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GEOMETRY

GEOMETRY . 2013-2014. Geometry Tools. August 27, 2013 Toolbox Automaticity . G.CO.13. G-CO.13 . Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. GEOMETRY . Geo: earth Metres : measure Geometry began as the study of earth measure. .

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GEOMETRY

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  1. GEOMETRY 2013-2014

  2. Geometry Tools August 27, 2013 Toolbox Automaticity

  3. G.CO.13 • G-CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

  4. GEOMETRY • Geo: earth • Metres: measure • Geometry began as the study of earth measure.

  5. GEOMETRIC CONSTRUCTIONS • "Construction" in Geometry means to draw shapes, angles or lines accurately. • These constructions use only compass, straightedge (i.e. ruler) and a pencil. • This is the "pure" form of geometric construction - no numbers involved!

  6. Inscribed In: • Inscribed Polygon • Inscribed Circle • A polygon is inscribed in a circle if the vertices of the polygon are on the circle. • A circle is inscribed in a polygon if the sides of the polygon are tangent to the circle.

  7. Equilateral Triangle: • An equilateral triangle is a triangle whose sides are all congruent

  8. REGULAR HEXAGON: • A hexagon (six sided polygon) with congruent sides and angles

  9. Creating an Inscribed Hexagon • This is one of the easiest constructions ever. • The radius of a circle can be struck around a circle exactly six times. • Lets watch: http://www.mathopenref.com/constinhexagon.html • Time to Try!!

  10. Using Geometry in Design:

  11. MANDALA

  12. QUESTIONS

  13. Having Kittens Work out whether this number of descendants is realistic. Here are some facts that you will need:

  14. HAVING KITTENS • Can you make a diagram or table to show what is happening? • Can you now look systematically at what happens to her kittens? And their kittens? • Do you think the first litter of kittens will have time to grow and have litters of their own? Then what about their kittens? • What have you assumed here?

  15. Collaborative Activity • Work in groups of two. • Produce a solution that is better than your individual solution. • Take turns to explain how your did the task and how you now thing it could be improved, then put your individual work aside. Try to produce a joint solution to the problem.

  16. Assessing Sample Student Responses Your task is to correct the work and write comments about its accuracy and organization. • What has the student done correctly? • What assumptions has he or she made? • How could the solution be improved?

  17. Sample Response: Alice

  18. Sample Response: Wayne

  19. Sample Response: Ben

  20. Reviewing Work • I have selected the important facts and used them to solve the problem. • I am aware of the assumptions I have made and the effect these assumptions have on the result. • I have used more than one method • I have checked whether my results make sense and improved my method if need be. • I have presented my results in a way that will make sense to others.

  21. G.CO.12 • G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

  22. TODAY WE WILL • Copy a segment. • Bisect a segment.

  23. UNDEFINED TERMS • Undefined terms are basic terms we need to describe the shape and size of objects. • There are three undefined terms in geometry: • Point, line, and plane

  24. POINT • Point (0-D): A location. • Example: The following is a diagram of points A, B, C, and Q:

  25. SPACE • Space: the set of all points.

  26. LINE • Line (1-D): A series of points that extend in two directions without end. • Example: The following is a diagram of two lines: line AB and line HG. • The arrows signify that the lines drawn extend indefinitely in each direction.

  27. PLANE • Plane (2-D): a flat, two-dimensional object. A plane must continue infinitely in all directions and have no thickness at all. • A plane can be defined by at least three non-collinear points or renamed by a script capital letter.

  28. GEOMETRIC CONSTRUCTIONS • "Construction" in Geometry means to draw shapes, angles or lines accurately. • These constructions use only compass, straightedge (i.e. ruler) and a pencil. • This is the "pure" form of geometric construction - no numbers involved!

  29. B A SEGMENT • Line Segment: part of a line containing two endpoints and all points between them. AB or BA

  30. Getting Started • In your notes draw two segments. Similar to the ones below.

  31. COPYING A SEGMENT • Given: • Construct: so that  • http://www.mathopenref.com/constcopysegment.html

  32. PRACTICE #1: • Given: • Construct: =

  33. PRACTICE #2: • Given: & • Construct: = + 2.

  34. PRACTICE #3: • Given: & • Construct: = 3- 2.

  35. Bisect • To divide into two equal parts. • You can bisect lines, angles, and more. • The dividing line is called the "bisector

  36. Midpoint • Midpoint: The point of a line segment that divides it into two parts of the same length. • 

  37. PERPENDICULAR BISECTORs • Perpendicular Lines: two lines that intersect to form right angles. (SYMOBOL ) • Perpendicular Bisector of a Segment: a line, segment, or ray that is perpendicular to the segment at its midpoint, thereby bisecting the segment into two congruent segments.

  38. BISECTING A SEGMENT • http://www.mathopenref.com/constbisectline.html

  39. Practice #4: • Given: • Construct:  at the midpoint M of .

  40. Practice #5: • Given: • Construct: Divide AB into 4 congruent segments.

  41. QUESTIONS

  42. G.CO.12 • G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

  43. TODAY WE WILL • Copy an angle. • Bisect an angle

  44. RAY B • Ray: part of a line that consists of one endpoint and all points of the line extending in one direction A AB , not BA

  45. ANGLES () • An angle is two rays meeting at a common vertex.

  46. ANGLES () • Angle: a pair of rays that share a common endpoint. • The rays are called the sides of the angle. • The common endpoint is called the vertex of the angle.

  47. A A ANGLES () • Obtuse angle 90 < mA < 180 • Right angle m  A = 90 • Acute angle 0 < mA < 90 A

  48. COPYING AN ANGLE • The angle RPQ has the same measure as BAC • http://www.mathopenref.com/constcopyangle.html

  49. Practice #1 • Given Acute Angle: ABC • Duplicate ABC, so that ABC = A’B’C’

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