## Fun with Baseball Stats

• Lesson
6-8
2

The following grades 6-8 activities allow students to explore statistics surrounding baseball. They are exposed to connections between various mathematical concepts and see where this mathematics is used in areas with which they are familiar. This lesson plan is adapted from the May 1996 edition of Mathematics Teaching in the Middle School.

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 baseball-card 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 three-digit 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 game-card 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.

 Baseball Stats Activity Sheet

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

FractionDecimalSquares
HR/AB32/393.08148
3B/AB1/393.00250
2B/AB20/393.05095
H/AE97/393.246825

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 game-card 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 three-inning game which is approximately twenty to twenty-five 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 time-out 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 teacher-facilitated 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.

Extensions

1. To extend this lesson, institute a player draft before the four phases begin. To begin the draft, students are divided into two groups and given a brief introduction. The groups are then given the opportunity to discuss their drafting strategy and are encouraged to interpret the statistics on the back of each card in terms of that player's potential contribution to the team. Students will disagree concerning which player is the best selection. For example, using the statistics from the Topps 1993 card, one student might argue that Rob Deer's thirty-two home runs make him the best selection. Another student might contend, however, that Shawon Dunston's substantially higher batting average makes him more valuable. Anyone who selects the rookie probably does not have a good understanding of the statistics being considered!

2. Enhancements can also be added during the playing of the game to make it more exciting. One possibility would be allowing runners on second base to attempt to score on a single by tossing a die; if the die is odd, they score, but if the die is even, they are out at the plate! Students will feel more ownership for the game if they are allowed to create their own extensions and subsequent rules.

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### Learning Objectives

Students will:

• Work with decimals, fractions, and percentages in the context of baseball statistics
• Develop skills in mathematical reasoning and computations and apply those skills to everyday life

### Common Core State Standards – Mathematics

• CCSS.Math.Content.7.NS.A.3
Solve real-world and mathematical problems involving the four operations with rational numbers.

• CCSS.Math.Content.7.EE.B.3
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or$2.50, for a new salary of \$27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

• CCSS.Math.Content.6.NS.A.1
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?