Today, soccer is one of the most “big-time” sports games which is viewed and played by a lot if people. Along with soccer, basketball’s NBA, American Football’s Super Bowl and Major League Baseball is also one of the of the primetime games that we watch and talk about.

The game of soccer is affected by a lot of factors. Some of these are: player skill, team play, strength of player, speed of player, technique, strategy, training and many more. All these factors affect how a player or a team can perform well when the game of soccer is being played. Unknown to some of us, the game of soccer is also largely related and affected by mathematics and physics.

Geometry in mathematics and physics, in particular, has a lot of effect in harnessing the capabilities of a player. Simple things like the right angle to kick a ball, the right amount of force to be applied and many different measurements and instant calculation are included and integrated with soccer which has a lot to do about mathematics. Geometry will be the main focus of this paper. How geometry affects soccer and what are the different interrelationships that can be found in the game will be discussed.

An introduction to geometry and a brief discussion of some topics in geometry will also be included. Some quick samples of how math is largely involved in soccer are enumerated as follows: Reliance on numbers, dimension and graphs of coaches and players in order to help improve player statistics and training. Coaches make use of framework and line references and other portions of geometry to inform the members of the team on what position in the field they should be. The goalie relies on the knowledge of angles to know where to position himself when defending the team’s goal.

Algebra helps a coach a lot because it gives him the information about which team is more likely to decline its playing performance in the next half. Each team utilizes data analysis and probabilities to help them improve their training and help each member of the team. Each player can assess and have an idea of what angle and what amount of force should he kick the ball for him to score a goal or to pass it to his teammate.

### Soccer and Geometry

Geometry is perhaps the most elementary of the sciences that enable man, by purely intellectual processes, to make predictions (based on observation) about physical world.

The power of geometry, in the sense of accuracy and utility of these deductions, is impressive, and has been a powerful motivation for the study of logic in geometry. According to Coxeter, Geometry helps us to make more accurate and impressive deductions and calculations.

As a branch of mathematics, geometry is involved in the manipulations and organization of different lines, shapes, angles and points. Geometry gives a lot of mathematical meaning to the meaningless shapes, lines, points and symmetry .Knowledge of these angles, lines, points and symmetries will give rise to a deeper understanding of geometry . These different understandings will help a lot in visualizing and learning geometry and how to relate and use it in soccer.

One of the common topics in geometry which is very much related to soccer is angle bisection. In geometry, bisection is the division of something into two equal parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are segment bisectors and angle bisectors. An angle bisector divides the angle into two equal angles.

An angle only has one bisector. Each point of an angle bisector is equidistant from the sides of the angle. The interior bisector of an angle is the line or line segment that divides it into two equal angles on the same side as the angle. The exterior bisector of an angle is the line or line segment that divides it into two equal angles on the opposite side as the angle.

The following is a classic example of how bisection is applied in soccer: The task of a goalkeeper also involves geometry: Think of the following problem: if an attacking player is approaching the goalkeeper, where does the goalkeeper need to stand to have the best chance of preventing a score? A good goalkeeper does not stand on the goal line too much of the time. He stays a bit in front of his goal, because it makes no sense to dive behind the goal line to “save” a ball.

When a single attacker is approaching him, he will try to be on the angle bisector of the lines from that player to the goal posts. He will turn his body toward the approaching player, so that when he dives to the side to stop a shot, he is as far from the left hand side as from the right hand side.

The shortest distances from the goalkeeper (GK, on the not-drawn angle bisector) to the lines from the attacker to the goal posts are perpendicular to these lines (“drawn” as ). So the goalkeeper will not dive to the side, but to maximize his reach he will always dive slightly forward.

This is a simplification, because in the sketch above the attacker finds more space to curve the ball on the right hand side of the goalkeeper (seen from the Attacker’s viewpoint) than he finds on the left hand side. There also are interesting dilemmas for the goalkeeper about how far he should stand in front of his goal: when he is far from his goal the goalkeeper can get a single player’s ball more easily, but the attacking player can also more easily lob the ball over the goalkeeper. And when a second player is coming at the same time, that player has a free path to the goal. The Soccer Ball’s Geometry Here are a some trivial and great ideas stored in a soccer ball.

