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Chapter 12

Chapter 12. Geometric Shapes. 12.1 Recognizing Geometric Shapes. The van Hiele Theory Level 0: Recognition Some relevant attributes of shape, straightness of lines, may be ignored. Some irrelevant attributes, such as orientation on a page, may be stressed.

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Chapter 12

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  1. Chapter 12 Geometric Shapes

  2. 12.1 Recognizing Geometric Shapes The van Hiele Theory Level 0: Recognition Some relevant attributes of shape, straightness of lines, may be ignored. Some irrelevant attributes, such as orientation on a page, may be stressed. Recognizes shapes holistically without noticing component parts.

  3. The van Hiele Theory Level 1: Analysis Focus is on parts of figures, such as sides and angles. Relevant attributes are understood and differentiated from irrelevant ones. Analytical thinkers may not believe a figure can belong to several classes, and have several names. A square is a rhombus and also a rectangle.

  4. Level 2: Relationships A child does understand abstract relationships among figures. A child can also use deduction to justify observations made at level 1. Level 3: Deduction Reasoning involves the study of geometry as a formal mathematical system. A child who reasons at level 3 understands the notions of postulates and theorems and can write formal proofs of theorems.

  5. Level 4: Axiomatics Level 4 is highly abstract and does not necessarily involve concrete or pictorial models. Theorems and postulates are at the center of interest, and are the subject of intense scrutiny. This level of study is found typically at the college level.

  6. Recognizing Geometric Shapes Children are taught the basic shapes, and many achievement tests ask them to identify the square, triangle, etc. Too often, children have only seen the “regular” shapes, and thus, do not have a complete idea of the important attributes a shape must have to represent a general type.

  7. Making new shapes by rearranging other shapes is a good way for students to develop visualization skills. How many different rectangles are formed by the heavy line segments? Seven vertical rectangles, and two horizontal rectangles, for a total of nine.

  8. Defining Common Geometric Shapes • Line segment; congruent line segments • Angle; congruent angles, equiangular • Vertex (vertices) • Quadrilateral; parallelogram, rhombus, rectangle, square, kite • Right angle • Perpendicular • Triangle; Isosceles, equilateral, scalene • Trapezoid; isosceles trapezoid

  9. 12.2 Analyzing Shapes Symmetry Reflection: If a figure can be folded across a line so that one side of figure completely matches the other side, the figure has reflection symmetry. This line is called the line of symmetry.

  10. Rotation Symmetry If there is a point around which the figure can be rotated, less than 360°, and the image matches the original figure perfectly, the figure is said to have rotational symmetry.

  11. Perpendicular Line Segments Test Let P be the point of intersection of l and m. Fold l at point p so that l folds across P onto itself. Then l and m are perpendicular if and only if m lies along the fold line.

  12. Parallel Line Segments Test Fold so that l folds onto itself. Any fold line can be used except for l itself. Then l and m are parallel if and only if m folds onto itself or an extension of m.

  13. Regular Polygons A simple closed curve in the plane is a curve that does not cross itself, and has the same starting and stopping point. A polygon is a simple closed curve comprised of line segments. A regular polygon is a polygon with all sides congruent.

  14. Concave vs. Convex A convex shape is a simple closed curve which can completely contain a line segment within its interior. A concave shape is a simple closed curve which cannot contain a line segment within its interior.

  15. Pentagon Exterior Angle Central Angle Vertex Angle

  16. Circles A circle is the set of all points in the plane that are a fixed distance away from a given point. The given point is called the center, and the fixed distance is called the radius. Radius Center Diameter

  17. 12.3 Properties of Geometric Shapes An infinitely large flat surface is called a plane. Points are locations within a plane. Connecting two points in a plane forms a line. Lines are straight, and extend infinitely long in each direction. Points that lie on the same line are collinear points. Two lines in the plane are parallel if they do not intersect. Three or more lines that contain the same point are concurrent lines.

  18. Angles A D B C E F Obtuse Acute

  19. Transversals 1 3 2 4

  20. Parallel Lines 1 3 5 2 4 6

  21. Theorem: Suppose that lines l and m are cut by a transversal t. Then l || m if and only if alternate interior angles formed by l, m, and t are congruent.

  22. 12.4 Regular Polygons and Tessellations Angle Measures A regular n-gon is both equilateral and equiangular. What about vertex angles?

  23. Vertex Angles

  24. Tessellations A polygonal region is a polygon together with its interior. An arrangement of polygonal regions having only sides and vertices in common that completely covers the plane is a tessellation or tiling. Theorem: Only regular 3-gons, 4-gons and 6-gons form tessellations of the plane by themselves.

  25. 12.5 Describing Three-Dimensional Shapes A dihedral angle is formed by the union of polygonal regions in space that share an edge. The regions forming the angle are called faces. Non-intersecting, non-parallel lines are called skew lines. Two lines in 3-space can be parallel, can intersect, or can be skew lines.

  26. Polyhedra A polyhedron is the union of polygonal regions, any two of which have at most a side in common, such that a connected finite region in space is enclosed without holes. A polyhedron is convex if every line segment joining two of its points is contained inside the polyhedron or is on one of its faces

  27. Face Edge Vertex

  28. General Types of Polyhedra Prisms are polyhedra with one pair of opposite faces that are identical. These faces are called bases, and the lateral faces are parallelograms. Pyramids are polyhedra formed by using a polygon for the base and a point on the plane of the base, called the apex, that is connected to each vertex of the base with a line.

  29. A regular polyhedron is one in which all faces are identical regular polygonal regions. There are exactly five regular convex polyhedra, called the Platonic Solids. Cube Tetrahedron Octahedron Dodecahedron Icosahedron

  30. Semiregular polyhedra have several different regular polygonal regions, with the same arrangement of polygons at each vertex.

  31. Curved Shapes in Three Dimensions Cylinders Right Cylinder Oblique Cylinder

  32. Apex Cones Right circular cone Oblique circular cone

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