Mathematical Methods Notes

Problem-Solving and Mathematical Modelling

• Problem Understanding: Students must first understand the problem's requirements.
• Plan Design: Design a strategy to solve the problem.
• Mathematical Translation: Translate the problem into a mathematical representation.
  • Determine applicable mathematical knowledge.
  • Identify and justify important assumptions, variables, and observations based on the logic of a proposed solution/model.
• Mathematisation: Formulating a model involves mathematisation, which is the transition from the real world to the mathematical world.
• Solution Application: Select and apply previously learned mathematical knowledge.
  • Approaches include:
    • Synthesizing and refining existing models.
    • Generating and testing hypotheses using primary or secondary data.
• Solution Methods: Solutions can be found using:
  • Algebraic, graphic, arithmetic and/or numeric methods
  • Utilizing technology where appropriate
• Reasonableness Consideration: Consider the solution's reasonableness and the model's utility.
• Verification and Evaluation: Verify results and evaluate the solution in relation to the original problem.
  • Explore strengths and limitations.
  • Refine the solution/model iteratively if necessary.
• Real-World Solution Check: Ensure the model provides a complete solution to the real-world problem.
• Methodological Rigour: Emphasize that problem-solving and mathematical modeling are iterative, not linear, processes.
• Communication and Justification: Solutions/models must be clearly communicated and justified.
  • Use mathematical and everyday language.
  • Draw conclusions, discuss results, strengths, and limitations.
  • Offer further explanations, justifications, and/or recommendations in the context of the initial problem.

Teaching Problem-Solving and Mathematical Modelling

• Teaching For vs. Learning Through: Consider teaching for and learning through problem-solving and mathematical modelling.
  • Teaching For: Teaching specific mathematical rules, definitions, procedures, problem-solving strategies, and critical model elements.
  • Learning Through: Presenting problems that require applying previously taught knowledge and skills to develop new mathematical understanding.
• Explicit and Connected Approach: Requires fluency of critical facts and processes at each step.
• Three Approaches (Based on Galbraith (1989))
  • Dependent:
    • Teacher explicitly demonstrates and teaches concepts and techniques.
    • Students solve and evaluate/verify.
    • Teaching For
  • Guided:
    • Teacher influences concept and technique choices.
    • Guidance is provided through all stages
    • Moving towards Learning Through
  • Independent:
    • Teacher cedes control, students work independently.
    • Students choose their own solutions/models.
    • Learning Through.
• Approach Exclusivity: These approaches are not mutually exclusive.
• Independent Approach as Extension: An independent approach can extend from a dependent or guided activity.
• Foundational Understanding: Students need relevant foundational understanding and skills before independent work.
• Progress Monitoring: Teachers should closely monitor student progress.

Strategies for Retaining and Recalling Information

• The Spacing Effect:
  • Recall and revisit information at intervals for better retention.
  • Multiple exposures over spaced intervals solidify long-term memory.
  • Plan teaching sequences to revisit previously taught information at intervals.
  • Repeated opportunities allow for formative feedback.
• The Retrieval Effect:
  • Practice remembering through regular, low-stakes questioning/quizzes.
  • More effective than searching through notes.
  • Inability to remember should be seen as a learning opportunity.
  • Trying to recall strengthens memory and identifies learning gaps.
  • More difficult retrieval practice is better for long-term learning.
• Interleaving:
  • Interspersing concepts, categories, skills, or question types during revision.
  • Contrasted with blocking (grouping elements together).
  • Example:
    • Blocking: AAAAA BBBBB CCCCC
    • Interleaving: ABCBC ABACA CBAB
  • Interleaving leads to better long-term recall.
  • Ensures spacing occurs.
  • Enhances inductive learning by highlighting differences between related concepts.
  • Spacing without interleaving does not appear to benefit this type of learning.
  • Interleaving can feel counterintuitive, but testing performance indicates greater learning.

