Topic 3: Geometry and Trigonometry
Geometry and trigonometry allow us to quantify the physical world, enhancing our spatial awareness in two and three dimensions. This branch provides us with the tools for analysis, measurement and transformation of quantities, movements and relationships.
Content-specific conceptual understandings:
• The properties of shapes are highly dependent on the dimension they occupy in space.
• Volume and surface area of shapes are determined by formulae, or general mathematical relationships or rules expressed using symbols or variables.
• The relationships between the length of the sides and the size of the angles in a triangle can be used to solve many problems involving position, distance, angles and area.
• Different representations of trigonometric expressions help to simplify calculations.
• Systems of equations often, but not always, lead to intersection points.
• In two dimensions, the Voronoi diagram allows us to navigate, path-find or establish an optimum position.
The sections of the unit will include:
3.1: The distance between two points in threedimensional space, and their midpoint. Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids. The size of an angle between two intersecting lines or between a line and a plane.
3.2: Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles. The sine rule,The cosine rule, Area of a triangle.
3.3: Applications of right and non-right angled trigonometry, including Pythagoras’ theorem. Angles of elevation and depression. Construction of labelled diagrams from written statements.
3.4: The circle: length of an arc; area of a sector.
3.5: Equations of perpendicular bisectors.
3.6: Voronoi diagrams: sites, vertices, edges, cells. Addition of a site to an existing Voronoi diagram. Nearest neighbour interpolation. Applications of the “toxic waste dump” problem.