Active Maths 4 Book 1 Teacher’s Handbook - Welcome -...

Active Maths 4 Book 1:Teacher’s Handbook

2 A C T I V E M AT H S 4 , B O O K 1 : T E A C H E R ’ S H A N D B O O K

CHAPTER 1 REAL NUMBERSCHAPTER 1 REAL NUMBERS

Chapter Design and Learning OutcomesChapter Design and Learning OutcomesThis chapter has been designed to facilitate the teaching and learning of the following learning outcomes:

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3.1 3.1 Number SystemsNumber Systems

n Consolidate their understanding of factors, multiples, prime numbers in �.

n Express numbers in terms of their prime factors.

n Revisit the operations of addition, multiplication, subtraction and division in the following domains: � of natural numbers, � of integers, � of rational numbers, � of real numbers and represent these numbers on a number line.

n Appreciate the order of operations, including brackets.

n Recognize irrational numbers and appreciate that � ≠ �.

n Work with irrational numbers.

n Geometrically construct √__

2 and √__

3 .

n Prove that √__

2 is not rational.

3.3 3.3 ArithmeticArithmetic

n Check a result by considering whether it is of the right order of magnitude and by working the problem backwards; round off a result.

3.5 3.5 Synthesis and problem-solving skillsSynthesis and problem-solving skills

Suggested Progression Through ChapterSuggested Progression Through Chapter1.1 1.1 Factors, Multiples and Prime Factors Factors, Multiples and Prime Factors This section reviews the Natural number system and introduces the student to important properties

of the prime numbers including the Fundamental Theorem of Arithmetic. HCF and LCM are also reviewed. Activity 1.1 deals with the Fundamental Theorem of Arithmetic (FTA). Activity 1.2 shows the student the technique for finding LCM and HCF using the FTA. Proof by contradiction is introduced and the infinitude of the prime numbers is established by contradiction (Activity 1.3). Students are introduced to the factorial notation.

1.21.2 Integers and Rational NumbersIntegers and Rational Numbers Here the set of integers and the set of rational numbers are reviewed. Some basic axioms of

the integers and rational numbers are listed and the axioms are used to prove some important properties (Worked Examples 1.7 and 1.8).

1.31.3 Irrational Numbers Irrational Numbers This section introduces irrational numbers. A proof of the irrationality of √

__ 3 is given; the student is

asked to prove that √__

2 is not rational in the corresponding Exercise. Geometric constructions of √__

2 and √

__ 3 are given.

1.4 1.4 Rounding and Significant FiguresRounding and Significant Figures Review of Junior Certificate material

1.51.5 Orders of Magnitude and Scientific Notation Orders of Magnitude and Scientific Notation Review of Junior Certificate material

3A C T I V E M AT H S 4 , B O O K 1 : T E A C H E R ’ S H A N D B O O K

Exercise DesignExercise DesignExercise 1.1Easy: 1–2, 8–11Medium: 3–4, 12Hard: 5–7, 13–15

Exercise 1.2Easy: 1–2Medium: 3–10Hard: 11–14

Exercise 1.3Easy: 1–3Medium: 4Hard: 5–6

Exercise 1.4Easy: 1–10Medium:Hard:

Revision ExercisesEasy: 1, 5–11, 15Medium: 2–3, 12–13Hard: 4, 14, 16–18


CHAPTER 2 ALGEBRA ICHAPTER 2 ALGEBRA I


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4.1 4.1 ExpressionsExpressions

n Evaluate expressions given the value of the variables.

n Expand and simplify expressions.

n Add and subtract expressions of the form, (ax + by + c) ± … ± (dx + ey + f ) and (ax2 + bx + c) ± … ± (dx2 + ex + f ), where a, b, c, d, e, f ∈ �.

n Use the associative and distributive properties to simplify expressions of the form (bx + cy + d) + … + (fx + gy + h), (x ± y)(w ± z).

n Factorize expressions of order 2.

n Add and subtract expressions of the form, a ______ bx + c

± q ______ px + r , where a, b, c, p, q, r ∈ �.

n Perform the arithmetic operations of addition, subtraction, multiplication and division on polynomials and rational algebraic expressions paying attention to the use of brackets and surds.

n Apply the binomial theorem.


Suggested Progression Through ChapterSuggested Progression Through Chapter2.12.1 Expressions Expressions In this section students are introduced to the terminology associated with algebra. Addition and

multiplication of algebraic terms is covered. The binomial theorem is introduced through Pascal’s triangle. Activity 2.1 is a very good discovery-based Activity to help students understand Pascal’s triangle. Addition, subtraction and multiplication of algebraic expressions are reviewed.

2.22.2 Factorising Factorising Expressions of order 2 are covered and also the sum and difference of two cubes. Activity 2.2

covers the factorisation of the sum and difference of two cubes.

