Pre-Visit Batting Average

Home , Batting average, Elvis Andrus, First baseman, Matt Holliday, Melky Cabrera, Michael Bourn

Statistics: Batter Up! - Level 3

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

Special Thanks to Thomas E. Campbell, 6-12 Math Teacher & Dean of Faculty at 4 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. Statistics: Batter Up! - Level 3

Applicable Common Core State Standards :

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 5 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. Statistics: Batter Up! - Level 3

Lesson

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 .

Special Thanks to Thomas E. Campbell, 6-12 Math Teacher & Dean of Faculty at 6 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. Statistics: Batter Up! - Level 3

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.

Special Thanks to Thomas E. Campbell, 6-12 Math Teacher & Dean of Faculty at 7 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. Statistics: Batter Up! - Level 3

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

Special Thanks to Thomas E. Campbell, 6-12 Math Teacher & Dean of Faculty at 8 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. Statistics: Batter Up! - Level 3

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.

Special Thanks to Thomas E. Campbell, 6-12 Math Teacher & Dean of Faculty at 9 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. Statistics: Batter Up! - Level 3

Activity

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.

Special Thanks to Thomas E. Campbell, 6-12 Math Teacher & Dean of Faculty at 10 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. Statistics: Batter Up! - Level 3

Linear Equations Activity Cards

Jose Reyes Ryan Braun Victor Martinez

181 = .337 X = .332 178 = .330 X 563 X

Matt Kemp Lance Berkman Hunter Pence

X = .324 147 = .301 190 = .314 602 X X

Joey Votto Carlos Beltran Nelson Cruz

X = .309 X = .300 125 = .263 599 520 X

Albert Pujols Aramis Ramirez Derek Jeter

173 = .299 173 = .306 X = .297 X X 546

Melky Cabrera Matt Holliday Alex Avila

X = .305 X = .296 137 = .295 658 446 X

Austin Jackson Jose Bautista Michael Bourn

147 = .249 155 = .302 X = .294 X X 656

Vladimir Guerrero Justin Upton Jason Bay

X = .290 X = .289 109 = .245 562 592 X

Special Thanks to Thomas E. Campbell, 6-12 Math Teacher & Dean of Faculty at 11 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. Corey Hart Seth Smith Miguel Montero

140 = .285 135 = .284 X = .282 X X 493

Adam Jones Elvis Andrus Placido Polanco

X = .280 X = .279 130 = .277 567 587 X

Dexter Fowler Carlos Lee Neil Walker

128 = .266 161 = .275 X = .273 X X 596

Special Thanks to Thomas E. Campbell, 6-12 Math Teacher & Dean of Faculty at 12 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. Statistics: Batter Up! - Level 3

Activity Cards Answer Key

Jose Reyes Ryan Braun Victor Martinez

181 187 178 537 563 540

(.337) (.332) (.330) Matt Kemp Lance Berkman Hunter Pence

195 147 190 602 488 606

(.324) (.301) (.314) Joey Votto Carlos Beltran Nelson Cruz

185 156 125 599 520 475

(.309) (.300) (.263) Albert Pujols Aramis Ramirez Derek Jeter

173 173 162 579 565 546

(.299) (.306) (.297) Melky Cabrera Matt Holliday Alex Avila

201 132 137 658 446 464

(.305) (.296) (.295) Austin Jackson Jose Bautista Michael Bourn

147 155 193 591 513 656

(.249) (.302) (.294)

Special Thanks to Thomas E. Campbell, 6-12 Math Teacher & Dean of Faculty at 13 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. Vladimir Guerrero Justin Upton Jason Bay

163 171 109 562 592 444

(.290) (.289) (.245) Corey Hart Seth Smith Miguel Montero

140 135 139 492 476 493

(.285) (.284) (.282) Adam Jones Elvis Andrus Placido Polanco

159 164 130 567 587 469

(.280) (.279) (.277) Dexter Fowler Carlos Lee Neil Walker

128 161 163 481 585 596

(.266) (.275) (.273)

Special Thanks to Thomas E. Campbell, 6-12 Math Teacher & Dean of Faculty at 14 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. Statistics: Batter Up! - Level 3

Lesson 2 – Pre-Visit Slugging Percentage

Objective : Students will be able to: • Set up and solve equations for batting average and slugging percentage. • Review prior knowledge of conversion between fractions, decimals, and percentages. • Compare sets of statistical data using graphs. • Draw conclusions from data.

