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Mathematics and Sports

Mathematics and Sports

Edited by Joseph A. Gallian
Volume: 43
Copyright Date: 2010
Edition: 1
Pages: 342
  • Cite this Item
  • Book Info
    Mathematics and Sports
    Book Description:

    This book is an eclectic compendium of the essays solicited for the 2010 Mathematics Awareness Month web page on the theme of Mathematics and Sports. In keeping with the goal of promoting mathematics awareness to a broad audience, all of the articles are accessible to college level mathematics students and many are accessible to the general public. The book is divided into sections by the kind of sports. The section on football includes an article that evaluates a method for reducing the advantage of the winner of a coin flip in an NFL overtime game; the section on track and field examines the ultimate limit on how fast a human can run 100 meters; the section on baseball includes an article on the likelihood of streaks; the section on golf has an article that describes the double-pendulum model of a golf swing, and an article on modeling Tiger Wood's career. The articles provide source material for classroom use and student projects. Many students will find mathematical ideas motivated by examples taken from sports more interesting than the examples selected from traditional sources.

    eISBN: 978-1-61444-200-4
    Subjects: Mathematics, Statistics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Preface
    (pp. vii-viii)
    Joe Gallian
  3. Table of Contents
    (pp. ix-xii)
  4. I Baseball

    • CHAPTER 1 Sabermetrics: The Past, the Present, and the Future
      (pp. 3-14)
      Jim Albert

      Baseball fans have always had a love for statistics. Even when professional baseball began in 1876, counts of the basic statistics such as hits, doubles, triples, home runs, walks, strikeouts, and runs were recorded. Pitchers and hitters have always been ranked with respect to measures such as the batting average, the number of home runs, and the average runs allowed. Currently, one of the most prestigious achievements for hitting is the Triple Crown, when a player simultaneously has the highest batting average, slugging percentage, and number of home runs. (The last player to obtain the Triple Crown was Carl Yastrzemski...

    • CHAPTER 2 Surprising Streaks and Playoff Parity: Probability Problems in a Sports Context
      (pp. 15-22)
      Rick Cleary

      One of the most appealing aspects of watching sports events is the possibility that a viewer may see something especially noteworthy, perhaps even unprecedented. Sports fans who have the good fortune to be in attendance when something spectacular happens have vivid memories and wonderful stories. Baseball fans love to describe the time they saw a no-hitter, a player hit for the cycle, a triple play, or some other rare event. A typical response to such descriptions is, “Wow, what are the chances of that?”

      Those of us who love sports and math frequently are asked, “What are the chances of...

    • CHAPTER 3 Did Humidifying the Baseball Decrease the Number of Homers at Coors Field?
      (pp. 23-30)
      Howard Penn

      Coors Field, where the Colorado Rockies play, has always been regarded as a home run friendly ballpark. It is at an altitude of approximately 5280 feet. According to thePhysics of Baseball[1], a batted ball travels approximately 10% farther at that altitude than at sea level. Figure 3.2 compares the flight of two batted balls with the same initial speed, angle of elevation, and wind speed. The lower curve represents the path if the ball is hit at sea level and the upper curve shows the path at an altitude of 5280 feet. In the latter case the ball...

    • CHAPTER 4 Streaking: Finding the Probability for a Batting Streak
      (pp. 31-52)
      Stanley Rothman and Quoc Le

      In baseball, a player can gain instant fame by duplicating or exceeding one of the fabled types of batting streaks. The most well known is Joe DiMaggio’s (1941) streak of hitting in 56 consecutive games. There are also other batting streaks such as Ted Williams’ (1949) 84-game consecutive on-base streak, Joe Sewell’s (1929) 115-game streak of not striking out in a game, and the 8-game streak of hitting at least one home run in each game, held by three players. Other streaks include most consecutive plate appearances with a hit (the record is 12 held by Walt Dropo (1952)), most...

  5. II Basketball

    • CHAPTER 5 Bracketology: How can math help?
      (pp. 55-70)
      Tim Chartier, Erich Kreutzer, Amy Langville and Kathryn Pedings

      Every year around the beginning of March, students and faculty at 65 schools along with a large portion of the United States get excited about the prospects of March Madness, the Division I NCAA Men’s Basketball Tournament. At the start of the tournament each of the 65 teams has a chance of ending the season crowned the champion. With the excitement of watching basketball comes the adventure of attempting to predict the outcome of the tournament by filling out a bracket. The NCAA estimates that 10% of the nation will fill out a bracket [4]. Be it participating in a...

