Tin Ka Ping Secondary School F.5 Mathematics Module 2 Teaching Syllabus

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1 Tin Ka Ping Secondary School F. Mathematics Module Teaching Syllabus Chapter 11 Indefinite Integration (I) Time To recognise the concept of Concept of indefinite integration. Concept of Indefinite Indefinite Integration Integration 11. To learn how to use the Teachers should help students Basic Integration integration formulas of understand the basic Formulas for algebraic, exponential integration formulas and their Elementary functions to find integrals. proofs. Functions To understand the basic properties of indefinite integrals. 11. To learn how to use integration For the integrand in the form Integration by by substitution to find f ( g( x)) g ( x), teachers should Integration by Substitution Substitution indefinite integrals. explain why 4 f ( g( x)) g ( x) dx can be written as f ( g( x)) d[ g( x)] without introducing the substitution u = g(x) To learn how to use indefinite For daily-life applications, Solving Problems integration to solve teachers may also explain the Geometrical Applications of by Indefinite geometrical problems. actual meanings of the Indefinite Integration 6 Integration To learn how to apply integration constants. indefinite integration to physics and other real-life or mathematical contexts. 1

2 Chapter 1 Indefinite Integration (II) Time 1.1 Learn how to use basic Integration Formulas integration formulas to find the for Trigonometric indefinite integrals of Functions trigonometric functions. 1. Learn how to find integrals Ask students to prove new Integration of involving trigonometric formulas introduced in this Trigonometric functions by simple section. Functions by substitution. Help students revise product Substitution Learn how to find integrals to sum formulas. using product to sum formulas. Learn how to find integrals in the form sin m xcos n x. Learn how to find integrals in the form tan m xsec n x or cot m xcosec n x. 1. concept of Students should understand More about inverse trigonometric functions why we More about Integration by. Integration by Substitution and their principal values. Learn how to find integrals substitute x asin, atan and asec for integrals involving Substitution involving a x, a x a x, a x and x a. and x a respectively. 1.4 Learn how to find indefinite Teachers should consider Integration by Parts integrals using integration by going through all examples in Integration by Parts parts. this section as the examples show different techniques of using integration by parts.

3 Chapter 1 Definite Integration Time To recognise the concept of Teachers may skip the proof of Definite Integration definite integration. the Fundamental Theorem of Concept of Definite To understand the properties Calculus. Integration of definite integrals. Animation: To recognise the Concept of Definite Fundamental Theorem of Integration Calculus. 1. To evaluate definite integrals The main concept of finding Evaluating Definite of algebraic functions, definite integral by substitution Evaluating Definite Integrals Integrals by trigonometric functions, is to transform the integrand to by Substitution Substitution exponential functions and another form that the primitive logarithmic functions by function can be found with substitution. basic integration rules. 1. To evaluate definite integrals Techniques of using Evaluating Definite of algebraic functions, integration by parts in this Evaluating Definite Integrals Integrals Using trigonometric functions, section are similar to those in Using Integration by Parts Integration by Parts exponential functions and Chapter 1. logarithmic functions using integration by parts. 1.4 To understand the properties Proofs of the theorems in this More about Definite of the definite integrals of odd, section are not difficult. Definite Integrals of Odd and Integrals even and periodic functions. Teachers may explain the Even Functions To evaluate definite integrals proofs given in the appendix of odd, even and periodic on p.e.. Definite Integrals of Periodic functions using these Functions properties.

4 Chapter 14 Applications of Definite Integration Time To use definite integration to When students find the area Finding Plane Areas find the areas of plane figures. bounded by a certain curve, the Finding Plane Areas by by Definite graph of the curve sometimes Definite Integration Integration is not given. Teachers should remind students to sketch the curve first so that they know the position of the bounded region. 14. To use the disc method to Teachers may use the Finding Volumes of find the volume of a solid animation provided in the Disc Method Solids of Revolution revolved about one of the Teaching CD-ROM so that Animation: by the Disc Method coordinate axes. students can visualize the Finding Volumes of Solids of To use the disc method to concept of the Disc Method. Revolution Disc Method find the volume of a solid revolved about a straight line parallel to one of the coordinate axes. 14. To use the shell method to Teachers can explain the Shell Finding Volumes of find the volume of a solid Method with the website Shell Method Solids of Revolution revolved about one of the below. by the Shell Method coordinate axes. To use the shell method to hdemos/shellmethod/gallery/ga find the volume of a solid llery.html revolved about a straight line So, students can visualize the parallel to one of the concept of the Shell Method. coordinate axes. 4

