File - Project Maths Made Easy

Report
4 Week Modular Course in Geometry & Trigonometry
Strand 2
Module 1
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Junior Certificate Synthetic Geometry
Axioms
Concepts
Set, plane, point, line, ray, angle, real number, length, degree, triangle,
right-angle, congruent triangles, similar triangles, parallel lines,
parallelogram, area, tangent to a circle, subset, segment, collinear
points, distance, midpoint of a segment, angle, ordinary angle, straight
angle, null angle, full angle, supplementary angles, vertically-opposite
angles, acute angle, obtuse angle, angle bisector, perpendicular lines,
perpendicular bisector of a segment, ratio, isosceles triangle,
equilateral triangle, scalene triangle, right-angled triangle, exterior
angles of a triangle, interior opposite angles, hypotenuse, alternate
angles, corresponding angles, polygon, quadrilateral, convex
quadrilateral, rectangle, square, rhombus, base and corresponding
apex and height of triangle or parallelogram, transversal line, circle,
radius, diameter, chord, arc, sector, circumference of a circle, disc, area
of a disc, circumcircle, point of contact of a tangent, vertex, vertices (of
angle, triangle, polygon), endpoints of segment, arms of an angle,
equal segments, equal angles, adjacent sides, angles, or vertices of
triangles or quadrilaterals, the side opposite an angle of a triangle,
opposite sides or angles of a quadrilateral, centre of a circle.
In addition for JCHL – concurrent lines
Syllabus pages 68 &69
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Recall and use of:
1. Two point axiom
2. Ruler axiom
3. Protractor Axiom
4. Congruent triangles
5. Axiom of parallels
Constructions
Ordinary Level:
13 Constructions
Higher Level:
15 Constructions
Theorems
Application of theorems, converses and
corollaries to solve problems
Ordinary Level:
10 Theorems
1 Converse
1 Corollary
Higher Level:
14 Theorems
5 Formal proofs
1 Converse
5 Corollaries
Ready Reckoner
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Resources
All available from:
http://www.projectmaths.ie
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Concepts
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Theorems: A Discovery Approach
“Theorems are full of potential for surprise and delight. Every theorem can be
taught by considering the unexpected matter which theorems claim to be true.
Rather than simply telling students what the theorem claims, it would be helpful if
we assumed we didn’t know it… it is the mathematics teacher’s responsibility to
recover the surprise embedded in the theorem and convey it to the pupils. The
method is simple: just imagine you do not know the fact. This is where the teacher
meets the students”.
Van Hiele
Model
of
Geometric
Taught
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Definition 25
Given two lines AB and CD, and a transversal AE of them, as in fig (a), the angles
 EAB and  ACD are called corresponding angles, with respect to the two lines
and the given transversal.
E
B
A
D
C
fig (a)
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Theorem 5 (Corresponding Angles)
Two lines are parallel if and only if for any transversal, corresponding angles are
equal.
Approaches:
1. Drawing
Students discover the theorem using ruler and protractor.
2. Geostrips
Let’s examine the use of Geostrips for theorems in the classroom
3. Student CD
How can the Student CD and Student Activity Sheet help?
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Application of Theorem 5
Periscope
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Theorem 14 (Pythagoras)
In a right-angled triangle the square of the hypotenuse is the sum of the squares of
the other two sides.
Monument to Pythagoras in his birthplace, Samos.
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Theorem 14 (Pythagoras)
In a right-angled triangle the square of the hypotenuse is the sum of the squares of
the other two sides.
Approaches:
1. Drawing
Construction on squared paper.
2. Cut - out
Two activity sheets.
3. Perigal’s Dissection
One of many illustrations of Pythagoras Theorem.
4. Student CD.
How can the Student CD and Student Activity Sheet help?
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Applications of Theorem 5 and its Converse
Marking out a Site
Checking for Square
On the left we have a sketch of say an "L" shaped building.
