How can transformations be represented trigonometrically? Horizontal and vertical dilations, translations, and/or reflections, and their effect on trigonometric functions and their graphs, including a brief treatment of all three reciprocal functions: Exploration, through the use of graphing technology and interactive geometry, of the effect of: vertical dilations on the equation and amplitude of trigonometric functions horizontal dilations on the equation and period of trigonometric functions vertical translations on the equation and range of trigonometric functions horizontal translations on the equation and range of trigonometric functions reflection in the on the equation and graph of trigonometric functions. The graphs and periodicity of the reciprocal trigonometric functions can also be considered using electronic technology. Subtopic 1.3: Trigonometric Identities Key Questions and Key Ideas Considerations for Developing Teaching and Learning Strategies What trigonometric identities for sine and cosine functions are used? Addition formulae Double angle formulae Students derive the formula for by the dot product (see Subtopic 13.9 of Stage 1 Mathematics). Students compare this derivation with the approach using the cosine rule (see Subtopic 14.7 of Stage 1 Mathematics). The other addition formulae are derived using the properties established in Subtopic 1.1. The identity could be considered at this point for later use in Topic 4: Calculus. Topic 2: Polynomials and Complex Numbers Subtopic 2.1: Complex Numbers Key Questions and Key Ideas Considerations for Developing Teaching and Learning Strategies
What is a parallelogram in three dimensions? Equality of vectors Coordinate systems and position vectors; components Recall that vectors are equal if they form opposite sides of a parallelogram. The notion of a parallelogram in three dimensions requires a discussion of when the opposite sides of a quadrilateral are coplanar so that it is not twisted; the simplest criterion is that its diagonals intersect. Applications (e.g. navigation and force) as encountered in Stage 1 Mathematics can readily be extended to three dimensions. Operations with vectors Addition of vectors, multiplication by scalars, length of vectors: both geometrically and algebraically Contexts and applications as mentioned above. Parallel vectors, collinearity, ratio of division of a line segment (internal and external) Can vectors be multiplied together? What is the meaning of the product? Scalar (dot) product and vector (cross) product: their properties and their interpretation in context Ratio of division is needed in the treatment of Bézier curves (see Topic 4: Calculus). These two operations on pairs of vectors provide important geometric information. The scalar and vector products should be treated in terms of coordinates and of length and angle. Conditions for perpendicularity and parallelism, and construction of perpendiculars. Cross-product calculation using the determinant Though determinants may not have been encountered previously, this is a suitable place for their introduction, since they arise naturally in the formulae for area and volume. (Note that the area of a parallelogram in two dimensions can be expressed as a determinant in terms of the coordinates of the defining vectors.) (Determinants are treated in Subtopic 3.4 of Stage 2 Mathematical Studies.) The cross-product of two vectors and in three dimensions is a vector, mutually perpendicular to and to whose length is the area of the parallelogram determined by and The rule determines its sense. The components and the vector itself may be expressed using determinants. (Applications of scalar and vector products include work and moments.) When and how is the scalar triple product used? Volume of a parallelepiped Test for coplanarity Scalar triple product as volume of a parallelepiped, again using determinant notation; test for coplanarity. Subtopic 3.2: Lines and Planes Key Questions and Key Ideas Considerations for Developing Teaching and Learning Strategies hand
Performance Standards for Stage 2 Specialist Mathematics - Mathematical Knowledge and Skills and Their Application Mathematical Modelling and Problem-solving Communication of Mathematical Information A Comprehensive knowledge of content and understanding of concepts and relationships. Appropriate selection and use of mathematical algorithms and techniques (implemented electronically where appropriate) to find efficient solutions to complex questions. Highly effective and accurate application of knowledge and skills to answer questions set in applied and theoretical contexts, especially proof. Development and effective application of mathematical models. Complete, concise, and accurate solutions to mathematical problems set in applied and theoretical contexts. Concise interpretation of the mathematical results in the context of the problem. In-depth understanding of the reasonableness and possible limitations of the interpreted results, and recognition of assumptions made. Development and testing of valid conjectures, with proof. Highly effective communication of mathematical ideas and reasoning to develop logical arguments, especially proof in applied and theoretical contexts. Proficient and accurate use of appropriate notation, representations, and terminology. B Some depth of knowledge of content and understanding of concepts and relationships. Use of mathematical algorithms and techniques (implemented electronically where appropriate) to find some correct solutions to complex questions. Accurate application of knowledge and skills to answer questions set in applied and theoretical contexts, including proof. Attempted development and appropriate application of mathematical models. Mostly accurate and complete solutions to mathematical problems set in applied and theoretical contexts. Complete interpretation of the mathematical results in the context of the problem. Some depth of understanding of the reasonableness and possible limitations of the interpreted results, and recognition of assumptions made. Development and testing of reasonable conjectures, with substantial attempt at proof. Effective communication of mathematical ideas and reasoning to develop mostly logical arguments, including proof in applied and theoretical contexts. Mostly accurate use of appropriate notation, representations, and terminology.