    • CHAPTER 6 Down 4 with a Minute to Go
      (pp. 71-84)
      G. Edgar Parker

      Play-by-play announcer : “The Hurrahs are down 81–77 with the ball coming out of this time-out and there are just 52 seconds left. I guess they’ll shoot the three. Right, Billy?”

      Expert analyst : No Jim, this is still a two-possession game. There is no need to panic; look for the easy two.

      This conversation shows up time after time on both college and NBA telecasts. Apparently, if the announcing “experts” reflect the collective thinking of the coaches, this is the strategy that most coaches intend to play in this situation. In what follows plausible strategies associated with being...

    • CHAPTER 7 Jump Shot Mathematics
      (pp. 85-90)
      Howard Penn

      Suppose a basketball player takes a 15 foot jump shot, releasing the ball from a height of 10 feet and an angle of elevation of 60 degrees. What is the initial speedV0needed for the shot to go in?

      If we neglect air resistance, this is a typical ballistic motion problem. The equations are [1]\[x(t)={{V}_{0}}\cos (\theta )t,\ y(t)=-\frac{g{{t}^{2}}}{2}+{{V}_{0}}\sin (\theta )t+{{h}_{0}}.\]

      For this problem we haveg= 32 ft/sec² andθ= 60º. Since the ball is released from the height of the basket, we can takeh0to be zero.

      The range is given by\[d=\frac{V_{0}^{2}\sin (2\theta )}{g}.\]

      If we setd= 15...

  6. III Football

    • CHAPTER 8 How Deep Is Your Playbook?
      (pp. 93-108)
      Tricia Muldoon Brown and Eric B. Kahn

      Many people have fond memories of participating in team sports in elementary school, high school, and even college. We recall the thrill of a close game and the joy or sadness after a win or loss. As we age, less of our time is devoted to playing sports as other responsibilities take priority and our bodies simply are not capable of competing at the same level. Our role in sport transforms from being an active participant to being a spectator. Many of us still go to the gym, play a game of racquetball, or pick up the old golf clubs...

    • CHAPTER 9 A Look at Overtime in the NFL
      (pp. 109-116)
      Chris Jones

      There is debate as to how NFL overtime games should be decided. In the current system a coin toss takes place and the team that wins has the choice to either kick-off or receive the ball. They can also defer this choice and choose an end of the field to defend. The game is played with regular NFL rules and the first team to score wins. In a regular season game the teams play for fifteen minutes, and if neither team scores the game is declared a tie. In the playoffs, where there must be a winner, they keep playing...

    • CHAPTER 10 Extending the Colley Method to Generate Predictive Football Rankings
      (pp. 117-130)
      R. Drew Pasteur

      Many algorithmic ranking systems in collegiate American football publish their results online each season. Kenneth Massey compares the results of over one hundred such systems (see [9]), and David Wilson’s site [14] lists many rankings by category. A variety of methods are used, and some are dependent on complex tools from statistics or mathematics. For example, Massey’s ratings [10] use maximum likelihood estimation. A few methods, including those of Richard Billingsley [3], are computed recursively, so that each week’s ratings are a function of the previous week’s ratings and new results. Some high-profile rankings, such as those of USA Today...

    • CHAPTER 11 When Perfect Isn’t Good Enough: Retrodictive Rankings in College Football
      (pp. 131-146)
      R. Drew Pasteur

      The highest division of collegiate football, the Football Bowl Subdivision (FBS) of the National Collegiate Athletic Association, formerly known as Division I-A, is the only NCAA team sport that does not determine an official national champion. Based on long-standing tradition, top teams compete in season-ending bowl games, and a national champion is unofficially chosen by polls of coaches and media members. Several major conferences have historical ties to particular bowl games. For example, the champions of the Big Ten and Pac-10 conferences have played in the Rose Bowl nearly every year since 1947. More often than not, the consensus top...

