using gsp in discovering a new theory dr. mofeed abu-mosa 20-3-2007 this paper 1. connects van hiele...
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Using GSP in Discovering a New TheoryDr. Mofeed Abu-Mosa
20-3-2007This paper1. Connects Van Hiele theory and its levels of
geometric thinking with the dynamic software program geometry sketchpad.
2. Offers an example of using GSP to discover a theory, which was developed from the Pythagorean Theorem by using GSP as a tool of thinking.
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k'''
j
Area NH'GE = 15.00 cm2
Area IL'KH = 15.00 cm2
Area DMLF = 15.00 cm2
H: (-5.00, -3.00)
K: (-5.00, 9.00)
L: (1.00, 11.00)
F: (9.00, -1.00) D: (5.00, 2.00)
M: (3.00, 5.00)
J: (-3.00, 3.00)
I: (-3.00, 0.00)
G: (-2.00, -5.00)
E: (6.00, -5.00)
O: (0.00, -2.00)
N: (2.00, -2.00)
C: (0.00, 3.00)
B: (2.00, 0.00)
A: (0.00, 0.00)
E
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K
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H'
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Through the study of geometry, students will learn:
• Geometric shapes and structures.
• How to analyzeanalyze shapes.
•Characteristics and relationshipsrelationships between shapes.
•Reasoning and justification skills.
• Through tools such as dynamic geometry softwaredynamic geometry software
which enables students to modelmodel, and have an interactive interactive experienceexperience with, a large variety of two-dimensional shapes NCTM 2000NCTM 2000
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k'''
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Area NH'GE = 15.00 cm2
Area IL'KH = 15.00 cm2
Area DMLF = 15.00 cm2
H: (-5.00, -3.00)
K: (-5.00, 9.00)
L: (1.00, 11.00)
F: (9.00, -1.00) D: (5.00, 2.00)
M: (3.00, 5.00)
J : (-3.00, 3.00)
I: (-3.00, 0.00)
G: (-2.00, -5.00)
E: (6.00, -5.00)
O: (0.00, -2.00)
N: (2.00, -2.00)
C: (0.00, 3.00)
B: (2.00, 0.00)
A: (0.00, 0.00)
E
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K
L
H'
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NO
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A B
How students learn specific mathematical domain or concept?
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k'''
j
Area NH'GE = 15.00 cm2
Area IL'KH = 15.00 cm2
Area DMLF = 15.00 cm2
H: (-5.00, -3.00)
K: (-5.00, 9.00)
L: (1.00, 11.00)
F: (9.00, -1.00) D: (5.00, 2.00)
M: (3.00, 5.00)
J : (-3.00, 3.00)
I: (-3.00, 0.00)
G: (-2.00, -5.00)
E: (6.00, -5.00)
O: (0.00, -2.00)
N: (2.00, -2.00)
C: (0.00, 3.00)
B: (2.00, 0.00)
A: (0.00, 0.00)
E
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K
L
H'
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Van Hiele theory a comprehensive theory yet formulated concerning geometry learning
The theory claims that when students learn geometry they progress
from one discrete level of geometrical thinking to another
This progress is discontinuous and the levels are sequential and hierarchical
The Van Hiele theory also suggests phases of instruction that help students progress through the levels.
Levels of Geometric Thinking Distributed According to Geometric SkillsLevels of Geometric Thinking Distributed According to Geometric Skills
Level skill
Recognition Analysis Deduction
Visual
Recognize geometric shapes by
its' picture without knowing the shapes prosperities
Recognize the relationship between different kinds of geometric shapes
Uses information about a geometric shape to deduce more information
Descriptive
Naming a geometric shape. Explain statements that describe geometric shape
Describes the relationships between geometric shapes. Defines geometric concepts clearly.
Understand the difference between
the definition, postulate and theorem
Logical
Understand the meaning of shape reservation in different situations.
Uses the prosperities of geometric shapes to identify the subset relation
Uses logic to prove and being able to deduce new knowledge from given facts
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k'''
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Area NH'GE = 15.00 cm2
Area IL'KH = 15.00 cm2
Area DMLF = 15.00 cm2
H: (-5.00, -3.00)
K: (-5.00, 9.00)
L: (1.00, 11.00)
F: (9.00, -1.00) D: (5.00, 2.00)
M: (3.00, 5.00)
J: (-3.00, 3.00)
I: (-3.00, 0.00)
G: (-2.00, -5.00)
E: (6.00, -5.00)
O: (0.00, -2.00)
N: (2.00, -2.00)
C: (0.00, 3.00)
B: (2.00, 0.00)
A: (0.00, 0.00)
E
L'
K
L
H'
G
H
F
I
J
M
D
NO
C
A B
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6
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k'''
j
Area NH'GE = 15.00 cm2
Area IL'KH = 15.00 cm2
Area DMLF = 15.00 cm2
H: (-5.00, -3.00)
K: (-5.00, 9.00)
L: (1.00, 11.00)
F: (9.00, -1.00) D: (5.00, 2.00)
M: (3.00, 5.00)
J: (-3.00, 3.00)
I: (-3.00, 0.00)
G: (-2.00, -5.00)
E: (6.00, -5.00)
O: (0.00, -2.00)
N: (2.00, -2.00)
C: (0.00, 3.00)
B: (2.00, 0.00)
A: (0.00, 0.00)
E
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K
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H'
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Level (3)
Logical
Level (2)Descriptive
Level (1)Visual
Recognize Shapes
Recognize relation between Shapes
Deduce more information
Describe Shapes
Describe relation between Shapes
Deduce more information
reasoning and justification skills, culminating in work with proof in the secondary grades.
Example Using GSP to construct a squareconstruct a square. • Draw segments (parallel and perpendicular) and try to make them congruent
by daggering the points (level (1)).
• Construct a grid and join between points on the grid level (1)
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•Construct a circle and perpendicular and parallel lines level (2)• Using transformations to do the construction level (2).•Use the measure tool to justify your work level (3)•Do the same construction in other ways level (3)•Ask the student to prove the construction in an abstract way level (3).
By using the Custom Tool teachers can follow the thinking of every student and assess the level he (she) reaches
Suggestion
Curriculum experts can maximize the use of dynamic software.
Rebuilding the geometric content is needed to change the traditional way curriculums are written.
Area KQP = 2.59 cm2
Area MNO = 2.59 cm2
Area RJL = 2.59 cm2
Area JKN = 2.59 cm2
Area AHG = 3.00 cm2
Area ICD = 3.00 cm2
Area EBF = 3.00 cm2
Area CAB = 3.00 cm2
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AH
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FG
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A B NK
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Ron's Theorem
Area GFNI = 10.00 cm2
Area OJKH = 10.00 cm2
Area DEML = 10.00 cm2
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My Extension
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Area C''''''C''A''A''' = 12.00 cm2
Area C''''A'''B''B'' = 2.50 cm2
Area CA''B = 0.50 cm2
Idea of the proof
AREA OF
TRIANGLE TRAPEZOID1 TRAPEZOID2 TRAPEZOID3 TRAPEZOID4 TRAPEZOID5
1 5 24 115 551 2640
2 10 48 230 1102 5280
3 15 72 345 1653 7920
4 20 96 460 2204 10560
5 25 120 575 2755 13200
6 30 144 690 3306 15840
7 35 168 805 3857 18480
Try to discover relation between the area of the origin triangle and the area of trapezoids
the geometric pattern can be converted into algebraic one
Thanks
Most of old theorem can lead our students to
new ones