Write the following questions on the chalkboard or overhead:
- I might make more money if I was in business for myself; should I quit my
- An earthquake might destroy my house; should I buy insurance?
- My mathematics teacher might collect homework today; should I do it?
Ask students to share their responses to each of these scenarios.
Ask students why their responses may be different from their
classmates. Ideally the class discussion will mirror some of the
concepts which follow.
Every day each of us must make choices like those described above.
The choices we make are based on the chance that certain events might
occur. We informally estimate the probabilities for events by using a
variety of methods: looking at statistical information, using past
experiences, asking other people's opinions, performing experiments,
and using mathematical theories. Once the probability for an event has
been estimated, we can examine the consequences of the event and make
an informed decision about what to do.
Making the connection between choice and chance is basic to
understanding the significance and usefulness of mathematical
probability. We can help middle school students make this connection by
giving them experiences wherein choice and change come into play
followed by tasks that cause them to think about, and learn from, those
The game of SKUNK presents middle-grade students with an experience
that clearly involves both choice and chance. SKUNK is a variation on a
dice game also known as "pig" or "hold'em." The object of SKUNK is to
accumulate points by rolling dice. Points are accumulated by making
several "good" rolls in a row but choosing to stop before a "bad" roll
comes and wipes out all the points. SKUNK can be played by groups, by
the whole class at once, or by individuals. The whole-class version is
described following an explanation of the rules.
The Game of SKUNK
To start the game each player makes a score sheet like this:
Each letter of SKUNK represents a different round of the game; play
begins with the "S" column and continue through the "K" column. The
object of SKUNK is to accumulate the greatest possible point total over
five rounds. The rules for play are the same for each of the five
- At the beginning of each round, every player stands. Then, a pair
of dice is rolled. (Everyone playing uses that roll of the dice; unlike
other games, players do not roll the dice for just themselves.)
- A player gets the total of the dice and records it in his or her column, unless a "one" comes up.
- If a "one" comes up, play is over for that round and all the player's points in that column are wiped out.
- If "double ones" come up, all points accumulated in prior columns are wiped
out as well.
- If a "one" doesn't occur, the player may choose either to try for more
points on the next roll (by continuing to stand) or to stop and keep what he or she has accumulated (by sitting down).
Note: If a "one" or "double ones" occur on the very first
roll of a round, then that round is over and each player must take the
Playing SKUNK with the Whole Class
The best way to teach SKUNK to the class is to play a practice
Draw a SKUNK score sheet on the chalkboard or overhead
transparency on which to record dice throws. Have all students make
their own score sheets on their own scrap paper. Have all students
stand up next to their chairs. Either you or a student rolls the dice.
Suppose a "four" and a "six come up, total 10. Record the outcome of
the roll in the "S" column on the chalkboard:
On the first roll, all the players get a total of the dice or a zero
if any "ones" come up. Kerry and Lisa are standing up, so they also
write "10" in their score sheets.
After each roll, players may choose either to remain standing or to
sit down. Those who are standing get the results of the next dice roll;
those who sit down keep the score they have accumulated for that round
regardless of future dice rolls. Once someone sits down, that person
may not stand up again until the beginning of the next round.
The sample game continues on the SKUNK Further Examples Sheet.
SKUNK Further Examples Sheet
Instead of focusing on a single class winner, more students will
be drawn into thinking about a strategy for doing well in this game by
emphasizing personal goals. When playing the game for the second and
third time, ask students to focus on trying to better their own
previous score. After each game ask for a show of hands of those who
did better than last time.
Thinking about SKUNK
Although playing SKUNK is fun, thinking about SKUNK is essential
for student understanding of the underlying concepts. In groups of two
or three, students should complete the questions on the handout.
Thinking About SKUNK Activity Sheet
Groups of students could organize whole-class experiments to
find answers to problems 4, 5, 6. As a class, share results and
solutions to the questions posed.Suggested solutions and discussion points
For question 1, the chance part of SKUNK is the dice roll and choice part is the decision to sit down or remain standing.
Many games can be listed for question 2. Games of pure chance
include Candy Land and Bingo. Games involving almost pure choice,
disregarding who goes first and your opponent's ability, include chess
and tic-tac-toe. Most games, such as hearts, basketball, or Monopoly,
involve both choice and chance. The game of Uno is mostly chance no
matter what choices are made. But poker can be either mostly chance or
mostly choice depending how is it played. Strategies are useful only in
games that allow for choices. But even games that have choices can be
mostly chance for a player who makes choices without any strategy.
Question 3 can lead to class discussions that involve interesting
probabilities and decisions from students' lives. Some events that a
thirteen-year-old would ascribe mostly to chance include these: you
find a $20 bill, you calculator is stolen, having a bad acne outbreak,
your cousin becomes a famous musician, your best friend has to go to a
different high school then you, and the like. Some typical events
resulting from a thirteen-year-old's choices might include these: a
girl dances with you because you asked her, you flunk a quiz because
you didn't study, you get paid your allowance because you do your
chores, and so on.
Questions 4, 5, and 6 can be done either by experimenting or making
theoretical arguments. For example, for question 5, dice could be
rolled many times and the points noted. Then the points could be
totaled and the average value per time calculated. One theoretical
approach is to list the equally likely outcomes for rolling a pair of
dice where SKUNK points are accumulated. Twenty-five equally likely
outcomes yield points. Such a list of outcomes is shown in Table 1. Rolls including a "one" are not shown
because no points are accumulated on the rolls.
The average of all the equally likely values is 8. This value can be either
calculated or observed from the symmetry of the table.
This lesson is adapted from an
article by Dan Brutlag, "Choice and Chance in Life: The Game of SKUNK,"
which appeared in Mathematics Teaching in the Middle School, Vol. 1, No. 1 (April 1994), pp. 28-33.