First, look carefully at a soccer ball and you’ll see that it’s the intersection of two Platonic solids – the icosahedron and the dodecahedron. In fact, like the dodecahedron it has 12 5-sided faces, and like the icosahedron it has 20 6-sided faces. You might look at Euler’s formula, which relates the number of faces, edges, and vertices of a solid. For example, a cube has 6 faces, 12 edges, and 8 vertices (corners). The tetrahedron has 4 faces, 6 edges, and 4 vertices.

### Conclusion

The use of geometry in soccer and the deeper understanding of it will further fuel the desires of students in learning mathematics. Different interrelationships between soccer and mathematics will help students to understand the importance and such subject. Teaching math will be much easier to professors if they combine topics like these to quickly get the attention of their students.

Lastly, mathematics will always be renowned to be a part of soccer and its role in the strategy of the game will always be recognized and used.

### Works Cited

- G. D. Birkhoff and R. Beatley, Introduction to Geometry, AMS Chelsea Publ., 2000, 3rd edition
- D. A. Brannan, M. F. Esplen, J. J. Gray, Angles and Bisection, Cambridge University Press, 2002
- Changkyu Seol, and Kyungwhoon Cheun. “A low complexity Euclidean norm approximation.(Technical report).” IEEE Transactions on Signal Processing 56.4 (April 2008): 1721(1726).
- Expanded Academic ASAP. Gale. California State Univ, Northridge. 22 Apr. 2008;http://find.galegroup.com.libproxy. csun.edu:2048/itx/start.do?prodId=EAIM;.D.
- Hilbert, Analyzing Geometry, Open Court, 1999
- F. Klein, Uses of Geometry, Dover, 2004D. Pedoe, Practical Geometry, Dover, 1988Shillingsburg, Peter L. “A Primer of Textual Geometry.
- (Book review).” Papers of the Bibliographical Society of America 102.1 (March 2008): 113(3). Expanded Academic ASAP. Gale. California State Univ, Northridge. 22 Apr. 2008;http://find.galegroup.com.libproxy.csun. edu:2048/itx/start.do?prodId=EAIM;.S. Roberts, Sports and Math, Walker ; Company, 2006
- Munier, Valerie, Claude Devichi, and Helene Merle. “A physical situation as a way to teach angle.(teaching geometry).” Teaching Children Mathematics 14.7 (March 2008): 402(6).
- Expanded Academic ASAP. Gale. California State Univ, Northridge. 22 Apr. 2008;http://find.galegroup.com.libproxy. csun.edu:2048/itx/start.do?prodId=EAIM;Riley, Mark T. “Euclid.
- ” Great Thinkers of the Western World. HarperCollins Publishers, 1992. 36(4).
- Expanded Academic ASAP.Gale. California State Univ, Northridge. 22 Apr. 2008;RILEY, MARK T.
- “ARCHIMEDES.” Great Thinkers of the Western World. HarperCollins Publishers, 1999. 44.
- Expanded Academic ASAP. Gale. California State Univ, Northridge. 22 Apr. 2008

## Problems With Study – Math Anxiety

IntroductionMany students are hesitant to learn mathematics because of the implication that this subject matter is difficult. With just the thought of numbers and figures to be solved, most students feel uncomfortable and experienced physiological responses like sweating of the palms which sometimes leads to mental block (Kogelman and Warren 12). This is generally referred to as math anxiety.

Math anxiety is experienced by each and every individual. Even those people who are regarded as math wizards also experience this emotional response. Such emotional response is not limited to persons termed as less fortunate for mathematical skills.Often times, problems arise because of the presence of intense anxiety.

One can not do math upon the presence of this emotional feeling. This usually interferes with the ability of the mind to concentrate, retain information and attend to details. As a result, students find it hard to understand lessons on mathematics and develop panic. They tend to give up in trying solving and are unable to comprehend the basics of mathematics.

