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Chapter 12 Geometric Shapes

Chapter 12 Geometric Shapes. Section 12.1 Recognizing Shapes. 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)

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Chapter 12 Geometric Shapes

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  1. Chapter 12 Geometric Shapes

  2. Section 12.1 Recognizing Shapes 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. Component parts and their attributes are used to describe and characterize figures. Relevant attributes are understood and are differentiated from irrelevant attributes. Level 2 (Relationships) There are two types of thinking at this level. First, a child understands abstract relationships among figures. For example, a square is both a rhombus and a rectangle. Second, a child can use informal deductions to justify observations made at level 1. For instance, a rhombus is also a parallelogram.

  3. Section 12.1 Recognizing Shapes The van Hiele Theory Level 3 (Deduction) Reasoning at this level includes the study of geometry as a formal mathematical system. A student at this level can understand the notions of mathematical postulates and theorems. Level 4 (Axiomatics) Geometry at this level is highly abstract and does not necessarily involve concrete or pictorial models. The postulates or axioms themselves become the object of intense, rigorous scrutiny. This level of study is only suitable for university students.

  4. Describing Common Geometric Shapes

  5. Describing Common Geometric Shapes

  6. Describing Common Geometric Shapes Picture on next slide

  7. Steel bridge in Portland

  8. Describing Common Geometric Shapes

  9. Describing Common Geometric Shapes

  10. Summary Quadrilateral Kite Trapezoid Parallelogram Isosceles Trapezoid Rectangle Rhombus Square

  11. Section 12.2 Analyzing Shapes Category 2 Category 1 What is the mathematical property that separates these two categories of shapes?

  12. 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. 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. The “fold line” just described is call the figure’s line (axis) of symmetry.

  13. Lines of symmetry for the following common figures.

  14. 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. (click to see animation) This equilateral triangle has 2 (non-trivial) rotation symmetries, 120° and 240° respectively. Since every figure will match itself after rotating 360°, we do not consider a 360° rotation as a rotation symmetry.

  15. Rotation symmetries of common figures We don’t count the trivial 360° rotation symmetry here. Rectangle (1 symmetry) Square (3 symmetries) Diamond (1 symmetry) Parallelogram (1 symmetry) Trapezoid (no symmetry) Regular Pentagon (4 symmetries)

  16. Polygons The word "polygon" derives from the Greek poly, meaning "many", and gonia, meaning "angle". Regular pentagon n = 5 Equilateral triangle n = 3 Square n = 4 Regular hexagon n = 6 Regular octagon n = 8 Regular heptagon n = 7

  17. Polygons and their nomenclature A Triangle (from Latin) has 3 sides A Quadrilateral (from Latin) has 4 sides A Pentagon (from Greek) has 5 sides A Hexagon (from Latin) has 6 sides A Heptagon (from Greek) (or a Septagon from Latin?) has 7 sides

  18. In fact, “Septagon” is not an official word for the 7-gon, it is not even in a dictionary. It was invented by some elementary school teachers to make it easier to remember. The Latin word septem means 7 and September means the seventh month. The old Roman calendar began the year in January, (named after the Roman god of fortune, Janus), and September was the seventh month. Afterwards, Julius Augustus (46 BC) named two more then-29 day periods after himself and September came to be as we know it in the Gregorian Calendar, the ninth month.

  19. An Octagon (from Greek) has 8 sides A Nonagon (from Latin) has 9 sides. A Decagon (from Greek) has 10 sides. A polygon with more than n (>10) sides is usually just called an n-gon. More names of polygons

  20. Convex and Concave Shapes A figure is convex if a line segment joining any two points inside the figure lies completely inside the figure.

  21. Angles in a polygon In a regular pentagon: the measure of a central angle is 360°/5 = 72° the measure of an exterior angle is also 360°/5 = 72° the measure of a vertex angle is 180° – 72° = 108°

  22. Circles A Circle is the set of all points in the plane that are at a fixed distance from a given point called the center. The distance from any point on the circle to the center is called the radius of the circle. The length of any line segment whose endpoints are on the circle and which contains the center is called the diameter of the circle. The segment is also called a diameter of the circle.

  23. Circles Circles have the following 3 properties that make them very useful. 1. They are highly symmetrical, hence they have a sense of beauty and are often used in designs. eg. dinnerware.

  24. Circles Circles have the following 3 properties that make them very useful. 1. They are high symmetrical, hence they have a sense of beauty and are often used in designs. eg. dinnerware.

