Geometry

6 Key excerpts on "Geometry"

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  eBook - ePub

  Fundamental Concepts of Geometry

  • Bruce E. Meserve(Author)
  • 2014(Publication Date)
  • Dover Publications

    (Publisher)

  CHAPTER 1 FOUNDATIONS OF Geometry The word “Geometry” is derived from the Greek words for “earth measure.” Since the earth was assumed to be flat, early geometers considered measurements of line segments, angles, and other figures on a plane. Gradually, the meaning of “Geometry” was extended to include the study of lines and planes in the ordinary space of solids, and the study of spaces based upon systems of coordinates, as in analytic plane Geometry, where points are represented by sets of numbers (coordinates) and lines by sets of points whose coordinates satisfy linear equations. During the last century, Geometry has been still further extended to include the study of abstract spaces in which points, lines, and planes may be represented in many ways. We shall be primarily concerned with the fundamental concepts of the ordinary high-school Geometry — euclidean plane Geometry. Our discussion of these concepts is divided into three parts: the study of the foundations of mathematics (Chapter 1), the development of euclidean plane Geometry from the assumption of a few fundamental properties of points and lines (Chapters 2 through 6), and a comparison of euclidean plane Geometry with some other plane geometries (Chapters 7 through 9). The treatment of the second part —the development of euclidean plane Geometry — forms the core of this text and emphasizes the significance of the assumptions underlying euclidean Geometry. Together, the three parts of our study enable us to develop an understanding of and appreciation for many fundamental concepts of Geometry. 1–1 Logical systems. We shall consider geometries as logical systems. That is, we shall start with certain elements (points, lines,. . .) and relations (two points determine a line,. . .) and try to deduce the properties of the Geometry

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  Mathematics Content for Elementary Teachers

  • 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. As the series of dots in Fig. 4.2 get smaller and smaller, we observe that the dimensions are diminishing—but any dot that we can see has some dimension, even the period at the end of this sentence. The fact that a point has no physical existence does not limit its usefulness, either in Geometry or everyday activities. Although a dot covers an infinite number of points, it represents the approximate location of a distinct point so well that we forget the difference and freely identify the dot as a pinpointed location. In mathematics, we label the point represented by a dot with a printed capital letter, for example, the points in Fig. 4.3 . The set of all possible fixed locations is called space

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  eBook - ePub

  Primary Mathematics for Trainee Teachers

  • Marcus Witt(Author)
  • 2014(Publication Date)
  • Learning Matters

    (Publisher)

  It is also important to remember that having a facility with number does not automatically lead to great prowess in visual and spatial thinking and vice versa. When embarking on a sequence of lessons on Geometry, it may well be worth reconsidering your classroom groupings (if you have them). Otherwise it is likely that some children will be under-challenged and others may be given work which is overly challenging. This area of mathematics is a real opportunity for some children, for whom maths is not usually their forte, to shine. If you can spot these children and acknowledge their prowess publically in this area of maths (asking them to explain their thinking, putting their work up on the walls, etc.), you may be able to encourage a more positive attitude towards the areas of mathematics that they find less easy.

  Children beginning school in September 2014 will leave full-time education in 2026 and therefore the education they receive throughout their formal time in school needs to be fit for purpose and equip them with the skills they may need for jobs and a lifestyle that no one really has knowledge of today. There are a few things we can be almost sure of: life will be lived in 3-D, there will be some form of currency, there will be a form of measurement and time will pass in one form or another. So although number facts are important, shape, space and measures should not be forgotten. Indeed, children are surrounded by shape a long time before number comes into their lives.

  Curriculum Link

  The 2013 National Curriculum (DfE, 2013) changed emphasis from the previous curricula and used the term ‘Geometry’. Furthermore, it separated ‘Shape, space and measure’ into: ‘Measurement’ and ‘Geometry’, which is further sub-divided into ‘properties of shapes’ and ‘position and direction’.

  What is Geometry? Atiyah (2001) writes:

  spatial intuition or spatial perception is an enormously powerful tool and that is why Geometry is actually such a powerful part of mathematics – not only for things that are obviously geometrical, but even for things that are not. We try to put them into geometrical form because that enables us to use our intuition. Our intuition is our most powerful tool

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  Fostering Children's Mathematical Power

  An Investigative Approach To K-8 Mathematics Instruction

  • Arthur Baroody, Arthur J. Baroody, Ronald T. Coslick(Authors)
  • 1998(Publication Date)
  • Routledge

    (Publisher)

  chapter 16 . In brief, “children who develop a strong sense of spatial relationships and who master the concepts and language of Geometry are better prepared to learn number and measurement ideas as well as other advanced topics “ (NCTM, 1989, p. 48).

