Heads Up on Entropy
By Charles Cox
And Maxcy Dimmick
1) Goal: Students will understand the Second Law of Thermodynamics which states,
“Nature favors entropy over order.”
2) Mathematical topics addressed are the following:
a) Conduct a simulation;
b) Organize data;
c) Interpret data and make inferences.
3) Map strand addressed: Mathematics 3: Data analysis, probability and statistics.
4) Grade Level: 8-9 grades.
5) Prerequisite skills: Students should be able to make a box plot.
6) Materials:
a) Fifty pennies per group;
b) Data recording sheets;
c) Written instructions for simulation;
d) Graph paper.
7) Procedure for teacher:
a) State the Second Law of Thermodynamics and write it on the board or overhead.
b) Give real-life examples such as the following:
i) Ask students what their rooms would look like after one week if they did not
spend any time straightening them.
ii) Ask students to predict what would happen to an old barn that was never
repaired or given a fresh coat of paint.
iii) Explain how the diffusion of air freshener is dependent on entropy.
c) Introduce the simulation:
i) Our simulation will begin with a high degree of order or very low entropy.
ii) With each trial we will observe the amount of disorder introduced to the
arrangement of the pennies.
d) Conduct the simulation according to instructions. Allow students to work in
groups of two or three to complete the simulation.
e) Reconvene the class and collect all of the trials as one set.
f) Briefly review how to construct a box plot and then allow students time to
construct a box plot of the data just collected.
g) Consider the following discussion questions:
i) Which is easier, tossing and trying to get all of the pennies to land on the same
side, or tossing and allowing coins to fall randomly on either side? (Answer:
allowing pennies to fall randomly on either side.)
ii) What is required to reestablish order? (Answer: work or effort.)
iii) We drill oil wells and pump 100,000’s of gallons from an oil well. Does the
oil have a high degree of entropy or a high degree of order? (Order.)
iv) Given time, will it be easier or more difficult to find oil? (More Difficult)
v) Will we ever run out of oil? (No, but it gets increasingly difficult to retrieve.)
8) Procedure for simulation:
a) Turn all 50 pennies “heads up” in the tray.
b) Make one toss of the tray and allow pennies to fall on table top or on floor.
c) Record the number of pennies that landed “heads up.”
d) Repeat for a total of 10 trials.
9) Assessment: This project is worth 30 points:
a) 10 points: Students will be assessed on the completeness/accuracy of their
answers to the following discussion questions.
i) What are two synonyms for entropy? (randomness, disorder)
ii) Does tossing the coins in the experiment lead to more entropy? Explain.
(Yes. There are now coins facing both directions.)
iii) Is the manufacturing of steel from ore an example of increasing entropy or of
increasing order? Why? (Increasing order. The steel is a more concentrated
form of the ore.)
b) 10 points: Students will be assessed on their participation in the discussions and
the data collection activity. The following scoring guide is suggested.
Activity:
Participates in
collecting,
collating data
Level 1
Does not
participate.
Contributes to
discussion
Does not
offer any
comments.
Level 2
Participates but
does not
complete
required tasks.
Offers comments
that are
reinforcing ideas
already
presented.
Level 3
Participates and
completes
required task.
Offers
comments that
are insightful/
perceptive.
c) 10 points: Students will be assessed on the accuracy, neatness and completeness
of their graphs. The following scoring guide is suggested.
Characteristic:
Graph contains 5
essential elements:
Title, labels, scales
Information
displayed in the
graph is neat and
complete.
Level 1
Graph
contains one
or two of the
elements.
Incomplete
data is
displayed.
Level 2
Graph contains
three or four of
the elements.
Level 3
Graph contains
all five of the
elements.
Complete data
is displayed in a
haphazard or
disorganized
manner.
Complete data
is displayed in a
neat and orderly
manner.
10) Extension Questions:
a) Will we ever run out of energy? (Sample Answer: No. The first law of
thermodynamics states that we will never destroy matter. The second law states
that it will become less and less useful.)
b) What is needed to retrieve its usefulness? (Sample Answer: Work or effort or
order.)
c) Ultimately, what does our future depend on – natural resources or energy?
11) References:
a) Christensen, John W. Global Science. (Kendall/Hunt Publishing Company:
Dubuque, IA) 1991, pp. 84-89.
b) Web page, Second Law of Thermodynamics:
http://www.secondlaw.com/two.html
Simulation Instructions and Data Recording Sheet
1) Turn all 50 pennies “heads up” in the tray.
2) Make one toss of the tray and allow pennies to fall on table top or on floor.
3) Record the number of pennies that landed “heads up.”
4) Repeat for a total of 10 trials.
Trial
Number
1
2
3
4
5
6
7
8
9
10
# of coins with heads up.
HEADS UP ON ENTROPY
Statistics Project
By Charles Cox
and Maxcy Dimmick
QUESTION:
What degree of entropy is introduced to the set of 50 pennies
with one toss?
SIMULATION
This simulation is an attempt to observe the Second Law of
Thermodynamics, which states, “Nature favors entropy over
disorder.” For the purpose of this experiment, we are defining
entropy as a measure of disorder and we are modeling entropy
with randomness.
The materials needed for the simulation are 50 pennies, a
tray or dish, paper, pencil, and graph paper.
The procedure for the simulation is as follows:
1. Turn all 50 pennies “heads up” in the tray.
2. Make one toss of the tray and allow pennies to fall on
the tabletop or on the floor.
3. Record the number of pennies that landed “heads up.”
4. Repeat steps 1-3 for a total of 20 trials.
The following scale is used to assess the degree of randomness
or entropy in the set of fifty coins:
21-30 heads: high degree of entropy
11-20 or 31-40 heads: moderate degree of entropy
0-10 or 41-50 heads: low degree of entropy
RAW DATA AND STATISTICS FROM SIMULATION
Data Set: 18, 28, 23, 24, 29, 26, 30, 32, 22, 31, 9, 21, 21, 41, 32,
34, 22, 34, 24, 24
Mean: 26.25
Median: 24
Mode: 24
Lower Extreme: 9
Upper Extreme: 41
Range: 32
Lower Quartile: 22
Upper Quartile: 31.5
Interquartile Range: 9.5
Number of Trials: 20
Box Plot of the Number of Heads Generated by One Toss
8
10
12
14
16
18
20
22
24
26
28
Number of Heads
30
32
34
36
38
40
Stem-And Leaf Plot
Number of Heads Generated by One Toss
0 9
1 8
2 1 2 2 3 4 4 4 6 8 9
3 0 1 2 2 4 4
4 1
1 8 = 18 Heads
Frequency Table
Number of Heads
Generated by One Toss
Number
Frequency
1-10
1
11-20
1
21-30
12
31-40
5
41-50
1
Histogram
Number of Heads Generated from One
Toss
14
Frequency
12
10
8
6
4
2
0
1 -- 10
11 -- 20
21 -- 30
31 -- 40
Number of Heads
41 -- 50
CONCLUSION
The results of this experiment suggest that a high degree of
entropy is introduced to the set of coins by one toss. This
conclusion is supported by several statistical elements. Both
the median (24) and the mean (26.5) fall near the optimum
number of heads for true randomness, which is 25. Also, a
histogram of the data grouped by tens (roughly corresponding
to the scale groupings) shows more trials falling in the “high
degree of entropy” range than in all of the other categories
combined.