Thus, whenever there is an exam or a board work to be done, they are likely to have this sense of frustration and declined self esteem. Such bad experiences in mathematics cause a student to avoid mathematics. It is but natural for a student to have this feeling of avoidance because this would only bring back memories of bad experiences. Sad to say, mathematics is one of the things that is hard to stay away from because it is a part of everyday life.

The only escape to shun this area is to either not to do them, or have someone do it (44). However, avoiding mathematics would only remind an individual of things that he can not do therefore developing a sense of inadequacy in this area of study.Causes of Math AnxietyAlthough there is no definite cause of math anxiety, researchers claim that pressure of timed tests and threats of public embarrassment are some of the sources of unproductive tension among many students. In addition, common practices inside the classroom like imposed authority by the teacher, public exposures and time deadlines also contributes to intensify the anxiety of students towards mathematics.

Studies also show that math myths and misconceptions set by the society causes an individual to be math anxious. Some of these myths includes beliefs that aptitude for math is inborn, being good in calculating means being good at math and men are far better than women in terms of mathematical thinking (Russell 1). These myths and misconceptions taken as true and applicable by most individuals lead to the discouragement of a person to study mathematics and because of this, math anxiety arises.The belief that the aptitude for math is inborn can be regarded as the most natural thing in the world.

This belief comes into picture because of the comparison of the field of mathematics to other fields like music and athletics. Some individuals are just born to be more talented in the field of music and athletics and to some extent these talents must be inborn. People tend to generalized this thought and consider that good mathematical skills must be innate. According to Russell, mathematics is indeed inborn; however, it is born in each and every individual.

Mathematics is a trait shared by the entire race thus, no one should be doubtful about his own capability to do math.To be good at calculating does not mean being good at mathematics. It is to be understood that mathematics is a science of ideas and not merely of calculation. Understanding basic concepts in math is needed than just doing exercises in calculations.

Figures in mathematics are representations of the topics in this subject matter but the success in mathematics does not rely on being a wiz at figures.Differences of men and women to do math typically has no basis. Claims regarding this gender bias belief continue to shade people’s attitude and develop a feeling of being inferior as oppose to the opposite sex. Interrelating this misconception to the first one that had been discussed, individuals regardless of there gender should bear in mind that the aptitude for mathematics is inborn in each and every individual.

Uncertainty on this ability is only brought by the society’s myths and misconceptions.The approach in learning and teaching mathematics also contributes to the development of anxiety (qtd. in Curtain-Philips 1). Most schools and institutions used the behaviorist approach in teaching which merely emphasizes learning through memorization and repetition.

Mastery on the steps in solving problems in mathematics is the main feature of this approach thus, the student tends to set aside the concepts behind solving these problems. Knowing the steps in solving a problem without understanding the concepts would bring the student to a state of loss if problems given in exams or seat works are different from what he memorizes. In these circumstances, panic comes in. Other factors in the learning place which causes math anxiety is the presence of an insensitive and punitive environment (Kogelman and Warren 18).

Teachers are supposed to help students to learn and love mathematics, but what usually happens inside the classroom is that they terrorize students and tend to embarrass them in front of other students.Family members also affect an individual’s viewpoint towards mathematics. Competition and comparison on the mathematical skills among siblings are immobilizing (19). An individual develops a sense of worthlessness when being compared to other family members who are more likely to be good in mathematics and as a result, one tends to stop pursuing to learn the subject.

Parents and teachers sometimes lack the reinforcement to push a student to do math and this also results to the decrease of enthusiasm of an individual to learn math. Development of math anxiety can be seen as a collective result of different factors working hand in hand with bad experiences.Overcoming Math AnxietyKnowing the different causes of math anxiety, researchers develop ways on how to address this problem. It was said earlier that being anxious about mathematics is experienced by everyone therefore, people should recognized this fact.

Students having great anxiety towards mathematics should accept the feeling and realize that it is not unusual (Kogelman and Warren 13). When an individual begins to accept this fact, it would be easier for him to process ideas on how would he outdo anxiety. Most of the time, when an individual fails to recognize that math anxiety is a common feeling shared by everyone, the brain’s capability to process and understand ideas are suppress.Myths and misconceptions about mathematics should be analyzed first before accepting.

