1. Geometric properties of
Triangles Quadrilaterals Polygons

2. Scope of Presentation Presented by Yan Hee Cheong
Lawrence Yong Shao Ping
Damien Yeo Tat Sheng
Tan Ing Keat
Elvin Yeo Boon Heng

3. The Egyptian Pyramid

5. The US Pentagon

6. 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

7. 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.

8. The greatest mathematical. textbook of all time is the
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

9. Basic elements of Geometry
URL : Basic elements of geometry

10. Basic elements of Geometry
Points, Lines and Angles can be manipulated to form various types of geometrical shapes and sizes

11. Van Hiele Theory of Geometric Thought
The Model (5 distinct levels)
level 0 – Visualization
level 1 – Analysis
level 2 – Informal Deduction
level 3 – Deduction
level 4 – Rigor
2. Properties of the Model
Sequential
3. Phases of Learning
Each level separated by a learning phase

12. Level Description Identify an object by its appearance 0 Visualization
1 Analysis
2 Informal Deduction
3 Deduction
4 Rigor
Identify an object by its appearance
Phases of learning
(Lower Primary)
Identify properties of a class of figures
(Upper Primary)
Formulates and uses definitions
(Lower Secondary)
Ability to state proofs
(Upper Secondary)
Analyzes various deductive systems
(JC, University)

13. 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

14. Activity 1
Level 2
Informal 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.

15. Equilateral Isosceles Equilateral Isosceles
1. Using toothpicks
No. of toothpicks 3 4 5 6 7
Is triangle possible?
No. of triangles
Kind of triangles
Y N Y Y Y
Equilateral Isosceles Equilateral Isosceles

16. Activity Two : Exploring quadrilaterals
Quadrilaterals are four-sided figures

17. Activity 2 : Classification of quadrilaterals
Match the following property cards to the following figures
4 sides equal
Diagonals not congruent
4 right angles
Diagonals congruent
opposite sides equal
Opposite sides parallel
opposite angles equal

18. RECTANGLE SQUARE PARALLELOGRAM RHOMBUS 4 sides equal
opposite sides equal
opposite sides equal
RECTANGLE
4 right angles
4 right angles
SQUARE
opposite angles equal
opposite angles equal
Diagonals congruent
Diagonals congruent
opposite sides parallel
opposite sides parallel
4 sides equal
PARALLELOGRAM
opposite sides equal
opposite sides equal
opposite angles equal
opposite angles equal
RHOMBUS
Diagonals not congruent
opposite sides parallel
opposite sides parallel

19. 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….

20. Stealth Technology
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)

21. 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.

22. Folding Activities
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…

23. 2)
Step 1: Obtain a square piece of paper as shown in the diagram on the right q p S R q R
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. p q S R z

24. What is special about the triangle RSZ?q S R z
Triangle RSZ is an isosceles triangle. Can you verify for yourself?……
(Side ZR = Side ZS)

25. 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’

26. Other interesting problems…
Interactive Geometry

27. Errors and Misconceptions
Is there such thing as a 2 sided Figure?
No. In general, N-sided figures are joined by lines and not curves.

28. Errors and Misconceptions
What is the difference between the two figures?
There is no difference and there is no such thing as an inverted triangle.

29. Errors and Misconceptions
Can this be considered a quadrilateral?
Yes, this is a four sided figure joined by lines at its end.

30. Errors and Misconceptions
Is sum of exterior always equals to ?

31. Errors and Misconceptions
Lets consider a regular convex octagon.
Sum of exterior =
450 x 8 = 3600
450
1350

32. Errors and Misconceptions
1200
1500
If there are reflex angles,
Sum of exterior =
=5400
2400
600
300
1200
300
It seems that this is only true for convex figures?

33. Errors and Misconceptions
1200
1500
-600
There must be consistency in
measuring exterior angles
2400
600
300
Sum of exterior
= (-60)0
=3600
1200
300
Thus it is also true for convex figures!
Hence it is true for all polygons!

34. Errors and Misconceptions
No. it can also have up to 2 parallel sides.
Is it true that the trapezium has only one parallel sides?
So, is a parallelogram a trapezium?
YES
Is a trapezium a parallelogram? NO

35. Testing your concepts (True or False)
Squares, rectangles, rhombus, parallelogram
and trapeziums are all quadrilaterals.
True
A square is a rectangle but a rectangle is not a square.
True

36. Grouping Quadrilaterals
Grouping four sided figures using a Venn diagram
ε as the universal set for quadrilaterals ε
square
Rectangle
Rhombus
square
Rectangle
Parallelogram
Trapezium

37. 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
A square is a rhombus but a rhombus is not a square.
True
A rhombus is a rectangle.
False

38. Errors and Misconceptions
Grouping four sided figures using a Venn diagram
ε as the universal set for quadrilaterals ε
Parallelogram
Trapezium
square
Rectangle
Rhombus
Rhombus
square
Rhombus
square
Rectangle

39. Testing your concepts (True or 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.
Squares, rectangles, rhombus are parallelograms
True
A parallelogram is a trapezium but a trapezium
is not a parallelogram.
True
Squares, rectangles, rhombus are trapeziums
True

40. Errors and Misconceptions
Grouping four sided figures using a Venn diagram
ε as the universal set for quadrilaterals ε
Rhombus
square
Rectangle
Trapezium
Parallelogram
Parallelogram
Trapezium

41. Questions and Answers