Time Required : 1 class period

Materials Needed : - Baseball cards (non-pitchers) – enough for each student to have one - Copies of the “Hall of Fame Hitters” worksheet (included) – 1 for each student - Graphing calculators or software with graphing capabilities such as Excel - Graph Paper - 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 Ratio - The relationship in quantity, amount, or size between two or more things (expressed as a quotient) Slugging Percentage - A measure of the power of a hitter, calculated as the number of total bases divided by the number of times at bat Statistics - A branch of mathematics dealing with the collection, analysis, interpretation, and presentation of numerical data

15 Statistics: Batter Up! - Level 3

Applicable Common Core State Standards :

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

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

CCSS.Math.Content.HSS-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

CCSS.Math.Content.HSS-ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

CCSS.Math.Content.HSS-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

16 Statistics: Batter Up! - Level 3

Lesson

1. Begin the lesson by reviewing that statistics are very important in baseball, so much so that hitters and pitchers often come to be identified by their hitting and pitching stats.

2. Explain that just like batting average , slugging percentage is a comparison statistic. Slugging percentage (SLG) measures a batter’s ability to hit with power.

3. Write down the formula for slugging percentage on the board:

Slugging Percentage = Total Number of Bases Number of At Bats

4. The number of total bases can be calculated by finding the sum of the following: • Number of singles x 1 base • Number of doubles x 2 bases • Number of triples x 3 bases • Number of home runs x 4 bases

5. Give students the following practice problem: • Ryan Howard comes up to bat 7 times in a doubleheader. He hits two singles, one double, no triples, and two home runs. What is his slugging percentage for the two games?

6. Go over the practice problem with students. • 2 singles x 1 = 2 bases • 1 double x 1 = 2 bases • 0 triples x 3 = 0 bases • 2 home runs x 4 = 8 bases • 12 total bases/7 at bats = 1.714

7. Explain that just like batting averages, slugging percentages have to be rounded to the thousandths place.

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8. Provide each student with a baseball card. Ask students, “Given the information on the back of the card, do you have enough data to be able to determine the player’s slugging percentage?”

9. Guide students to understand that all data necessary for calculating slugging percentage is on the back of the card. While the numbers of doubles, triples, and home runs are listed, the number of singles is not. To find the number of singles, the number of 2B, 3B, and HR is subtracted from the number of hits (H). Once the number of singles is determined, slugging percentage can be calculated.

10. Introduce the activity.

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Activity

1. For this activity, students may work individually or in pairs. Provide each student or pair with a “Hall of Fame Hitters” worksheet (included).

2. Have students complete the worksheet.

3. Facilitate an in-class discussion in which students interpret the data they have just calculated. Use the following questions to help guide discussion: - What is the relationship between a player’s hits and his batting average? - What is the relationship between a player’s hits and slugging percentage? - Which player should be considered more valuable to a team: a batter who has a high batting average but few home runs, or a batter who has a modest average but hits many home runs? Why?

4. Discuss the type of graph that would be most useful to compare these hitting statistics: scatter plot, box plot, histogram, etc.

5. Have each student create 2 graphs (class choice) - one showing the data for batting average and one showing the data for slugging percentage. You may choose to have students create their graphs on graph paper, or using software with graphing capabilities such as Excel.

6. Have students identify any patterns they see in the graphs.

Conclusion: To complete this lesson and check for understanding, for homework, have students research the statistics for two baseball players of their choice. Compare their performances and determine which of the two had a better year statistically. Students should write an analysis that justifies their position.

19 Statistics: Batter Up! - Level 3

Hall of Fame Hitters

Name: ______Date:______

All the players listed below have been inducted into the National Baseball Hall of Fame. Answer the questions on Page 2 of this worksheet.

