Figures
In mathematics, "figures" refer to geometric shapes or forms, such as circles, squares, triangles, and other polygons. These shapes are defined by their specific attributes, such as sides, angles, and dimensions. Figures are often used in geometry to study properties, relationships, and measurements of these shapes.
7 Key excerpts on "Figures"
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)
...p.48 2 GEOMETRIC Figures AND THEIR REPRESENTATIONS 2.1. Introduction The notion of geometric figure, while defined loosely as well as restrictedly by Euclid (“14. A figure is that which is contained by any boundary or boundaries”, Euclid, 1956, p. 153), is central to Euclid’s Elements : Postulates enable possible figure constructions, and propositions assert properties of those Figures or demonstrate that other Figures can be constructed. Later expositions of geometric knowledge have varied in the extent to which they center on the notion of figure, with modern treatises either making figure a centerpiece as an application of set theory (“by figure we mean a set of points”, Moise, 1974, p. 37) or making figure a derived notion within a more general consideration of geometry as the study of transformations of space onto itself (see for example Guggenheimer, 1967; see also Jones, 2002; Usiskin, 1974). The notion of figure has also played a crucial role in scholarship on the teaching and learning of geometry (e.g., Duval, 1995; Fischbein, 1993). This chapter considers how scholarship from various disciplines has influenced our community’s thinking about geometric Figures. It brings perspectives from mathematics and mathematicians, from the history and philosophy of mathematics, from cognitive science and semiotics, from technology and from mathematics education proper. Influenced by those perspectives we elaborate on the role of the geometric figure in geometry teaching and learning. We articulate some conceptions of figure related to its various representations in the context of making a curricular proposal on which to found research and development: To conceptualize the study of geometry in secondary schools as a process of coming to know Figures as mathematical models of the experiential world...
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Chapter 7 The Shape of Things This chapter is about geometry. Every day, you see many things that have to do with geometry and you use geometric principles, even though you don’t think of them as geometry. Tires are circles, and they had better be attached at the exact center of the circle to function properly. Honeycombs are made up of hexagons (six-sided Figures). Even the truss on a bridge is a trapezoid, and bridges are made up of many triangles because the triangles create rigidity. A lot of understanding geometry is knowing the words that describe a shape. Pay particular attention to the definitions in the following sections, although they are words you probably already know. Two words that pertain to all two-dimensional closed geometric Figures are perimeter and area. (Closed means all the corners are connected.) The perimeter is the distance around a figure, or the sum of the lengths of all of its sides. A typical perimeter is a fence around a plot of land. Area is a term used for the space enclosed by any closed figure. It is expressed in square units (in 2, ft 2, and so forth) and is found by various formulas, some of which are on the GED ® test formula sheet. Typical areas that we see every day are a rug or a plot of land enclosed by a fence. Lines and Angles Geometric shapes have everything to do with lines and angles, so you must understand them first. Even circles, which themselves have no straight lines or angles, have straight lines and angles within them that tell, for example, the size of the circle as well as parts of the circle. A line actually goes on forever in both directions, or we say, “It goes on to infinity (∞) in both directions.” If we want to concentrate on a part of a line, we call that a line segment, and we show which line segment we mean by stating its endpoints. So if we are interested in a line that goes from the 1-inch to the 5-inch measure, we mean a 4-inch line segment...
eBook - ePub- Christine Hopkins, Ann Pope, Sandy Pepperell(Authors)
- 2013(Publication Date)
- David Fulton Publishers(Publisher)
...1998 A History of Mathematics: an introduction Harlow: Addison-Wesley. Kline, M. 1972 Mathematics in Western Culture Harmondsworth: Penguin. Royal Society/JMC 2001 Teaching and Learning Geometry 11–19 London: Royal Society. 4.2 PROPERTIES OF SHAPE Much of the power of mathematics comes from making statements that are true for a whole set of objects such as all even numbers or all quadrilaterals. An important stage in the process is agreeing on useful ways of classifying types of shapes and types of numbers. Consider the shapes below: these are plane shapes. They could be sorted into: They could also be sorted into: POLYGONS The closed shapes that have only straight edges are known as polygons. ‘Poly’ means many and ‘gons’ means knees or angles. One way of classifying polygons is by the number of sides they have: When there are more than twelve sides the polygon can be named informally, for example a polygon with 15 sides can be referred to as a 15-gon. There are many ways of sorting polygons: All the concave shapes have at least one of their interior angles greater than 180°. Polygons with all interior angles less than 180° are convex. Two regular polygons have special names: a regular triangle has three equal sides and three equal angles, it is called an equilateral triangle a regular quadrilateral has four equal sides and four equal angles, it is called a square. PROPERTIES The properties of any geometric shape are those features which remain invariant for that shape. For example a triangle always has three sides and the sum of the interior angles is 180°. The lengths of the sides and the sizes of the interior angles can vary. TRIANGLES Triangles, with just three sides, are the simplest polygons, they can be classified: either by the size of the largest angle or by the lenghts of their side An equilateral triangle is an example of a regular polygon. Two shapes which may differ in size but are otherwise identical are called similar. E.g...