Oftentimes, these misconceptions and myths cause one’s enthusiasm to decline. It is not reasonable to believe in such myths and misconceptions because some of which are vague and has no basis at all. The problems on developing math anxiety in a traditional classroom condition could be solved by providing a much friendly environment to students. This is for the part of the learning institution to be done.

Approach used in teaching should also be re-examined drawing more emphasis on teaching methods which include less lecture, more student directed classes and more discussion. The learning institution should also develop a strategy of teaching which will make students realize the importance and benefits of mathematics. There should be a motivating factor for the students to overcome the anxiety on this subject matter like more job options and high paying job opportunities would be obtained if students would be likely to love mathematics. The importance of mathematics in everyday living should be presented to the students in order for them to develop a positive outlook.

It is a task of an educator also to be a good role model to their students. Mentioning bad experiences in mathematics to the class would not do any help.For the part of the students, developing a positive attitude in learning math would reduce the level of anxiety. Nevertheless, educators also play a vital role developing positive attitudes, because this goes hand in hand with quality teaching.

Students should also have the determination to understand math. If the lesson presented is unclear, questions and a clear illustrations or demonstrations should be asked. One should focus more on the concepts being taught rather on the steps of solving problems. Although practicing regularly especially in lessons which seem to be difficult would help decrease the level of anxiety of an individual.

Studying in group or hiring a tutor would also help a student t strengthen her confidence in doing math. Being in groups or asking help from a skilled person provide students a chance to exchange ideas, to ask questions freely, to clarify ideas in meaningful ways and to express feelings about their learning. Discussion of wrong answers can be useful in helping other people to look at the problem. Creative learning strategies in understanding mathematics would also be helpful in overcoming math anxiety.

It is also helpful for students to take down notes on lessons presented and reviewing these lessons after class. By doing this, a student would be able to recognize what ideas are not clear.Family encouragement would also prevent the development of a student’s anxiety towards mathematics. Parents, more often than not, influence their children in career choices.

If a child would be likely to see his parents’ perseverance and dedication to the path of career that they take, then he would be likely to pursue more challenging courses and careers and most of these challenging courses are math-inclined.A collective support from the parents, teachers and the student would aid in overcoming mathematics anxiety. There should be a good interaction within the three persons involve in order to succeed in solving the problem regarding math anxiety.ConclusionMath anxiety is an emotional response which causes an individual’s cognitive processing to be disrupted.

Most students suffer from math anxiety because of the interrelation of different factors. The classroom environment, teachers, myths and misconceptions learned from the society, parents as well as other family members has a significant role in the development as well as prevention of math anxiety.Students who refuse to overcome his attitude towards anxiety tend to acquire a job with less compensation. A person’s ability to explore and go into deeper and challenging careers is suppressed by the fear of math.

In order to avoid this, one should develop a scheme or approach towards his learning studies in mathematics. A little push from family members as well as educators would be of great help in overcoming math anxiety.As early as possible, math anxiety should be somehow reduce in the mindset of students. If a student experienced a negative implication about mathematics, he would be more likely to avoid taking courses having numerous math subjects.

If this happens, he’s capability of finding a high wage job and opportunity in expanding his career would be loss thus, he would be likely to end up with lower math competence and achievement.Most students become discourage because of the mistakes that they committed in solving math problems. These mistakes should somehow be viewed as an approach to further develop one’s skill in mathematics. After all, it is by committing mistakes that we learn.

Students should not view this negatively but instead consider this as a positive experience to pursue doing mathematics.Understanding this emotional response alone is not enough, there must be a thorough grasps of the many facets of relations between feelings and effort to do math. people should always keep in mind that math anxiety is a learned emotional response, therefore there is always be ways to manage and if possible shun this feeling.Work CitedKogelman, Stanley and Joseph Warren.

Mind Over Math. New York, N.Y. McGraw-Hill Professional, 1979.

Russell, Deb. Math Anxiety. 1 May 2008.<http://math.

about.com/od/reference/a/anxiety.htm>.Scarpello, Gary.