2.3 2.3 AlgebAlgebraic Fractraic Fractions I: Addition and Subtraction ions I: Addition and Subtraction Students learn how to add and subtract algebraic fractions. Activity 2.3 takes the student through a

step by step approach for summing algebraic fractions.

2.42.4 AlgAlgebraic Fractions II: Multiplication and Divisionebraic Fractions II: Multiplication and Division Students learn how to multiply and divide algebraic fractions. Activity 2.4 deals with complex

expressions such as 4 _ 5 – 3 _ 4

_____ 2 _ 3 – 1 _ 2

. Activity 2.5 introduces the student to long division.


Exercise DesignExercise DesignExercise 2.1Easy: 1–9Medium: 10–13Hard: 14–16

Exercise 2.2Easy: 1–13, 26–27, 30–38Medium: 14–25, 28–29, 39–40Hard: 41–52

Exercise 2.3Easy: 1–21Medium: 22–25, 28–30Hard: 26–27, 31–34


Exercise 2.5Easy: 1–6Medium: 7–14Hard:

Revision ExercisesEasy: 1 (a), (b), (c), 2–6Medium: 7–9Hard: 10–12


CHAPTER 3 ALGEBRA IICHAPTER 3 ALGEBRA II


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4.14.1 Expressions Expressions

n Rearrange formulae.

4.2 4.2 Solving EquationsSolving Equations

n Select and use suitable strategies (graphic, numeric, algebraic, mental) for finding solutions to equations of the form: f(x) = g(x) with f(x) = ax + b, g(x) = cx + d where a, b, c, d ∈ �.

n Select and use suitable strategies (graphic, numeric, algebraic, mental) for finding solutions to simultaneous linear equations with two unknowns and interpret the results.

n Select and use suitable strategies (graphic, numeric, algebraic, mental) for finding solutions to simultaneous linear equations with three unknowns.

n Select and use suitable strategies (graphic, numeric, algebraic, mental) for finding solutions to equations of the form: f(x) = k with f(x) = ax2 + bx + c (and not necessarily factorisable) where a, b, c ∈ � and interpret the results.

n Select and use suitable strategies (graphic, numeric, algebraic, mental) for finding solutions to one linear equation and one equation of order 2 with two unknowns and interpret the results.

n Form quadratic equations given whole number roots.

n Use the Factor Theorem for polynomials.

n Select and use suitable strategies (graphic, numeric, algebraic, mental) for finding solutions to cubic equations with at least one integer root and interpret the results.


Suggested Progression Through ChapterSuggested Progression Through Chapter3.1 3.1 Solving Linear EquationsSolving Linear Equations Review of Junior Certificate material through a multi representational approach. Activity 3.1

introduces the student to the graphical approach. Some of the questions in Exercise 3.1 require a trial and error approach, while the remaining questions require either a graphical or algebraic solution.

3.2 3.2 Solving Simultaneous Linear Equations in Two VariablesSolving Simultaneous Linear Equations in Two Variables Review of Junior Certificate material through a multi representational approach. Activity 3.2

introduces the student to the graphical approach. Some of the questions in Exercise 3.2 require a trial and error approach, while the remaining questions require either a graphical or algebraic solution.

3.33.3 Solving Simultaneous Equations in Three Variables Solving Simultaneous Equations in Three Variables In this section students learn the technique for solving simultaneous equations in three variables.

3.43.4 Solving Quadratic Equations Solving Quadratic Equations Again a multi representational approach is used to solve quadratic equations. In Activity 3.3

students are asked to draw the graph of a quadratic function and hence solve an associated equation. Activity 3.4 teaches the student how to solve a quadratic by completing the square and Activity 3.5 uses the quadratic formula. Exercise 3.4 consists of questions that require the student to find the solutions by either algebraic, graphical or trial and error approaches.


3.5 3.5 Simultaneous Equations: One Linear and One Non-LinearSimultaneous Equations: One Linear and One Non-Linear The two approaches used here are graphical and algebraic. Activity 3.6 introduces the student to

the graphical approach.

3.63.6 The Factor Theorem The Factor Theorem At the outset a polynomial function is defined. Students are introduced to the factor theorem

through Worked Examples. Activity 3.7 and 3.8 show the student how to use graphs to find the roots of polynomial functions.

3.73.7 Manipulation of Formulae Manipulation of Formulae Review of Junior Certificate material.

3.83.8 Unknown Coefficients Unknown Coefficients In this section students are given either quadratic or cubic functions with unknown coefficients and

some other information, such as the roots of the equation. The task is to find the unknown coefficients.

