Geometry/Introduction

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Introduction to High School Geometry[edit]

The word geometry comes originally from Greek, meaning literally, to measure the earth. It is an ancient branch of mathematics, but its modern meaning depends largely on context.

  • To the elementary or middle school student (ages six to thirteen in the U.S. school system), geometry is the study of the names and properties of simple shapes (e.g., the defining properties of triangles, squares, rectangles, trapezoids, circles, prisms, etc., along with formulas for their areas or volumes).
  • To the high school student (ages fourteen to seventeen in the U.S. system), geometry has two flavors: synthetic and analytic. Synthetic geometry uses deductive proof to study the properties of points, lines, angles, triangles, circles, and other plane figures, roughly following the plan laid out by the Greek textbook writer Euclid around 300 B.C. Analytic geometry follows the pioneering work of the French mathematicians René Descartes (1596-1650) and Pierre Fermat (1601-1665) to impose a coordinate grid on the plane, making it possible to study geometric objects (e.g., lines, parabolas, and circles) by means of algebra (e.g., linear equations and quadratic equations) and vice versa.
  • To the mathematical researcher geometry is a subject that has grown far from its roots, and he may refer to his field as modern geometry to distinguish it from the school subject. Modern geometry comes in two main flavours: algebraic geometry (Algebraic curves and surfaces, algebraic varieties), and differential geometry (Riemannian geometry, Spin geometry, Lie groups and algebras). Differential geometry is used in many ways; to describe the universe in the theory of general relativity (Lorentzian 4-manifolds), to describe soap bubble films (minimal surfaces) and as the building blocks of particle physics (Lie groups).

The first part of this wiki textbook aspires to be a high school geometry text adequate to satisfy the California curriculum content standards.

Why invest the energy to learn geometry?[edit]

Basic geometry is a very powerful practical problem solver. It was used by the ancient Egyptians and Greeks for solving most problems and was the proto-type for rational thinking. Where we use algebra today the Greeks used geometry then. It is still very current in all the building and fabrication trades. Before building something big and expensive it is better to work out the bugs in a small scale model. Before expending a lot of energy in making a model it is best to do a drawing. With geometry the drawings become very accurate and can be used to predict measurements and costs. Geometry can be easy to master; the proofs are more fun than sudoko; and its applications are as practical as a hammer and saw. It will give you a sophisticated visual intuition and a strong sense of rational proof and a jumping-off place for some of the most abstract areas of pure mathematics. It’s hard to imagine a mathematical education without geometry.

What should I know before attempting this text?[edit]

  • You should be able to do arithmetic up to addition of fractions without a calculator (1/4 + 2/5 = 13/20); and solve simple algebra problems (2x -1= 0 then x= 1/2) and (X^2-16=0). Often algebra and geometry are taught together and the students would know enough algebra by the time they needed it. It wouldn't hurt to read the first few chapters in both arithmetic and beginning algebra while doing geometry. Mathematics builds on itself. This means that only if you work all the examples and learn the definitions will later pages make sense to you. Keep a notebook and make lots of drawings. Talking to others about what you learn is invaluable. You do your part and a good text book will get you there.

How difficult? Intuition vs. rigor:[edit]

Fifteen little boys sit in front of their first soccer coach, knowing nothing about the game. The coach stands up to teach them soccer. How does he begin?

Perhaps he lectures for several hours, explaining the rules of soccer along with common strategies and tactics. The little boys grow bored. Having no experience of the game, they have trouble understanding the rules and even more trouble remembering them. By the end they have no sense of what a soccer game feels like.

On the other hand, perhaps the coach divides the boys into two teams, takes them to a soccer field, throws them a ball, tells them, "Try to get the ball into the other team's goal and keep it out of your own," and then lets them play. Knowing no rules, the boys play a game little like soccer. They pick the ball up and run with it, paying no attention to the boundaries of the field. When they finally get the idea to kick the ball, they have no skill at dribbling, passing, or working together as a team. They have some fun, but again they end up not knowing what a soccer game feels like.

The first approach (teaching all the rules before playing) is what mathematicians might call a rigorous approach to soccer while the second (just playing without worrying about rules) is what they might call an intuitive approach. Rigor emphasizes the rules. Intuition emphasizes the play. Good teaching and good learning need a balance of both. A good soccer coach starts by teaching the boys some of the most important rules, making them practice some basic drills, and letting them play a little. At first he probably emphasizes the intuitive, letting them play more, so that they get a sense of how the game works and how much fun it can be. Later, as they develop a taste for the game, he focuses more on skills and drills, on rules, strategy, and tactics.

The teacher of beginning geometry students faces the same question as the soccer coach: emphasize rigor or emphasize intuition? Following the example of the good coach, this first chapter emphasizes intuition while introducing a little rigor. Specifically this chapter introduces many of the elementary concepts and much of the terminology of geometry in the context of an intuitive treatment of Euclidean geometry. Later chapters revisit these topics with greater rigor.

So, for instance, this chapter addresses Euclidean geometry without explaining what makes it Euclidean and without mentioning other sorts of geometry. Again, since elementary geometers think of points as dots, this chapter explains that a point is like an infinitely small dot. It is a helpful mental image that has served geometers well for over 2000 years. Rigorous approaches to geometry leave the term point undefined, but this text reserves that subtle and confusing convention for a later chapter. Again, this text assumes that its readers already understand something about area and volume and the units for measuring them, so it does not try to define and explain standard units of area and volume.


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