The Middle School Math Project

Fantasy Baseball Part I (Number and Number Relations)

Objective Students will study a thematic unit based on the game of baseball to investigate number and number relations.

Overview of the Lesson The following is an overview of the unit Fantasy Baseball - Parts I and II: Students play a few innings of baseball as an introductory activity to this thematic unit based on the game of baseball. The unit is centered around the analysis of the statistical data contained on professional baseball trading cards. These cards are first used to familiarize students with some of the basic terminology associated with the game. Next, students are directed to focus their attention on the actual statistical data included on the card and how to utilize this statistical data to determine the overall rating of a player’s past performance. Additionally, students form work with decimals and ratios to obtain key statistical data. This key statistical data, such as the number of home runs to the number of cumulative times at bat, are essential to the next phase of the unit: building a nine-player baseball team. Student pairs act as co-managers of a baseball team. Each pair receives a pack of baseball cards. They compute and analyze the key statistical data of the given players, make decisions on who they want to keep and who they want to trade, arrange their lineups, and play simulated games. Students use the key statistical data to construct individual player spinners, determine the line up, and play a simulated baseball game. The ultimate goal is to be the manager of the winning team of the Fantasy Baseball World Series.

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Materials Per Student ❶ Fantasy Baseball Pre-Assessment Worksheet ❷ Baseball Card Glossary ❸ Player Guidelines ❹ Copy of Baseball Statistics or Baseball Cards for 2 players (Robin Ventura is included in materials) ❺ Player Comparison Chart Worksheet ❻ Calculator ❼ Cutout tag board bases labeled Home, 1’st, 2’nd. and 3’rd

Procedure Describe the Fantasy Baseball unit to the students: During this unit, you will become managers of your very own baseball team. You will learn how to use decimals and ratios to analyze key statistical data, draft players, make trades, construct a lineup, and play simulated games, all the way to the Fantasy Baseball World Series. All of this will occur as we study the mathematics of baseball. Baseball trading cards will be used as our source of information. Note: A copy of a Fantasy Baseball Pre-assessment is included. If used, students should complete this activity prior to the beginning the unit. It is important that all students have a general knowledge of how the game of baseball is played before they study this unit. Therefore, as an introduction, have students play a few innings of baseball. If this is not possible, show a video clip of a few innings of a baseball game and discuss some of the plays and players. Place the four tag board bases around the room. Each student should receive a copy of the Robin Ventura baseball card (or a baseball card of a player of your choosing ) and a copy of the “Baseball Card Glossary.” Point out that the terminology in the glossary matches the terminology on the baseball card. Have students demonstrate their understanding of the definitions of the terms in the glossary by playing a simulated game. These terms include: hit (H), double (2B), triple (T), homerun (HR), runs batted in (RBI), base on balls (BB), strikeout (SO), and stolen base (SB). The other terms which are included in the glossary should also be discussed at this time. These additional terms will probably be very easy for students to understand MSMP — Fantasy Baseball-Part 1: Lesson Guide • http://www.pbs.org/mathline

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PBS MATHLINE®

and master. They include year (YR), TEAM, games (G) and official at bats (AB). Average (AVG) will be discussed later. Place students in small groups where they further discuss the glossary terms. Have them closely scrutinize the data of the sample player based on the definitions of these terms. This should strengthen their familiarity with some of the baseball terminology, while at the same time making them keenly aware of how an understanding of decimals, ratios, and statistics enable them to make decisions based on mathematics. To prepare students to be able to make informed decisions about trades, line-ups etc., it is essential to have them focus on some comparative relationships which are called key statistics. To begin this process, review ratios and proportions. Following this review, distribute the Player Comparison Chart and Player Guidelines. The Player Guidelines should be used to assist students in the evaluation and discrimination of players based on their interpretation of the data. Systematically discuss and compute the following key statistics. Make sure that students understand why these statistics are so important and the theoretical implications embedded in their values. Batting Average (AVG)

H AB

On-Base Percentage (OB%)

(H + BB) (AB + BB)

Cumulative at Bats (CUM. AB)

