Mathematics
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.
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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)
In the following we review the most important points from that research that contribute to ground our way of thinking about the geometry curriculum. We then consider how it is possible to use those existing conceptions to develop in students a theoretical stance toward geometric Figures, and eventually a knowledge of geometry as a mathematical theory. 2.4. The Geometric Diagram in the Literature In this section we weave contributions from the literature on diagrams to suggest that the work of moving from the four basic conceptions to more sophisticated conceptions of figure can be anchored in activities that use and further problematize the representations of those basic conceptions of figure. Among those representations, a notable one on which there is substantive scholarship is the geometric diagram, but we take this review as possibly relevant to other comparable representations such as maps, replicas, photographs, and dynamic sketches. We lead this review with the question: What are the affordances and constraints of diagrams in the study of geometric Figures? 2.4.1. The Geometric Diagram in Mathematics Education and Cognitive Science Many mathematics education scholars have contributed to scholarship on diagrams and their relation to geometric Figures, sometimes referring to them as “pictures” or “sensory representations” (e.g., Duval, 1995; Fischbein, 1993). For more comprehensive reviews of this literature we encourage the reader to see Battista (2007), Clements and Battista (1992), and Sinclair et al. (in press). Here we concentrate on some that have been particularly helpful to make our point. 2.4.1.1. Seeing vs Knowing: How Diagrams Preserve and Lose Information In his studies of students’ creation and interpretation of representations of three-dimensional Figures, Parzysz (1988) argued for the need to distinguish qualitatively the two dimensional diagrams of plane Figures from the two dimensional diagrams of three-dimensional Figures
eBook - PDF- Peter Dale(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
59 4 The Geometry of Common Shapes 4.1 TRIANGLES AND CIRCLES Geometry is the study of constructible shapes. In this chapter we will review some of the shapes that occur in geomatics and GIS that occupy ordinary two- or three-dimensional “Euclidian” space. Euclid was a Greek mathematician of the 3rd cen-tury bc who worked out a series of axioms or postulations concerning points, lines, angles, surfaces, and volumes. From these, he derived 465 theorems. His basic axioms included such statements as that for any two distinct points there is only one straight line that passes through them and if three distinct points are not on a straight line then there is only one plane that will pass through them. Euclid identified 10 axioms but subsequently a further one was added, namely that only one straight line can be drawn parallel to a given line through any point not on that line. In Euclidian space, the shortest distance between two points is a straight line. Two lines that are either parallel or intersect form a plane—in fact parallel lines may be said to intersect at a point at infinity, an important consideration when drawing images of three-dimensional (3D) objects in perspective on a plane (2D) surface, as discussed in Chapter 10. The triangle is the simplest shape that is made up of straight lines. In fact, all 2D shapes can be regarded as being made up from a series of triangles, just as every curve can be thought of as a series of short straight lines. Although this can give rise to a number of errors, for example, when calculating an area enclosed by a curved line, the approximation can be adequate for many practical purposes. Triangles come in all sorts of shapes and sizes but the basic fact is that the angles of a plane triangle add up to half of a complete turn or 180 ° ; the angles of a spherical triangle, which is one drawn on the surface of a sphere, add up to more than 180 ° . For the present, we will only consider plane triangles.
eBook - PDF- (Author)
- 2015(Publication Date)
- For Dummies(Publisher)
If you recognise squares, rectangles, triangles and circles, you’re off to a good start. Add in cubes, cuboids (box shapes), spheres (ball shapes), cylinders and pyramids, and I think you’re good on the ‘recognising shapes’ 234 Part III: Picturing and Measuring: Shapes, Weights and Graphs front. The shape of an object doesn’t depend on its size or orientation — if you twist a square around, the shape is still a square, even if it looks like a diamond. Regardless of how big a shape is or which way around you draw it, a shape’s properties and name stay the same. You also need to know the difference between two-dimensional and three- dimensional shapes. The ‘D’ in a 3D movie stands for ‘dimensions’. The dimensions make the movie seem as if everything isn’t just flat on a screen but has depth. Similarly, in maths, two-dimensional objects are flat (you can draw them on paper) and three-dimensional Figures come out of the page. Sussing out shapes you know You need to recognise the following two-dimensional — or flat — shapes, which I also show in Figure 11-1: 6 Square: A shape with four equal-length straight sides arranged at right-angles to each other. 6 Rectangle: A shape with four straight sides at right-angles to each other. The sides aren’t necessarily all the same length, but sides opposite each other are always the same length. 6 Triangle: A shape with three straight sides. As you progress with maths, most of the geometry you do is based on triangles. 6 Circle: This is the only curved shape you really need to know about. The technical definition is ‘a shape with all of the points a fixed distance from the centre’, but you’ll recognise a circle when you see one. Figure 11-1: A square, a rectangle, a triangle and a circle. 235 Chapter 11: Shaping Up Three-dimensional Figures are objects that don’t lie flat.
