111020 analytic geometry

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Understanding versus Procedure

Analytic Geometry

Martin Zacharias

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Understanding? Comprehension?

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Everyday life in schoolGiven are to lines and

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: txg

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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. …

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Everyday life in school?

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• 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.

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Vertical lines

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Distance

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Circle (0|0) with radius 4

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Descartes Dictionary

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•We need activities for the pupils.

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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

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Crossing section on a cube

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Diagonals

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2D of real objects

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Forms of a roof

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Read out the point

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S is the summit of a pyramid

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half of a cube

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Cross-sections by liquids

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Cross-section of a regular Hexagon

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Werkzeug -CDF:\Vortrag\Kiel_100922\CD_AnaGeo\start.htm

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Schrägbilddarstellungen

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Corner of a cube

C=(8|3|6), F=(2|9|6), G=(4|5|10), H=(0|1|8)

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Vector

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Vectors in objects

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Vectors in objects

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Regular Hexagon in a cube

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Regular Hexagon in an Oktaeder

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Rectangle in a cubeCenter of the cube – center of the rectangle.

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Cross-section

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Diagonals in the space

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Mittendreieck

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Lines in the space

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Lagebeziehung von Geraden im Raum

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Area in a roof (Walmdach)

Describe the blue area with lines.

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Line segments in a Pyramid

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„Haus des Nikolaus“

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Cube with a cut off corner

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Cut off cube (Kuboktaeder)

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Position of lines

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Point of intersection?

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

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Lagebeziehung mit GTR

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Lagebeziehung und GTR

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Ebenen im Raum

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experiment - search

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Ebene - experimentell

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Objektstudie

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What happens, when … ?

…B goes towards F with the angle beta?

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