WELCOME TO HONORS GEOMETRY Take your name card from the table Place it on your desk with your name facing the instructor Fill out the information form that is on your desk Sit quietly

HONORS GEOMETRY DR. ALAN L. BREITLER Class #1

Outline  Administrative details  Name, address, phone number, mailing address  Classroom procedures  Syllabus  Why study Geometry?  Objectives  Objective: Be able to apply the order of operations to a mathematical statement  Objective: Define/describe points, lines, segments, rays, planes, volumes, equidistant, collinear, coplanar  HW Assignment

Administration  Classroom procedures  One person talks at a time  If you wish to be called on, raise your hand  You may be called on to answer a question, even if your hand is not raised  Complete information form

Classroom Emergency Procedures  Immediate attention  Explain chart on wall

What’s missing?

Zero  Europeans first used zero around 1200 CE  By the 1600's zero began to come into widespread use but still only after encountering a lot of resistance.  Of course there are still signs of the problems caused by zero. Recently many people throughout the world celebrated the new millennium on 1 January Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified. Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the 21 st century begin on 1 January Zero is still causing problems!

Euclid  A Greek mathematician, often referred to as the "father of geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BCE). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor.

Basic, self evident assertions  Postulates  It is possible to draw a straight line from any point to any other point.  It is possible to extend a line segment continuously in both directions.  It is possible to describe a circle with any center and any radius.  It is true that all right angles are equal to one another.  It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles  less than the two right angles.  Common notions  Things which are equal to the same thing are also equal to  one another.  If equals are added to equals, the wholes are equal.  If equals are subtracted from equals, the remainders are equal.  Things which coincide with one another are equal to one another.  The whole is greater than the part.

Why Geometry?  Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry. Although Euclid's Elements contains a book on number theory, it is based on geometry. In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines.

Why Geometry?  Studying geometry helps students improve logic, problem solving and deductive reasoning skills. The study of geometry provides many benefits, and unlike some other complex mathematical disciplines, geometry has many practical and daily applications. It is used in art, engineering, sports, cars, architecture and much more.

Why Geometry?  Since geometry deals with space and shapes, it is easy to see why it has many applications in the life of an average person as opposed to algebra or calculus, which are typically only used by those going into math-related fields.  One reason geometry is studied is to improve visual ability. Most people think in terms of shapes and sizes, and understanding geometry helps improve reasoning in this area.

Why Geometry?  The ability to think in three-dimensional terms is another reason to study geometry. The idea behind two and three dimensions as they relate to shapes is something derived from the study of geometry. The shapes people most often deal with in the world are three-dimensional shapes. Understanding the size and space of these shapes is something taught by geometry.  At some point, nearly everyone has to figure out the size of a three-dimensional shape. Rooms, houses, vehicle trunk space, furniture and yards are all examples of three-dimensional shapes encountered in everyday life

Challenge  Name an occupation or profession that does not use geometry

Order of operations  * 9/4 – 6 * (8 – 2)  Perform operations within ( )  * 9/4 – 6 * 6  Perform multiplications  /4 – 36  Perform divisions  – 36  Perform additions and subtractions  17.25

Dimensions  A point has 0 dimensions  Moving a point through space traces out a line, which has 1 dimension (length)  Moving a line through space traces out a plane, which has 2 dimensions (length and width)  Moving a plane through space traces out a volume of space, which has 3 dimensions (length, width, height)  Moving a volume of space through space traces out another volume of space!

HW Assignment  Pages 3-4 in textbook, Written Exercises 1-10, due September 6