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1 WORK PROGRAM Chapter 3 Geometry Strand: Space and geometry Substrands and outcomes: Two-dimensional space Angles Properties of geometrical figures Properties of geometrical figures Properties of geometrical figures Deductive geometry Section Are you ready? (page 84) SGS3.2b Measures, constructs and classifies angles SGS4.2 Identifies and names angles formed by the intersection of straight lines, including those related to transversals on sets of parallel lines, and makes use of the relationships between them SGS4.3 Classifies, constructs and determines the properties of triangles and quadrilaterals SGS4.4 Identifies congruent and similar two-dimensional figures stating the relevant conditions SGS5.2.1 Develops and applies results related to the angle sum of interior and exterior angles for any convex polygon SGS5.3.1 Constructs arguments to prove geometrical results GC tips, Investigations, History of mathematics, Maths Quest challenge, 10 Quick Questions, Code puzzles, Career profiles SkillSHEETs, WorkSHEETs, Interactive games, Test yourself, Topic tests (CD-ROM) SkillSHEETs (page 84) 3.1: Naming angles 3.2: Classifying angles 3.3: Complementary and supplementary angles 3.4: More angle relations 3.5: Angles and parallel lines 3.6: Angles in a triangle Technology applications (CD-ROM) Learning outcomes SGS3.2b classifying angles as right, acute, obtuse, reflex, straight or a revolution SGS4.2 labelling and naming angles using A and XYZ notation identifying adjacent angles, vertically opposite angles, straight angles and angles of complete 2 revolution, embedded in a diagram using the words ‘complementary’ and ‘supplementary’ for angles adding to 90 and 180º respectively, and the terms ‘complement’ and ‘supplement’ establishing and using the equality of vertically opposite angles identifying and naming the alternate angle pairs, the corresponding angle pairs and the cointerior angle pairs for two lines cut by a transversal recognising the equal and supplementary angles formed when a pair of parallel lines are cut by a transversal using angle properties to identify parallel lines SGS4.3 justifying informally 3 Angle review (page 85) Ex 3A Angle review (page 88) Code puzzle (page 91) SkillSHEET 3.1: Naming angles (page 88) SkillSHEET 3.2: Classifying angles (page 88) SkillSHEET 3.3: Complementary and supplementary angles (page 88) SkillSHEET 3.4: More angle relations (page 88) SkillSHEET 3.5: Angles and parallel lines (page 89) SkillSHEET 3.6: Angles in a triangle (page 90) SkillSHEET 3.7: Angle sum of a quadrilateral (page 90) that the interior angle sum of a triangle is 180º, and that any exterior angle equals the sum of the two interior opposite angles Cabri geometry: Vertically SGS3.2b opposite angles (page 86) classifying angles as Cabri geometry: Parallel right, acute, obtuse, lines (page 87) reflex, straight or a Cabri geometry: revolution Corresponding angles SGS4.2 (page 87) labelling and naming Cabri geometry: Alternate angles using A and angles (page 87) XYZ notation Cabri geometry: Co using the common interior angles (page 87) conventions to indicate Mathcad: Angle review right angles and equal (page 88) angles on diagrams Mathcad: Angle sum identifying and naming (page 90) adjacent angles (two Cabri geometry: Angle angles with a common sum of a triangle vertex and a common (page 90) arm), vertically opposite angles, straight angles and angles of complete revolution, embedded in a diagram using the words ‘complementary’ and ‘supplementary’ for 4 angles adding to 90º and 180º respectively, and the terms ‘complement’ and ‘supplement’ establishing and using the equality of vertically opposite angles identifying and naming alternate angle pairs, corresponding angle pairs and co-interior angle pairs for two lines cut by a transversal recognising the equal and supplementary angles formed when a pair of parallel lines are cut by a transversal using angle properties to identify parallel lines using angle relationships to find unknown angles in diagrams SGS4.3 using the common conventions to mark equal intervals on 5 Other polygons (page 92) WE 1, 2a–b, 3 Ex 3B Other polygons (page 95) Investigation: Sum of angles in a polygon (page 92) Investigation: Angles in a regular polygon (page 94) Investigation: Regular polygons (page 96) SkillSHEET 3.8: Angle sum of a polygon (page 95) WorkSHEET 3.1 (page 96) Mathcad: Angles in polygons (page 95) Cabri geometry: Angle sum of a polygon (page 95) Cabri geometry: Exterior angles of a polygon (page 96) Cabri geometry: Regular diagrams justifying informally that the interior angle sum of a triangle is 180º, and that any exterior angle equals the sum of the two interior opposite angles using a parallel line construction to prove that