  7. IV Golf

    • CHAPTER 12 The Science of a Drive
      (pp. 149-156)
      Douglas N. Arnold

      “Math and science are everywhere.” With those words, championship golfer Phil Mickelson began a public service television advertisement produced by ExxonMobil and premiered during the 2007 broadcast of the Masters Golf Tournament. I had the privilege to serve as the mathematical consultant for the ad and for the accompanying website,The Science of a Drive, from which the title of this article is taken. Figure 12.1 displays a still frame taken from the advertisement and another taken from the website.

      The golf drive does indeed provide numerous examples of the ways mathematics elucidates common physical phenomena. Many aspects of it...

    • CHAPTER 13 Is Tiger Woods a Winner?
      (pp. 157-168)
      Scott M. Berry

      Tiger Woods is one of those rare athletes who accomplish feats in their sport that are freakish. In this small group are guys such as Babe Ruth, Wayne Gretzky, Wilt Chamberlain, Barry Bonds, and Jack Nicklaus. Woods dominates a sport where the population of players are all very good–and very tightly bundled in their ability. To win one tournament, beating 100+ of these players is incredibly difficult. To average one tournament victory a year for 10 years is a Hall of Fame type accomplishment. Tiger Woods has won 71 of 253 (28.1%) official PGA Tour tournaments around the world...

    • CHAPTER 14 G. H. Hardy’s Golfing Adventure
      (pp. 169-178)
      Roland Minton

      There are very few sentences in print that contain both the word “golf” and the name G.H. Hardy. Hardy (1877–1947) was one of the most prolific and influential mathematicians of the early twentieth century. His bookA Mathematician’s Apology[1] makes the case for mathematics as a pure discipline of austere beauty and uncompromising standards. He wrote, “The mathematician’s patterns, like those of the painter’s or the poet’s, must be beautiful, the ideas, like the colours or the words, must fit together in a harmonious way. There is no permanent place in the world for ugly mathematics.” He found...

    • CHAPTER 15 Tigermetrics
      (pp. 179-186)
      Roland Minton

      Viewed one way, this article is a brief report on the most enjoyable data mining project that I can imagine. A more global, though possibly overstated, view is that this is an announcement that golf statistics are about to change dramatically. Data sets exist to do for golf what sabermetrics has done for baseball.

      The data come from the PGA Tour’s ShotLink system. ShotLink is a system of lasers and volunteers who record the location of every shot, including qualitative information such as lie (rough or not, uphill or not, and so on) and quantitative information such as distance to...


    • CHAPTER 16 Can Mathematics Make a Difference? Exploring Tire Troubles in NASCAR
      (pp. 189-200)
      Cheryll E. Crowe

      Trouble with tires has long plagued the National Association for Stock Car Auto Racing (NASCAR). Difficulties at various speedways across America have contributed to frustration among drivers and fans. At the 2008 Indianapolis Motor Speedway Brickyard 400 race, tires lasted only 12 laps, the equivalent of 30 miles instead of the normal 80 miles (32 laps). A record setting 52 of the 160 laps were run under caution due to the disintegration of tires.[1] Fans of the race were displeased, to say the least. With track capacity of approximately 350,000, attendance at the 2009 race was significantly low with an...

  9. VI Scheduling

    • CHAPTER 17 Scheduling a Tournament
      (pp. 203-216)
      Dalibor Froncek

      Suppose we have four teams named 1, 2, 3, 4 and we want to schedule a three-day round robin tournament with each team playing one game on each day. All we need to do is to choose an opponent for team 1 on the first day and another opponent for team 1 on the second day. All other games are determined by these choices.

      Say we choose the game 1–2 for Friday and 1–3 for Saturday. Of course, the remaining teams must meet on both days—teams 3 and 4 on Friday and teams 2 and 4 on...

  10. VII Soccer

    • CHAPTER 18 Bending a Soccer Ball with Math
      (pp. 219-224)
      Tim Chartier

      Aerodynamics in sports has been studied ever since Newton commented on the deviation of a tennis ball in his paperNew theory of light and colourspublished in 1672. Today, the field of computational fluid dynamics (CFD) studies the effect of aerodynamics in such sports as soccer and NASCAR racing. See Figure 18.1.

      Soccer matches are filled with complex aerodynamics as evidenced in the way balls curve and swerve through the air. World class soccer players such as Brazil’s Roberto Carlos, Germany’s Michael Ballack, and England’s David Beckham exploit such behavior, especially in a free kick.

      According to research by...