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 - ePub- P.K. Jayasree, K Balan, V Rani(Authors)
- 2021(Publication Date)
- CRC Press(Publisher)
...i.e., it deals with measurement of various parameters of geometric Figures. It is all about the method of quantifying. It is done using geometric computations and algebraic equations to deliver data related to depth, width, length, area, or volume of a given entity. However, the measurement results got using mensuration are mere approximations. Hence actual physical measurements are always considered to be accurate. There are two types of geometric shapes: (1) 2D and (2) 3D. 2D regular shapes have a surface area and are categorized as circle, triangle, square, rectangle, parallelogram, rhombus, and trapezium. 3D shapes have surface area as well as volume. They are cube, rectangular prism (cuboid), cylinder, cone, sphere, hemisphere, prism, and pyramid. 3.2.1 Mensuration of Areas 3.2.1.1 Circle For a circle of diameter d as shown in Figure 3.1 having circumference C, Area, A = 1 4 π d 2 (3.1a) = π r 2 (3.1b) = 0.07958 C 2 (3.1c) = 1 4 C × d (3.1d) Circumference, C = π d (3.2a) = 3.5449 area (3.2b) Side of a square with the. same area, A = 0.8862 d (3.3a) = 0.285 C (3.3b) Side of an inscribed square = 0.707 d (3.4a) = 0.225 C (3.4b) Side of an inscribed equilateral triangle = 0.86 d (3.5) Side of a square of equal periphery as a circle = 0.785 d (3.6) 3.2.1.2 Square Area = side 2 = 1.2732 × area of inscribed circle (3.7) Diagonal = √ 2 × side (3.8) Circumference of a circle circumscribing a square = 4.443 × side of square (3.9) Diameter of a...
eBook - ePubTeaching Mathematics with Insight
The Identification, Diagnosis and Remediation of Young Children's Mathematical Errors
- Anne D. Cockburn(Author)
- 2005(Publication Date)
- Routledge(Publisher)
...Given their importance, therefore, it is essential that geometry and spatial reasoning receive greater attention in instruction and in research. (Clements and Battista, 1992, p. 457) At a less esoteric level Ben-Chaim et al. (1989) point out that graphical representations of three-dimensional shapes … are commonly used in a great number of practical situations and disciplines for conveying spatial information, for example maps, diagrams, flow-charts, and scientific or technical descriptive drawings… Providing all pupils with opportunity to explore a variety of types of representations of spatial and geometric information, as well as to communicate such representations should be a basic educational objective. (p. 121) Clements and Battista (1992) have expressed concern that young children in the United States are not receiving a sufficient grounding in basic geometric concepts and cite Stigler et al.’s work (cited in Clements and Battista, 1992) that shows that their performance on visualisation and paper-folding tests is far poorer than that of their Japanese counterparts. I have not found comparative studies of pupils in the United Kingdom but, with a growing emphasis on numeracy and a tendency to teach shape in whole class sessions, I suspect we have little room for complacency. I would suggest that early years work on shape is, conceptually, relatively straightforward. Therefore, rather than launching into a mathematical explanation of the topic —which, in any event, others can do better than I can (see, for example, Haylock, 1995; Dickson et al., 1984; Williams and Shuard, 1994)—I will move on to a consideration of the potential problems without further ado. Brainstorming As with previous topics, I suggest you begin by brainstorming the potential problems associated with the teaching of shape to young children. Then, if you find it helpful, categorise them into what the source of the problem might be (e.g...
eBook - ePubMathematics in the Primary School
A Sense of Progression
- Sandy Pepperell, Christine Hopkins, Sue Gifford, Peter Tallant(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...Now are there any more possibilities? This requires children to imagine examples and calls on skills of visualising and logical thinking about what might be possible. • Can you find a way of checking that you have them all? Developing a system for checking requires more logical thinking; children may vary the angles within one shape at a time. This means keeping one property fixed (the relationship between the length of the sides) and systematically varying another (the angles). This is an important way of thinking mathematically. • Decide on a way of making groups of shapes that go together, and label your collection. Depending on their experience, children may classify shapes according to a range of properties – shapes which: • remind you of shapes in the environment, for example house shapes; • have one or more axes of symmetry; • have right angles; • have angles larger or smaller than right angles – obtuse or acute angles; • have pairs of parallel sides; • have pairs of equal sides. Older children can be asked to make some rules for other groups to identify as true or false, along the lines of: • All quadrilaterals with an axis of symmetry … • You cannot make a shape with parallel sides with … If children are challenged to explain why statements are true then they are encouraged to reason mathematically and to provide arguments which may verge on proof. This kind of investigation can result in the children discovering some important geometrical relationships. In working with shapes made out of geostrips children can observe what changes and what stays the same...
