Rectangle (1 symmetry) Square (3 symmetries) Diamond (1 symmetry) Parallelogram (1 symmetry) Trapezoid (no symmetry) Regular Pentagon (4 symmetries)ġ6 Polygons The word "polygon" derives from the Greek poly, meaning "many", and gonia, meaning "angle". We don’t count the trivial 360° rotation symmetry here. Since every figure will match itself after rotating 360°, we do not consider a 360° rotation as a rotation symmetry. (click to see animation) This equilateral triangle has 2 (non-trivial) rotation symmetries, 120° and 240° respectively. The “fold line” just described is call the figure’s line (axis) of symmetry.ġ3 Lines of symmetry for the following common figures.ġ4 Rotation Symmetry A 2D figure has rotation symmetry if there is a point around which the figure can be rotated, less than a full turn, so that the image matches the original figure perfectly. Reflection Symmetry (also called folding symmetry) A 2D figure has reflection symmetry if there is a line that the figure can be “folded over” so that one-half of the figure matches the other half perfectly. Paper cup (silhouette) Isosceles Trapezoid Trapezoid whose non-parallel sides are of the same length.ġ0 Summary Quadrilateral Trapezoid Parallelogram Kite Isosceles Trapezoid Rectangle Rhombus SquareĬategory 1 Category 2 What is the mathematical property that separates these two categories of shapes?ġ2 Symmetries In formal terms, we say that an object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation does not change the object or its appearance. Model Abstraction Description Trapezoid Quadrilateral with exactly one pair of parallel sides. Kite Quadrilateral with two non-overlapping pairs of adjacent sides that are of the same length Model Abstraction Description Railing Parallelogram Quadrilateral with 2 pairs of parallel sides Diamond Rhombus Quadrilateral with 4 sides the same length. Door Rectangle Quadrilateral with 4 right angles. Tile Square Quadrilateral with all sides the same length and 4 right angles. Model Abstraction Description Steel frame of a bridge Right triangle Triangle with one right angle. Yield sign Equilateral triangle Triangle with all 3 sides the same length. Teepee Isosceles triangle Triangle with 2 sides the same length. Model Abstraction Description Bike frame Scalene triangle Triangle with all 3 sides different. Vertical flag pole Right angle Angle formed by two lines, one vertical and one horizontal. Model Abstraction Description Top of a window Line segment Open pair of scissors Angle The union of two line segments with a common endpoint. This level of study is only suitable for university students. The postulates or axioms themselves become the object of intense, rigorous scrutiny. Level 4 (Axiomatics) Geometry at this level is highly abstract and does not necessarily involve concrete or pictorial models. A student at this level can understand the notions of mathematical postulates and theorems. The van Hiele Theory Level 3 (Deduction) Reasoning at this level includes the study of geometry as a formal mathematical system. For instance, a rhombus is also a parallelogram. Second, a child can use informal deductions to justify observations made at level 1. For example, a square is both a rhombus and a rectangle. First, a child understands abstract relationships among figures. Level 2 (Relationships) There are two types of thinking at this level. Relevant attributes are understood and are differentiated from irrelevant attributes. Component parts and their attributes are used to describe and characterize figures. The van Hiele Theory Level 0 (Recognition) At this lowest level, a child recognizes certain shapes holistically without paying attention to their components Level 1 (Analysis) The child focuses analytically on the parts of a figure, such as its sides and angles.
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