Fundamentals of Geometry
The fundamentals of geometry encompass the basic principles and concepts that form the foundation of the study of shapes, sizes, and properties of space. This includes understanding points, lines, angles, and shapes, as well as concepts such as congruence, similarity, and symmetry. Geometry also involves the study of spatial relationships and the application of geometric principles to solve real-world problems.
6 Key excerpts on "Fundamentals of Geometry"
eBook - ePub- Douglas K. Brumbaugh, Peggy L. Moch, MaryE Wilkinson(Authors)
- 2004(Publication Date)
- Routledge(Publisher)
...4 Geometry Focal Points Undefined Terms Angles Simple Closed Curves, Regions, and Polygons Circles Constructions Third Dimension Coordinate Geometry Transformations and Symmetry You might be surprised about how many real-life concepts are included in the study of geometry. Young children experiment with ideas such as over versus under, first versus last, right versus left, and between, without realizing that they are studying important mathematics concepts. Additionally, early attempts at logic, even the common everyone else gets to do it arguments that are so popular with children, are geometry topics. A cube is a three-dimensional object with six congruent faces and eight vertices—often called a block. You may also use the term block to identify a prism that is not a cube, but, although you may not think of it very often, you can probably identify the differences between a cube and a prism that is not a cube, as shown in Fig. 4.1. Fig. 4.1. In this chapter, you will review, refine, and perhaps, extend your understanding of geometry. When Euclid completed a series of 13 books called the Elements in 300 BC, he provided a logical development of geometry that is unequaled in our history and is the foundation of our modern geometry study. Geometry is a dynamic, growing, and changing body of intuitive knowledge. We will let you explore conjectures and provide opportunities for you to create informal definitions. There will be some reliance on terms and previous knowledge, especially when we get to standard formulas. Undefined Terms Some fundamental concepts in geometry defy definition. If we try to define point, space, line, and plane, then we find ourselves engaging in circular (flawed) logic. The best we can do is accept these fundamental concepts as building blocks and try to explain them. A fixed location is called a point, which is a geometric abstraction that has no dimension, only position. We often use a tiny round dot as a representation of a point...
eBook - ePubUnderstanding Mathematics for Young Children
A Guide for Teachers of Children 3-7
- Derek Haylock, Anne D Cockburn(Authors)
- 2017(Publication Date)
- SAGE Publications Ltd(Publisher)
...9 Understanding Geometry A Mathematical Experience I was watching two boys in the nursery class chasing each other round the playground on tricycles. At high speed they were weaving their way skilfully around all the various obstacles lying around the playground, judging which gaps were large enough to get through, staying within what they understood to be the boundaries for the game and simultaneously relating their route and position to those of the other boy. I asked myself, ‘Were they experiencing mathematics?’ Were they? How important is this kind of informal and intuitive experience of space and shape? In this Chapter In this chapter we endorse the validity and significance of this kind of experience for young children, as providing a foundation for the later development of geometric thinking. We explain how number work and geometric thinking are linked through the two fundamental processes of transformation and equivalence that are at the heart of thinking mathematically. We then provide an analysis of what children will learn about the geometry of space and shape using these two key concepts: looking at all the different ways in which shapes can be transformed, and all the ways in which shapes can be recognized as being in some sense the same, or equivalent. Number and Shape: Two Branches of Mathematics In our view, the two boys on their tricycles described above were undoubtedly engaging in mathematics. Geometry is about describing position and movement in space and recognizing the properties of two- and three-dimensional shapes. Life in a well-equipped nursery is full of such crucial experience of space and shape: building models; playing with construction materials; packing away the toys; putting things in the right place where they fit the available spaces on shelves or in boxes; creating patterns with shapes; rearranging the furniture; moving some objects by pushing and others by rolling; and so on...
eBook - ePubMathematical Reasoning
Patterns, Problems, Conjectures, and Proofs
- Raymond Nickerson(Author)
- 2011(Publication Date)
- Psychology Press(Publisher)
...13 CHAPTER Foundations and the “Stuff” of Mathematics In spite, or because, of our deepened critical insight we are today less sure than at any previous time of the ultimate foundations on which mathematics rests. (Weyl, 1940/1956, p. 1849) The “Euclidean Ideal” As already noted, for many centuries the prevailing view among mathematicians appears to have been consistent with what Hersh (1997) calls “the Euclidean ideal,” according to which one starts with self-evident axioms and proceeds with infallible deductions. Plato, Aristotle, and other Greek philosophers of their era considered the axioms of mathematics to be self-evident truths. It was the beyond-doubt intuitive obviousness of certain assertions—two points determine a unique line; three points determine a unique plane—that qualified them to be used as axioms from which less intuitively apparent truths could then be deduced. Geometry (literally “earth measurement”) was rooted in the properties of three-dimensional space. Lakoff and Núñez (2000) associate the Euclidean view with a widely held folk “theory of essences,” which they liken to Aristotle’s classical theory of categories, according to which all members of a category were members by virtue of a shared essence. The essence of a member of category X was the set of properties that were necessary and sufficient to satisfy the criteria for membership. They contend that Euclid brought the folk theory of essences into mathematics by virtue of claiming that a few postulates characterized the essence of plane geometry. “He believed that from this essence all other geometric truths could be derived by deduction—by reason alone! From this came the idea that every subject matter in mathematics could be characterized in terms of an essence—a short list of axioms, taken as truths, from which all other truths about the subject matter could be deduced” (p. 109)...