5 Chapter 1 Matrices Time Section Teaching Objective Teaching Guide IT Teaching 1.1 Apart from introducing Matrices basic concepts and different kinds of matrices, Basic Concepts and Notation notation of matrices. teachers should mention the of Matrices Recognise the concept of equality of 1 special kinds of matrices. matrices such as zero matrices, square matrices, identity matrices, etc. 1. Teachers should emphasize Matrix operations of that matrix operations are Addition, Subtraction and Operations matrices such as different from real number Scalar Multiplication of addition, subtraction operations. Matrices and scalar Teachers may also compare multiplication. properties of operations of Transpose of Matrix matrices with that of real properties of numbers. Multiplication of Matrices transpose of In this section, some Animation: matrices. questions require students to Matrix Multiplication prove a matrix formula using properties of mathematical induction. multiplication of Teachers may help students matrices. review the related concepts if necessary.

6 Chapter 16 Determinants and Inverse of Square Matrices Time Section Teaching Objective Teaching Guide IT Teaching 16.1 Recognise the When evaluating determinants of Determinants concept of order, students must learn how to Determinants determinants of order choose the most efficient method, and order. Learn to expand determinants of order by Sarrus rule. Sarrus rule or cofactor expansion. Cofactor Expansion of a Determinant Recognise the concept of the minor and the cofactor of an element in a determinant. Learn to expand determinants of order by cofactor expansion. 16. Recognise the Many students mix up the Properties of properties of properties of matrices and Properties of Determinants Determinants determinants. determinants. Teachers should emphasize that their properties are completely different. In this section, teachers should not let students use Sarrus rule. Students should apply theorems when expanding a determinant. 6

7 Time Section Teaching Objective Teaching Guide IT Teaching 16. Teachers should make sure that Inverses of concept of singular students know how to find the Inverses of Matrices Square matrices and inverse of a square matrix. In Software Demonstration: Matrices non-singular Chapter 17, students need to solve Finding Inverses of Square matrices. systems of linear equations by Matrices method of inverse matrix. concept of the Many students may carelessly treat inverse of a matrix. properties of the inverse of a square 4 concept of cofactor matrix as common properties of all matrices. Teachers should remind matrices and adjoint students to check whether the given matrices. square matrix is non-singular whenever applying Theorem computation of the inverse of a matrix. properties of the inverse of a matrix. 7

8 Chapter 17 System of Linear Equations Time Recognise the concepts Teachers may give more examples of Basic Concepts of a system of m linear consistent or inconsistent systems of of Systems of equations in n unknowns. linear equations if needed. Linear Equations concepts of a system of homogeneous linear equations and a system of non-homogeneous linear equations. Determine whether a system of linear equations is consistent or inconsistent. 17. Recognise the concept Teachers may point out that Method of of a matrix equation AX = Theorem 17. does not imply Method of Inverse Inverse Matrix B and convert a system of AX = B has no solution if and only if Matrix linear equations to its A is singular. In fact, if A is matrix equation. singular, AX = B has either no solution or infinitely many solutions. technique of solving a system of linear equations of order or by the method of inverse matrix. Determine whether a system of linear equations of order or has a unique solution or not. 8

