Math and Sports Paul Moore April 15, 2010 Math in Sports? Numbers Everywhere – Score keeping – Field/Court measurements Sports Statistics – – – Batting Average (BA) Earned Run Average (ERA) Field Goal Percentage (Basketball) Fantasy Sports Playing Sports – Geometry – Physics Outline Real World Applications – Basketball Velocity & angle of shots Physics equations and derivation – Baseball Pitching Home run swings Stats – Soccer Angles of defense/offense – Math in Education Math in Basketball Score Keeping – point, point shots – Free throws 94’ by 50’ court Basket 10’ off the ground Ball diameter 9.5” Rim diameter 18.5” point line about 24’ from basket Think of any ways math can be used in basketball? Math in Basketball Basketball Shot At what velocity should a foul shot be taken? Assumptions/Given: – Distance About 14 feet (x direction) from FT line to middle of the basket – Height 10 feet from ground to rim – Angle of approach Close to 90 degrees as possible Most are shot at 45 degrees – Ignoring air resistance Math in Basketball Heavy Use of Kinematic Equations – Displacement: s = s0 + v0t + ½at s = final position s0 = initial position v0 = initial velocity t = time a = acceleration This is 490….where did this equation come from? Math in Basketball By definition: Average velocity vavg = Δs / t = (s – s0) / t Assuming constant acceleration vavg = (v + v0) / Combine the two: (s – s0) / t = (v + v0) / Δs = ½ (v + v0) t Math in Basketball Δs = ½ (v + v0) t By definition: Acceleration a = Δv / t = (v – v0) / t Solve for final velocity: v = v0 + at Substitute velocity into Δs equation above Δs = ½ ( (v0 + at) + v0) t s – s0 = ½ ( 2v0 + at ) t = v0t + ½at s = s0 + v0t + ½at ! a D a T Math in Basketball Displacement Function s = s0 + v0t + ½at Break into x and y components (sx): x = x0 + v0xt + ½at (sy): y = y0 + v0yt + ½at Displacement Vectors: sy s sx Math in Basketball (sx (sx): x = x0 x0 + v0x v0xtt + ½ax ½axt (sy (sy): y = y0 y0 + v0y v0ytt + ½ay ½ayt Need further manipulation for use in our real world application Often will not know the time (like in our example here) or some other variable Here: – ax = 0, x0 = – ay = -32 ft/sec2 (sx (sx): x = v0x v0xtt (sy (sy): y = y0 y0 + v0y v0ytt + (-16)t Math in Baseball Previous examples not incorporate drag or lift Graphs with equations including drag and lift: Optimal realistic angle: about 35 degrees Stats in Baseball Baseball produces and uses more statistics than any other sport Evaluating Team’s Performance Evaluating Player’s Performance Coaches and fantasy players use these stats to make choices about their team 2010 Season Stats SPLITS G AB R H 2B 3B HR RBI BB SO SB CS AVG OBP SLG OPS Season 23 8 2 348 516 696 1.212 619 2146 271 631 158 93 394 208 303 13 ? 358 500 858 18 4 0 333 455 611 1.066 162 466 162 162 41 41 101 162 41 20 348 516 696 1.212 Career Last days Projected Stats in Baseball Some Important Stats: Batters – – – – Batting Average (BA) Runs Batted In (RBI) Strike Outs (SO) Home Runs (HR) Pitchers – – – Earned Run Average (ERA) Hits Allowed (per innings) (H/9) Strikeouts (K) Stats in Baseball Batting Average (BA) – Ratio between of hits to “at bats” – Method of measuring player’s batting performance – Format: 348 – “Batting 1000” e s i c Exer ≈ Hits BA( AVG ) = AtBats Stats in Baseball Runs Batted In (RBI) – Number of runs a player has batted in Earned Run Average (ERA) – Mean of earned runs given up by a pitcher per nine innings Earned _ Runs _ Allowed ERA = ⋅ Innings _ Pitched Hits Allowed (H/9) – Average number of hits allowed by pitcher in a nine inning period (hits _ allowed ⋅ 9) H /9 = innings _ pitched Soccer “Soccer is a game of angles” Goaltending vs Shooting Angles in Soccer Goaltending As a keeper, you want to give the shooter the smallest angle between him and the two posts of the goal Player θ A B Goal Able to cut off a significant amount of shots like this Where should goalie stand to best defend a shot? Angles in Soccer Penalty Kicks This is why during penalty kicks, goalies are required to stand on the goal line until the ball is touched If they were able to approach the ball before, the goalie would significantly decrease angle of attack Player A θ B Goalie Angles in Soccer May think it best to stand in a position that bisects goal line Gives shooter more room between goalie and left post, than right post Angles in Soccer Instead would be better to bisect the angle between shooter and two posts Goalie should also stand square to the ball Angles in Soccer As distance from goal increases, the angle bisection approaches the goal line bisection Angles in Soccer Shooting On the opposite end, shooter wants to maximize angle of attack What path should they take? http://illuminations.nctm.org/ActivityDetail.aspx?ID=158 Sports & Math Education Incorporation and application of math in sports is a creative, and wildly successful method of teaching mathematics Professors, University of Mississippi taught fantasy football to 80 student athletes Before, 38% received A’s on a pretest After, 83% received A’s on a postest http://www.fantasysportsmath.com/ Sports & Math Education Innovative way to get students doing math Even if some are not interested, they’re able to understand the practicality and application of mathematical concepts Discussion What sports did you all play? Can you think of any other ways math is involved in sports? Do you think incorporating sports is an effective method of teaching mathematics? – Why or why not?