Player Games At Hits 2B 3B HR Batting Slugging Bats Average Percentage Orlando 2124 7927 2351 417 27 379 Cepeda

Rod Carew 2469 9315 3053 445 112 92

Ty Cobb 3034 11434 4189 724 295 117

Joe DiMaggio 1736 6821 2214 389 131 361

Hank Aaron 3298 12364 3771 624 98 755

Tony Gwynn 2440 9288 3141 543 85 135

Mickey Mantle 2401 8102 2415 344 72 536

Dave Winfield 2973 11003 3110 540 88 465

Ted Williams 2292 7706 2654 525 71 521

Babe Ruth 2503 8399 2873 506 136 714

20 Statistics: Batter Up! - Level 3

Hall of Fame Hitters – Page 2

1. Based on the information given, determine each player’s batting average and slugging percentage. *Remember* You will first have to determine the player’s number of singles before you can determine Slugging Percentage.

2. Transfer the data to a spreadsheet on a computer or to a graphing calculator.

3. Determine the following for each player: - The percentage of home runs per game - The percentage of home runs per at bat - The percentage of home runs per hit

4. Rank the 10 players in descending order of the ratio of home runs per game.

5. Rank the 10 players in descending order of the ratio of home runs per at bat.

6. Rank the 10 players in descending order of the ratio of home runs per hit.

7. Rank the 10 players in descending order of batting average.

8. Rank the 10 players in descending order of slugging percentage.

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Hall of Fame Hitters Answer Key

Player Games At Hits 2B 3B HR Batting Slugging Bats Average Percentage Orlando 2124 7927 2351 417 27 379 .297 .499 Cepeda

Rod Carew 2469 9315 3053 445 112 92 .328 .429

Ty Cobb 3034 11434 4189 724 295 117 .366 .512

Joe DiMaggio 1736 6821 2214 389 131 361 .325 .579

Hank Aaron 3298 12364 3771 624 98 755 .305 .555

Tony Gwynn 2440 9288 3141 543 85 135 .338 .459

Mickey 2401 8102 2415 344 72 536 .298 .557 Mantle

Dave 2973 11003 3110 540 88 465 .283 .475 Winfield

Ted Williams 2292 7706 2654 525 71 521 .344 .634

Babe Ruth 2503 8399 2873 506 136 714 .342 .690

Determine the following for each player : - The percentage of home runs per game - The percentage of home runs per at bat - The percentage of home runs per hit

Cepeda: 17.8% HR/G; 4.7% HR/AB; 16.1% HR/H Carew: 3.7% HR/G; 0.9% HR/AB; 3.0% HR/H Cobb: 3.9% HR/G; 1.0% HR/AB; 2.8% HR/H DiMaggio: 20.8% HR/G; 5.3% HR/AB; 16.3% HR/H Aaron: 22.9% HR/G; 6.1% HR/AB; 20.0% HR/H

22 Statistics: Batter Up! - Level 3

Gwynn: 5.5% HR/G; 1.5% HR/AB; 4.3% HR/H Mantle: 22.3% HR/G; 6.6% HR/AB; 22.2% HR/H Winfield: 15.6% HR/G; 4.2% HR/AB; 15.0% HR/H Williams: 22.7% HR/G; 6.7% HR/AB; 19.6% HR/H Ruth: 31.2% HR/G; 8.5% HR/AB; 24.9% HR/H

Rank the 10 players in descending order of the ratio of home runs per game. Ruth, Aaron, Williams, Mantle, DiMaggio, Cepeda, Winfield, Gwynn, Cobb, Carew

Rank the 10 players in descending order of the ratio of home runs per at bat. Ruth, Williams, Mantle, Aaron, DiMaggio, Cepeda, Winfield, Gwynn, Cobb, Carew

Rank the 10 players in order of the ratio of home runs per hit. Ruth, Mantle, Aaron, Williams, DiMaggio, Cepeda, Winfield, Gwynn, Carew, Cobb

Rank the 10 players in descending order of batting average. Cobb, Williams, Ruth, Gwynn, Carew, DiMaggio, Aaron, Mantle, Cepeda, Winfield

Rank the 10 players in descending order of slugging percentage. Ruth, Williams, DiMaggio, Mantle, Aaron, Cobb, Cepeda, Winfield, Gwynn, Carew

23 Statistics: Batter Up! - Level 3

Lesson 3 – Pre-Visit Teams & Players by the Numbers

Objective : Students will be able to: • Review how to find the mean, median and mode of a data set. • Calculate the standard deviation of a data set. • Evaluate data using a stemplot.

