The Paul Molitor Simulation
The “Investigative Process” (as defined by GAISE*)
Content and Process Targets
*2.0 Use a variety of mathematical and technological tools to determine
measurements directly (ruler, protractor, compass) and to check reasonableness
of answers (calculators and computers). Apply appropriate technology, e.g.
computers and graphing calculators, to interpret appropriateness of solutions.
B.6 Determine reasonableness of answer.
D.2 Select and use tools with appropriate degree of precision to determine
measurements directly.
*3.0 Use data in a variety of formats to summarize, predict, and analyze applications.
E.1
Organize, display, compare and interpret data in a variety of ways in
mathematical and real-world contexts, e.g. histograms, line graphs, stem-andleaf plots, scatter plots, box and whiskers, bar charts, Venn diagrams, tables,
circle graphs.
E.2
Interpret, analyze and make predictions from organized and displayed data,
e.g. measures of central tendency, measures of variation such as standard
deviation, mean, median, mode, range, dispersion, outliers, line of best fit,
percentiles.
*4.0 Use a variety of counting methods to identify all possible occurrences of events.
Apply this information to predict, analyze, and evaluate mathematical
occurrences.
E.3
Analyze, evaluate and critique methods and conclusions of statistical
experiments, e.g. randomness, sampling, techniques, surveys.
E.4
Determine the likelihood of occurrence of simple and complex events, e.g.
combinations and permutations, fundamental counting principle, experimental
versus theoretical probability and independent, dependent and conditional
probability.
Developed by Henry Kranendonk for use at the Milwaukee Mathematics Partnership (MMP)
Fourth Annual Math Teacher Leader Kickoff, August 2007. The MMP is supported by the
National Science Foundation under Grant No. 0314898.
Outline of Lesson:
Step 1: Formulate the question(s)
(This is an excellent opportunity to start the
investigation with a basic question, namely
“What is probability?” Remember, this is a
difficult concept for everyone!)
The question for this particular investigation is:
What is the probability or likelihood that a
baseball hitter with a batting average similar to
Paul Molitor will have at least 1 hit in 39
consecutive games?
The details of his accomplish that involve a good
discussion are these:
(1)
(2)
During most of the time this near
record was made, Molitor had a batting
average of .333 (or 1 hit for 3 times at
bat).
In addition to the above, for each game
he played, Molitor was at bat 3 or more
times per game. (Forget the walks and
other baseball details some students
will bring up!)
Ask the question: “So, Mr. Molitor is in a game
and during that game he is at bat at least 3 times.
Do you think he will get a hit? (Use the visual.)
Explain.
Probability is essentially a “rating scale” that
something (an “event”) might or might not occur.
For some students, developing this idea with the
following visual might illustrate this point:
If I carry out this “event,” the outcome I
am looking for is:
Very
Could or
Very
unlikely to could not likely to
occur.
occur.
occur.
Review with students what is an expectation and
how a “rating scale” from 0 to 1 (or 0% to 100%)
was developed to quantify the expectation.
The primary question around Paul Molitor’s
accomplishment is based on essentially students
not having a good sense of what this outcome
might be. (Could it be close to 1? Could it be
close to 0? And, whatever your estimate, what is
the basis of making this estimate?)
The challenge is also to explain enough of the
details of baseball to have this situation make
sense. For a non-baseball person, this requires at
least explaining the following sports related terms:
at bat, hit, out, a game, batting average, and streak.
(Streak is the most challenging to explain.)
Next question: “So, Mr. Molitor got a hit. Do
you think he will get a hit in the next game?”
Next question: “So, what about getting a hit in
the next 39 games?”
Developed by Henry Kranendonk for use at the Milwaukee Mathematics Partnership (MMP)
Fourth Annual Math Teacher Leader Kickoff, August 2007. The MMP is supported by the
National Science Foundation under Grant No. 0314898.
Step 2: Collect data to answer
the question
Within this step is the challenge of how to collect the data. The
goal will be to involve a simulation – or, a process that attempts
to “mimic” the same outcomes as the investigation, but carried
out in a way that it is possible to repeat the event numerous
times.
What ways can we mimic this event?
How can we repeat it numerous times?
(I anticipate you will need to lead students to develop a
simulation process. This is okay as a major goal of this lesson is
to understand what is a simulation and how can students carry it
out.)
The simulation designed will need to mimic:
• an “at bat”
• a “hit” or an “out”
(based on an batting average of
.333)
• a game
• a streak
This demonstration will involve at least 2 different ways to
mimic the event. There are some assumptions in each –
challenge the students to think what assumptions (or what
ideas) are used to mimic the event. In the last step, these
assumptions will be reviewed.
Carrying out a simulation is often (and maybe be in this case
also) boring. Assigning groups and collecting results from the
groups is the most effective. This process, however, can lead to
a great discussion why technology has taken on the simulation
of events to answer these types of questions.
Developed by Henry Kranendonk for use at the Milwaukee Mathematics Partnership (MMP)
Fourth Annual Math Teacher Leader Kickoff, August 2007. The MMP is supported by the
National Science Foundation under Grant No. 0314898.
Step 3: Analyze the data
After repeated results are obtained from the simulation,
organize the data.
Possible formats for organizing the data are:
• line plot
• frequency table (starting with 0)
• simplified histogram
Summaries could also include the mean of the streak values, or
the balance point on the line plot of the display. The mean of
the streak values should be interpreted as the “expected value”
of the outcome of this event.
The language of probability is introduced as the rate of the
expected outcome of the investigative questions. Returning to
several of the details of this event, the data collected from the
simulation can now provide more insights into the likelihood of
this event occurring.
Possible extensions of this investigation: Did the different
methods of collecting data produce different results? This is
clearly a more advanced topic for analysis. All students,
however, can use the data to ask the questions.
Step 4: Interpret Results
The summarized results/displays of the data are used to answer
the investigative question. In addition, the results are used to
summarize the process. Do you think the event of getting a
streak of this type was properly simulated? Why or why not?
What data might we collect to further investigate what happens
in baseball?
(The lesson outlined in the Navigation book indicates that
students could collect data indicating the current streak values
of baseball players.)
The primary question that some students are ready to ask and
address is: “Is getting at least 1 hit in a game independent or
dependent on what happened in the previous game?”
Let students explore other investigations that might be
answered with a simulation.
GAISE or “Guidelines for Assessment and Instruction in Statistics Education”
Paul Molitor problem:
Probability of at least 1 hit:
2
1–( )
3
Developed by Henry Kranendonk for use at the Milwaukee Mathematics Partnership (MMP)
Fourth Annual Math Teacher Leader Kickoff,
August 2007. The MMP is supported by the
!
National Science Foundation under Grant No. 0314898.