the interior angle sum of a triangle is 180º proving, using a parallel line construction, that any exterior angle of a triangle is equal to the sum of the two interior opposite angles establishing that the angle sum of a quadrilateral is 360º SGS5.2.1 applying the result for the interior angle sum of a triangle to find, by dissection, the interior sum of polygons with 4, 5, 6, 7, 8, sides defining the exterior angle of a convex 6 polygons (page 96) Constructing triangles (page 97) WE 4, 5, 6 Ex 3C Constructing triangles (page 99) 10 Quick Questions 1 (page 101) Maths Quest challenge: Q1–2 (page 101) SkillSHEET 3.9: Constructing angles with a protractor (page 99) Game time 001 (page 100) Cabri geometry: Three sides (page 99) Cabri geometry: Two angles and a side (page 99) Cabri geometry: Two sides and an angle between (page 100) polygon establishing that the sum of the exterior angles of any convex polygon is 360º applying angle sum results to find unknown angles expressing in algebraic terms the interior angle sum of a polygon with n sides (Communicating) finding the size of the interior and exterior angles of regular polygons with 5, 6, 7, 8, … sides (Applying strategies) solving problems using angle sum of polygon results (Applying strategies) SGS4.3 recognising and classifying types of triangles on the basis of their properties (acute-angled, rightangled, obtuse-angled, scalene, isosceles and equilateral triangles) 7 Further sketching and constructing (page 102) WE 7, 8, 9, 10, 11, 12 Ex 3D Further sketching and constructing (page 107) Maths Quest challenge: Q1 (page 108) Investigation: Fractals (page 109) 10 Quick Questions 2 (page 111) WorkSHEET 3.2 (page 108) Cabri geometry: Two sides and an angle between (page 107) constructing various types of triangles using geometrical instruments, given different information sketching and labelling triangles and quadrilaterals from a given verbal description (Communicating) recognising that a given triangle may belong to more than one class (Reasoning) recognising that the longest side of a triangle is always opposite the largest angle (Applying strategies, Reasoning) SGS4.2 applying angle results to construct a pair of parallel lines using a ruler and a protractor or a ruler and a pair of compasses (Applying strategies) constructing a pair of perpendicular lines using a ruler and a pair 8 of compasses (Applying strategies) SGS4.3 constructing various types of triangles using geometrical instruments, given different information constructing various types of quadrilaterals bisecting an angle by applying geometrical properties (Applying strategies) bisecting an interval by applying geometrical properties (Applying strategies) drawing a perpendicular to a line from a point on the line by applying geometrical properties (Applying strategies) drawing a perpendicular to a line from a point off the line by applying geometrical properties (Applying strategies) using ruler and compasses to construct 9 Constructing polygons (page 112) WE 13, 14, 15, 16, 17 Ex 3E Constructing polygons (page 118) Game time 002 (page 119) WorkSHEET 3.3 (page 132) Cabri geometry: Star polygons (pages 116, 119) Cabri geometry: Circumcentre (page 118) Cabri geometry: Centroid (page 118) Cabri geometry: Incentre (page 118) angles of 60º and 120º by applying geometrical properties (Applying strategies) SGS4.4 using dynamic geometry software to investigate the properties of geometrical figures (Applying strategies, Reasoning) SGS4.3 constructing various types of triangles using geometrical instruments, given different information constructing various types of quadrilaterals bisecting an interval by applying geometrical properties (Applying strategies) drawing a perpendicular to a line from a point on the line by applying geometrical properties (Applying strategies) drawing a perpendicular to a line 10 Geometry in architecture, design and art (page 119) WE 18, 19, 20a–b Ex 3F Geometry in architecture, design and art (page 124) Summary (page 126) Chapter review (page 127) Investigation: The Golden Ratio (page 122) SkillSHEET 3.10: Using the quadratic formula (page 125) ‘Test yourself’ multiple choice questions (page 128) Cabri geometry: Golden rectangle (page 125) Cabri geometry: Tessellations (page 125) from a point off the line by applying geometrical properties (Applying strategies) SGS5.3.1 solving Euclidean geometry problems solving problems using deductive reasoning (Reasoning, Applying strategies) SGS4.4 recognising congruent figures in tessellations, art and design work (Reflecting) SGS 5.3.1 solving Euclidean geometry problems solving problems using deductive reasoning (Reasoning, Applying strategies) make, refine and test conjectures (Questioning, Communicating, Applying strategies, Reasoning) 11 Topic tests (2)