No longer available |Learn more- Tom Bassarear, Meg Moss(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
8 Geometry as Shape SECTION 8.1 Basic Ideas and Building Blocks SECTION 8.2 Two-Dimensional Figures SECTION 8.3 Three-Dimensional Figures From a basic perspective, geometry is the study of shapes, their relationships, and their properties. What is the value of geometry? Who needs to know geometry and who has needed to know geometry? How has understanding of geometry shaped our lives? These are some of the questions that will be addressed in this and the next two chapters. According to the Common Core State Standards, the study of geometry begins in kindergarten where students identify, describe, analyze, and compare two-dimensional and three-dimensional shapes. Reasoning with shapes and their attributes continues throughout elementary school. In fourth grade, angles and lines are explored, and in fifth grade the coordinate plane ( x and y axes) is included in the geometry strand. In sixth through eighth grades the geometry strand continues with area, surface area and volume, transformations, and finally the Pythagorean theorem. Here are a few examples where people have needed geometry and where this need has produced greater understanding: • People in ancient Egypt developed methods to determine the boundaries of their land. In fact, the word geometry literally means “to measure the earth.” • Explorers needed maps to show where they had been. • House builders have used many shapes, from Native American teepees, which look like cones but are more like many-sided pyramids, to African houses resembling cylinders, to the more familiar house structures with triangles and quadrilaterals, to geodesic domes with equilateral triangles. • Surveyors and planners needed to ensure that tunnels or railroad tracks built from both ends would actually meet in the middle. • Artists wanted to portray convincingly what they saw with their eyes or visualized in their minds. • Architects and builders needed buildings that would be both beautiful and strong.
eBook - PDFNumber Treasury 3: Investigations, Facts And Conjectures About More Than 100 Number Families (3rd Edition)
Investigations, Facts and Conjectures about More than 100 Number Families
- Margaret J Kenney, Stanley J Bezuszka(Authors)
- 2015(Publication Date)
- World Scientific(Publisher)
Chapter 3 Plane Figurate Numbers I think I have always had a basic liking for the natural numbers. We can conceive of a chemistry which is different from ours, or a biology, but we cannot conceive of a different mathematics of numbers. What is proved about numbers will be a fact in any universe. Julia Bowman Robinson (1919–1985) Problems and activities with plane Figures come alive when you use manipulatives. Chips, circular discs, or washers on a geoboard are effective tools for modeling as are square and triangular dot paper. The topic of plane figurate numbers is ideal for blending arithmetic, algebra and geometry. Polygons A polygon is a closed plane figure formed by three or more line segments called sides . Each segment intersects exactly two other segments, one at each endpoint. No two line segments with a common endpoint are on the same line. Each end-point is called a vertex of the polygon. Each of these Figures is a polygon. None of these Figures is a polygon. 73 74 Number Treasury 3 A regular polygon is a polygon with all sides congruent and all angles congruent. Equilateral triangle 3-gon Square 4-gon Regular pentagon 5-gon Regular hexagon 6-gon EXERCISE 37 1. Use a straightedge or ruler to draw some polygons with 4, 5, and 6 sides that are different from those shown above. 2. What is the angle measure of each angle in a regular a) 3-gon? b) 4-gon? c) 5-gon? d) 6-gon? 3. Use a ruler and protractor to draw a regular 8-gon, or octagon. Figurate Numbers While the Pythagoreans probably were familiar with both perfect and amicable numbers, they are credited with originating figurate numbers. These numbers are important because they form a link between geometry and arithmetic. Figurate or polygonal numbers are classes of numbers that can be represented by dots arranged in specific geometrical or polygonal patterns. Triangular Numbers Triangular numbers are figurate numbers that can be represented by a triangular array of dots.
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