AB + BB

HR : CUM. AB (unit ) Ratio

HR = AB + BB

1 x

SO : CUM. AB (unit) Ratio

SO = AB + BB

1 x

Note: The HR : CUM. AB and the SO : CUM. AB are computed to represent unit ratios so that there is a clearer picture of what is happening in the homerun and strikeout categories. Therefore, if the unit ratio is known, the manager will have some theoretical idea of what performances to expect from a player based on his past performances. To assess students’ interpretation of these ratios, have them determine the most favorable performances of this player and why. When assessing the HR: CUM. AB (unit) Ratio, is it more favorable to have the unit compared to a larger number or smaller number? Likewise, when assessing the SO : CUM. AB (unit) Ratio, is it more favorable to compare the unit to a larger number or smaller number?

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PBS MATHLINE®

For practice, provide students with the data for a another baseball player. Again, students should go through the process of computing the key statistical data. The results should be analyzed to determine the possible benefits or handicaps in having this player on their team.

Resources Scheidt, Tim. Fantasy Baseball, 1994. Giant Step Press. Solvang, California

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Ideas for Online Discussion (Some ideas may apply to more than one standard of the NCTM Professional Standards for Teaching Mathematics.)

Standard 1: Worthwhile Mathematical Tasks ❶ “The teacher of mathematics should pose tasks that are based on knowledge of students’ understandings, interests, and experiences.” (p. 25) A unit that builds around a theme, is different from one that is structured around a mathematical topic. Comment on any strengths or concerns you might have about each way of organizing. ❷ The knowledge of so many mathematical skills is needed to successfully complete this unit. Do you feel that the skills should be taught in advance and then applied to the situation or should the skill be taught in the context of the application? Standard 4: Tools for Enhancing Discourse ❸ In this thematic unit, students played and dramatized a game, used real data from baseball cards, chose when to use a calculator and computer, and constructed spinners to simulate a game. When planning a unit, how do you insure that a variety of tools for enhancing discourse are used? Standard 6: Analysis of Teaching and Learning ❹ It is essential for teachers to “make plans, both short- and long-range.” (p. 63) Using thematic units as an integral part of the instructional program provides the environment for both. What affect has the use of thematic units had on your students and on you?

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FANTASY BASEBALL Player Comparison Chart Team Name _______________________________ Managers __________________________

_____________________________

Complete the chart below for each of the eligible players you have received. Be sure to calculate the batting average and on-base percentage to the nearest thousandth and record the HR: Cum. AB ratio and SO: Cum. AB ratio as unit ratios rounded to the nearest whole number. Pos./YR

Name

PBS MATHLINE® MSMP — Fantasy Baseball-Part 1: Activity Sheet

AB

Cum. AB

Avg.

OB%

HR: SO: Cum.AB Cum.AB

Name _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Date

_______________________

Period _ _ _ _ _ _ _ _ _ _ _ _ _ _

Fantasy Baseball Pre-Assessment -- New Kids In Town Imagine that two new kids have just moved into your neighborhood and are looking to play on a local baseball or softball team. You have been told by your coach that one of these students will be joining your team. Below are the statistics that they accumulated for their previous teams. Analyze these statistics to help you determine which player your would like to have on your team. On a separate sheet of paper, explain who you would select, Player A or Player B. Be sure to support your selection with the statistics that are given. Player A AB 37

H 13

2B 4

3B 0

HR 5

BB 3

SO 9

HR 2

BB 9

SO 7

Player B AB 61

H 24

2B 8

3B 3

PBS MATHLINE® MSMP — Fantasy Baseball Part 1: Activity Sheet

BASEBALL CARD GLOSSARY YR

the year the player played for a given team.

TEAM

the team played for during the year. Major League teams are represented by the team name (Tigers, Giants, etc.). Minor League teams are represented by the city in which the tram resides (Nashville, Eugene, etc.)

AVG

this number represents the player's batting average. This average is the decimal equivalent of the ratio of hits to official at bats.

G

the number of games the player appeared in for the team.

