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© Project Maths Development Team – Draft Module 1 4 Week Modular Course in Geometry & Trigonometry Strand 2

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© Project Maths Development Team – Draft Axioms Recall and use of: 1. Two point axiom 2. Ruler axiom 3. Protractor Axiom 4. Congruent triangles 5. Axiom of parallels Axioms Recall and use of: 1. Two point axiom 2. Ruler axiom 3. Protractor Axiom 4. Congruent triangles 5. Axiom of parallels Theorems Application of theorems, converses and corollaries to solve problems Ordinary Level: 10 Theorems 1 Converse 1Corollary Higher Level: 14 Theorems 5 Formal proofs 1 Converse 5 Corollaries Theorems Application of theorems, converses and corollaries to solve problems Ordinary Level: 10 Theorems 1 Converse 1Corollary Higher Level: 14 Theorems 5 Formal proofs 1 Converse 5 Corollaries Constructions Ordinary Level: 13 Constructions Higher Level: 15 Constructions Constructions Ordinary Level: 13 Constructions Higher Level: 15 Constructions 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 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 Junior Certificate Synthetic Geometry

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© Project Maths Development Team – Draft Ready Reckoner

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© Project Maths Development Team – Draft All available from: http://www.projectmaths.ie Resources

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© Project Maths Development Team – Draft Concepts

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© Project Maths Development Team – Draft “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”. Theorems: A Discovery Approach Van Hiele Model of Geometric Taught

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© Project Maths Development Team – Draft 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. fig (a) Definition 25

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© Project Maths Development Team – Draft Two lines are parallel if and only if for any transversal, corresponding angles are equal. Theorem 5 (Corresponding Angles) 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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© Project Maths Development Team – Draft Periscope Application of Theorem 5

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© Project Maths Development Team – Draft 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. Theorem 14 (Pythagoras)

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© Project Maths Development Team – Draft Theorem 14 (Pythagoras) 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? In a right-angled triangle the square of the hypotenuse is the sum of the squares of the other two sides.

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© Project Maths Development Team – Draft 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. Marking out a SiteChecking for Square Applications of Theorem 5 and its Converse

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© Project Maths Development Team – Draft Cable Stayed Bridges Cable Stay Bridges

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© Project Maths Development Team – Draft 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? 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? Practical Problem 1Practical Problem 2

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© Project Maths Development Team – Draft Can you see Pythagoras in the floor of this Italian restaurant? http://www.qedcat.com/archive/32.html 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. A Greek in an Italian RestaurantSome More Applications

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© Project Maths Development Team – Draft Towards Geometric Proofs Two Points Axiom. Axiom 1 Ruler axiom 2. Axiom 2 Protractor axiom 3 Axiom 3 Definition 1: Line segment Definition 2: Collinear Definition 3: Triangle Definitions 4 - 17 Definition 18: Complementary Angles Theorem 1: Vertically opposite Angles Congruency Axiom 4 Definition 19 - 20 Definition 21 Isosceles Definition 24: Alternate Definition 22 - 23 Axiom of parallels Axiom 5 Theorem 2: Isosceles triangles Theorem 3: Alternate Angles Theorem 4: Angle sum 180 o Definition 25: Corresponding Definition 26: Exterior Angle Theorem 5: Corresponding 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.

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© Project Maths Development Team – Draft Approaches: 1.Proof Interactive Proof. 2.Student CD. How can the Student CD and Student Activity Sheet help? The exterior angle of a triangle is equal to the sum of the interior remote angles. Theorem 6

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© Project Maths Development Team – Draft Other ShapesPythagorean Triples

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© Project Maths Development Team – Draft Ordinary Level: 13 Constructions. Higher Level: 15 Constructions Constructions

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© Project Maths Development Team – Draft Ready Reckoner

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© Project Maths Development Team – Draft 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? Line parallel to given line, through a given point. Construction 5

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Acute angles are < 90 0 Obtuse angles are > 90 0 Right angles are = 90 0 Supplementary angles total to 180 0. Complementary angles total to.

Acute angles are < 90 0 Obtuse angles are > 90 0 Right angles are = 90 0 Supplementary angles total to 180 0. Complementary angles total to.

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