Lesson 1: Pre-Visit Batting Average

Objective: Students will be able to:

• Calculate a batting average.
• Review prior knowledge of conversion between fractions, decimals, and percentages.
• Use basic linear algebra to solve for unknown variables in batting average equations.

Time Required:

1 class period

Materials Needed:

- Baseball cards (non-pitchers) – enough for each student to have one
- “Linear Equations Activity Cards” (included), printed and cut out
- Calculators
- Scrap Paper
- Pencils

Vocabulary:

Batting Average – A measure of a batter’s performance, calculated as the number of hits divided by the number of times at bat
Statistics - A branch of mathematics dealing with the collection, analysis, interpretation, and presentation of numerical data

Lesson

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1. To begin this lesson, discuss that in almost every sport, players are evaluated or judged using statistics. Ask students, “What are statistics?”

2. Guide students to understand that the term “statistics” refers to a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of numerical data.

3. In baseball, statistics are a big part of the game. Numerical data is collected on every player and every team in the major leagues. This data is organized and interpreted by everyone from sportswriters to managers who then draw conclusions from the data.

4. Give each student one baseball card and have students examine the information on the back of each card. Ask, “What sort of statistical information about a player is available on a baseball card?”

5. Students may or may not be familiar with the code letters used for different statistics. Review that BA = batting average, G = games played, AB = at bats, R = runs, H = hits, 2B = doubles, 3B = triples, HR = home runs, RBI = runs batted in, SB = stolen bases.

6. Ask students if they know how any of these statistics are calculated. Many of these statistics, such as number of games and number of hits, are simply counts. Other statistics require mathematical formulas to figure out.

7. Discuss the concept of average and how it is usually calculated – by adding together the outcomes of a given undertaking and dividing by how many times outcomes were observed.

8. Explain that today students will be looking more closely at one of the most common baseball statistics: batting average.

9. Discuss that this statistic is used to describe the proportion of time that a batter gets a hit (single, double, triple, home run) when he or she gets a chance to bat. One complication is that many times that a batter goes up to bat, he is not given a chance to get a hit. Sometimes the player is walked or gets hit by a pitch, and sometimes the player is asked to make an out to benefit his team by helping a teammate advance around the bases (a “sacrifice bunt” or “sacrifice fly”).

10. Explain that a batting average is calculated by first counting the number of times that a batter reaches base by getting a hit. This number of hits is then divided by the number of times that he gets a chance to hit (an “At Bat”).

11. Write down the formula for batting average on the board: Hits (H)/At Bats (AB).

12. In a typical season, a good player, who plays in most of his or her team’s games, might get about 180 hits in about 600 at bats. This would give the player a batting average of 180/600 or .300.

13. Batting average is usually rounded off to the nearest thousandth (three digits after the decimal) and most people don’t bother writing the leading zero. In fact, most baseball statisticians do not mention the decimal point. If a player has a batting average of 0.256, we would say that he or she is a “two-fifty-six hitter.”

14. Discuss that although we call it “batting average,” this statistic could also be called a “batting percentage.” The data shows us what percent of the time the batter was successful.

15. Write down the average .275 on the board. Ask students, “How would this average be converted to a percentage?”

16. Using the example of .275, demonstrate that in order to change an average to a percentage, the decimal is moved two places to the right. Thus, .275 becomes 27.5%

17. Discuss that for a major league player, a .275 average is pretty good. However this means that the batter was successful just over 25% of the time. Nearly 73% of the time, he didn’t get a hit! This demonstrates just how difficult it is to be a major league batter.

18. Now review how to turn a percentage into a decimal. Write down the percentage 32% on the board. Ask students, “If we know that a player hit successfully 32% of the time, what is his batting average?”

19. Using the example of 32%, demonstrate that in order to change a percentage to an average, the decimal is moved two places to the left. Thus, 32% becomes a .320 average.

20. Next, challenge students to determine a player’s number of hits or at bats using algebra. Ask students, “Let’s say we know that Derek Jeter went to bat 8 times during a double header. He hit successfully 62.5% of the time. How many hits did he get?”

21. If necessary, explain the process of solving the problem:

o First, convert the percentage to a decimal.
62.5% becomes .625

o Now, place that information in the formula for batting average.
H/AB = Average
H/AB = .625

o The problem also tells us how many times Jeter went to bat. Place that information in the equation as well.
H/8=.625

o To solve a linear equation, you have to add, subtract, multiply, or divide both sides of the equation by numbers and variables, so that you end up with a single variable on one side and a single number on the other. Any operation done on one side must be done on the other.

o In this case, in order to get H by itself, multiply each side by 8.
H/8 x 8 = .625 x 8

o We now have the answer: H = 5

19. Try a similar problem, this time solving for at bats. “Let’s say we know that Prince Fielder got 7 hits during a 3-game series. He hit successfully 63.6% of the time. How many times did Prince Fielder bat?”

o Again, start by converting the percentage to a decimal.
63.6% becomes .636

o Place that information in the formula for batting average.
H/AB = Average
H/AB = .636

o Place Fielder’s number of hits into the equation.
7/AB = .636

o This time, in order to solve for AB, we need to first multiply by AB.
7/AB x AB = .636 x AB
7 = .636AB

o Now we need to get AB by itself, so we divide by .636 on each side.
7/.636 = .636AB/.636

o We now have the answer: 11 = AB

20. Remind students that when solving for hits or at bats, the answer must be a whole number. No one gets 6.5 hits in a game. Therefore the answer must be rounded to the nearest whole.

21. Introduce the activity.

Activity

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1. Pass out “Linear Equations Activity Cards” (included), one to each student in the class. Some students will solve for number of hits, some students will solve for number of at bats.

2. Have students solve their equations, then convert the player’s batting average to a percentage.

3. Once every student has finished, have students calculate a collective batting average and batting percentage for the entire class.

Conclusion:
Explain that the average for each player listed in a box score is a cumulative average for the season to date. For a particular game, students can calculate the batting average for an entire team by dividing total hits by total at bats. To complete this lesson and check for understanding, for homework, have students use the sports section of a newspaper (or go online) to locate box scores from 3 games. Students should calculate the batting averages of the 6 teams that played. Compare results to determine which team had the best batting average.

Common Core Standards

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CCSS.Math.Content.HSN-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

CCSS.Math.Content.HSN-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.

CCSS.Math.Content.HSA-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

CCSS.Math.Content.HSA-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Special Thanks to Thomas E. Campbell, 6-12 Math Teacher & Dean of Faculty at Waynflete School in Portland, ME – and – Daniel T. Crocker Math Teacher at Hall-Dale Middle School in Farmingdale, ME for their contributions to this lesson.