Reporting Standards (A-E)

• Reporting standards are summary statements that describe typical performance at each of the five levels (A–E).
• A:
  • Recalls, uses, and communicates comprehensive mathematical knowledge from Algebra, Functions, relations and their graphs, Calculus and Statistics in simple familiar, complex familiar, and complex unfamiliar situations.
  • Evaluates reasonableness of solutions, justifies procedures and decisions, and solves mathematical problems in those situations.
• B:
  • Recalls, uses, and communicates thorough mathematical knowledge from Algebra, Functions, relations and their graphs, Calculus and Statistics in simple familiar and complex familiar situations.
  • Evaluates reasonableness of solutions, justifies procedures and decisions, and solves mathematical problems in those situations.
• C:
  • Recalls, uses, and communicates mathematical knowledge from Algebra, Functions, relations and their graphs, Calculus and Statistics in simple familiar situations.
  • Evaluates the reasonableness of solutions, justifies procedures and decisions, and solves mathematical problems in simple familiar situations.
• D:
  • Recalls, uses, and communicates partial mathematical knowledge from Algebra, Functions, relations and their graphs, Calculus and Statistics in simple familiar situations.
  • Sometimes evaluates reasonableness, justifies procedures/decisions, and solves problems in simple familiar situations.
• E:
  • Recalls, uses, and communicates isolated mathematical knowledge from Algebra, Functions, relations and their graphs, Calculus and Statistics in simple familiar situations.
  • Rarely evaluates reasonableness and infrequently justifies procedures/decisions in simple familiar situations.

Unit Reporting

• Units 1 and 2:
  • Schools judge individual assessment instruments using school-determined methods (reporting standards or ISMG).
  • Marks aren't required for unit result reporting to QCAA.
  • Unit assessment program comprises instruments allowing student demonstration of objectives.
  • A–E unit judgment is made using reporting standards.
  • Schools report student results to QCAA as Satisfactory (S), Unsatisfactory (U) or Not Rated (NR).
• Units 3 and 4:
  • Schools mark internal assessments using ISMGs.
  • Provisional mark by criterion reported to QCAA for each internal assessment.
  • QCAA confirms results and combines them with external assessment results.
  • QCAA determines subject result as mark out of 100 and A–E grade.

Unit 1: Surds, Algebra, Functions and Probability

• Topics:
  • Surds and quadratic functions
  • Binomial expansion and cubic functions
  • Functions and relations
  • Trigonometric functions
  • Probability
• Overview: Working with surds, relationships between variable quantities, binomial theorem, quadratic/cubic/reciprocal functions, graphs of relations, trigonometric functions, inferential statistics, conditional probability and independence.
• Unit Objectives:
  1. Recall mathematical knowledge.
  2. Use mathematical knowledge.
  3. Communicate mathematical knowledge.
  4. Evaluate the reasonableness of solutions.
  5. Justify procedures and decisions.
  6. Solve mathematical problems.

Topic 1: Surds and Quadratic Functions

Sub-topic: Surds (4 hours)

• Concept: Understand a surd as an irrational number represented using a square root or radical sign.
• Simplification: Simplify square roots of natural numbers with perfect square factors.
  • Example: $(\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9}\sqrt{5} = 3\sqrt{5})$
• Rationalization: Rationalize the denominator of fractional expressions.
  • Example: $(\frac{\sqrt{7}}{\sqrt{3}} = \frac{\sqrt{7}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{7}\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}} = \frac{\sqrt{21}}{3})$
• Operations: Use four operations to simplify surds.
  • Example: $(\sqrt{5} - 2\sqrt{5} + 4\sqrt{5} = 3\sqrt{5})$ and $(2\sqrt{3} \times 5\sqrt{11} = 10\sqrt{33})$

Sub-topic: Quadratic Functions (7 hours)

• Features of Graphs: Recognize and determine features of graphs $y = x^2$, $y = ax^2 + bx + c$, $y = a(x - h)^2 + k$, and $y = a(x - x<em>1)(x - x</em>2)$. Includes parabolic nature, turning points, symmetry axes, and intercepts.
• Algebraic Solutions: Solve quadratic equations using factorization, quadratic formula, completing the square, and technology.
• Graphing: Sketch graphs of quadratic functions with/without technology.
• Discriminant: Use the discriminant to determine the number of solutions.
• Turning Points/Zeros: Determine turning points and zeros of quadratic functions.
• Modeling: Model and solve problems involving quadratic functions.

