent to all radioactive isotopes is that the rate at which they decay is proportional to the number of isotopes that exist at any time ‘ t
.’ This property will help stimulate the idea behind the experiment that that will be performed.
Basis for the Experiment
This portion of the report is intended as a guide – a ‘lab manual,’ if you will – for the secondary students.
To establish the principle of the experiment to be performed, imagine that we are able to collect some radioactive isotopes. Let us say that we each collect the same type of isotope and the same number (48) and place them in separate containers. We know that, since the isotopes are identical, they are all decaying at the same rate, as explained above. Also, we will assume that we have measured this decay rate and have found that, after one minute, half of the isotopes have decayed. This property is defined as the half life of the isotope and, like the decay rate, is a constant.
Theoretically, then, at t = one minute, 24 isotopes will have decayed and 24 will still exist. Likewise, at t = two minutes, twelve isotopes will remain whereas the other twelve will have decayed. Following this pattern, we can say that after three minutes, six isotopes remain; after four minutes, three isotopes remain, and so forth 3 . Understand that this description of radioactive decay is based on many decays occurring over the ‘long run’ with millions of isotopes.
3 Theoretically, the number of atoms never completely equals zero; it approaches zero as t goes to infinity.
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IV. Description of the Experiment / Activity
Radioactive decay of isotopes can be modeled in an experiment with pennies.
Imagine that we let 48 pennies represent the 48 radioactive isotopes. By placing the pennies
‘heads-up’ in a box, we can pretend that they are 48 ‘un-decayed’ isotopes at time t = zero minutes (the start of the experiment). Upon closing the box and shaking it a few times, we can expect – from the laws of probability – that approximately half of the pennies will be in a
‘heads-up’ position and half will be in a ‘tails-up’ position. Let the number of pennies in the
‘heads-up’ position represent the number of un-decayed isotopes. Record this number and the numbers from the other students in your group in the appropriate boxes for Trial 1 on page 1 of the Activity Worksheet (note: Trial 1 corresponds to an elapsed time of t = one minute, the half-life of the isotope/penny). Theoretically, your number for Trial 1 would equal 24, however, this experiment represents a single event. Recall that the concepts of radioactive decay and half-life are based on what occurs in the long run with many isotopes and many decays.
Next, remove the ‘tails,’ i.e., the ‘decayed’ isotopes, from the box and repeat the shaking process with the heads-up pennies remaining from Trial 1. You will then have a result for time t = two minutes. Record this number in the box for Trial 2. Again, theory predicts that half of the un-decayed pennies from t = one minute will have decayed into a tails-up position. However, this experiment is only a model; do not be alarmed if your results are different from theoretical results. Repeat the process for a total of seven trials (seven
‘half-lives’), or until all the pennies are tails-up – whichever comes first. This portion of the experiment should take roughly seven to ten minutes.
Using the empty graph on page one of the Activity Worksheet, plot your group’s results. Draw straight lines between your data points. Compare your plot with the plot titled
“Theoretical Result” at the bottom of page one. The shape of the line in the theoretical plot is known as an ‘exponential decay.’ Use the empty space on the worksheet to briefly comment on the comparison between the plot of your data and the theoretical plot: does your plot resemble an exponential decay? This part of the experiment can be done in-class; it should not take more than a few minutes. Comments can be added afterwards as homework.
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Another interesting aspect of our experimentation will be to calculate the mean number of heads, h
1
, and the associated standard deviation,
 h
1
, after the first trial. Using the data from all the groups in the class will accomplish this. Part II on page one of the
Activity Worksheet contains a table for recording this data and the results of your calculations. Use a calculator and the appropriate equations for h
1
and
 h
1
for this part of the worksheet. Theoretically, that is, in the ‘long run,’ what would you expect h
1
to be?
Note: data must be collected from the other groups during class; the calculations can be carried out as homework. This portion of the experiment should take a three to five minutes, at maximum.
Page two of the worksheet contains a probability tree for an activity identical to the one we performed; however, instead of using 48 pennies and six trials, this tree is based on an experiment with three pennies and three trials. The result is that there are 20 possible outcomes. Each branch of the tree is labeled with its conditional probability. For example, the conditional probability that two heads ( 2 h ) remain after the second trial ( II ), given that two heads remained after the first trial ( I ), is 1/3. Moreover, the probability that two heads are in the box after the first trial and two heads are in the box at the end of the second trial is
(1/4)*(1/3) = 1/12; this result is based on the multiplication rule for independent events. Your job is to calculate the probabilities for each of the 20 possible outcomes. What would you expect their sum to be?
Students may start on this part of the activity in-class; however, it is doubtful that they would have time to complete it. It will then have to be finished later as homework. Overall, the total time necessary to perform the experiment and establish a good start on the worksheet should take approximately 20 to 25 minutes.
Summary of Tasks to Perform:
Page One:
1.
Plot your group’s data and draw a line between the points.
2.
Briefly comment on the shape of your plot and compare it to the ‘theoretical’ plot.
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3.
Calculate and record h
1
and
 h
1
for the entire class; predict what h
1 would be in the
‘long run.’
Page Two:
1.
Calculate and record the probabilities for each of the 20 possible outcomes on the probability tree.
2.
Calculate and record the sum of the 20 probabilities; does it equal one? Should it?
V. Assessment Tool: Activity Worksheet
Attached at the end of this report is the two-page Activity Worksheet for the secondary students. Part I of page one (Data/Plot) is designed for the students to be done as a group. Part II of page one (Calculations) and all of page two (Probability Tree) are designed to be done on an individual basis.
Possibilities for further investigation would include, but not be limited to:
1.
For the probability tree, students could calculate the four probabilities associated with: three heads remaining after the three trials, two heads remaining, one head remaining, and zero heads remaining.
2.
For students interested in the decay equation, which is given as
N ( t )

