For my project, I chose a topic that is very near and dear to my heart, which is field hockey. I played field hockey for many years with various teams and clubs, however, I had to stop playing due to injury. I am now a coach for my old middle school team, which I enjoy very much because I can stay involved in the sport I love. I thought it would be interesting to examine the math that goes into field hockey because I never have thought about it extensively before. First off I would like to provide a description of field hockey and it is a “game played by two opposing teams of 11 players each who use sticks curved at the striking end to hit a small, hard ball into their opponent’s goal.” (britannica.com). The game consists of two 30-35 minute halves and there is a 5-10 minute halftime between the two. The length of the game depends on what age and level of play a team is at. For example, an elementary or middle school team may not play a full 60-minute game, compared to most high school, college, and international teams. During the game, a coach can call one timeout per half which lasts about 2 minutes in length. Coaches are allowed an unlimited number of substations during the match, which makes it easier for players to rest and hydrate. Two umpires officiate the match and there is usually a third umpire sitting at the scorer’s table to keep track of how many goals, assists, fouls, and cards, etc were recorded in a game.
The scoring system of field hockey works very similar to soccer; 1 goal equals 1 point. Whatever team scores the most points or goals during the match wins. Although, “games tied at the end of regulation time will have up to two 10-minute “sudden victory” periods to determine the winner.” (ncaa.org). During the sudden victory periods, teams must reduce the number of players to seven per side which allows a quicker and more open game. However, if the score remains tied after the two periods of overtime penalty strokes may be used to determine a winner. In a penalty stroke competition, teams must select 5 players to take alternating penalty strokes against the opposing goalkeeper. Whatever team scores the most stroke wins the game.
Field hockey is played on a rectangular field also called a pitch, that is made of either grass or turf. The length of the field measures 100 yards with a width of 60 yards. The pitch is also divided into four smaller rectangles, that each measure about 25 yards long. The design of the field is very math-heavy because it holds 16 right angles, 10 parallel sets, and 2 semi-circles that are called the striking zone or the “D” which each measure 180 degrees. At the end of both sides of the field sits a goal that “is 7ft tall (2.14m) by 12ft wide (3.66m).” (realbuzz.com).
To play the game of field hockey each player must carry a “stick” that normally measures between 31-38 inches. Shorter or longer sticks are available, and the best way to see if a stick is an appropriate size for you is if the stick measures up directly to your hip. All sticks must be right-handed, left-handed sticks are prohibited due to the increase in danger and potential injury. Sticks are commonly made out of carbon fiber, fiberglass, and composite blends, however, a younger player will most likely play with a wooden stick, due to it being heavier and able to hit the ball harder. All sticks have a rounded handle and a J-shaped hook at the bottom to hit the ball. Every stick is flattened on the left side and rounded on right and it is illegal to hit the ball with the rounded side of your stick. The top third of the stick is called the handle, and it is wrapped in a material similar to that used on a tennis racket. However, a standard field hockey ball is “made of solid plastic, weighing between 5 1/2 ounces and 5 3/4 ounces with a circumference of 8 13/16 inches to 9 1/4 inches.” (balls.com). Game balls are usually white or orange so, the players can easily see the ball among the green grass or turf. Some field hockey balls are covered with indentations to reduce aquaplaning that can cause inconsistent speeds on wet surfaces. A player with a strong strike could hit and propel the ball up to speeds of 70mph.
Field hockey uses a three-tier penalty card system of warnings and suspensions. The three cards are a green card which is shaped like a triangle, a yellow card which is shaped like a rectangle, and a red card which is shaped like a circle. There are different shapes and colors so players can easily identify the type of card they have received. When shown a green card it indicates an official warning when a minor offence has occurred. This is the least intense warning a player will receive, and they will have to leave the field for 2 minutes for committing the foul. When shown a yellow card it indicates an official suspension for either a recurring or dangerous foul. A player can be suspended from the match from anywhere between 5 to 10 mins, depending on the intensity of the foul. When shown a red card it indicates a permanent exclusion from the rest of the game, without substitution. The player must also leave the pitch and surrounding area until the game is over. Usually, a player is banned for a certain period of time or number of matches after shown a red card. The coach and fans can also receive cards during the match if they are being inappropriate or disruptive to the players and refs.
Penalty corners are awarded when there is a foul inside the circle that goes against the defending team. If an attacker commits a foul in their opponent’s circle then the other team will receive the ball and they are then able to hit the ball from the 16-yard line. During a penalty corner, “the ball is placed on the goal line at least 10 yards from the nearest goal post. One attacking player hits the ball to a teammate just outside the striking circle line. No shot on goal may be taken until the ball is stopped or come to rest on the ground outside the circle.” (momsteam.com). There is no limit to how many attacking players are involved in a corner, However, there can only be 5 defenders involved in a corner and that includes the goalkeeper. However, all attackers must be outside the circle before the hit is taken and if they aren’t then a foul is called against them and they can potentially lose their corner opportunity. Defenders have to be behind the goal line before the hit is taken but they can emerge when the ball is inserted. Other remaining defenders must be positioned at half field and they can then run back to help their team once the ball is inserted.
There is a lot of math involved in the positioning of a player. Players are recommended to stay around five yards away from their teammates to create better angles and chances of an attack. Players want to try to stay in triangle, diamond, see-saw, and block formations so, it is easier to attack and defend the ball at all times. There is a rule that there can not be three players touching the ball at the same time. Two players from opposing teams can battle for the ball, however, if another player interferes it is considered a third-party and the ball is automatically given to the team who did not commit the foul. Players want to stay in “standard position” or “active position” throughout the match so, they can have faster dribbles and stronger tackles. To execute active positioning you want to keep your feet staggered, with your left foot forward and your right slightly back, which should create a triangle between your feet and the ball/stick. It is important to remember that when trapping a fastball a player’s stick should be at a 120-degree angle. This angle can decrease depending on the speed and accuracy of the ball. Also if a player swings their stick very high, and their follow-through travels measures more than 90 degrees, the player will get called for a foul and potentially be carded for that dangerous act.
Math and angles are very important when passing and receiving the ball. When passing the ball you always want your left foot and shoulder pointing in the direction of your target. Players are again recommended to stay 5 yards away from each other and in their diamond, triangle, and block formations, to create better chances of attack. There are around 6 ways to pass the ball effectively though there are more ways a pass/hit can be executed. Players should always be a position to support their teammates. They should call for a pass when on a flat which is a horizontal pass or a through which is a vertical pass. The most common type of pass is called the sweep, and the goal is to hit the ball with the flat side of your stick by turning the handle backward approximately 180 degrees. A sweep can be done in reverse too, however, the stick would be about 90 degrees away from a normal sweep.
Now that I have discussed how math works in an attacker’s game, I want to share how math is involved with a defenders game. There are 3 main types of tackles used: block, jab, and reverse. To execute a tackle, a player’s feet must be parallel or facing outward 30 degrees. Their knees should be bent at a 120-degree angle to maximize the force and power behind the tackle. A clockwork structure is used to successfully execute a tackle against an opponent. A player will have the most successful if they tackle at either a 2, 3, 9, or 10 o’clock angle. A 2 and 3 o’clock angle would be a block tackle and a 9 or 10 o’clock angle would be reverse tackle. Overall, field hockey is a very math-heavy sport and every aspect of the game has to do with some degree of the subject.