Topic 2: Binomial Expansion and Cubic Functions

Sub-topic: Binomial Expansion (3 hours)

• Combinations: Understand a combination as an unordered set of $r$ objects taken from a set of $n$ distinct objects.
• Pascal's Triangle: Link Pascal's triangle and the notation $(n \choose r)$.
• Binomial Theorem: Use the binomial theorem $(x + y)^n = x^n + {n \choose 1}x^{n-1}y + … + {n \choose r}x^{n-r}y^r + … + y^n$ to expand expressions.
  • Example: $(2x - 1)^3$

Sub-topic: Cubic Functions (9 hours)

• Polynomial Identification: Identify coefficients and degree of a polynomial.
• Expansion: Expand quadratic and cubic polynomials from factors.
• Features of Graphs: Recognize and determine features of graphs of $y = x^3$, $y = a(x - h)^3 + k$, and $y = a(x - x<em>1)(x - x</em>2)(x - x_3)$. Includes shape, intercepts, and behavior as $x \rightarrow \infty$ and $x \rightarrow -\infty$.
• Solving: Solve cubic equations using technology and algebraically when factorized.
• Graphing: Sketch graphs of cubic functions with/without technology.
• Modeling: Model and solve problems involving cubic functions.

Topic 3: Functions and Relations

Sub-topic: Introduction to Functions and Relations (5 hours)

• Relation Concept: Understand a relation as a mapping between sets, a graph, and a rule/formula that defines one variable in terms of another.
• Distinction: Distinguish between functions and relations using the vertical line test.
• Notation: Recognize and use function notation, domain/range, independent/dependent variables.
• Piece-wise Functions: Use piece-wise functions as combinations of sub-functions with restricted domains.
• Modeling: Model and solve problems involving piece-wise functions.

Sub-topic: Graphs of Relations (4 hours)

• Circular Shapes: Recognize and determine features of the graphs of $x^2 + y^2 = r^2$ and $(x - h)^2 + (y - k)^2 = r^2$, including circular shapes, centers, and radii.
• Parabolic Shapes: Recognize and determine features of the graph of $y^2 = x$, including parabolic shape and symmetry.
• Square Root Functions: Recognize and determine features of graphs of $y = a\sqrt{x - h} + k$, including shape, intercepts, and behavior as $x \rightarrow \infty$ and $x \rightarrow -\infty$.
• Sketching: Sketch graphs of relations.
• Modeling: Model and solve problems involving relations.

Sub-topic: Reciprocal Functions (2 hours)

• Hyperbolic Shapes: Recognize the hyperbolic shape, intercepts, asymptotes, and behavior as $x \rightarrow \infty$ and $x \rightarrow -\infty$ of the graphs $y = \frac{1}{x}$ and $y = \frac{a}{x-h} + k$
• Modeling: Model and solve problems that involve reciprocal functions.

Topic 4: Trigonometric Functions

Sub-topic: Circular Measure and Radian Measure (2 hours)

• Radian Measure: Define, use, and understand the relationship between radian and degree measure.
• Arc Lengths/Areas: Calculate lengths of arcs and areas of sectors in circles.

Sub-topic: Introduction to Trigonometric Functions (8 hours)

• Unit Circle: Understand the unit circle definition of $\cos(\theta)$, $\sin(\theta)$, and $\tan(\theta)$ and periodicity using radians.
• Exact Values: Understand and use exact values of $\cos(\theta)$, $\sin(\theta)$, and $\tan(\theta)$ at integer multiples of $\frac{\pi}{6}$ and $\frac{\pi}{4}$.
• Graphing: Sketch graphs of $y = \sin(x)$, $y = \cos(x)$, and $y = \tan(x)$ on extended domains.
• Parameter Effects: Recognize and determine the parameters $a$, $b$, $h$, and $k$ effect on the graphs of $y = a \sin(b(x - h)) + k$, $y = a \cos(b(x - h)) + k$.
• Sketching Parametric Trig Functions: Sketch the graphs of $y = a \sin(b(x - h)) + k$, $y = a \cos(b(x - h)) + k$.
• Solving Trig Equations: Solve trigonometric equations, including using the Pythagorean identity $sin^2(A) + cos^2(A) = 1$
• Modeling: Model and solve problems that involve trigonometric functions.

Topic 5: Probability

Sub-topic: Language of Events and Sets (4 hours)

• Outcomes/Sample Spaces/Events: Use concepts and language of outcomes, sample spaces, and events as sets of outcomes.
• Set Notation: Use set language and notation for events, including $A'$ for complement of A, $A \cap B$ for intersection, and $A \cup B$ for union. Recognize mutually exclusive events.
• Illustrations: Use everyday occurrences to illustrate set descriptions/representations and set operations using Venn diagrams.