9 Time 17. Recognise the concept of Teachers should mention that Cramer s Rule Cramer s Rule. both Method of Inverse Matrix Cramer s Rule Solve a system of linear and Cramer s Rule only help us equations of order and by find a unique solution as the Cramer s Rule. required coefficient matrix must be a non-singular square matrix. To raise students attention, teachers may point out that there is a method for finding not only unique solution but also infinitely many solutions, and such method will be discussed in next section Recognise the concept of Students may easily treat the Gaussian converting a system of linear elementary row operations as Gaussian Elimination Elimination equations to its augmented certain properties of matrices or matrix. determinants. Teachers should Recognise the concept of the clarify that the main role of three types of transformations elementary row operations is to involved in solving a system of transform systems of equations. linear equations. Teachers should guide Recognise the concept of the students to transform the corresponding three types of augmented matrix into row elementary row operations echelon form. Meanwhile, involved in simplifying an teachers should emphasize that augmented matrix. the process involved is not Recognise the concept of unique. Row Echelon form and understand the method of back substitution. Learn to solve a system of linear equations by Gaussian elimination. Learn to use Gaussian elimination. 9

10 Time 17. Recognise the concept of a Students may mix up the System of system of homogeneous linear meanings of unique solution, System of Homogeneous equations. trivial solution, non-trivial Homogeneous Linear Linear Equations Recognise the concept and solutions and infinitely many Equations condition for trivial solution solutions. Teachers should give and non-trivial solutions of a more examples and help system of homogeneous linear students distinguish those equations. keywords. Learn to find the non-trivial solutions of a system of homogeneous linear equations. 10

11 Chapter 18 Introduction of Vectors Time Section Teaching Objective Teaching Guide IT Teaching 18.1 Apart from introducing scalars Basic concepts of scalars and vectors, teachers should Basic Concepts of Vectors Concepts of and vectors. stress the concept of equality of Vectors Recognise the vectors. notation of vectors. 1 equality of vectors. concepts of negative vector, zero vector and unit vector. 18. Understand basic Teachers should emphasize that Operations operations of vectors, vector operations are different Operations and Properties and Properties including addition, from real number operations. of Vectors of Vectors subtraction and Students must consider both Software Demonstration: scalar multiplication. magnitudes and directions of Finding the sum and vectors when performing vector difference of two vectors properties of the addition and scalar multiplication of vectors. operations. Teachers may also compare properties of operations of vectors with that of real numbers. Teachers should make sure that students understand the characteristics of parallel vectors and collinear points. 18. Teachers should clearly explain Vectors in concept of position the concept of position vectors Vectors in -dimensional -dimensional vectors. with respect to different Space Space representation of a reference points so that students should not constantly treat the vector on a point named O as the only rectangular reference point. coordinate plane. 11

12 Time Section Teaching Objective Teaching Guide IT Teaching 18.4 Students may fail to recognise Vectors in -dimensional vectors in -dimensional space. Vectors in -dimensional -dimensional rectangular Teachers should give more Space Space coordinate system. examples and explain them in representation of a details. vector on a -dimensional rectangular coordinate system. 18. Understand how to Teachers should help students Division of a Line Segment find the point of division using understand Theorem 18.7 for vectors in both D and D Division of a Line Segment vectors. spaces. 1

13 Chapter 19 Scalor products and Vector products with Applications Time 19.1 Scalar definition of scalar Product of product of two vectors. Two Vectors properties of scalar product. scalar product of vectors in rectangular coordinate system. application of vectors to plane geometry. 19. Vector definition of vector Product of product of two vectors. Two Vectors properties of vector Teachers may let students prove some Scalar Product of Two Vectors properties of scalar products in Theorem 19.1 by definition. Teachers should explain the geometric meaning of scalar product in details. Teachers should discuss scalar products of vectors in both D and D spaces. In this section, teachers should guide students to analyse geometric problems by using vectors only. Teachers should explain the geometric meaning of Vector Product of Two Vectors vector product in details. Animation: Teachers may help Vector Product of Two Vectors students review properties product of two vectors. of determinants if needed. vector Teachers may let product of vectors in students prove Theorems -dimensional coordinate by using system. determinants. scalar triple product in -dimensional coordinate system. 1

14 Time More about Scalar Products definition of projection of a vector onto another vector. Learn to find areas of Teachers should discuss projection of vectors in both D and D spaces. Teachers should explain the geometric meaning of and Vector plane figures using scalar triple product in Products vectors. Learn to find volumes of solids using vectors. details. More about Scalar Products and Vector Products Teaching Tool: Scalar Product of Two Vectors 14