Sunday, October 5, 2014

Geometry

Geometry
1.
Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th Century BC). By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. In the classical world, both geometry and astronomy were considered to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.

The introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures such as plane curves could now be represented analytically in the form of functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry. The subject of geometry was further enriched by the study of the intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.

In Euclid's time, there was no clear distinction between physical and geometrical space. Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation and raised the question of which geometrical space best fits physical space. With the rise of formal mathematics in the 20th century, 'space' (whether 'point', 'line', or 'plane') lost its intuitive contents, so today we have to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meaning) and abstract spaces. Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. These spaces may be endowed with additional structure which allow one to speak about length. Modern geometry has many ties to physics as is exemplified by the links between pseudo-Riemannian geometry and general relativity. One of the youngest physical theories, string theory, is also very geometric in flavour.

While the visual nature of geometry makes it initially more accessible than other mathematical areas such as algebra or number theory, geometric language is also used in contexts far removed from its traditional, Euclidean provenance (for example, in fractal geometry and algebraic geometry).

2.
The word "geometry" literally means "earth measure." Well, explorers, cartographers, and topographers have taken care of all that stuff for us, so what's left for geometers? Yes, that's a real word.

Well, a whole lot, actually! Geometry is a way to measure things in the world as opposed to Planet Earth itself. (Can you imagine the size of the ruler you'd need?) It's about using visible figures and mathematical concepts to understand the physical world around us. Sounds thrilling, doesn't it?

Don't worry. It's not nearly as bad as it seems. Comparing shapes and applying formulas? Piece of cake. Finding side lengths and analyzing angles? Child's play. Proving theorems and performing constructions? Bring it on.

In fact, compared to the painstakingly exact measurements that cartographers and topographers have to take, geometry is practically a Hawaiian vacation so come on and join the luau! (Make sure to BYOL: bring your own lei.)

The basic elements of geometry are objects you see every day but probably never think about (unlike your Nintendo DS, which you think about every day and still can't seem to find). We're talking about lines, angles, and shapes—and lots of 'em.

That being the case, geometry will require you to draw sometimes. In fact, one of the best things to do if you're ever unsure about a geometry problem is to draw a picture. You don't have to be the next Claude Monet or Andy Warhol, but we'd advise you to steer clear of the Pablo Picasso Salvador Dali neck of the woods.

You'll also use proofs, fact-based arguments that lead to a logical conclusion, to dissect and discover the properties of these shapes. Chances are good that you've probably never written a proof before, so we'll cover exactly what proofs are and how best to tackle them. (Hint: get a running start.)

While geometry does primarily work in the visible arena, we'll still need the math tools we've been gathering so far. Hopefully addition, subtraction, multiplication, and division go without saying, but we said them anyway just to be safe. Basic algebra will definitely come in handy also—especially using linear equations and manipulating variables.

We'll also touch on coordinates and the x-y plane (and even the x-y-z plane), so we're hoping you kept the distance formula in a safe somewhere. If not, don't worry your pretty little head since it comes from the Pythagorean theorem anyway.

A lot of tools and concepts in geometry aren't that popular in other areas of math. Sure, graphs are handy and you might see a triangle thrown into a math lesson every once in a while, but when was the last time you saw a cone in an algebra class? No, that leaning tower of ice cream you snuck in on the last day of school doesn't count.

Geometry is really the awkward third wheel of math courses. If they were siblings, then algebra, trigonometry, and calculus would be a pop rock band sensation and geometry would be the Bonus Jonas. Underrated, yet full of hidden talent and possibility.

And honestly, geometry has something that none of the other math fields can claim: a solid application to the real world.

Seriously! How often are you going to find the degree of a polynomial in real life? When have you ever solved a real-world problem by taking the indefinite integral?

Geometry, on the other hand, has real life applications all over the place! All around us are lines, shapes, angles, distances, symmetries, congruencies, circles, squares, triangles, volumes, and areas! Textiles and clothing, video game design, and architecture are just a few of the fields that use geometry and apply it to the world (or virtual world) around us.

And if you've ever helped put together a piece of Ikea furniture, you know how useful geometry (and an Allen wrench) can be.

Lots of TV show themes—especially the good ones—get stuck in your head all day long and make you want to do nothing other than sit back, relax, and watch Dorothy, Rose, Blanche, and Sophia plow into their umpteenth cheesecake.

Well, we believe that geometry deserves its own theme song. We could probably make it happen with some music lessons, but for now we'll have to settle for a slightly less rhythmic method.

If you want rhythm, try reading these like the Fresh Prince of Bel-Air.

When you were a wee little child, your parents would take you to the neighborhood park to play. You'd swing and slide and climb on the jungle gym. Once in a blue moon, you brought colored chalk with you and played tic-tac-toe with your friends on the sidewalk. That was the closest you ever got to merging logic and fun.

We aren't going to claim that the logic in geometry is as fun as a playground. Very few things in life beat the sheer joy of making it to the other side of the monkey bars. What we will claim is that logic can be fun. Just ask any member of the United States Chess Federation.

In fact, the logic we use in geometry is like a Vulcan kid's playground. It's reasonable, but not overly complex. It gets us piecing together logical statements and eventually building arguments in the form of proofs. And it's fun…for Vulcans, anyway.

3.
Geometry stands for:
geo which means earth and metria which means measure. (Greek)
A major contributor to the field of geometry was Euclid - 325 BC who is typically known as the Father of Geometry and is famous for his works called The Elements. As one progresses through the grades, Euclidian geometry (Plane Geometry) is a big part of what is studied. However, non-Eucledian geometry will become a focus in the later grades and college math.
Simply put, geometry is the study of the size, shape and position of 2 dimensional shapes and 3 dimensional figures. However, geometry is used daily by almost everyone. In geometry, one explores spatial sense and geometric reasoning. Geometry is found everywhere: in art, architecture, engineering, robotics, land surveys, astronomy, sculptures, space, nature, sports, machines, cars and much more.

When taking geometry, spatial reasoning and problem solving skills will be developed. Geometry is linked to many other topics in math, specifically measurement and is used daily by architects, engineers, architects, physicists and land surveyors just to name a few. In the early years of geometry the focus tends to be on shapes and solids, then moves to properties and relationships of shapes and solids and as abstract thinking progresses, geometry becomes much more about analysis and reasoning.

Geometry is in every part of the curriculum K-12 and through to college and university. Since most educational jurisdictions use a spiraling curriculum, the concepts are re-visited throughout the grades advancing in level of difficulty. Typically in the early years, learners identify shapes and solids, use problem solving skills, deductive reasoning, understand transformations, symmetry and use spatial reasoning. Throughout high school there is a focus on analyzing properties of two and three dimensional shapes, reasoning about geometric relationships and using the coordinate system. Studying geometry provides many foundational skills and helps to build the thinking skills of logic, deductive reasoning, analytical reasoning and problem solving to name just a few.

Some of the tools often used in geometry include: Compass, protractors, squares, graphing calculators, Geometer's Sketchpad, rulers etc.

The grade by grade goals section will give more specific details of the geometry concepts expected in grades Pre-K to 12.