Time Required : 1-2 class periods

Materials Needed : - Copies of the “Batting Stats by Team for the 2011 Season” table (included) – 1 for each student - Copies of the “MLB Scouting Reports” packet (included) – 1 for each student or pair of students - Calculators - Scrap Paper - Pencils - Internet access for student research

Vocabulary: Mean – The average of a set of data Median - The middle number in a set of data when the data is arranged in numerical order Mode - The number that appears the most in a data set Range - The difference between the largest and smallest number in a data set Standard Deviation - A number that measures how far away each number in a data set is from the mean Statistics - A branch of mathematics dealing with the collection, analysis, interpretation, and presentation of numerical data Stemplot - A device for presenting quantitative data in a graphical format so that it is easy to see the shape of a distribution

24 Statistics: Batter Up! - Level 3

Applicable Common Core State Standards :

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

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

CCSS.Math.Content.HSS-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

CCSS.Math.Content.HSS-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

25 Statistics: Batter Up! - Level 3

Lesson

1. To begin this lesson, provide students with the “Batting Stats by Team for the 2011 Season” table (included).

2. Have students spend a minute or two examining the information available on the table. Ask students, what type of data is available on this table? Is there anything that stands out to you? Is it easy or difficult to try to compare teams just by looking at the numbers?

3. Explain that teams could be compared using every column of data on this table, but since that would take a very long time, start by comparing teams using a single data set: number of hits (H). Explain that it is often helpful to start evaluating data by creating a graph. For this type of data, a stemplot will allow for easy comparison.

4. Demonstrate how to create a stemplot. A basic stemplot contains two columns separated by a vertical line. The left column contains the “stems” and the right column contains the “leaves.” For hits, it will be easiest to create a stem using the ones and tens places, and a leaf of the hundreds and thousands places. For example, Boston had 1600 hits. The stem and leaf would look like this:

16|00

5. Next write down all of the possible stems (12-16), and record each hit total by writing down the leaf on the line corresponding to the stem.

6. Work with students to create a stemplot for all 30 teams. Have students copy this example into their notebooks.

12|8463 13|8457248094873095455819572527 14|523829227709342342 15|99401360 16|00

26 Statistics: Batter Up! - Level 3

7. Once the stemplot is finished, encourage students to draw conclusions from the shape of the data on the plot. For example, although they haven’t calculated it yet, students should be able to determine that the mean and median will fall in the 1300-1400 range. Students should also be able to identify the spread of data as well as any outliers.

8. Now, have students go over the stemplot again, this time replacing the leaf with an abbreviation for the team’s name (using different colored pens or pencils for each league helps).

9. Have students draw further conclusions from this stemplot. With the team names in place of numbers, it’s easier to see that there was a greater spread between hits for AL teams than hits for NL teams. Other observations might include: - Since the “Batting Stats” table is ranked by Runs, we can determine that Toronto was able to achieve the 6 th most overall runs with a lower number of hits than other top run-scoring teams. It could be that Toronto was better able to get players into scoring position. Also, a quick examination of Toronto’s HR data will show that the team had the 5th highest HR total, and a .413 team SLG. Although they had fewer hits than many teams, it appears that they had more powerful ones.

10. Review the definitions for mean , median , and mode . - Mean is the average of a set of data. To calculate the mean, find the sum of the data and then divide by the number of data. - Median is the middle number in a set of data when the data is arranged in numerical order. First arrange the data in numerical order. Then find the number in the middle or the average of the two numbers in the middle. - Mode is the number that appears the most. Sometimes a data set will have more than one mode (bimodal), and sometimes there is no mode in a data set (when all numbers occur only once).

11. Have students figure out the mean, median, and mode for the “hits” column on the 2011 Batting Stats table. Answers: Mean = 1408.9, Median = 1394.5, Mode = 1357

27 Statistics: Batter Up! - Level 3

12. While it is useful to know that the average number of hits by all teams in Major League Baseball, the mean does not reveal anything about the distribution or variation of the data set. So we need to come up with some way of measuring not just the average, but also the spread of the distribution of data.

