Portfolio. Axiomatic Systems & Different Geometries. Mathematics Essay
Course Description:
In this course we will learn about the following topics: 1. Axiomatic Systems
2. Different Geometries: neutral, euclidean, hyperbolic, elliptic. 3. Models of geometries
A student who successfully completes this course will (among other things):
• Give rigorous constructions of plane figures using various sets of tools.
• Explain what an axiomatic system is and how axiomatic systems encode intuition.
• Distinguish between theorems of an axiomatic system and properties of particular models. • Prove facts about neutral, Euclidean, and hyperbolic geometry.
Portfolio
A portfolio is a collection of student work that tells the story of a student’s efforts, progress, or achievement during the course. It consists of a number of pieces of work produced by individual students and assembled for a specific purpose.
Some of the goals of a portfolio are:
• Help students learn the skills of reflection and self-evaluation; • Assist the instructor with grading and assessment
Entries
Entries in a portfolio must be selected by the students. Some examples are: • Homework problems
• Student created problems
• Problems the student felt where especially interesting and challenging
• Problems that required multiple solution approaches or strategies
• Samples of problems that reveal student strengths or evidence of persistence • Sample problems that show progress over time
• Reflections!!!
Reflection
Reflection is when students articulate their thinking about some aspect of the mathematical process and their progress as a mathematician.
For example,
• Reasons for selecting a particular problem,
• Aspects of the solution that the student especially liked
• Things that the student will do differently they next time they encounter a similar problem • Areas for improvement,
• Strengths as mathematicians
Portfolio
Author Name
Institution Affiliation
In this course, I have learned the basics of geometry, which is a branch of mathematics concerned with the properties of space and questions of size, shape and relative position of different figures. We were taught that geometry arose independently in varying cultures and was previously considered a way of dealing with areas, volumes, and lengths. With time, new geometrical techniques were developed and geometers (people who study geometry at the advanced level) started seeing elements of formal mathematical science emerging all over the world.
In the 3rd century, geometry arose in India, when different texts were formulated that could provide rules for geometric constructions (Caticha, 2015). Islamic mathematicians played a significant role in preserving Greek ideas of geometry in the 17th century. Since then, geometry has expanded into manifolds and non-Euclidean geometry and has described spaces that are present beyond the range of human experience. We also learned that geometry is closely associated with fields like art, physics, and architecture.
Homework Problems
During the course, we studied that Euclidean geometry is an important branch of mathematics. It is basically the study of figures (such as angles, lines, and points) which are constructed using the compass and the straightedge. Problems related to this field were quite challenging, as we were asked to determine the diameters and areas of two-dimensional figures again and again. In order to determine the geometric designs, we used four different tools: straightedge, compass, ruler, and protractor. Technically speaking, a true geometric construction is possible only with a straightedge or a compass, but we were told that the rest of the tools were also in use by the ancient Greeks and that they could also give reliable results, so we had to try all of them one-by-one. Today, the study of geometry has become an integral part of the training of professionals like architects, draftspersons, mathematicians, physicists, and engineers.
A detailed overview of the basic tools we used is as follows:
1 Straightedge – This is a comprehensive geometric tool used for constructing straight lines. It has no marks, and sometimes we can use a ruler as a straightedge by ignoring its measuring marks.
2 Ruler – Another important geometric tool we have learned about is a ruler, which we primarily use to measure the length of line segments. Rulers contain visible marks and are largely used to measure centimeters or inches. We used a ruler by placing the zero mark on the point where the measurement had to begin. Then it lied over the point at which the measurement was to end.
3 Compass – This is a V-shaped tool that we use for constructing arcs of circles or circles. On one side, it can hold a pencil, and on the other side, there is a point that anchors the tool at a specific locat...