AB

this represents the number of official at bats the player had during the season. Official at bats (AB) do not include walks (BB) or sacrifice hits (bunts, sacrifice files). Sacrifice hits do not count as at bats because the player makes an expected out in order to advance the runner(s).

R

the number of runs the player scored (times he crossed home plate).

H

the number of hits a player got during the season. This number represents the total singles, doubles, triples and home runs the player accumulated during the season.

2B

the number of doubles or times the player reached second base safely due to a hit.

3B

the number of triples or times the player reached third base safely due to a hit.

HR

the number of home runs the player hit during the season.

RBI

the number or runs batted in that the player was credited with during the season. This means that other players scored runs due to the player's hitting performance.

BB

the number of walks (also known as bases on balls) the player received during the season.

SO

the number of strikeouts the player had during the season.

SB

the number of stolen bases the player had during the season.

PBS MATHLINE® MSMP — Fantasy Baseball Part 1: Activity Sheet

Fantasy Baseball Player Guidelines Use the following guidelines to help you determine whether or not you would like to keep certain players for your team. Although these guidelines do not represent everything you should consider, they'll be helpful when you receive your initial draft of players. BATTING AVERAGE below .220

— weak hitter: you won't want to keep unless their other statistics (OB%, HR : Cum., AB ratio, SO : Cum. AB ratio) are considered favorable.

.220 - .250

— low average to average hitter: again, check their other statistics and weigh your choices.

.250 - .300

— good, solid hitter; more than likely you will want to keep a player of this caliber unless other statistics are unfavorable or you are able to trade for a player with better overall statistics.

.300 & up

— excellent hitter: this could be your superstar. Look over the other statistics but expect to keep this player.

ON-BASE PERCENTAGE The opinion of a group of student experts: "We like to see a player that has an on-base percentage that is 60 to 100 points higher than their batting average. We also feel that this percentage should be .340 or higher: The higher the percentage, the better!" HR : CUM. AB (UNIT) RATIO The guidelines for this statistic generated by the same group of students experts: "We consider a HR : Cum. AB ratio of 1:30 to be very good. We feel that this ratio or one that has a smaller number deserves strong consideration. The smaller the number, the better!" SO : CUM. AB (UNIT) RATIO The guidelines for this statistic, also generated by the group of students experts: "We consider a SO : Cum. AB ratio of 1:8 to be very good. We feel that this ratio or one that has a larger number deserves strong consideration. In this case, the larger the number, the better!" PBS MATHLINE® MSMP — Fantasy Baseball Part-1: Activity Sheet

The Middle School Math Project

Fantasy Baseball Part II (Number and Number Relations)

Objective Students will study a thematic unit based on the game of baseball to investigate number and number relations.

Overview of the Lesson Following is the overview of Fantasy Baseball - Parts I and II: Students play a few innings of baseball as an introductory activity to a thematic unit based on the game of baseball. The unit is centered around the analysis of the statistical data contained on professional baseball trading cards. These cards are first used to familiarize students with some of the basic terminology associated with the game. Next, students are directed to focus their attention on the actual statistical data included on the card and how to evaluate these statistics to determine the overall rating of a player’s past performance. Additionally, students form ratios using selected data to obtain key statistical data. This key statistical data, such as the number of home runs to the number of cumulative times at bat (unit ratio), are essential to the next phase of the unit: building a nine-player baseball team. Student pairs act as co-managers of a baseball team. Each pair receives a pack of baseball cards. They compute and analyze the key statistical data of the given players, make decisions on who they want to keep and who they want to trade, arrange their lineups, and play simulated games. The simulated games are played using player spinners which are configured based on the statistical data of each player. The ultimate goal is to be the manager of the winning team of the Fantasy Baseball World Series.

