In grades 5-8, the mathematics curriculum should include
explorations of probability in real-world situations so that students
- model situations by devising and carrying out experiments
or simulations to determine probabilities;
- model situations by constructing a sample space to determine
- appreciate the power of using a probability model by comparing
experimental results with mathematical expectations;
- make predictions that are based on experimental or theoretical
- develop an appreciation for the pervasive use of probability
in the real world.
Probability theory is the underpinning of the modern world.
Current research in both the physical and social sciences cannot
be understood without it. Today's politics, tomorrow's weather
report and next week's satellites all depend on it. (Huff
and Geise 1959)
An understanding of probability and the related area of statistics
is essential to being an informed citizen. Often we read statements
such as, "There is a 20 percent chance of rain or snow today."
"The odds are three to two that the Cats will win the championship."
"The probability of winning the grand prize in the state lottery
is 1 in 7 240 000." Students in the middle grades have an intense
interest in the notions of fairness and the chances of winning games.
The study of probability develops concepts and methods for investigating
such situations. These methods allow students to make predictions
when uncertainty exists and to make sense of claims that they see
The study of probability in grades 5-8 should not focus on developing
formulas or computing the likelihood of events pictured in texts.
Students should actively explore situations by experimenting and
simulating probability models. Such investigations should embody
a variety of realistic problems, from questions about sports events
to whether it will rain on the day of the school picnic. Students
should talk about their ideas and use the results of their experiments
to model situations or predict events. Probability is rich in interesting
problems that can fascinate students and provide settings for developing
or applying such concepts as ratios, fractions, percents, and decimals.
Probability, the measure of the likelihood of an event, can be
determined theoretically or experimentally. Students in the middle
grades must actively participate in experiments with probability
so that they develop an understanding of the relationship between
the numerical expression of a probability and the events that give
rise to these numbers (e.g., 2/5 as it relates to the probability
of choosing a red marble from a hat). Students must not only understand
the relationship between the numerical expression and the probability
of the events but realize that the measure of certainty or uncertainty
varies as more data are collected.
Students have many misconceptions and poor intuitions about probabilistic
situations. In order to bring these ill-formed notions to the conscious
level so that they can be confronted, students should be asked to
guess what will happen next or what the result of the experiment
will show. An unexpected result has a much greater potential to
cause students to rethink their basic assumptions if they have articulated
their ideas before their experiment or analysis of the situation.
To see how the predictions we hear and see every day are based
on probability, students must use their knowledge of probability
to solve problems. In modeling problems, conducting simulations,
and collecting, graphing, and studying data, students will come
to understand how predictions can be based on data. Mathematically
derived probabilities can be determined by building a table or tree
diagram, creating an area model, making a list, or using simple
counting procedures. Students develop an appreciation of the power
of simulation and experimentation by comparing experimental results
to the mathematically derived probabilities. For example, students
can conduct experiments with two dice to determine the experimental
probability of rolling a 4, construct a table to establish the theoretical
probability, and then compare the two results. The following activity
includes all these concepts and an analysis of fairness.
Arrange students in pairs and give each pair three chips: one
chip with an "A" on one side and a "B" on the
other; a second with "A" on one side and "C"
on the other; and the third with "B" on one side and "C"
on the other. One student tosses all three chips simultaneously
onto the desk. Player 1 wins if any two chips match; Player 2 wins
if all three chips are different.
After the students have decided whether to be Player 1 or Player
2 and have tossed the chips many times, they might want to revise
their choices. When the class has completed the experiment and discussed
the results, the theoretical probability can be analyzed by completing
a tree diagram (fig. 11.1).
Fig. 11.1. Tree diagram
Player 1 can win in six ways, but Player 2 can win in only two
ways; hence, the probability of Player 1 winning is 6/8 and the
game is clearly unfair.
Students should also understand that some probability problems
do not have theoretical solutions. Given a set of thumbtacks, what
is the probability that one thumbtack will land "point up"
when tossed? Many students will guess that the answer is 1/2. Through
experimentation, they will discover that the probability changes
with the size of the head of the tack and the length of the shank.
At the K-4 level, students can flip coins, use spinners, or roll
number cubes to begin their study of probability. At the middle
school level, such experiments should be extended so that students
can determine the probabilities inherent in more complex situations
using simple methods. For example, if you are making a batch of
6 cookies from a mix into which you randomly drop 10 chocolate chips,
what is the probability that you will get a cookie with at least
3 chips? Students can simulate which cookies get chips by rolling
a die 10 times. Each roll of the die determines which cookie gets
a chip. The same type of simulation can help students determine
how many boxes of cereal they should expect to purchase to receive
at least 1 each of 6 types of prizes given away randomly, 1 to a
box. With a computer or set of random numbers, this problem can
be extended to simulate an industrial quality-control situation
or to analyze the number of defective items that might occur under
certain conditions on an assembly line. All these problems, experiments,
and simulations can be easily studied, performed, and analyzed by
middle school students.
Once students have experimented with a problem, a computer can
generate hundreds or thousands of simulated results. It is important
that the computer simulation follow active student exploration.
This follow-up broadens students' understanding and provides them
with an opportunity to observe how a greater number of trials can
refine the probability model.
The nature of probability encourages a systematic and logical
approach to problem solving. Throughout their experimentation and
simulation, students should be making hypotheses, testing conjectures,
and refining their theories on the basis of new information. Probability
also can be applied to data analysis. Students can use charts, graphs,
and plots to make predictions; this activity reinforces their interpretation
of the information and their derivation of other useful information.
The table in figure 11.2 gives the
record for Joan Dyer's last 100 times at bat during the softball
season. She is now coming up to bat. Use the data to answer the
Fig. 11.2. Softball stats
What is the probability that Joan will get a home run?
What is the probability that she will get a hit?
How many times can she be expected to get a walk in her next
14 times at bat?
Probability connects many areas of mathematics. For example, fraction
concepts play a critical role in the study of probability. Topics
such as equivalent fractions, comparison of fractions, addition
and multiplication of fractions, as well as whole number operations
and the relationships among fractions, decimals, and percents can
be reinforced through the study of probability.
Dividing the area of a rectangle into fractional parts to model
a probability problem provides an excellent opportunity for students
to identify the relationship between concepts in geometry and operations
In the maze in figure 11.3, Tom is
to pick a path at random. Use the grid in figure
11.4 to determine the probability that he will enter room A
or room B.
Fig. 11.3. Maze
Fig. 11.4. Grid
To calculate the probabilities directly, students would have to
multiply probabilities and then add the results. However, the use
of an area model makes the problem more accessible. By using fractional
parts of a region, a student can represent the sequence of possible
choices for paths and then add simple fractions to determine the
probabilities. P (A) = 1/6 + 1/6 = 2/6.
The study of probability in grades 5--8 actively engages students
in exploring events and situations relevant to their daily lives.