3.93.9 Problem-Solving Using Algebra Problem-Solving Using Algebra..

Exercise DesignExercise DesignExercise 3.1Easy: 1–19Medium: 20–24Hard:



Exercise 3.4Easy: 1–8, 16–17Medium: 9–12Hard: 13–15, 18

Exercise 3.5Easy: 1–9Medium: 10–18Hard: 19

Exercise 3.6Easy: 1–8, 11Medium: 9–10Hard: 12–23




Revision ExercisesEasy: 1–3, 5Medium: 4Hard: 6–13


CHAPTER 4 ALGEBRA IIICHAPTER 4 ALGEBRA III


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4.3 4.3 InequalitiesInequalities

n Select and use suitable strategies (graphic, numeric, algebraic, mental) for finding solutions to inequalities of the form: g(x) ≤ k, g(x) ≥ k, g(x) < k, g(x) > k, where g(x) = ax + b and a b, k ∈ �.

n Select and use suitable strategies (graphic, numeric, algebraic, mental) for finding solutions to inequalities of the form: g(x) ≤ k, g(x) ≥ k, g(x) < k, g(x) > k, where g(x) = ax2 + bx + c or

g(x) = ax + b ______ cx + d

and a, b, c, d, k ∈ �, x ∈ �.

n Use notation |x|.

n Select and use suitable strategies (graphic, numeric, algebraic, mental) for finding solutions to inequalities of the form: |x – a| < b, |x – a| > b and combinations of these, where a, b ∈ �, x ∈ �.

Suggested Progression Through ChapterSuggested Progression Through Chapter4.1 4.1 Surd EquationsSurd Equations The technique for solving equations involving surds is introduced.

4.24.2 Linear Inequalities Linear Inequalities This section begins with a review of the number systems, �, � and � and their representation

on the numberline. Simple linear inequalities are then explained through Worked Example 4.4 followed by an explanation of compound linear inequalities through Worked Examples 4.5 and 4.6.

4.34.3 Quadratic and Rational Inequalities Quadratic and Rational Inequalities Quadratic and rational inequalities are solved with the aid of a graph. Activity 4.1 provides a

step by step approach to solving quadratic inequalities. The Exercise set also contains a practical application of inequalities.

4.44.4 Absolute Value (Modulus) Absolute Value (Modulus) The absolute value of a real number is defined. Equations involving absolute values are solved

using algebraic and graphical techniques. Activity 4.2 provides a step by step approach to solving modulus inequalities. The Exercise set also contains a practical application of inequalities.

4.54.5 Discriminants Discriminants The connection between the discriminant and the nature of the roots of a quadratic is introduced

in Activity 4.3. Worked Examples 4.15 and 4.16 deal with applications of this knowledge.

4.64.6 Inequalities: Proofs Inequalities: Proofs Formal proofs of abstract inequalities.








Revision ExercisesEasy: 1–3Medium: 4–6Hard: 7–12


CHAPTER 5 ARITHMETICCHAPTER 5 ARITHMETIC


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3.33.3 Arithmetic Arithmetic

n Make and justify estimates and approximations of calculations; calculate percentage error and tolerance.

n Accumulate error (by addition or subtraction only).

n Solve problems involving: costing: materials, labour and wastage.

n Solve problems involving: income tax and net pay (including other deductions).


Suggested Progression Through ChapterSuggested Progression Through Chapter5.15.1 Approximation, Percentage Error and Tolerance Approximation, Percentage Error and Tolerance The ideas of percentage error and tolerance are explained. Practical Worked Examples and many

practical questions in the Exercise set reinforce understanding.

5.25.2 Costing: Materials, Labour and Wastage Costing: Materials, Labour and Wastage Direct, indirect, variable and fixed costs are explained. Worked Examples 5.4 and 5.5 calculate the

cost of particular orders for a company using a tabular approach.

5.35.3 Income Tax Income Tax A comprehensive explanation of all the terminology used in relation to income tax is given at the

outset. Worked Examples 5.8 and 5.9 cover complete income tax calculations for two individuals.

5.45.4 VAT: Value-Added Tax VAT: Value-Added Tax An explanation of VAT and VAT rates is given at the beginning of the section. Worked Example

5.10 explains how to calculate VAT given the cost and the rate. Worked Example 5.11 explains how to calculate VAT given the cost inclusive of VAT and the rate. Worked Example 5.12 explains how to calculate the rate of VAT given the sale price and the amount of VAT.