  In this chapter, we examine the value of Geometry and the nature of geometric learning (Unit 14 • 1) and then focus on how Geometry can be taught in a stimulating manner (Unit 14 • 2).

  What the NCTM Standards Say

  Grades K-4 “In grades K-4, the mathematics curriculum should include two- and three-dimensional Geometry so that students can:

  • describe, model, draw, and classify shapes;
  • investigate and predict the results of combining, subdividing, and changing shapes;
  • develop spatial sense;
  • relate geometric ideas to number and measurement ideas;
  • recognize and appreciate Geometry in their world” (p. 48).

  Grades 5–8 “In grades 5–8, the mathematics curriculum should include the study of the Geometry of one, two, and three dimensions in a variety of situations so that students can:

  • identify, describe, compare, and classify geometric figures;
  • visualize and represent geometric figures with special attention to developing spatial sense;
  • explore transformation of geometric figures;
  • represent and solve problems using geometric models;
  • understand and apply geometric properties and relationships;
  • develop an appreciation of Geometry as a means of describing the physical world” (p. 112).

  14 • 1  Geometry and Children’s Geometric Thinking

  Both literally and figuratively, we live in a geometric world. We encounter parallel lines on ruled paper, printed pages, railroad tracks, and so forth; and we think about our world—including our personal relationships—in terms of geometric analogies (see Figure 14.1

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  eBook - ePub

  Space

  • Peter Merriman(Author)
  • 2022(Publication Date)
  • Routledge

    (Publisher)

  On the one hand, this can be seen in the close links between Geometry and different understandings of the human body. On the other hand, contemporary social science and humanities scholars are finding it increasingly difficult to bracket off mathematics from their understandings of space and spatiality at a time when computer systems, code, and algorithms are fundamental to modern conceptions of space and spatiality and to the everyday infra-structuring and governance of many environments (Thrift and French 2002 ; Kitchin and Dodge 2011 ; Amoore and Piotukh 2016; Amoore 2020). Section one provides a necessarily abbreviated outline of the emergence of Geometry and arithmetic, examining how and when mathematical figurations of space and spacing came to shape and dominate European thought, and how Geometry emerged from embodied apprehensions and measurements of the earth. In section two, I pick up from my discussion of Descartes and Newton in Chapter 2, to examine the development of modern Geometry and mechanics in the seventeenth and eighteenth centuries, before tracing the emergence of non-Euclidean geometries in the nineteenth and twentieth centuries. In section three, I outline the adoption of positivist scientific methodologies and the incorporation of mathematical and statistical approaches to space in the spatial social sciences in the twentieth century, focusing on the development of social physics in the 1940s and 1950s, and spatial science and regional science in the 1950s, 1960s and 1970s. By doing this, I aim to show how the efforts of scholars to develop a science of space and spatial relations was highly varied and often highly creative, drawing upon different branches of mathematics, physics, and statistics, including principles from topology, chaos theory, and catastrophe theory

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  Philosophy of Science

  The Link Between Science and Philosophy

  • Philipp Frank(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  A′C′ must coincide with We can see, in this way, how, by introducing somewhat more complicated axioms, we can prove the theorems of congruence in a completely logical way without referring to the physical idea of “bringing into coincidence.” We did not give any physical interpretation of the terms; we simply stated certain axioms in which the terms occur. From this, it is clear, we cannot tell anything about the external world. It follows only that if some congruence exists, other congruences exist also, but we do not know “what congruence is.”

  On the other hand, we learn that Geometry provides us with laws about the properties of physical bodies. If we make a triangle from rigid steel, we check, by real measurements, that the sum of the angles is approximately 180°. Now the problem arises: How can the logical system of mathematical Geometry, e.g., Hilbert’s system of axioms, help us to obtain the physical laws about triangles of steel or wood? This connection will be discussed in the following section.

  10. Operational Definitions in Geometry

  We have seen that the system of mathematical Geometry, if properly formalized, becomes independent of the meanings of terms such as straight lines and points. The whole system can then be considered as a definition of these terms, inasmuch as it gives all the properties of them. Axiom I, for example, can be formulated as follows: “Points” and “straight lines” are such objects, and “coincidence” is such a property that one and only one straight line can coincide with two given points. This is an “implicit definition” of geometric terms. Axiom I (Section 8) expresses the same thing in a different form. We call it an “implicit definition”of points and straight lines. These definitions are, as every definition is, arbitrary. Whatever happens in the world of experience, no one can prevent us from formulating these definitions. They are neither true nor false; they stipulate rules according to which the geometrical terms, “point,” “straight line,” “coincidence,” etc., are to be connected with one another;

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