  11. VIII Tennis

    • CHAPTER 19 Teaching Mathematics and Statistics Using Tennis
      (pp. 227-240)
      Reza Noubary

      The difficulties faced by educators teaching mathematics and statistics are well known. To help, many textbooks try to motivate students by introducing varied applications. This addresses both students’ desire to see the relevance of their studies to the outside world and also their skepticism about whether mathematics and statistics have any value. This idea works mostly with students who are committed to a particular academic or career field. For typical students, applied examples may fail to motivate if they are not of immediate concern to them or they do not occur in their daily lives.

      Fortunately, students have some common...

    • CHAPTER 20 Percentage Play in Tennis
      (pp. 241-256)
      G. Edgar Parker

      One of the intriguing supplements to television commentary for tennis matches these days is the computer analysis from software packages that compile the results from the charting of matches. Charting is not new, but the facility that the computer provides in creating, tabulating, and cross-referencing data from the charts has made more information easily accessible. The computer can use the charts to provide ready information on the progress of a match or data for the analysis of a player’s strengths and weaknesses over several or many matches. Furthermore, software that can provide such data is readily available. A fundamental question...

  12. IX Track and Field

    • CHAPTER 21 The Effects of Wind and Altitude in the 400m Sprint with Various IAAF Track Geometries
      (pp. 259-278)
      Vanessa Alday and Michael Frantz

      Track and field meets include many events, among them the 400m sprint. In a standard International Association of Athletics Federations (IAAF) track, there are eight lanes, and a maximum of eight runners in a race. Although each IAAF track has the same dimensions, questions have arisen as to the effect that wind and altitude have on the runners’ performances, regardless of the event. Several models have been created to describe their effects on the 100m sprint, the 200m sprint, and the 4 × 100m relay. Modeling these performances proved to be relatively simple, but until 2004 no one had ever...

    • CHAPTER 22 Mathematical Ranking of the Division III Track and Field Conferences
      (pp. 279-286)
      Chris Fisette

      In the National Collegiate Athletic Association (NCAA), there are no ranking systems for the track and field conferences in any division. The only rankings in track and field are the school rankings at the national championship meet. Because of the way they are made, the national champion school may not even be the strongest team in its conference.

      At the national meet, it is possible to win the national title with only a few top athletes. In 2008, the outdoor national title was won with a total of only 35 points. These could have been earned with three first place...

    • CHAPTER 23 What is the Speed Limit for Men’s 100 Meter Dash
      (pp. 287-294)
      Reza Noubary

      At the August, 2009 world track and field competitions in Berlin, Usain Bolt, the Jamaican sprinting sensation put on some amazing performances, shattering his own records in both the 100 and 200 meter lowering both by 0.11 seconds to an amazing 9.58 seconds in the 100 meter and 19.19 in the 200 meter. The man is certainly on another level.

      His time is the greatest improvement in the 100 meter record since electronic timing began in 1968. Bolt is not done yet and who knows how fast he can run. In fact, he thinks he can do better. He said...

    • CHAPTER 24 May the Best Team Win: Determining the Winner of a Cross Country Race
      (pp. 295-314)
      Stephen Szydlik

      Collegiate cross country running in the upper Midwest is highly competitive. In the 36 years (through the fall of 2008), that the National Collegiate Athletic Association (NCAA) has sponsored a Division III (non-scholarship) men’s team championship, the top team has come from Illinois, Wisconsin, or Michigan 26 times [6]. The University of Wisconsin-Oshkosh had won three championships in the years leading up to the fall of 2001, when this story begins. The Oshkosh team that year had a great mix of talent and experience, and I had hopes that they would contend for the national title. One early-season test for...

    • CHAPTER 25 Biomechanics of Running and Walking
      (pp. 315-328)
      Anthony Tongen and Roshna E. Wunderlich

      Running speed is essential for many sports, whether it is the ability to beat a defender, run faster than an opponent, or develop enough take-off velocity to achieve distance or height on a jump. Running tends to occur at faster speeds than walking, although speed walkers can achieve speeds of up to 4.6 meters per second using an unusual gait in which the hip is dropped each step. Running is defined as a gait in which there is an aerial phase, a time when no limbs are touching the ground. Aside from wind resistance and gravity, there are no external...

  13. About the Editor
    (pp. 329-329)