Revision ExercisesEasy: 1–4Medium: 5–9Hard: 10


CHAPTER 6 LENGTH, AREA AND VOLUMECHAPTER 6 LENGTH, AREA AND VOLUME


Strand 36.16.1 Length, Area and Volume Length, Area and Volume

n Select and use suitable strategies to find the length of the perimeter and the area of the following plane figures: parallelogram, trapezium, and figures made from combinations of these.

n Select and use suitable strategies to find the surface area and volume of the following solid figures: cylinder, right cone, right prism and sphere.

n Use the trapezoidal rule to approximate area.

n Solve problems involving the length of the perimeter and the area of plane figures: disc, triangle, rectangle, square, parallelogram, trapezium, sectors of discs, and figures made from combinations of these.

n Investigate the nets of prisms (polygonal bases), cylinders and cones.

n Solve problems involving surface area and volume of the following solid figures: rectangular block, cylinder, right cone, triangular-based prism (right angle, isosceles and equilateral), sphere, hemisphere, and solids made from combinations of these.


Suggested Progression Through ChapterSuggested Progression Through Chapter6.16.1 Two-Dimensional (2D) Shapes Two-Dimensional (2D) Shapes The area and perimeter of the rectangle, the triangle, the parallelogram, the trapezium and the

circle are covered. The Exercise set contains many real-life questions.

6.26.2 Rectangular Solids and Prisms Rectangular Solids and Prisms This section covers the volume and surface area of cuboids and more generally prisms. Nets are

introduced and used to find the surface area of the prism.

6.36.3 Cylinders, Cones, Spheres and Hemispheres Cylinders, Cones, Spheres and Hemispheres The volume and surface area of the cylinder, sphere, hemisphere and cone are treated in this

section. The Exercise set contains many practical applications.

6.46.4 Trapezoidal Rule Trapezoidal Rule The trapezoidal rule for approximating the area under a curve is introduced. Worked Example 6.8

uses the trapezoidal rule to find the area under a curve. Worked Example 6.9 uses the trapezoidal rule to find the length of an offset.







CHAPTER 7 INDICES AND LOGARITHMSCHAPTER 7 INDICES AND LOGARITHMS


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3.13.1 Indices Indices

n Solve problems using the rules for indices (where a, b ∈ �; p, q ∈ �; ap, aq ∈ �; a, b ≠ 0):

n ap aq = ap + q

n ap __ aq = ap–q

n a0 = 1

n (ap)q = apq

n a 1 __ q =

q

√__

a , q ∈ �, q ≠ 0, a > 0

n a p __ q =

q

√__

ap , q ∈ �, q ≠ 0, a > 0

n a–p = 1 __ ap

n (ab)p = ap bp

n ( a __ b ) p = a

p __

bp

n Solve problems using the rules of logarithms

n loga (xy) = loga x + loga y

n loga ( x __ y ) = loga x – loga y

n loga xq = qloga x

n loga a = 1 and loga 1 = 0

n logb x = loga x _____ loga b


Suggested Progression Through ChapterSuggested Progression Through Chapter7.17.1 Indices (Exponents) Indices (Exponents) The exponential function is introduced using Activities 7.1 and 7.2. The natural exponential function is

introduced as an infinite series in Activity 7.3. The first six questions in Exercise 7.1 cover this material.

7.27.2 Laws of Indices (Exponents) Laws of Indices (Exponents) The laws of indices are derived in Activity 7.4. Questions 6 to 19 test understanding of the laws of indices.

7.37.3 Equations with x as an index Equations with x as an index In this section the student learns how to solve equations of the type ax = b, where b can be written

as a power of a.


7.47.4 Surds Surds Surds are defined and the laws of surds are derived in Activity 7.5.

7.57.5 Logarithms Logarithms Logarithms are defined. Activity 7.6 reinforces the definition. The laws of logarithms are derived in

Activities 7.7 to 7.11.

7.67.6 Using Logarithms to Solve Practical Problems Using Logarithms to Solve Practical Problems This section explores some practical problems that can only be solved with knowledge of indices

or logarithms.





Exercise 7.5Easy: Medium: 1–6Hard: 7–8



CHAPTER 8 FUNCTIONSCHAPTER 8 FUNCTIONS


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5.1 5.1 FunctionsFunctions

n Recognize that a function assigns a unique output to a given input.

n Graph functions of the form:

• ax where a ∈ �, x ∈ �,

• ax + b where a, b ∈ �, x ∈ �,

• ax2 + bx + c where a, b, c ∈ �, x ∈ �.

n Interpret equations of the form: f(x) = g(x) as a comparison of the above functions.

n Use graphical methods to find approximate solutions to f(x) = 0, f(x) = k, f(x) = g(x) where f(x) and g(x) are of the above form.

n Form composite functions.


