2-Dimensional Figures
2-dimensional figures are shapes that exist in two dimensions, typically on a flat surface. They have length and width, but no depth. Common examples include squares, rectangles, circles, triangles, and polygons. These figures are fundamental to geometry and are used to study properties such as area, perimeter, and angles.
6 Key excerpts on "2-Dimensional Figures"
eBook - ePubThe Path To Early Math
What Preschool Teachers Need to Know
- Ingrid Crowther(Author)
- 2021(Publication Date)
- Gryphon House Inc.(Publisher)
...CHAPTER 8: Two-and-Three Dimensional Geometric Shapes “Investigating and exploring shape and space is an inherent part of children’s mathematical development. If you observe the young child playing freely in an outdoor environment, their ability to navigate obstacles, lift, move, or stack objects, or push and pull items are all examples of geometry that come naturally to them.” —Juliet Robertson, Messy Maths: A Playful Outdoor Approach Geometric Shapes Defined I n defining geometric shapes, we consider reasoning about the shape, thinking about how the shapes relate to one another, and thinking about how shapes are perceived in space. Reasoning about geometric shapes means looking at the physical attributes of different shapes: The number of vertices or points, such as three points in a triangle Straight or curved lines, such as those in rectangles or ovals The number of sides a shape has, such as eight sides to an octagon Whether the shapes are flat or curved (2-D or 3-D), such as a circle or a sphere The second component, thinking about how shapes relate to each other, involves looking at them and seeing how they fit together. For example, two identical squares when placed together form a rectangle, and two identical isolate triangles (triangles with two equal sides) form a rhombus when placed together. It’s also important to look at how shapes are perceived in space. This requires children to learn some of the spatial terminology that supports these perceptions...
- 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 - 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...
- Fuchang Liu(Author)
- 2017(Publication Date)
- Routledge(Publisher)
...In other words, their being congruent doesn’t disqualify such a shape from being a rectangle. No definition of the rectangle ever says that the two pairs of opposite sides must be of different lengths in order to be a rectangle. When they are of the same length, it’s still a rectangle: It’s simply a special rectangle. We call this special rectangle a square. The relationship between rectangles and squares can be illustrated with the diagram in Figure 10.3. This diagram indicates that the relationship between these two categories is such that one is a subset of the other. They aren’t mutually exclusive of each other. Specifically, all squares are rectangles, but not all rectangles are squares. A squares is just a special kind of rectangle. What’s Wrong with Saying “Triangles, Rectangles, Squares, and Hexagons”? Although Jane realized that a square is a special rectangle, in practice she often regarded these two shapes, unwittingly, as being two different categories. For example, before presenting geometric solids, Jane wanted her children to review some common 2-D shapes so that when it came to a face on a solid, they could easily recognize that shape. She drew several figures on the board (see Figure 10.4) and said, “We have learned different two-dimensional shapes such as triangles, rectangles, squares, and hexagons, and now we are going to move on to three-dimensional shapes.” Figure 10.4 Geometric Shapes That Aren’t Mutually Exclusive of Each Other The mistake here is juxtaposing rectangles and squares, and giving children the impression that each type of shape is independent of the other. In fact, this is the root for the common misconception that a rectangle has two longer sides and two shorter sides, as discussed in the previous section. When a list of things is enumerated, each member is usually independent of any other member in the list. In other words, they should be mutually exclusive of each other...
