The document discusses the importance of understanding geometric concepts through real-world objects and examples, rather than just focusing on procedures and calculations. It provides several examples of using lines, planes, and shapes within objects to help students visualize and comprehend geometric relationships over just using equations and finding points of intersection. The examples include discussing lines and planes within cubes, pyramids, roofs and other objects. The goal is to help students progress from visualizing real situations, to drawing representations, to then expressing things analytically to build conceptual understanding.

1. Understanding versus Procedure Analytic Geometry Martin Zacharias

2. Understanding? Comprehension?

3. Everyday life in school Given are to lines and a) Which situation do they have? Find the points of intersection. b) Describe the plane E, which includes the lines g and h. c) Find a line, which has no points of intersection with g and h. Give the reason for your choice. d) Find the points of intersection to the basic-planes (xy-plane, xz-plane and yz-plane) and draw the lines g and h. Draw the point of intersection of both lines. e) Find the angle between g and h. f) Find the angle between the basic-planes. …

4. Everyday life in school?

5. The exercise isn’t to bad. You have to argue, you have do draw. But the finding of terms and equations and the calculation with equations is dominant. Geometry and basis ideas don’t exist.

7. Vertical lines

8. Distance

9. Circle (0|0) with radius 4

10. Descartes Dictionary

11. We need activities for the pupils.

12. Concept by Objects From doing to thinking in three steps to comprehend the real model (situation) to represent und describe it in a drawing to represent and describe it in the analytic model

13. Crossing section on a cube

14. Diagonals

16. 2D of real objects

17. Forms of a roof

18. Read out the point

19. S is the summit of a pyramid

20. half of a cube

23. Cross-sections by liquids

24. Cross-section of a regular Hexagon

25. Werkzeug -CD F:\Vortrag\Kiel_100922\CD_AnaGeo\start.htm

26. Schrägbilddarstellungen

27. Corner of a cube C=(8|3|6), F=(2|9|6), G=(4|5|10), H=(0|1|8)

28. Vector

29. Vectors in objects

30. Vectors in objects

31. Regular Hexagon in a cube

32. Regular Hexagon in an Oktaeder

33. Rectangle in a cube Center of the cube – center of the rectangle.

34. Cross-section

35. Diagonals in the space

36. Mittendreieck

39. Lines in the space

40. Lagebeziehung von Geraden im Raum

42. Area in a roof (Walmdach) Describe the blue area with lines.

43. Line segments in a Pyramid

44. „ Haus des Nikolaus“

45. Cube with a cut off corner

46. Cut off cube (Kuboktaeder)

47. Position of lines

48. Point of intersection? No, it doesn’t exist a point of intersection.

49. Lagebeziehung mit GTR

50. Lagebeziehung und GTR

51. Ebenen im Raum

52. experiment - search

54. Ebene - experimentell

55. Objektstudie

57. What happens, when … ? … B goes towards F with the angle beta?