13. Explain that the standard deviation is a number that measures how far away each number in a data set is from the mean. If the standard deviation is large, it means the numbers are spread out from their mean. If the standard deviation is small, it means the numbers are close to their mean.

14. Walk students through the process of determining the standard deviation for hits in the American League in 2011. Have students copy down this example into their notebooks for future reference.

15. Listed below is the number of hits for all American League teams. Step 1 - Determine the mean. Answer: (20004/14) Mean = 1428.9

H 1600 1452 1599 1540 1384 1560 1434 1324 1380 1394 1387 1330 1357 1263

28 Statistics: Batter Up! - Level 3

16. Step 2 – For each team’s hit total, determine the distance from the mean.

H Distance from Mean 1600 171.1 1452 23.1 1599 170.1 1540 111.1 1384 -44.9 1560 131.1 1434 5.1 1324 -104.9 1380 -48.9 1394 -34.9 1387 -41.9 1330 -98.9 1357 -71.9 1263 -165.9

17. Step 3 - Square each distance.

H Distance Distance from Mean Squared 1600 171.1 29275.21 1452 23.1 533.61 1599 170.1 28934.01 1540 111.1 12343.21 1384 -44.9 2016.01 1560 131.1 17187.21 1434 5.1 26.01 1324 -104.9 11004.01 1380 -48.9 2391.21 1394 -34.9 1218.01 1387 -41.9 1755.61 1330 -98.9 9781.21 1357 -71.9 5169.61 1263 -165.9 27522.81

29 Statistics: Batter Up! - Level 3

18. Step 4 - Add up the sum of the distances squared. Answer = 149157.7

19. Step 5 - Divide by (n-1) where n represents the amount of numbers you have. (In this case 14)

149157.7 = 11473.7 (14-1)

20. Step 6 - Take the square root of the average distance. √ 11473.7 = 107.1

21. The standard deviation for this data set is 107.1.

22. Have students work on their own to determine the standard deviation for hits in the National League.

Teacher’s Reference for NL:

H Distance Distance from Mean Squared 1513 121.6 14786.56 1438 46.6 2171.56 1429 37.6 1413.76 1357 -34.4 1183.36 1422 30.6 936.36 1477 85.6 7327.36 1409 17.6 309.76 1423 31.6 998.56 1395 3.6 12.96 1345 -46.4 2152.96 1358 -33.4 1115.56 1319 -72.4 5241.76 1442 50.6 2560.36 1325 -66.4 4408.96 1284 -107.4 11534.76 1327 -64.4 4147.36

30 Statistics: Batter Up! - Level 3

Sum of Distances Squared = 60301.96

60301.96/(16-1) = 4020.1

√4020.1 = 63.4 Standard Deviation = 63.4

23. Have students compare the standard deviations of the two data sets. AL = 107.1, NL = 63.4.

24. Encourage students to draw conclusions from the standard deviation of hits. Basically, the standard deviation of this data set tells us that National League teams are more equally balanced in terms of hits than teams in the American League.

25. Introduce the activity.

31 Statistics: Batter Up! - Level 3

Activity

1. For this activity, students may work in pairs or individually.

2. Explain the instructions for this activity. Each student (or pair of students) has been hired as a scout for a Major League Baseball team. The team owner is in the market for a new first baseman. As a scout, you are responsible for evaluating two candidates for the position of first baseman. You are to create a scouting report on each candidate and make a recommendation for which of the two candidates the team owner should hire.

3. Explain that for each player, a scouting report must include the following: - A stemplot showing the player’s batting average over the course of his career - The mean, median, and mode of his average - The standard deviation of his average

4. Player statistics can be easily found at http://espn.go.com/mlb/ or at http://www.baseball-reference.com/ . You may choose to assign students two current first basemen to evaluate or have students choose for themselves.

5. Finally, students must choose which of the two players they would like to hire for their teams and explain their reasoning using the player’s statistics as evidence.

6. Provide students with the “MLB Scouting Reports” packet (included). Have students complete this project in class or as a homework assignment.

