Geometry
Geometry is a branch of mathematics that deals with the study of shapes, sizes, and properties of space. It explores concepts such as points, lines, angles, surfaces, and solids, and their relationships and measurements. Geometry is used to solve problems related to spatial reasoning, design, architecture, and various scientific fields.
5 Key excerpts on "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 - 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 - 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 - ePubMathematics in the Primary School
A Sense of Progression
- Sandy Pepperell, Christine Hopkins, Sue Gifford, Peter Tallant(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...As the spatial aspect of mental mathematics, it is increasingly being seen as of central importance to children’s learning. Valuing the space and shape curriculum is an important start to developing children’s spatial mathematical ability. Work on shape and space can sometimes lack a sense of purpose and progression, degenerating into just the learning of names of shapes and transformations. (Of course, there are technical terms to be learned, and there are definitions in the Glossary at the end of Section 3.2 for reference.) Coherence can be given to shape and space work, by ensuring that children are making and monitoring decisions to solve spatial problems, developing the ability to communicate mathematically verbally and visually, and above all reasoning and generalising about spatial properties in increasingly sophisticated ways. The activities in this section are therefore of an exploratory or problem-solving nature and emphasise: • reasoning; • communicating; and • visualising. Since the reasoning, communicating and visualising in these activities are about the properties of shapes and transformations, children will necessarily also develop their understanding of how shapes are classified and their technical vocabulary. It is important to present children with a variety of examples for a technical word, otherwise they may get a limited idea of what the word refers to (see Figure 3.1). Many children think that a ‘tilted’ square is not a square, because they have only learned to match one image to the word rather than being encouraged to identify the key properties and to confront tricky examples. Similarly some children think that only equilateral triangles can be triangles, refusing to recognise irregular triangles or even ‘upside down’ ones, because they have only learned to attach the name to one kind, and that always with a ‘horizontal’ base...
