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Presented by
Yan Hee Cheong
Lawrence Yong Shao Ping
Damien Yeo Tat Sheng
Tan Ing Keat
Elvin Yeo Boon Heng
How would you describe these structures ??
Mathematically, they are structures of very unique
and magnificient ‘Geometric’ designs
This lead us to our topic for today : Geometric properties of Triangles, Quadrilaterals and Polygons
Some History of Geometry
• Egyptians ( 2000 – 500 B.C.)
Ancient Egyptians demonstrated practical knowledge of geometry through surveying and construction of projects
• Babylonians ( 2000 – 500 B.C.)
Ancient clay tablets reveal that the Babylonians knew the Pythagorean relationships.
• Greeks ( 750 – 250 B.C.)
Practiced centuries of experimental geometry.
The greatest mathematical textbook of all time is the Elements – written by Euclid of Alexandria (320 to 260 B.C.). The book had dictated the study of geometry for > 200 years
Euclid
Basic elements of Geometry
Points, Lines and Angles can be manipulated to form various types of geometrical shapes and sizes
Van Hiele Theory of Geometric Thought
1. The Model (5 distinct levels)level 0 – Visualizationlevel 1 – Analysislevel 2 – Informal Deductionlevel 3 – Deductionlevel 4 – Rigor
2. Properties of the Model Sequential
3. Phases of Learning Each level separated by a learning phase
0 Visualization
1 Analysis
2 Informal Deduction
3 Deduction
4 Rigor
Level Description
Identify an object by its appearancePhases of
learning
Formulates and uses definitions
Ability to state proofs
Identify properties of a class of figures
Analyzes various deductive systems
(Upper Secondary)
(Upper Primary)
(Lower Secondary)
(Lower Primary)
(JC, University)
Phases of Learning
Teaching-learning act Teacher stimulate students to learn and
construct meaning in their learning
Phase 1: Inquiry/Information
Phase 2: Directed Orientation
Phase 3: Explication
Phase 4: Free Orientation
Phase 5: Integration
Activity 1
Level 2Informal Deduction: A network of relations begins to form
• You are given a pile of toothpicks all the same size.• First pick three toothpicks.• Can you form a triangle using all three toothpicks
placed end to end in the same plane?• Can a different triangle be formed? • What kinds of triangles are possible? • Now take four toothpicks and repeat the questions.
Then repeat with five toothpicks, and so on.
No. of
toothpicks 3 4 5 6 7
Is triangle possible?
No. of triangles
Kind of triangles
1. Using toothpicks
Y N Y Y Y
1 0 1 1 2
Equilateral Isosceles Equilateral Isosceles
Activity 2 : Classification of quadrilateralsMatch the following property cards to the following figures
4 sides equal
4 right angles
Opposite sides parallel
Diagonals congruent
opposite sides equal
opposite angles equal
Diagonals not congruent
4 sides equal
4 sides equal
4 right angles 4 right angles
Diagonals congruent Diagonals congruent
opposite sides equal
opposite sides equal opposite sides equal
opposite angles equal opposite angles equal
opposite angles equal opposite angles equal
Diagonals not congruent
opposite sides parallel opposite sides parallel
opposite sides parallelopposite sides parallel
SQ
UA
RE
RE
CT
AN
GL
E
RH
OM
BU
S
PA
RA
LL
EL
OG
RA
M
opposite sides equal
Interesting Problems
• What’s the largest rectangle that can be inscribed in an equilateral triangle ?
Hint: The first task is to maximize the dimensions of the inscribed rectangle….
• Stealth Technology makes use of geometrical properties of polygons
• Stealth Technology being used in airplanes, objective being
-To make an airplane invisible to radar waves
Egs of such airplanes are F-117A
(hexagonal shape)
Stealth Technology
F-117A NightHawk (Triangular shape)
Because of their geometrical shapes, they are able to reflect radar signals and thus able to serve its function as a warplane efficiently.