“Helping Students Get Past Math Anxiety”. Techniques: Connecting Education & Careers 82 (2007): 34-35.

## Math: Systems Of Linear Equations In Three Variables

In real life we sometimes face such times of tasks solution of which need the knowledge of solving systems of linear equations in three variables. First of all you should understand what is an equation in three variables. For example, you may have such an object: x + 3y = 6.

This object we call “linear equation in two variables”. Here x, y are two variables. The word “equation” in this name means that we have equality between x + 3y (left side) and 6 (right side). The word “linear” means that the object defines a particular straight line on plane xy.

The equation x + 3y + z = 6 is similar in form, and so it is a linear equation in three variables. An equation in three variables is graphed in three-dimensional coordinate system. The graph of a linear equation in three variables is a plane, not a line. Generally, if A, B, C, and D are real numbers, with A, B, and C not all zero, then Ax + By + Cz = D

(1) is called a linear equation in three variables.

You may notice, that there are many (telling the truth, infinitely many) set of numbers x, y, z that satisfy linear equation in three variables. For example we can easily check that sets x = 0, y = 1, z = 3x = 3, y = 0, z = 3x = 3, y = 1, z = 0 are solutions to the equation x + 3y + z = 6. Hence, a solution to an equation in three variables is an ordered triple such as (0, 1, 3), where the first coordinate is the value of x, the second coordinate is the value of y, and the third coordinate is the value of z. The other two solutions we shall present in form (3, 0, 3) and (3, 1, 0).

When you have to solve a system of three linear independent equations (system of three linear independent equations consists of three equations of type (1), you need to find such triples that satisfy all three equations simultaneously. In overall there are three possible cases:

1) The system may have a single solution (we call it ordered triple);

2) The system may have infinitely many solutions;

3) The system may have no solutions.

Now I will teach you haw to solve the system with a single solution and I will show you by examples how we can find the solution for such system of equations. I want you to solve the equation using the addition method.

I have three sets of equations that I want you to use.In the addition method we eliminate a variable by adding the equations. So how it works? Let us take the first set of equations that is Example

1 x + 3y + z = 6

3x + y – z = -2

2x + 2y – z = 1

The addition property of equality allows you to add the same number to each side of an equation. You can also use the addition property of equality to add the two left sides and the two right sides.

In this particular case, it is the easiest to add the first and second equations and this operation allows you to eliminate z-term (as far as you add two z-terms one of which is with a positive sign and another with negative). On the left side of the derived equation, you need to add separate terms that include x, y and zThe addition property of equality allows you to add the same number to each side of an equation. You can also use the addition property of equality to add the two left sides and the two right sides. When adding you should use the distributive property to combine all like terms involved in a sum.

In the derived equation you notice that z-term is now absent.

x + 3y + z = 6

3x + y – z = -2

4x + 4y = 4

Add Now in derived equation divide each side by a common factor. In our case it is 4. x + y = 1.

Now you need to write the derived equation instead of the first one in the set and rewrite the second and third equation again. In this way, you will receive a new system of equations equivalent to the initial one. It is equivalent in terms of the same solution, however, it is simpler.

x + y = 1

3x + y – z = -2

2x + 2y – z = 1.

Now you have to subtract the third equation from the second one and again in such a way you will eliminate z-term. After you subtracted left side of equation from the left side of the equation and the right side of equation from the right side of equation you should use the distributive property to combine all like terms involved in a difference. As you notice z-term now disappeared again. For convenience, you write the derived equation instead of the second one and rewrite the first and third equations without changes.

x + y = 1

x – y = -3

2x + 2y – z = 1.

As the net step you need to add the first and second equation (as you did before) and write the received result instead of the first equation and then subtract the second equation from the first one and write received result instead of the second equation. You should leave the third equation without changes.

2x = -2

2y = 4

2x + 2y – z = 1.

Now in the first and second equations divide each side by 2 and rewrite third equation without changes.

x = -1

y = 2

2x + 2y – z = 1.

Now you can easily solve the third equation for z. Write down the result instead of the third equation, and leave the first and second equation without changes.

x = -1

y = 2

z = 2x + 2y – 1.