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Conclusion : To conclude this lesson and check for understanding, review the results of each student’s (or pair of students’) scouting reports. Students should share their statistical findings as well as their recommendation for which player to hire. After all students have had an opportunity to share, discuss the results with the class. Which players were more likely to be recommended? Players with consistently good averages over their careers, or players who show variance in their careers, but appear to be improving? Why?

For homework, have students write journal entries in which they address the importance of statistics. Do statistics tell a manager or an owner everything he or she needs to know about a player? What are some skills that can’t be revealed through statistics?

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MLB Scouting Reports

Name(s):______Date:______

Instructions: You have been hired as a scout for a Major League Baseball team. The owner of your team is in the market for a new first baseman. As a scout, you are responsible for evaluating two candidates for the position of first baseman. You are to create a scouting report on each candidate and make a recommendation for which of the two candidates the team owner should hire.

For each player, your scouting report must include the following: - A stemplot showing the player’s batting average over the course of his career - The mean, median, and mode of his average - The standard deviation of his average

Player statistics can be easily found at http://espn.go.com/mlb/ or at http://www.baseball-reference.com/ .

34 Statistics: Batter Up! - Level 3 Player 1: Name: ______Height: ______Weight: ______Bats: ______Throws: ______

1. List this player’s batting average for each year of his career.

2. Draw a stemplot using the player’s batting average for each year of his career.

3. What conclusions can you draw from this stemplot? ______

4. What is this player’s mean average? ______

5. What is this player’s median average?______

6. What is the mode?______

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7. Determine the standard deviation for this player’s batting average. Show your work.

Standard Deviation:______

8. What conclusions can you draw based on the standard deviation of this player’s average? ______

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Player 2: Name: ______Height: ______Weight: ______Bats: ______Throws: ______

1. List this player’s batting average for each year of his career.

2. Draw a stemplot using the player’s batting average for each year of his career.

3. What conclusions can you draw from this stemplot? ______

4. What is this player’s mean average? ______

5. What is this player’s median average?______

6. What is the mode?______

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7. Determine the standard deviation for this player’s batting average. Show your work.

Standard Deviation:______

8. What conclusions can you draw based on the standard deviation of this player’s average? ______

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Scouting Report Recommendation:

Which player do you recommend hiring for your team? Why? Explain your reasoning using the player’s statistics as evidence.

______

39 Statistics: Batter Up! - Level 3

Batting Stats by Team for the 2011 Season

Red = American League Blue = National League

Team GP AB R H 2B 3B HR TB RBI AVG OBP SLG OPS Boston 162 5710 875 1600 352 35 203 2631 842 .280 .349 .461 .810 NY Yankees 162 5518 867 1452 267 33 222 2451 836 .263 .343 .444 .788 Texas 162 5659 855 1599 310 32 210 2603 807 .283 .340 .460 .800 Detroit 162 5563 787 1540 297 34 169 2412 750 .277 .340 .434 .773 St. Louis 162 5532 762 1513 308 22 162 2351 726 .273 .341 .425 .766 Toronto 162 5559 743 1384 285 34 186 2295 704 .249 .317 .413 .730 Cincinnati 162 5612 735 1438 264 19 183 2289 697 .256 .326 .408 .734 Colorado 162 5544 735 1429 274 40 163 2272 697 .258 .329 .410 .739 Arizona 162 5421 731 1357 293 37 172 2240 702 .250 .322 .413 .736 Kansas City 162 5672 730 1560 325 41 129 2354 705 .275 .329 .415 .744 Milwaukee 162 5447 721 1422 276 31 185 2315 693 .261 .325 .425 .750 NY Mets 162 5600 718 1477 309 39 108 2188 676 .264 .335 .391 .725 Philadelphia 162 5579 713 1409 258 38 153 2202 693 .253 .323 .395 .717 Baltimore 162 5585 708 1434 273 13 191 2306 684 .257 .316 .413 .729 Tampa Bay 162 5436 707 1324 273 37 172 2187 674 .244 .322 .402 .724 Cleveland 162 5509 704 1380 290 26 154 2184 671 .250 .317 .396 .714 LA Angels 162 5513 667 1394 289 34 155 2216 629 .253 .313 .402 .714 Chicago Sox 162 5502 654 1387 252 16 154 2133 625 .252 .319 .388 .706 Chicago Cubs 162 5549 654 1423 285 36 148 2224 610 .256 .314 .401 .715 Oakland 162 5452 645 1330 280 29 114 2010 612 .244 .311 .369 .680 LA Dodgers 161 5436 644 1395 237 28 117 2039 613 .257 .322 .375 .697 Atlanta 162 5528 641 1345 244 16 173 2140 606 .243 .308 .387 .695 Florida 162 5508 625 1358 274 30 149 2139 596 .247 .318 .388 .706 Washington 161 5441 624 1319 257 22 154 2082 594 .242 .309 .383 .691 Minnesota 162 5487 619 1357 259 25 103 1975 572 .247 .306 .360 .666 Houston 162 5598 615 1442 309 28 95 2092 579 .258 .311 .374 .684 Pittsburgh 162 5421 610 1325 277 35 107 1993 580 .244 .309 .368 .676 San Diego 162 5417 593 1284 247 42 91 1888 563 .237 .305 .349 .653 San Francisco 162 5486 570 1327 282 24 121 2020 534 .242 .303 .368 .671 Seattle 162 5421 556 1263 253 22 109 1887 534 .233 .292 .348 .640