• ax3 + bx2 + cx + d where a, b, c, d ∈ �, x ∈ �

• abx where a ∈ �, b, x ∈ �.

n Interpret equations of the form f(x) = g(x) as a comparison of the above functions.

n Use graphical methods to find approximate solutions to f(x) = 0, f(x) = k, f(x) = g(x), where f(x) and g(x) are of the above form, or where graphs of f(x) and g(x) are provided.

n Investigate the concept of the limit of a function.

n Recognize surjective, injective and bijective functions.

n Find the inverse of a bijective function.

n Given a graph of a function, sketch the graph of its inverse.

n Express quadratic functions in complete square form.

n Use the complete square form of a quadratic function to • find the roots and turning points • sketch the function.


• ax2 + bx + c where a, b, c ∈ �, x ∈ �

• abx where a, b ∈ �

• logarithmic

• exponential

• trigonometric.

n Interpret equations of the form f(x) = g(x) as a comparison of the above functions.


Suggested Progression Through ChapterSuggested Progression Through Chapter8.18.1 Introduction Introduction A function is defined and some important terminology is introduced. Examples of functions are given.

8.28.2 Functions as Mappings from One Set to Another Functions as Mappings from One Set to Another Closed and open intervals are explained and the notation used to write functions is introduced.


8.38.3 Composite Functions Composite Functions Composition of functions is explored in this section. Notation is introduced and three Worked

Examples on composition of functions are given.

8.4 8.4 Linear and Quadratic FunctionsLinear and Quadratic Functions Linear functions and quadratic functions are defined. Worked Example 8.6 shows how to graph a

linear and quadratic function.

8.58.5 Expr Expressiessing Quadratic Functions in Completed Square Form ng Quadratic Functions in Completed Square Form ((Vertex FormVertex Form)) Activities 8.1–8.3 guide the student to discover how to write a function in completed square form.

Worked Example 8.9 shows how to find the turning point of a quadratic function in completed square form.

8.68.6 Cubic Functions Cubic Functions A cubic function is defined. In Worked Examples 8.10 and 8.11 some cubic functions are graphed

and questions are answered on the graph.

8.7 8.7 Exponential FunctionsExponential Functions Exponential functions are defined and graphed. There are five Worked Examples in this section

one being a practical application of exponential functions. The Exercise set contains 17 questions, four of which are practical applications.

8.88.8 Graphing Logarithmic Functions Graphing Logarithmic Functions In this section the student learns how to graph logarithmic functions. Logarithmic functions to

different bases are used in the Exercise set.

8.9 8.9 Transformations of Linear FunctionsTransformations of Linear Functions The linear function is expressed in the form, y = mx + c. The student then learns how to rotate

the graph about a fixed point and how to find the image of the line by a translation.

8.108.10 Transformations of Quadratic Functions Transformations of Quadratic Functions Various transformations of the quadratic function are dealt with in this section including vertical

and horizontal translations.

8.11 8.11 Transformations of Cubic FunctionsTransformations of Cubic Functions This is similar to 8.10.

8.12 8.12 Transformations of Exponential FunctionsTransformations of Exponential Functions Transformations of the function, f(x) = ax under transformations such as, f(x + h) and f(x) + h are

covered.

8.13 8.13 Transformations of Logarithmic FunctionsTransformations of Logarithmic Functions Transformations of the function, f(x) = loga x under transformations such as, af(x), f(x + h) and

f(x) + h are covered.

8.14 8.14 Injective, Surjective and Bijective FunctionsInjective, Surjective and Bijective Functions Injective, surjective and bijective functions are defined. Tests for injectivity and surjectivity are

introduced.

8.158.15 Inverse Functions Inverse Functions The inverse of a function is defined and the inverse function of given bijective functions is found.

8.16 8.16 Graphs of Functions and their InversesGraphs of Functions and their Inverses The student learns how to graph the inverse of a given function.



Exercise 8.2Easy: 1–8Medium: 9–12, 14Hard: 13




Exercise 8.6Easy: 1Medium: 2–3Hard:

Exercise 8.7Easy: 1–2, 6Medium: 3–5, 7–11Hard: 12






Exercise 8.13Easy: Medium: 1–10Hard:



CHAPTER 9 NUMBER PAT TERNS, CHAPTER 9 NUMBER PAT TERNS, SEQUENCES AND SERIESSEQUENCES AND SERIES


Strand 33.13.1 Number Systems Number Systems

n Appreciate that processes can generate sequences of numbers or objects.

n Investigate patterns among these sequences.

n Use patterns to continue the sequence.

n Generate rules/formulae from those patterns.

n Generalize and explain patterns and relationships in algebraic form.

n Recognize whether a sequence is arithmetic, geometric or neither.

n Find the sum to n terms of an arithmetic series.

n Verify and justify formulae from number patterns.

n Investigate geometric sequences and series.

n Apply the rules for sums, products, quotients of limits.

n Find by inspection the limits of sequences such as: lim n → ∞

n ______ n + 1 ; lim n → ∞

r n,|r|<1.

n Solve problems involving finite and infinite geometric series including applications such as recurring decimals.