  25. 2. Every point on a circle bears the same distance from the center. This is called the equidistance property. Applications: wheels

  26. 3. For a given (fixed) perimeter, the circle has the largest area. Applications: soda cans, or any container for pressurized liquid are all cylindrical in shape.

  27. Section 12.3 Properties of Lines and Angles Definition Two different given lines L1 and L2 on a plane are said to be parallel if they will never intersect each other no matter how far they are extended.

  28. Angles An angle is the union of two rays with a common endpoint. side interior vertex side

  29. Degrees Angles are measured by a semi-circular device called a protractor. The whole circle is divided into 360 equal parts, each part is defined to have measure one degree (written 1°). Hence a semi-circular protractor has 180 degrees. One degree is divided into 60 minutes and one minute is further divided into 60 seconds. Notations: For instance, 27 degrees 35 minutes 41 seconds is written as 27°35’41”

  30. Names of angles A straight angle has 180 degrees An obtuse angle has measure between 90° and 180°. A right angle has exactly 90°. An acute angle has measure less than 90°.

  31. Definition Two angles are called vertical angles if they are opposite to each other and are formed by a pair of intersecting lines. A B Theorem Any pair of vertical angles are always congruent.

  32. More special angles Two angles are said to be supplementary if their measures add up to 180°. α β Two angles are said to be complementary if their measures add up to 90°. α β

  33. Perpendicular Lines Two lines are said to be perpendicular to each other if they intersect to form a right angle

  34. L1 L2 T Parallel Lines and Angles Definition Given two line L1 and L2 (not necessarily parallel) on the plane, a third line T is called a transversal of L1 and L2 if it intersects these two lines.

  35. a c L1   L2 T • Definitions • Let L1 and L2 be two lines (not necessarily parallel) on the plane, and T be a transversal. • a and form a pair of corresponding angles. • c and  form a pair of corresponding angles etc.

  36. c d L1   L2 T • Definitions • Let L1 and L2 be two lines (not necessarily parallel) on the plane, and T be a transversal. • c and form a pair of alternate interior angles. • d and  form a pair of alternate interior angles.

  37. a L1  L2 T • Definitions • Let L1 and L2 be two lines (not necessarily parallel) on the plane, and T be a transversal. • a and form a pair of alternate exterior angles. • b and  form a pair of alternate exterior angles.

  38. Angle Sum in a Triangle Draw an arbitrary triangle on a piece of paper and label all 3 angles. Next cut out the triangle, and then cut it into 3 parts (as indicated by the dashed lines) Arrange the 3 angles side by side, can you get a straight angle? b a c Conclusion: The angle sum in a triangle is always 180°

  39. Angle Sum in other Polygons What is the sum of all angles in a quadrilateral? Answer: 180 × 2 = 360 What is the sum of all angles in a pentagon? Answer: 180 × 3 = 540

  40. Angle Sum in other Polygons Conclusion: For a polygon with n sides, the angle sum is (n – 2) × 180°

  41. Classification of triangles according to their angles. A triangle with one right angle is called a right triangle. A triangle with one obtuse angle is called an obtuse triangle. A triangle with 3 acute angles is called an acute triangle.

  42. Classification of triangles according to their sides. A triangle with 3 different sides is called a scalene triangle. A triangle with 3 equal sides is called an equilateral triangle. A triangle with 2 equal sides is called an isosceles triangle.

  43. Angles & Angle Sums in Regular polygons For a regular pentagon, m(central angle) = central angle m(vertex angle) = (3 × 180) ÷ 5 = 108 center vertex angle

  44. Application of Degree Measure Angles can be used to indicate directions. The only difference is that the measure can be greater than 180º. In navigation, the direction can be any value between 0º and 360º.

  45. The Bearing System The exact (magnetic) North is defined to be 0 degree. Any other direction is defined to be the number of degrees away from exact North measuring in the clockwise direction. N 130º south east direction

  46. The Bearing System In particular, 90º is equal to exact East, N 90º = East

  47. The Bearing System and 180º is equal to exact South, N 180º = South

  48. The Bearing System and 270º is equal to exact West, N 270º = West

  49. Runway Numbers In any airport, each runway is assigned a number according to the direction it is pointing at – except that the units digit is omitted for simplicity. For example, runway 24 is actually pointing at 240º, and it means that during final approach, the aircraft is heading 240º - which is about south west.

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