Statistics: Batter Up! - Level 3

Applicable Common Core State Standards :

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

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

CCSS.Math.Content.HSS-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

CCSS.Math.Content.HSS-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

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Lesson & Activity

*Note* The goal of this activity is for students to apply the material they have learned in this unit as well as in class. Ideally, students will be able to track their fantasy baseball teams for several weeks of a semester. Since most of the professional baseball season takes place in the summer, students may be limited to April-June and August-September depending on your school calendar.

1. Introduce this activity by explaining how fantasy baseball will work in your classroom. Students will act as managers of pretend baseball teams. They will choose teams composed of real players, and they will earn points based on their players’ performances in real games.

2. Each week, students will use newspapers or online resources to locate their players’ statistics and calculate the number of points earned by their teams. Points will be awarded as follows:

OFFENSE

- Single (1 pt), Double (2 pts), Triple (3 pts), Home Run (4 pts) - Walk (1 pt), Hit By Pitch (1pt), Run (1 pt), RBI (1pt) - Stolen Base (2 pts) - Caught stealing (-1 pt), Strikeout (-0.5 pt)

PITCHING

- Win (7 pts), Save (7 pts) - Each inning pitched (3 pts) - Strikeout (0.5 pt) - Walk (-1 pt), Earned Run (-1 pt), Hit Allowed (-1 pt), Hit Batter (-1 pt) - Loss (-5 pts)

3. Set up a player draft. Students may either draft any current player in Major League Baseball for their teams, or they can be limited to a smaller pool of players of your choice (perhaps 50-75).

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4. No matter the complexity of the draft, make sure that students have time to examine their player options before a draft takes place. Discuss the values associated with different statistics as students are evaluating their potential draft picks.

5. During the player draft, students will need to choose players for the following positions:

- Hitters: C, 1B, 2B, 3B, SS, OF, OF, OF - Pitchers: Starting (5), Relief (2) - Bench: Five spots

6. For the sake of simplicity, students should keep the players they draft at the start of the season rather than attempting to negotiate trades.

7. You may choose to have students only keep track of their points throughout the season, or, you can set up weekly head-to-head competitions. In this case, pairs of student teams would compete against each other for one week at a time. The winner of each pair is the team with the most points at the end of the week. Teams are awarded wins and losses accordingly, and you can set up a chart showing classroom standings. With this method you can also set up class playoffs toward the end of the semester with the best teams going head-to-head until an overall winner is determined.

8. The league winner is the student whose team has the most points at the end of the season. Consider offering a small trophy or some other small prize for the winner.

Conclusion: At the conclusion of the fantasy league, students may graph and compare a wide variety of different statistics. For example, students may compare class data for RBIs per player, fewest strikeouts per player, stolen bases per player, home runs per player, etc. Students may also assess the players on their own teams. For example, students could calculate the median number of RBIs for the season, the mean batting average and slugging percentage, or the standard deviation of stolen bases.