Folding Activities
1) A sheet of paper is rectangular. How can I use folding to make a perfect square without making any measurements?
Explain why your method produces a perfect square…
qp
S R
q
R
p q
S R
z
2)
Step 1: Obtain a square piece of paper as shown in the diagram on the right
Step 2: Fold the square in half, so that PS lies exactly on top of QR. Crease carefully along the middle vertex
Step 3: Fold along the line through R and Z; crease and then unfold. Then fold along the line through S and Z to make a third crease line.
What is special about the triangle RSZ?
Triangle RSZ is an isosceles triangle. Can you verify for yourself?……
(Side ZR = Side ZS)
p q
S R
z
Extension of folding activities….
• Use folding to find the special point Z’ on the centre-fold for which the triangle RSZ’ is an equilateral triangle
Explain why the triangle RSZ’ is an equilateral triangle
S R
Z’
Errors and Misconceptions
• No. In general, N-sided figures are joined by lines and not curves.
• Is there such thing as a 2 sided Figure?
Errors and Misconceptions
• There is no difference and there is no such thing as an inverted triangle.
• What is the difference between the two figures?
Errors and Misconceptions
• Yes, this is a four sided figure joined by lines at its end.
• Can this be considered a quadrilateral?
Errors and Misconceptions
• Lets consider a regular convex octagon.
1350 1350
1350
1350
13501350
1350
1350
450
450
450
450
450450
450
450
450 x 8 = 3600• Sum of exterior =
• Sum of exterior
= 1500 + 1500 +1200 + 1200
=5400
Errors and Misconceptions
1200300 300
2400
600
300 300
• It seems that this is only true for convex figures?
1200
1500
15001200
• If there are reflex angles,
MISCONCEPTION
• Sum of exterior
= 1500 + 1200 +1500 + (-60)0
=3600
Errors and Misconceptions
1200300 300
2400
600
300 300
• Thus it is also true for convex figures!• Hence it is true for all polygons!
1200
1500
1500
-600
• There must be consistency in measuring exterior angles
Errors and Misconceptions
• No. it can also have up to 2 parallel sides.YES
NO
• Is it true that the trapezium has only one parallel sides?
• So, is a parallelogram a trapezium?
• Is a trapezium a parallelogram?
Testing your concepts (True or False)
• A square is a rectangle but a rectangle is not a square.
• Squares, rectangles, rhombus, parallelogram and trapeziums are all quadrilaterals. True
True
Grouping Quadrilaterals
• Grouping four sided figures using a Venn diagram
• ε as the universal set for quadrilateralsε
Rhombus
square
Rectangle
Parallelogram
Trapezium
square Rectangle
Testing your concepts (True or False)
True
False• A rhombus is a rectangle.
• A square is a rhombus but a rhombus is not a square.
• A square is a rectangle but a rectangle is not a square.
• Squares, rectangles, rhombus, parallelogram and trapeziums are all quadrilaterals. True
True
Errors and Misconceptions
• Grouping four sided figures using a Venn diagram
• ε as the universal set for quadrilaterals
Rhombus square
ε
Parallelogram
Trapezium
square Rectangle
Rhombus
Rhombus square Rectangle
Testing your concepts (True or False)
• A parallelogram is a trapezium but a trapezium is not a parallelogram.
• Squares, rectangles, rhombus are parallelograms True
True
False
• A square is a rectangle but a rectangle is not a square.
• A rhombus is a rectangle.
• A rhombus is a parallelogram but a parallelogram is not a rhombus.
• A square is a rhombus but a rhombus is not a square.
• Squares, rectangles, rhombus, parallelogram and trapeziums are all quadrilaterals. True
True
True
True
• Squares, rectangles, rhombus are trapeziums True
Errors and Misconceptions
• Grouping four sided figures using a Venn diagram
• ε as the universal set for quadrilateralsε
Rhombus square Rectangle
Parallelogram
TrapeziumParallelogram Trapezium