Suggested Progression Through ChapterSuggested Progression Through Chapter9.19.1 Patterns Patterns At the outset a definition of both a pattern and a sequence is given. The idea of linear patterns

is introduced and the general term of a linear pattern (arithmetic sequence) is derived through Activity 9.1.

9.29.2 Arithmetic Series Arithmetic Series The formula for the sum to n terms of an arithmetic sequence is derived in Activity 9.2.

The Exercise set contains many real-life questions on arithmetic series.

9.39.3 Non-Linear Sequences Non-Linear Sequences Here the student is introduced to quadratic, exponential and cubic sequences through Activities 9.3

and 9.4.

9.49.4 Geometric Sequences and Series Geometric Sequences and Series The student derives the sum to n terms of a geometric sequence in Activity 9.5. Exercise 9.4

contains many questions on the applications of geometric sequences and series to real-life situations. The properties of limits are listed and in Exercise 9.5 these properties are utilised to find the limits of various sequences.

9.59.5 Infinite Series Infinite Series In this section the sum to infinity of a geometric sequence is considered.


Exercise DesignExercise DesignExercise 9.1Easy: 1–2, 4–9, 14–15Medium: 3, 10–13, 16–26Hard: 27–32


Exercise 9.3Easy: 1–7Medium: 8–12 Hard: 13

Exercise 9.4Easy: 1–5 Medium: 6–15Hard: 16–20

Exercise 9.5Easy: 1Medium: 2–4Hard: 5–6




CHAPTER 10 FINANCIAL MATHEMATICSCHAPTER 10 FINANCIAL MATHEMATICS


Strand 3


n Solve problems involving finite and infinite geometric series including applications such as financial applications, e.g. deriving the formula for a mortgage repayment.

3.3 3.3 ArithmeticArithmetic

n Solve problems involving finding depreciation (reducing balance method), compound interest.

n Use present value when solving problems involving loan repayments and investments.


Suggested Progression Through ChapterSuggested Progression Through Chapter10.1 10.1 Present ValuePresent Value The formula for present value is derived in Activity 10.1. In Worked Examples 10.1, 10.2 and

10.3, three applications of present value are treated.

10.2 10.2 Compound Interest: Loans and InvestmentsCompound Interest: Loans and Investments APR and AER are dealt with in this section. Practical examples on loans and investments are

covered in the Worked Examples.

10.310.3 Depreciation (Reducing-Balance Method) Depreciation (Reducing-Balance Method) In Activity 10.2 the depreciation formula is derived. Worked Examples 10.8–10.11 cover:

the future value of an asset, a schedule of depreciation, the present value of an asset and a calculation of the number of years taken for an asset to depreciate.

10.4 10.4 Applications and Problems Involving Geometric SeriesApplications and Problems Involving Geometric Series This section deals with amortised loans and bonds. The amortisation formula is derived

in Activity 10.3.

Exercise DesignExercise DesignExercise 10.1Easy: 1–4Medium: 5–8Hard: 9






CHAPTER 11 PROOF BY INDUCTIONCHAPTER 11 PROOF BY INDUCTION


Strand 3


n Prove by induction, simple identities such as the sum of the first n natural numbers and the sum of finite geometric series.

n Prove by induction, simple inequalities such as n! > 2n, 2n > n2 (n ≥ 4) (1+ x)n ≥ 1 + nx (x > –1).


Suggested Progression Through ChapterSuggested Progression Through Chapter11.111.1 Introduction Introduction This section is a brief introduction to the meaning and importance of proof in mathematics.

11.211.2 Summations Summations Students are introduced to the sigma notation. Activity 11.1 explores some of the properties

of sums.

11.311.3 Proof by Induction Proof by Induction This section introduces the ideas behind proof by induction.

11.4 11.4 Proofs Involving SeriesProofs Involving Series Here the student learns to use induction to prove formulae for the sum to n terms of a given

series. Activities 11.2 and 11.3 ask the students to conjecture formulae for given series and then to prove the formulae using induction.

11.5 11.5 Divisibility ProofsDivisibility Proofs Here the student learns to prove that certain expressions, such as 7n – 1 are always divisible by a

given rational, in this case 6, for all values of n ∈ �.

11.6 11.6 InequalitiesInequalities In this section the student learns about the properties of inequalities and then uses induction to

prove inequalities such as 2n ≥ 1 + n, for all n ∈ �.

Exercise DesignExercise DesignExercise 11.1Easy: 1–3Medium: 4Hard: 5–6




Revision ExercisesEasy: 1–2, 5–6Medium: 3–4Hard: 7–10


CHAPTER 12 COMPLEX NUMBERSCHAPTER 12 COMPLEX NUMBERS


Strand 3

3.1 3.1 Number Systems Number Systems

n Investigate the operations of addition, multiplication, subtraction and division with complex numbers in rectangular form a + ib.

n Illustrate complex numbers on an Argand diagram.

n Interpret the modulus as distance from the origin on an Argand diagram and calculate the complex conjugate.

