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CONTENTS Beertino Yeong Hui Approaches/pedagogy A level syllabus Diophantus’s problem Pedagogical Consideration roots of function Multiplication Cubic Example and Division of complex numbers Complex conjugates Yu Jie A level syllabus Pedagogical considerations Basic definition & John Argand Diagram A Level syllabus Addition and Pedagogical considerations Subtraction of complex numbers Learning difficulties Uniqueness of Complex Numbers
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APPROACHES/PEDAGOGY Axiomatic Approach Common in textbooks. Start by defining complex numbers as numbers of the form a+ib where a, b are real numbers.Back to Table of contents Diophantus’s problem
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APPROACHES/PEDAGOGY Utilitarian Approach Briefly describe Complex Numbers lead to the theory of fractals It allows computer programmers to create realistic clouds and mountains in video games.Back to Table of contents Diophantus’s problem
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APPROACHES/PEDAGOGY Totalitarian Approach! ( Just kidding )Back to Table of contents Diophantus’s problem
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APPROACHES/PEDAGOGY Historical Approach Why this approach? Real questions faced by mathematicians. Build on pre-existing mathematical knowledge, Quadratic formula Roots.Back to Table of contents Diophantus’s problem
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APPROACHES/PEDAGOGY So, how does the approach goes? First, bring about the quadratic problem. Tapping on prior knowledge Quadratic formula. Roots of an equation. Followed with definition of root. Then sub value into root to get a cognitive conflict. Give another example, this time it’s cubic Tap on prior knowledge again Completing Square to get to Completing Cube (Cardano’s Method) Solve to get a weird answer. Show that weird answer is 4, and get another cognitive conflict.Back to Table of contents Diophantus’s problem
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DIOPHANTUS’S PROBLEM Diophantus Arithhmetica (C.E 275) A right-angled triangle has area 7 square units and perimeter 12 units. Find the lengths of its sides.Approaches/Pedagogy Back to Table of contents Root of Function
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SOLUTION AND PROBLEMApproaches/Pedagogy Back to Table of contents Root of Function
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ROOT OF FUNCTIONDiophantus’s problem Back to Table of contents Cubic Example
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ROOT OF FUNCTIONDiophantus’s problem Back to Table of contents Cubic Example
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CUBIC EXAMPLE Root of Function Back to Table of contents
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CUBIC EXAMPLE Root of Function Back to Table of contents
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LASTLY Root of Function Back to Table of contents
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A-LEVEL SYLLABUSBack to Table of contents Pedagogical considerations
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PEDAGOGICAL CONSIDERATIONS Operations in complex plane is similar but not exactly the same as vector geometry (see complex multiplication and division) Building on Prior Knowledge Rules-Based Approach vs Theoretical UnderstandingSyllabusBack to table of contents (Teaching) Basic Definition
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PEDAGOGICAL CONSIDERATIONS Multimodal Representation and usage of similarity in vector geometry for teaching of complex addition and subtraction Algebraic proof for uniqueness of complex numbers and should it be taught specifically No ordering in complex plane, not appropriate to talk about z1 > z2 ordering is appropriate for modulus, since modulus of complex numbers are real valuesSyllabusBack to table of contents (Teaching) Basic Definition
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BASIC DEFINITONS First defined by Leonard Euler, a swiss mathematician, a complex number, denoted by i, to be i2 = -1 In general, a complex number z can be written as where x denotes the real part and y denotes the imaginary partPedagogical considerationsBack to Table of contents Argand Diagram