N o e kt
, (1) where N(t) is the number of isotopes at any time t ; N o
is the number of isotopes at t = 0, and k is the ‘decay constant’ given by
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k

Ln ( 1
 half

/ 2 ) life
 , (in units of inverse time) (2) they could solve equation (1) for t : t


Ln

N ( t )
N o


. k
(3)
They would then be able to construct a spreadsheet with columns for N o
, half-life,
N(t ), and t . One interesting study would be insert different values for N o
and half-life in the columns and compare the times it would take for N o
to decay to various fractions of itself.
For example, let N(t ) equal .75
N o
, .5
N o
, and .25
N o
and observe the variations in the time ( t ) required for N o
to decay to these different amounts. Figure 1 is an example of this study.
N o half-life [min.]
2500 1
2500
2500
1
1
10000
10000
10000
0.5
0.5
0.5
N(t)
1875
1250
625
7500
5000
2500 t [min.]
0.4150
1
2
0.2075
0.5
1
Figure 1.
Example of a Computer-Generated Spreadsheet for Further Study.
Furthermore, the spreadsheet could be manipulated in several different ways; for example, students could be creative and choose to plot N(t ) versus t (the exponential decay curve) or generate a much larger spreadsheet with wider ranges of N o
and half-life.
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VI. Middle School Students’ Activity
The experiment performed with the secondary students can be modified to appeal to middle school students. For example, since the assumption is that they are familiar with the term probability as “the likelihood that an event occurs,” they should understand that there are two possible outcomes associated with flipping a penny -- and that both are equally likely.
They could each be given an equal number of pennies (six) and asked to flip them one at a time. We could then record on the blackboard the number of heads each student ended up with and use these results to draw a probability histogram. There is always the possibility that the histogram may not show a peak at three heads, however, the students will learn that in the long run, after many such events, the largest frequency will occur at three heads.
Time permitting, we may be able to extend this concept to that of decay – of course, on a much less advanced level as the secondary students’ experiment. I would perform this experiment for them, while they looked on and helped count decayed and un-decayed pennies, or, they could break up into groups and perform the experiment themselves. I believe it would very instructive to the students to see that an event with two equally likely outcomes
(i.e., probabilities of one-half for each outcome) can be viewed in a different manner than that of flipping one coin at a time. They may not realize, initially, that this is my intent. However,
I will explain to them the connection between flipping one coin at a time: “The outcome will be either heads or tails,” (i.e., “half and half”) and flipping a large number of coins at a time:
“Most likely, the outcome from this event will be that half of the pennies will land heads-up and half will land tails-up.” In this manner, it is my hope that they will be able to conceive the difference between the two events.
The activity with the middle school students can be accomplished in a time frame similar to that of the secondary students – approximately 20 to 25 minutes.
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