Strand 4

4.4 4.4 Complex Numbers Complex Numbers

n Use the Conjugate Root Theorem to find the roots of polynomials.

n Work with complex numbers in rectangular and polar form to solve quadratic and other equations including those in the form zn = a, where n ∈ � and z = r (cos q + isin q ).

n Use De Moivre’s Theorem.

n Prove De Moivre’s Theorem by induction for n ∈ �.

n Use applications such as nth roots of unity, n ∈ � and identities such as cos 3q = 4 cos 3q – 3 cos q.

3.5 3.5 Synthesis and problem-solving skills Synthesis and problem-solving skills


Suggested Progression Through ChapterSuggested Progression Through Chapter12.1 12.1 Complex NumbersComplex Numbers This section explains the need for complex numbers and the complex number i is defined.

12.2 12.2 Complex Numbers and the Argand DiagramComplex Numbers and the Argand Diagram The Argand diagram is introduced to allow the plotting of complex numbers. Activity 12.1

presents the idea of the modulus of a complex number.

12.3 12.3 AdditiAdditionon and Subtraction of Complex Numbers; Multiplication by a Real Number and Subtraction of Complex Numbers; Multiplication by a Real Number The rules for adding and subtracting complex numbers are introduced in Activity 12.2. Activity 12.3

explains how complex numbers can be used to do transformations. In particular addition or subtraction of complex numbers is the equivalent of a translation on the complex plane.

12.4 12.4 Multiplication of Complex Numbers Multiplication of Complex Numbers Multiplication of complex numbers is defined and Activity 12.4 explains how multiplication by a

complex number is the equivalent of a rotation followed by dilation on the complex plane.

12.5 12.5 DiviDivision of Comsion of Complex Numbersplex Numbers Division of complex numbers is defined.


12.6 12.6 QuadraticQuadratic Equations with Complex RootsEquations with Complex Roots Quadratic equations with complex roots are solved using the quadratic formula.

12.7 12.7 Polynomials with Complex RootsPolynomials with Complex Roots A polynomial is defined and an important consequence of the Fundamental Theorem of Algebra

is introduced viz. a polynomial of degree n will have exactly n roots. This fact coupled with the Conjugate Root Theorem is used to solve polynomial equations.

12.812.8 Polar Form of a Complex Number Polar Form of a Complex Number The argument of a complex number is defined and the student learns how to write a complex

number in polar form. Activity 12.5 derives the technique for multiplying and dividing complex numbers in polar form.

12.9 12.9 De MDe Moivoivre’s Theorem and Applicationsre’s Theorem and Applications The theorem of De Moivre is explained and a proof by induction is given for n ∈ �. Identities

such as cos 3q = 4 cos 3q – 3 cos q are covered in Exercise 12.9.

12.10 12.10 De Moivre’s Theorem for n De Moivre’s Theorem for n ∈ �� General polar form is explained through Activity 12.6 and the extension of De Moivre to n ∈ �

is treated.






Exercise 12.6Easy: Medium: 1–10Hard: 11

Exercise 12.7Easy: Medium: 1–9Hard: 10–16


Exercise 12.9Easy: 1–5Medium: 6Hard: 7–11



CHAPTER 13 DIFFERENTIAL CALCULUS ICHAPTER 13 DIFFERENTIAL CALCULUS I


Strand 3

n Apply the rules for sums, products, quotients of limits.

Strand 5

5.1 5.1 FunctionsFunctions

n Informally explore limits and continuity of functions.

5.2 5.2 CalculusCalculus

n Differentiate linear and quadratic functions from first principles.

n Differentiate the following functions:

• polynomial

• exponential

• trigonometric

• rational powers

• inverse functions

• logarithms.

n Find the derivatives of sums, differences, products, quotients and compositions of functions of the above form.


Suggested Progression Through ChapterSuggested Progression Through Chapter13.1 13.1 CalculusCalculus A historical introduction to the calculus.

13.213.2 Limits Limits and and Continuity Continuity The notion of a limit is introduced. Continuity at a point is defined. There are four Worked

Examples to increase the student’s understanding of the concepts.

13.3 13.3 Differentiation from First PrinciplesDifferentiation from First Principles In this section a formal definition of the derivative is given and students learn to differentiate from

first principles using the definition.

13.413.4 DifferentiatingDifferentiating Polynomial Functions and Functions with Rational Powers Polynomial Functions and Functions with Rational Powers Activity 13.1 derives the rule for differentiating a constant function. The properties of limits are

stated and Activity 13.2 derives the sum rule for derivatives. The derivative of a polynomial follows.

