High School Geometry and Proofs, Coursework Example

Geometrical Proofs in High School Classrooms

Geometrical proofs should be in the high school curriculum. In Elements of Geometry, Euclid laid the foundation for the axiomatic method of modern mathematics and geometric proofs. In Elements of Geometry, Euclid shows students how to calculate the properties of geometrical objects from a simple set of axioms and provides additional mathematical proofs. Knowledge of the work of the Pythagorean Theorem is required by the high school student.

Euclid’s textbook was used into the 20th century for the study of geometry. Euclid’s, Elements of Geometry has been the second most highly published book, second only to the Bible. His concepts paved the way for the invention of the telescope, and its contributions to man’s knowledge of the stars and the world. This textbook is still relevant to modern day high school student in his or her understanding of geometric proofs.

Elements of Geometry provides instructions on how to find the center of a circle, how to draw a line that is tangent to a circle, how to find angles in circles, and Thale’s theorem (an angle in a semi-circle is a right angle). Other parts of Elements of Geometry describe the construction of a given triangle for a regular pentagon, inscription and circumscription of circles on arbitrary triangles, discussion of Eudoxus’ abstract theory of ratio and proportion, geometric proportions and similar figures, basic number theory, geometric progressions, number theory, irrational magnitudes, basic solid geometry, measurement of solids, and how to construct a polyhedral.

The geometric concepts and mathematical proofs written by Euclid have paved the way for the invention of the telescope, and its contributions to high school students’ knowledge of the stars and the universe. Euclid understood the importance of geometric proofs and that is why his textbook on geometrical proofs should be included in the high school classroom. Geometric proofs enable the high school student to envision a world where mathematics plays the most important role. Euclid’s vision of geometrical principles and proofs live on to this day in the field of modern mathematics. Euclid’s geometry and his textbook, Elements of Geometry, have made a lasting contribution to understanding the physical universe, and mathematics.

Visual artists are also impacted by geometric proofs. Photographers, graphic designers, painters, sculptors, interior designers, etc, all are beneficiaries of the knowledge obtained in Euclid’s Elements of Geometry. Students who attend art school must learn the basics of Euclid geometry in order to create designs, and learn about dimensions. Architects must be knowledgeable of geometric proportions and similar figures, basic number theory, geometric progressions, number theory, irrational magnitudes, and basic geometry to construct buildings.

The Greeks were the first to introduce mathematical proof and deductive reasoning (Heath, 1963). The most ancient mathematical texts were related to the Pythagorean Theorem. The Pythagorean Theorem states that “in any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose side are the two legs.” The Pythagorean equation can be written as: a2 b2 = c2, where c = the hypotenuse, a = 1st side opposite the hypotenuse, and b = the 2nd side opposite the hypotenuse (Dantzig, 1955).

The Pythagorean Theorem should be, without question, a part of a high school student’s curriculum. The Pythagorean Theorem is based on lengths and areas, and is metric and area in interpretation. Both use geometric and algebraic techniques (Clagett, 1999). The theorem can be applied to other, non-triangular dimensions, including non-Euclidean spaces, objects with no right triangles, higher dimensional spaces, and non-dimensional solids (Clagett, 1999).

Geometric proofs preceded deductive logic and modern geometry. Modern mathematicians have used geometric proofs to make the universe seem more comprehensible to the average man. Without geometric proofs, the high school student could not understand how a sundial worked. He or she would be unable to comprehend patterns in space, ellipses, and patterns of the stars, lunar patterns, or planetary orbits. Geometric proofs in Euclid’s, Elements of Geometry, and the Pythagorean Theorem should be included in a high school mathematics curriculum.

Understanding geometric proofs is simply not learning equations. Mathematical education, at the high school level, should inspire the student to think deductively. Deductive logic will allow the student to excel in all classes, not only in geometry and mathematics. The modern world is in desperate need of people who can solve problems by deductive logic in every discipline. That is why it is imperative that high school students understand the concepts of geometric proofs.

Works cited

Clagett Marshal Ancient Egyptian Science: A Source Book [Book] = Memiors of the American Philosophical Society 232. – Philadelphia : American Philosophical Society, 1999. – Vol. 3.

Dantzig Tobias TheBehquest of the Greeks [Book]. – [s.l.] : Charles Scribner’s Sons, 1955. – p. 97.

Heath, Thomas L. (1956). The Thirteen Books of Euclid’s Elements, Vol. 1 (2nd ed.). New York: Dover Publications.