When builders are setting it out, after putting pegs in the three main
corners, they want to check for square before getting the rest of the
profiles fixed.
They work out the length of the red dotted line to check for a right
angle.
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Cable Stay Bridges
P ro b lem
Su p p o se a n e n gin e e r h a s to d e sign th e ca b le
su p p o rtin g th e b rid ge d e ck.
If th e co n cre te to w e r is 150m a b o ve th e
ro a d d e ck a n d th e su p p o rt p o in t fo r th e
Cable Stayed Bridges
m a in ca b le is 100 m a w a y fro m th e to w e r ,
h o w lo n g d o e s th e ca b le h a ve to b e ?
So lu tio n
c a b
2
2
2
a  100 m
b  150 m
c
a b
c
100  150
c
10000  22500
c
32500
2
2
2
c  180.28 m
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2
Practical Problem 1
Given that the second story window is 4.5 m above the
ground and the only sure footing for the ladder is at
least 2m from the house, can a window cleaner who
has a ladder of length 5m reach the window to clean it?
Practical Problem 2
The dimensions of your TV cabinet are 18" by 24".
You want to buy a TV with a 27" screen in the 16 : 9 format.
Check to see if it will fit in your cabinet before you go and
purchase it?
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A Greek in an Italian Restaurant
Can you see Pythagoras in the floor of this Italian restaurant?
http://www.qedcat.com/archive/32.html
Some More Applications
Triangulation helps to locate a cell phone making an emergency
call. It might use the Sine Rule to find the phone. The law of
cosines can also reduce to the Pythagorean theorem.
Pythagoras’ theorem is used in fractal geometry e.g. Movie
and Video game environments are drawn in 3-D using
triangles.
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Towards Geometric Proofs
Consider a geometry wall: the foundations of the wall consist of the early axioms and definitions.
Each conclusion builds upon previous knowledge.
The cement holding the wall together is the deductive logic that is used to prove the next theorem.
Definition 26:
Exterior Angle
Definition 25:
Corresponding
Theorem 2:
Isosceles triangles
Definition 21
Isosceles
Definitions 4 - 17
Two Points Axiom.
Axiom 1
Ruler axiom 2.
Axiom 2
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Theorem 3:
Alternate Angles
Definition 22 - 23
Definition 18:
Complementary
Angles
Theorem 5:
Corresponding
Definition 24:
Alternate
Theorem 1:
Vertically opposite
Angles
Protractor axiom 3
Axiom 3
Theorem 4:
Angle sum 180o
Definition 1:
Line segment
Axiom of parallels
Axiom 5
Congruency
Axiom 4
Definition 19 - 20
Definition 2:
Collinear
Definition 3:
Triangle
Theorem 6
The exterior angle of a triangle is equal to the sum of the interior remote angles.
Approaches:
1. Proof
Interactive Proof.
2. Student CD.
How can the Student CD and Student Activity Sheet help?
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Other Shapes
Pythagorean Triples
Ta ke a n y tw o w h o le n u m b e rs a n d fo rm :
1. Th e d iffe re n ce b e tw e e n th e ir sq u a re s.
2. Tw ice th e ir p ro d u ct.
3. Th e su m o f th e ir sq u a re s.
P ro o f :
Le t a  n  m , b  2n m a n d c  n  m , w h e re n  m
2
2
2
2
(n  m )  (2m n)  n  2n m  m  4n m
2
2 2
2
4
2
2
 n  2n m  m
4
2
 (n  m )
2
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2 2
2
4
4
2
2
Constructions
Ordinary Level: 13 Constructions.
Higher Level: 15 Constructions
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Ready Reckoner
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Construction 5
Line parallel to given line, through a given point.
Approaches:
1. Instructions
Use any method, animated or otherwise, to show the steps of the
construction.
2. Students using their mathematical instruments
Have instructions on board as they are working out their construction.
3. Student CD.
How can the Students CD and Student Activity Sheet help?
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