13.5 13.5 The Chain RuleThe Chain Rule The chain rule is derived and the Worked Examples contain some applications of the chain rule.


13.6 13.6 The Product Rule and Quotient RuleThe Product Rule and Quotient Rule Activities 13.3 and 13.4 derive the product and quotient rules. Worked Example 13.17 contains

an application of the product rule and an application of the quotient rule.

13.7 13.7 TrTrigigonometric Functionsonometric Functions The derivatives of sin x, cos x and tan x are stated and the Worked Examples contain applications

involving the chain rule and the product rule.

13.8 13.8 Differentiation of Inverse TrigDifferentiation of Inverse Trigonoonometric Functionsmetric Functions The derivatives of the inverse trigonometric functions are dealt with. Worked Example 13.22

derives the derivative of the function, f(x) = sin–1 x __ a .

13.9 13.9 The Exponential Function and the Natural Logarithm FunctionThe Exponential Function and the Natural Logarithm Function The derivative of the exponential function is explained and the derivative of f(x) = ln x is derived.







Exercise 13.7Easy: 1 Medium: 2–9Hard: 10–12




CHAPTER 14 DIFFERENTIAL CALCULUS IICHAPTER 14 DIFFERENTIAL CALCULUS II


Strand 5

5.2 5.2 Calculus Calculus

n Find first and second derivatives of linear, quadratic and cubic functions by rule.

n Associate derivatives with slopes and tangent lines.

n Apply differentiation to:

• rates of change

• maxima and minima

• curve sketching.


Suggested Progression Through ChapterSuggested Progression Through Chapter14.1 14.1 The Second DerivativeThe Second Derivative The second derivative is explained and examples of first- and second-order differential equations

are given. Worked Example 14.2 verifies the solution of a second-order differential equation.

14.2 14.2 Increasing and Decreasing FunctionsIncreasing and Decreasing Functions Increasing and decreasing functions are explained. A connection is made between the derivative

of a function and the domain where the function is either increasing or decreasing.

14.3 14.3 Stationary PointsStationary Points Stationary points are defined and the second derivative test is introduced. The concept of

concavity is explained and points of inflection defined. Activity 14.1 derives the conditions for points of inflection.

14.4 14.4 MMaximaximum and Minimum Problemsum and Minimum Problems In this section calculus is used to solve some real-life optimization problems.

14.5 14.5 Rates of ChangeRates of Change The idea of rate of change is introduced in Activity 14.2. Worked Examples contain real-life

applications of rates of change.








CHAPTER 15 INTEGRAL CALCULUS CHAPTER 15 INTEGRAL CALCULUS


Strand 5

5.2 5.2 Calculus Calculus

n Recognize integration as the reverse process of differentiation.

n Use integration to find the average value of a function over an interval.

n Integrate sums, differences and constant multiples of functions of the form:

• xa, where a ∈ �

• ax, where a ∈ �

• sin ax, where a ∈ �

• cos ax, where a ∈ �.

n Determine areas of plane regions bounded by polynomial and exponential curves.


Suggested Progression Through ChapterSuggested Progression Through Chapter15.1 15.1 Antiderivatives and the Indefinite IntegralAntiderivatives and the Indefinite Integral The idea that integration is the reverse process of differentiation is explained. In Worked

Example 15.1 a function, f(x) is differentiated and three antiderivatives of the derived function are found.

15.2 15.2 Integrating PolynomialsIntegrating Polynomials Activity 15.1 teaches the student how to integrate a polynomial based on the idea of

antiderivatives. Two Worked Examples consolidate the idea.

15.3 15.3 Integrating Exponential FunctionsIntegrating Exponential Functions Again the idea of antiderivatives is used to integrate the exponential function. Worked

Example 15.4 deals with the integration of functions of the type, ax, where a ∈ �.

15.4 15.4 Definite Integrals and AreaDefinite Integrals and Area The definite integral is defined and used to find the area under a graph. The area under a graph is

also found by summing the areas of rectangles under the graph. Activity 15.2 covers the technique for finding the area by summing rectangles.

15.5 15.5 Intersecting Curves and the Trapezoidal RuleIntersecting Curves and the Trapezoidal Rule In this section the area bounded by two curves is found. There is also a review of the trapezoidal

rule for finding the area under a curve.

15.6 15.6 Average Value of a FunctionAverage Value of a Function The formula for the average value of a function over a given interval is derived.





Exercise 15.4Easy: 1–6Medium: 7–8, 11–15Hard: 9–10, 16–20



Revision ExercisesEasy: 1–7Medium: 8, 10–14Hard: 9, 15–20

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