eBook - ePubThe Learning and Teaching of Geometry in Secondary Schools
A Modeling Perspective
- Pat Herbst, Taro Fujita, Stefan Halverscheid, Michael Weiss(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
...Thus, this chapter is, broadly speaking, a chapter about what there is to be learned in secondary school geometry. It plays on the theme that secondary geometry is the study of geometric figures and elaborates on that theme as it makes it more and more complex. We do not pretend that geometry is solely the study of figures but we do contend that a focus on geometric figures is at the core of any viable study of geometry in secondary school, and that any comprehensive reorganization of the subject matter (e.g., as a study of transformations of the plane or space) would require working out a transition from an earlier consideration of the geometric figure. From a conceptualization as the study of geometric figures, the secondary geometry course can also take care of some of the other goals traditionally ascribed to the teaching and learning of geometry, including learning to master space and learning to craft proofs. We assert this, in particular, because students do not encounter geometry for the first time in secondary school. Rather, they come to secondary schools with knowledge of geometry that has been building up since they started to interact with the world through movement, observation, play, and talk, and through their primary education. Much of that interaction has been enabled by things and indexed by signs that relate, in various ways, to geometric figures. Thus our first move is to argue that when students come to secondary geometry they already have some conceptions of figure even if they don’t necessarily use the word figure. These conceptions of figure are ways of making sense of their activity at various levels of spatial organization. Those conceptions of figure are, at the very least, prior knowledge upon which new geometric experiences will be built. Furthermore, the study of geometry in secondary school stands as a chance to challenge and improve those conceptions of figure...
eBook - ePub- Britannica Educational Publishing, Nicholas Faulkner(Authors)
- 2017(Publication Date)
- Britannica Educational Publishing(Publisher)
...M athematicians have long studied the logical and philosophical basis of mathematics, including whether the axioms of a given system ensure its completeness and its consistency. Because mathematics has served as a model for rational inquiry in the West and is used extensively in the sciences, foundational studies have far-reaching consequences for the reliability and extensibility of rational thought itself. A remarkable amount of practical mathematics, some of it even fairly sophisticated, was already developed as early as 2000 BCE by the agricultural civilizations of Egypt and Mesopotamia and perhaps even farther east. However, the first to exhibit an interest in the foundations of mathematics were the ancient Greeks. Arithmetic or Geometry Early Greek philosophy was dominated by a dispute as to which is more basic, arithmetic or geometry, and thus whether mathematics should be concerned primarily with the (positive) integers or the (positive) reals, the latter then being conceived as ratios of geometric quantities. (The Greeks confined themselves to positive numbers, as negative numbers were introduced only much later in India by Brahmagupta.) Underlying this dispute was a perceived basic dichotomy, not confined to mathematics but pervading all nature: is the universe made up of discrete atoms (as the philosopher Democritus believed) which hence can be counted, or does it consist of one or more continuous substances (as Thales of Miletus is reputed to have believed) and thus can only be measured? This dichotomy was presumably inspired by a linguistic distinction, analogous to that between English count nouns, such as “apple,” and mass nouns, such as “water.” As Aristotle later pointed out, in an effort to mediate between these divergent positions, water can be measured by counting cups. The Pythagorean school of mathematics, founded on the doctrines of the Greek philosopher Pythagoras, originally insisted that only natural and rational numbers exist...
eBook - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 6 Geometry Topics CHAPTER 6 GEOMETRY TOPICS Plane geometry refers to two-dimensional shapes (that is, shapes that can be drawn on a sheet of paper), such as triangles, parallelograms, trapezoids, and circles. Three-dimensional objects (that is, shapes with depth) are the subjects of solid geometry. TRIANGLES A closed three-sided geometric figure is called a triangle. The points of the intersection of the sides of a triangle are called the vertices of the triangle. A side of a triangle is a line segment whose endpoints are the vertices of two angles of the triangle. The perimeter of a triangle is the sum of the measures of the sides of the triangle. An interior angle of a triangle is an angle formed by two sides and includes the third side within its collection of points. The sum of the measures of the interior angles of a triangle is 180°. A scalene triangle has no equal sides. An isosceles triangle has at least two equal sides. The third side is called the base of the triangle, and the base angles (the angles opposite the equal sides) are equal. An equilateral triangle has all three sides equal.. An equilateral triangle is also equiangular, with each angle equaling 60°. An acute triangle has three acute angles (less than 90°). An obtuse triangle has one obtuse angle (greater than 90°). A right triangle has a right angle. The side opposite the right angle in a right triangle is called the hypotenuse of the right triangle. The other two sides are called the legs (or arms) of the right triangle. By the Pythagorean Theorem, the lengths of the three sides of a right triangle are related by the formula c 2 = a 2 + b 2 where c is the hypotenuse and a and b are the other two sides (the legs). The Pythagorean Theorem is discussed in more detail in the next section. An altitude, or height, of a triangle is a line segment from a vertex of the triangle perpendicular to the opposite side...