Now you need to substitute the result for x (x = -1) from the first equation and the result for y (y = 2) from the second equation for variables x and y in the third equation and get the solution of the system.

x = -1

y = 2

z = 1.

Solution: (-1, 2, 1).

Now let us look at the second set of equations that is Example 2:

2y + z = 7

2x – z = 3

x – y = 3

This system of linear equations is much simpler the previous one. First of all you need to combine the first and second equations by addition. Then write the derived result instead of the first equation and leave the second and third equations without changes.

2x + 2y = 10

2x – z = 3

x – y = 3

Now you need to simplify equation by dividing each side by 2, and permute the second and third equations.

x + y = 5

x – y = 3

2x – z = 3

Now you have to do the same transformations as you did with the first set of equations. More particular, you need to add the first and second equations and write the received result instead of the first equation and then subtract the second equation from the first one and write received result instead of the second equation. You should leave the third equation without changes.

2x = 8

2y = 2

2x – z = 3

Now simplify the first and the second equations by dividing each side by 2.

Then write the derived results instead of the first and second equations. Then solve third equation for z. And now you may write down the result instead of the third equation. x = 4 (1)y = 1 (2)z = 2x – 3 (3) Now substitute the result for x (x = 4) from the first equation for variable x in the third equation and get the solution of the system.

x = 4 (1)y = 1 (2)z = 5 (3) Solution: (4, 1, 5). And finally let us consider the last set of equations and call it Example 3: 4x + 5y + z = 6 (1)2x – y + 2z = 11 (2)x + 2y + 2z = 6 (3) To solve this system of equations you need as the first step to multiply each side of equation (1) by 2 and write down the derived result instead this equation. Then you have to multiply each side of equation (2) by -1 and again write down the derived result instead this equation. You should leave the equation (3) without changes.

8x + 10y + 2z = 12

-2x + y – 2z = -11

x + 2y + 2z = 6

Now you need to add equation (1) and equation (2) and write down the derived result instead of the equation (1). Here you will notice that by this operation you eliminated z-term the equation (1). Then you have to add equation (2) and equation (3) that will allow you to eliminate z-term. Similarly write down the derived result instead the equation (3).

Combine all like terms in derived equations. So you may now notice that z-term is absent in two derived equations. Leave the equation (3) without changes again.

6x + 11y = 12

-x + 3y = -5

x + 2y + 2z = 6

Now you need to multiply each side of the second equation by -11 and write down the derived result instead of this equation.

Then you have to multiply each side of the first equation by 3 and write down the derived result instead of this equation. Leave the equation (3) without changes.

18x + 33y = 3

11x – 33y = 55

x + 2y + 2z = 6

You need to add the equation (1) and the equation (2), combine x like terms and write down the derived result instead of the equation (1). This procedure allows you to eliminate y-term.

Simplify the equation (2) by dividing each side of this equation by 11. Leave the equation (3) without changes.

29x = 58

x – 3y = 5

x + 2y + 2z = 6

Simplify the equation (1) by dividing each side of this equation by 29. Leave the equation (3) without changes.

x = 2

– 3y = 5

x + 2y + 2z = 6

Solve the second equation for y. Then you have to write down the derived result instead this equation. Leave the equation without changes.

x = 2

y = (x – 5)/3

x + 2y + 2z = 6

Now substitute the result for x (x = 2) from the first equation for variable x in the second equation and get y.

Solve the third equation for z. Write down derived result instead this equation.

x = 2

y = -1

z = (6 – x – 2y)/2

Substitute the result for x (x = 2) from the first equation and the result for y (y = -1) from the second equation for variables x and y in the third equation and get the solution of the system.

x = 2

y = -1

z = 3

Solution: (2, -1, 3).

To sum up you can check all these solutions by substituting the derived values of variables into the equations and if the results will match it will mean that you solved these systems correctly. By such three simple examples, you learned how to solve the system of equations in three variables using the addition method. However, you have to know that the addition method is not the only possible way to solve such systems of equations. You can also use other methods like elimination, substitution or matrices methods and you will receive the same results.