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ARGAND DIAGRAM z=x+yi Im(z) x : Real Part P(x,y) y : Imaginary y Part Important aspect, common student error is forgetting Re(z) 0 x that x,y are both real valuedBasic DefinitionsBack to Table of contents Addition
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EXTENSION FROM REAL NUMBERS(ENGAGING PRIOR KNOWLEDGE) The Real Axis Im(z) (x-axis) represents the real number z line. y |z| In other words the real numbers just have the imaginary θ part to be zero. Re(z) 0 x e.g. 1 = 1 + 0 iBasic DefinitionsBack to Table of contents Addition
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ADDITION OF COMPLEX NUMBERS Complex Addition Addition of 2 complex numbers z1 = x1 + y1i, z2 = x2 + y2i z1 + z2 = (x1 + y1i) + (x2 + y2i) = (x1 + x2) + (y1 + y2) i Addition of real and imaginary portions and summing the 2 parts up Geometric Interpretation (vector addition)RationaleMultimodal Representation: Argand DiagramEngaging prior knowledge: Addition for Real NumbersArgand DiagramBack to Table of contents Subtraction
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MMR IN ADDITION Multimodal Representation used: Pictorial Geometric Interpretation Vector Addition Im(z) z1 z1+z2 z2 Re(z) 0Argand DiagramBack to Table of contents Subtraction
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SUBTRACTION OF COMPLEX NUMBERS Complex Subtraction Difference of 2 complex numbers z1 = x1 + y1i, z2 = x2 + y2i z1 - z2 = (x1 + y1i) - (x2 + y2i) = (x1 - x2) + (y1 - y2) i Subtraction of real and imaginary portions and summing the 2 parts up Geometric Interpretation (vector subtraction)RationaleMultimodal Representation: Argand DiagramEngaging prior knowledge: Subtraction for Real NumbersAdditionBack to Table of contents Uniqueness
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MMR IN SUBTRACTION Multimodal Representation used: Pictorial Geometric Interpretation Vector Subtraction Im(z) z1-z2 z1 Re(z) -z2 0AdditionBack to Table of contents Uniqueness
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UNIQUENESS OF COMPLEX NUMBERS If two complex numbers are the same, i.e. z1 = z2, then their real parts must be equal, and their imaginary parts are equal. Algebraically, let z1 = x1 + y1i, z2 = x2 + y2 i, if z1 = z2 then we have x1 = x2 and y1 = y2 Geometrically, from the argand diagram we can see that if two complex numbers are the same, then they are represented by the same point on the argand diagram, and immediately we can see that the x and y co- ordinates of the point must be the same.Subtraction Back to table of contents
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A-LEVEL SYLLABUSBack to Table of contents Pedagogical Considerations
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PEDAGOGICAL CONSIDERATIONS Operations in complex plane is similar but not exactly the same as vector geometry (see complex multiplication and division) Limitations in relating to Argand diagram (pictorial) for teaching of complex multiplication and division in Cartesian form Building on Prior Knowledge Rules-Based Approach vs Theoretical UnderstandingSyllabus Back to table of contents Multiplication
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PEDAGOGICAL CONSIDERATIONS Properties… of complex multiplication assumed (commutative, associative, distributive over complex addition) of complex division assumed (not associative, not commutative) of complex conjugates (self-verification exercise) Notion of identity element, multiplicative inverse Use of GC Accuracy of answersSyllabus Back to table of contents Multiplication
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MULTIPLICATION OF COMPLEX NUMBERS Complex multiplication Scalar Multiplication z = x + yi, k real number kz = k(x + yi) = kx + kyi Geometric Interpretation (vector scaling) k ≥ 0 and k < 0 Rationale Multimodal Representation: Argand Diagram Engaging prior knowledge: Multiplication for Real NumbersPedagogical considerations Back to Table of contents Division
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MULTIPLICATION OF COMPLEX NUMBERS i4n = I, i4n+1 = i, i4n+2 = -1, i4n+3 = -I for any integer n Explore using GC (Limitations) Extension of algebraic identities from real number system (z1 + z2 )(z1 – z2 ) = z12 – z22 (x + iy)(x – iy) = x2 – xyi + xyi + y2 = x2 + y2 ALWAYS real Rationale Engaging prior knowledge: Multiplication for Real Numbers Cognitive process: AssimilationPedagogical considerations Back to Table of contents Division
