Understanding the Cards and the Statistics
When the students enter the classroom, they find several piles of
baseball cards around the room. Each student is asked to select one
card to use throughout the lesson. Students familiar with baseball
usually seek out a player with whom they are familiar, whereas others
simply choose a player with an unusual name or a name that matches
their own. The teacher must ensure that all students are familiar with
the baseballcard statistics that will be used in this lesson. A sample
of necessary information was taken from the 1993 Topps baseball cards
of Detroit Tigers's outfielder Rob Deer, Chicago Cubs's shortstop
Shawon Dunston, and an unnamed rookie and is shown below. The selected
rows represent at bats (AB), hits (H), doubles (2B), triples (3B), home
runs (HR), slugging percentage (SLG), and batting average (AV),
respectively
Selected Information from Baseball Cards of
Three Players (Topps 1993)  Name  Deer  Dunston  Rookie  At bats (AB)  393  73  26  Hits (H)  97  23  5  Doubles (2B)  20  3  1  Triples (3B)  1  1  0  Home runs (HR)  32  0  1  Slugging percentage (SLG)  .547  .384  .346  Batting Average (AV)  .247  .315  .192 

To help students gain confidence, they should be asked to
determine the batting average of their player by finding the quotient
of hits (H) to at bats (AB) and rounding this decimal to the nearest
thousandth. Students generally react with enthusiasm when they find
that their calculations agree with the batting average (AB) printed on
their card. At this point the students should be told to fill in the
bottom row of their activity sheets with the batting average and the
information they used to calculate it (H and AB). They should also be
reminded that despite the common convention of stating a batting
average as a threedigit number, mathematical correctness predicates
that this statistic is actually a decimal rounded to the nearest
thousandth.
Calculations and Using the Activity Sheet
In the next phase of the lesson, the students will perform
calculations to determine the percent of at bats in which their player
obtains each of these possible outcomes: single, double, triple, and
home run. These percents will determine how many of the 100 squares on
their gamecard activity sheet should contain each of these outcomes.
Blank squares will correspond with outs. The number of squares to be
filled with each possible outcome can be obtained by converting a
fraction into a decimal, rounding it to the nearest hundredth, and
converting this decimal back into a fraction with a denominator of 100.
The numerator of this fraction will be the number of squares on the
game card that should correspond to that outcome. To avoid unnecessary
arguments concerning the effects of rounding, the same procedures
should be used in the creation of all game cards.
The foregoing procedure will be illustrated using Rob Deer's
statistics. He hit 32 home runs in 393 at bats. The first box in the
HR/AB row, therefore, should read 32/393. Students should use their
calculators to find 0.0814249 as the decimal equivalent of this ratio
and place it in the second box. The number to be placed in the
"squares" box corresponds with the percent of the time that the player
hits a home run. This number can be found by rounding the decimal to
the nearest hundredth, converting this decimal to a fraction whose
denominator is 100, and determining that the percent corresponds to the
numerator of this fraction. In this case:
0.0814249 = 0.08 = 8/100 = 8%.
At this point in the lesson, it may be helpful to show the following information on the chalkboard:
Player Name  Rob Deer 
Student Name  
 Fraction  Decimal  Squares 

HR/AB  32/393  .0814  8 
3B/AB  1/393  .0025  0 
2B/AB  20/393  .0509  5 
H/AE  97/393  .2468  25 
Thus, 8 squares on the game card will be filled with HR. Similarly,
the number of squares corresponding with triples and doubles can be
determined in the same way. One triple in 393 at bats implies that Rob
Deer triples so rarely that none of the 100 squares should indicate
this outcome because 1/393 rounded to the nearest percent would be 0.
Twenty doubles in 393 at bats yield:
20/393 = 0.05 = 5/100 = 5%.
This indicates that 5 squares should be labeled 2B.
Since the cards do not indicate the number of singles that a player
has gotten, an alternative strategy must be used to find the number of
squares to be filled with 1B. A brief discussion should culminate with
the students' realizing that the total number of squares filled can be
determined by the batting average. Deer's batting average of .247
indicates that 25 of the 100 squares should be filled with an outcome
that corresponds to some kind of hit. (Note: Students will notice that
baseball averages are printed without a leading zero, as they would get
on a calculator.) Since 13 squares are already taken by home runs and
doubles, 12, or 25  13, squares should be marked 1B. If the lesson is
being completed over two class periods, the teacher should collect the
activity sheets at this point and check them before the next day;
otherwise, an informal perusal by the teacher is sufficient to
guarantee that no grievous errors exist.
Creating the Game Board
The students are now ready to complete their game cards by
filling in the appropriate number of squares with home runs (HR),
triples (3B), doubles (2B), and singles (1B). Showing the completed
game card of Deer, which meets these criteria, can be helpful at this
juncture (see above.) A question that often arises is whether the
positioning of the favorable outcomes is important. Some students will
cluster all the favorable outcomes in the same rows or columns, whereas
others will spread them out across the board.
Play Ball!
Before beginning the game, the teacher should explain that the
result of an at bat will be determined by using the two spinners, which
are strategically placed on the gamecard activity sheet. The square
that corresponds to the ordered pair of numbers attained by the two
spins determines the outcome of that at bat. The location of the
spinners is particularly helpful to students who quickly recognize that
2 followed by 5 is different from 5 followed by 2.
To engage more students effectively, split the class into four
teams so that two games can be played simultaneously. The teacher marks
each of the classroom baseball diamonds by labeling appropriately
positioned desks as home plate and the other bases. The teacher
explains that from the results of an at bat, the student (1) moves
around the bases or (2) is "out" and must root for his or her teammates
while awaiting the next turn. For simplicity, each base runner advances
the same number of bases indicated by the type of hit made by the
batter. After a brief demonstration, the students will be capable of
continuing to play on their own. The game progresses in this fashion
for a certain number of innings or a predetermined time limit. One
possibility is to allow for a threeinning game which is approximately
twenty to twentyfive minutes long.
As the game is being played, the students quickly begin to
anticipate what numbers are favorable on the second spin (top, left
corner of game card) after the first spin (bottom, right corner of the
game card) has occurred.
For example, if the first spinner yields a 5 for Rob Deer, members
of his team immediately begin shouting for a 6 to come up on the second
spinner. However, students realize that the first spin for the player
shown below will completely determine whether the outcome is favorable.
When you feel that the students are aware of these phenomena, call a
timeout and consider the positioning of favorable outcomes. Most
students will realize that the first spin of the player in the previous
figure will yield certain success three out of ten times, whereas the
first spin of the player in is irrelevant in that regardless of the
outcome of that spin, success will occur on three out of ten of the
second spins.
Their experiences throughout the game, combined with
teacherfacilitated discussion, should help students realize that the
placement of favorable outcomes does not make any difference concerning
the probability of those outcomes' occurring. This realization reveals
students' development of a better informal sense of the laws of
probability.