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Materials Each Pair ❶ Pack of Baseball Trading Cards ❷ Baseball Card Glossary (from Part I) ❸ Player Guidelines (from Part I) ❹ Statistical Guide (from Part I) ❺ Player Comparison Chart (from Part I) ❻ Player Analysis Charts ( 2 worksheets ) ❼ How to Make A Player Wheel (Written Directions & Pictorial Model) ❽ Calculator ❾ Player Wheel materials: card stock scissors, markers, spinners (spinners can be made by spinning a bobby pin around the pencil point)

Procedure For the remainder of the unit, students should work in pairs, as co-managers of a baseball team. Distribute a set of professional baseball trading cards to each of the teams. Each team will need a name, so first have the co-managers decide on names for their teams. Once the names have been selected and recorded, the co-managers are to use the data from each of the baseball cards in their pack to complete the Player Comparison Chart. Instruct them to evaluate this information and based on these evaluations decide which players they want to keep and which players they want to trade. Remind them that they must have a player for each of the nine positions on the team: pitcher, catcher, first baseman, second baseman, third baseman, shortstop, right fielder, center fielder and left fielder. When all of the co-managers have completed this evaluation process and have made the decisions on who they want to trade, allocate appropriate time for the trade talks. At this time co-managers make the necessary deals to obtain the players they need/want to complete their nine player teams. The next phase involves the completion of the Player Analysis Charts. One chart should be completed for each player on the team. (There are 5 per page, so comanagers will need 2 pages.) Specific instructions for computing the additional data are included on the Statistical Guide sheet. The co-managers first use this information to decide on the batting order for their players. Suggest that they should think carefully about how to strategically position MSMP — Fantasy Baseball-Part 2 Lesson Guide • http://www.pbs.org/mathline

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PBS MATHLINE®

their players in the line-up, particularly their top hitters, to maximize the potential for scoring runs. Secondly, this information is used as the source of data for the player wheels. A player wheel must be constructed for each player. It’s construction is based on the probabilistic outcome of a player’s performance each time he comes up to bat. Generally when a player is at bat one of the following occurs: He either hits a single, double, triple, home run, or he walks, strikes out, flies out or grounds out. The player wheel is therefore divided into those eight sections. Based on the player’s record last season, the number of times each one of these events occurred is compared to the total number of times the player came up to bat. These ratios are renamed to their decimal equivalents and are then converted to degrees. A circle is then divided into the eight sections based on the number of degrees in each section. Using a spinning device, the player wheel becomes a tool to be used when students play the Fantasy Baseball games. (Instructions for constructing the player wheel are provided on the "How to Make a Player Wheel" sheet. A pictorial model is also provided.) Students are finally ready to begin the Fantasy Baseball tournament. Provide a model baseball diamond and four chips for each game. Co-managers will spin each wheel in the order of their line-up and move their players (chips) around the bases based on the label in the section where the spinner lands. During the games, they must keep a record of the number of outs and the number of runs. Because of the limiting features of playing a simulated game of baseball, stolen bases and certain plays such as an on-base player advancing two bases from his teammate’s single, will not be allowed Depending on the amount of time you can devote to the tournament, you may have to limit games to 2 or 3 innings. You may also want to divide the teams into 2 leagues where each team plays every team in its league. The team with the most wins in each league will play each other in the Fantasy Baseball World Series.

Extensions & Connections Have students investigate the role that mathematics plays in other sports such as basketball, soccer, tennis and football. If there are 20 baseball teams and every team played each other, and all of the winners played each other, and these winners played each other, in total how many games will have been played to determine the winner?

MSMP — Fantasy Baseball-Part 2 Lesson Guide • http://www.pbs.org/mathline

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PBS MATHLINE®

Resources Scheidt, Tim. Fantasy Baseball, 1994. Giant Step Press. Solvang, California.

Ideas for Online Discussion (Some ideas may apply to more than one standard of the NCTM Professional Standards for Teaching Mathematics.)

Standard 1: Worthwhile Mathematical Tasks ❶ Provide a brief overview of a thematic unit that you have used with your students. ❷ From your experience, does the thematic unit concept provide the missing structure needed to enhance the learning of the content? Standard 3: Students’ Role in Discourse ❸ Fantasy Baseball provided numerous opportunities for students to “try to convince themselves, and one another of the validity of particular representations, solutions, conjectures, and answers.” (p. 45) Describe how promoting and encouraging discussion and debate can enhance the student learning and increase the level comprehension. Standard 6: Analysis of Teaching and Learning ❹ How would you assess students’ knowledge of the mathematical content at the conclusion of this unit?