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DIVISION OF COMPLEX NUMBERS Solve simultaneous equations (using the four complex number operations) Finding square root of complex numberMultiplication Back to Table of contents Conjugates
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COMPLEX CONJUGATES Let z = x + iy. The complex conjugate of z is given by z* = x – iy. Conjugate pair: z and z* Geometrical representation: Reflection about the real axis Multiplication: (x + iy)(x – iy) = x2 + y2 Division: Realising the denominatorRationale Bruner’s CPA Recalling prior knowledge, Law of recencyDivisionBack to Table of contents Learning Difficulties
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COMPLEX CONJUGATES Properties: Exercise for students (direct verification) 1. Re(z*) = Re(z); Im(z*) = -Im(z) 7. (z1 + z2)* = z1* + z2* 2. |z*| = |z| 8. (z1z2)* = z1*z2* 3. (z*)* = z 9. (z1/z2)* = z1*/z2*, 4. z + z* = 2Re(z); z – z* = 2Im(z) if z2 ≠ 0 5. zz* = |z|2 6. z = z* if and only if z is realRationale Self-directed learningDivisionBack to Table of contents Learning Difficulties
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LEARNING DIFFICULTIES/COMMON MISTAKES In z = x + yi, x and y are always REAL numbers Solve equations using z directly or sub z = x + yi Common mistake: (1 + zi)* = (1 – zi) Confused with (x + yi)* = (x - yi)DivisionBack to Table of contents
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A-LEVEL SYLLABUSBack to Table of contents Pedagogical Considerations
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PEDAGOGICAL CONSIDERATION Start with a simple quadratic equation Example: x2 + 2x + 2 = 0. Get students to observe and comment on the roots.Rationale: Bruner’s CPA Approach: Concrete Engaging Prior KnowledgeSyllabusBack to Table of contents Learning Difficulties
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PEDAGOGICAL CONSIDERATION Direct attention to discriminant of quadratic equation What can we say about the discriminant?Rationale: Engaging Prior Knowledge: Linking to O-Level Additional Maths knowledge Involving students in active learning (Vygotsky’s ZPD)SyllabusBack to Table of contents Learning Difficulties
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PEDAGOGICAL CONSIDERATION Examples on Solving for Complex Roots of Quadratic Equations Expose students to different methods: Quadratic Formula Completing the Square MethodRationale:Making Connections between real case and complex caseGetting students to think activelySyllabusBack to Table of contents Learning Difficulties
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PEDAGOGICAL CONSIDERATION Different version: What if we are given one complex root?Example:If one of the roots α of the equation z2 + pz + q = 0 is 3 − 2i, and p, q ∈ ℝ , find p and q.Rationale:Understanding and applying concepts learntSyllabusBack to Table of contents Learning Difficulties
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PEDAGOGICAL CONSIDERATION Good to Fundamental Theorem of Algebra knowOver the set of complex numbers, every polynomial with real coefficients can be factored into a product of linear factors.Consequently, every polynomial of degree n with real coefficients has n roots, subjected to repeated roots.Rationale:Making Connections to Prior Knowledge in Real CaseSyllabusBack to Table of contents Learning Difficulties
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PEDAGOGICAL CONSIDERATION Good to Visualizing Complex Roots know Exploration with GeoGebraRationale:Stretch higher ability students to think furtherMotivates interest in topic of complex numbersSyllabusBack to Table of contents Learning Difficulties
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PEDAGOGICAL CONSIDERATION Extending from quadratic equations to cubic equations Can we generalize to any polynomial? Recall: finding conjugate roots of polynomials with real coefficientsRationale:Making sense through comparing and contrastingSyllabusBack to Table of contents Learning Difficulties
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LEARNING DIFFICULTIES/COMMON MISTAKES X: complex roots will always appear in conjugate pairs ‘No roots’ versus ‘No real roots’ Difficulty in applying factor theorem Careless when performing long division Application of ‘uniqueness of complex numbers’ does not occur naturallyPedagogical ConsiderationsBack to Table of contents
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