MSMP — Fantasy Baseball-Part 2 Lesson Guide • http://www.pbs.org/mathline

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FANTASY BASEBALL Player Analysis Chart (Examples) Year 1991 Cum.

Name

AB

H

IB

2B

3B

HR

BB

SO

686

172

RATIO

123 123 686

25 25 686

1 1 686

23 23 686

80 80 686

67 67 686

DECIMAL EQUIVALENT

.1793

.0364

.0014

.0335

.1166

DEGREES

65°

13°

0°

12°

Robin Ventura

Other Outs

FO

GO

183 183 686

184 184 686

.0976

.2667

.2682

42°

35°

96°

97°

FO

GO

155 155 646

155 155 646

367

YEAR 1985 Cum.

Name

Other Outs

AB

H

IB

2B

3B

HR

BB

SO

646

216

RATIO

162 162 646

26 26 646

18 18 646

10 10 646

34 34 646

86 86 646

DECIMAL EQUIVALENT

.2508

.0402

.0279

.0155

.0526

.1331

.2399

.2399

DEGREES

90°

14°

10°

6°

19°

48°

86°

86°

Willie McGee

PBS MATHLINE® MSMP — Fantasy Baseball-Part 2: Activity Sheet

310

Player Analysis Chart Cum. AB

Name

IB

2B

3B

HR

BB

SO

Other Outs

FO

GO

IB

2B

3B

HR

BB

SO

Other Outs

FO

GO

H

IB

2B

3B

HR

BB

SO

Other Outs

FO

GO

H

IB

2B

3B

HR

BB

SO

Other Outs

FO

GO

H

IB

2B

3B

HR

BB

SO

Other Outs

FO

GO

H

RATIO DECIMAL EQUIVALENT DEGREES Cum. AB

Name

H

RATIO DECIMAL EQUIVALENT DEGREES Cum. AB

Name

RATIO DECIMAL EQUIVALENT DEGREES Cum. AB

Name

RATIO DECIMAL EQUIVALENT DEGREES Cum. AB

Name

RATIO DECIMAL EQUIVALENT DEGREES PBS MATHLINE® MSMP — Fanatasy Baseball-Part 2: Activity Sheet

FANTASY BASEBALL How to Make A Player Wheel Year 1991 Name

Cum. AB

H

IB

2B

3B

HR

BB

SO

Robin Ventura

686

172

123 123 686

25 25 686

1 1 686

23 23 686

80 80 686

67 67 686

RATIO DECIMAL EQUIVALENT DEGREES

.1793 .0364 65°

13°

.0014 .0335 0°

12°

.1166 .0976 42°

35°

Other Outs 367

FO

GO

183 183 686

184 184 686

.2667

.2682

96°

97°

A.

Use a copy of the Player Wheel master or construct two (2) circles, 4 1/2" in diameter, on cardstock. Clearly mark the center point on the circle.

B.

Using your protractor as a straight edge, draw a radius anywhere in your wheel.

C.

Get out your completed Player Analysis Charts. Determine your preferred order of placement for each statistical category. Using your first category, construct the appropriate angle and draw another radius. Label this section of your wheel with the correct statistical abbreviation.

D.

Continue Step C until all of the angles have been measured and the statistical abbreviations recorded on the wheel.

E.

Write the player's full name on the back of the wheel.

F.

Cut out the player wheel.

PBS MATHLINE® MSMP — Fantasy Baseball-Part 2: Activity Sheet

FANTASY BASEBALL How to Make a Player Wheel (Pictorial model)

B

A

HR

BB

1B

SO 1B

Robin Ventura

GO 2B

C

D

E and F C H I C A G O

Robin Ventura BB HR SO 1B GO 2B Avg. 284

G PBS MATHLINE® MSMP — Fantasy Baseball-Part 2: Activity Sheet

H and I

W H I T E S O X