Sub-topic: Conditional Probability and Independence (7 hours)

• Probability Rules: Use the rules $P(A') = 1 - P(A)$ and $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
• Conditional Probability: Understand the notion of conditional probability and recognize language indicating conditionality.
• Notation and Formula: Use the notation $P(A|B)$ and the formula $P(A \cap B) = P(A|B)P(B)$.
• Independence: Understand and use the notion of independence of event A from event B, defined by $P(A|B) = P(A)$.
• Formula for Independent Events: Use the formula $P(A \cap B) = P(A)P(B)$ for independent events.
• Relative Frequencies: Use relative frequencies from data as point estimates of conditional probabilities or indications of possible event independence.
• Modeling: Model and solve problems that involve probability.

Unit 2: Calculus and Further Functions

• Topics:
  1. Exponential functions
  2. Logarithms and logarithmic functions
  3. Introduction to differential calculus
  4. Applications of differential calculus
  5. Further differentiation
• Overview: Exponential graphs and applications, logarithms, logarithmic laws and functions, rates of change, derivatives, calculus of power and polynomial functions, curve sketching, tangents and normals, rates of change, differentiation rules.
• Unit Objectives:
  1. Recall mathematical knowledge.
  2. Use mathematical knowledge.
  3. Communicate mathematical knowledge.
  4. Evaluate the reasonableness of solutions.
  5. Justify procedures and decisions.
  6. Solve mathematical problems.

Topic 1: Exponential Functions

Sub-topic: Indices and Index Laws (4 hours)

• Usage: Using indices (including negative and fractional indices) and the index laws.
• Conversion: Convert radicals to and from fractional indices.
• Scientific Notation: Understand and use scientific notation.

Sub-topic: Introduction to Exponential Functions (6 hours)

• Qualitative Features: Recognize, and determine the qualitative features of graph $y = r^x$ (where r > 0), including asymptote and intercept.
• Parameter Effects: Recognize and determine the effect of the parameters $h$, $k$, and $r$ on the graph of $y = r^{(x-h)} + k$ (where r > 0).
• Graphing: Sketch graphs of exponential functions.
• Solving Equations: Solve equations involving exponential functions.
• Modeling: Model and solve problems that involve exponential functions.

Topic 2: Logarithms and Logarithmic Functions

Sub-topic: Logarithms and Logarithmic Laws (5 hours)

• Definition: Define logarithms as indices, where $a^x = b$ is equivalent to $x = \log_a(b)$, and convert between both forms.
• Logarithmic Laws
  • $log<em>a(x) + log</em>a(y) = log_a(xy)$
  • $log<em>a(x) - log</em>a(y) = log_a(\frac{x}{y})$
  • $log<em>a(x^n) = n log</em>a(x)$
  • $log<em>a(x) = \frac{log</em>b(x)}{log_b(a)}$
  • $log_a(a) = 1$
  • $log_a(1) = 0$
• Solving Equations: Solve equations involving indices using logarithms.

Sub-topic: Logarithmic Functions (7 hours)

• Qualitative Features: Recognize and determine the qualitative features of the graph of $y = log_a(x)$ (where a > 1), including asymptote and intercept.
• Parameter Effects: Recognize and determine the effect of the parameters $a$, $h$ and $k$ on the graph of $y = log_a(x - h) + k$ (where a > 1).
• Graphing: Sketch graphs of logarithmic functions.
• Solving Equations: Solve equations involving logarithmic functions.
• Modeling: Model and solve problems that involve logarithmic functions (e.g. decibels in acoustics and the Richter scale for earthquake magnitude).

Topic 3: Introduction to Differential Calculus

Sub-topic: Rates of Change and the Concept of Derivatives (10 hours)

• Average Rate of Change: Determine average rate of change in practical contexts.
• Derivative from First Principles: Use the rule $f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ to determine the derivative of simple power and polynomial functions from the first principle.
• Instantaneous Rate of Change: Interpret the derivative as the instantaneous rate of change.
• Tangent Gradient: Interpret the derivative as the gradient of a tangent line of the graph of $y = f(x)$.
• Power Rule: Use the rule $\frac{d}{dx}x^n = nx^{n-1}$ for positive integers
• Derivative as Function: Understand the concept of the derivative as a function.
• Properties of Derivatives: Recognize and use properties of the derivative: $\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$
• Calculating Derivatives: Calculate derivatives of power and polynomial functions.

Topic 4: Applications of Differential Calculus

Sub-topic: Graphical Applications of Derivatives (12 hours)

• Instantaneous Rates of Change: Determine instantaneous rates of change.
• Equation of Tangent/Normal: Determine the equation of a tangent and a normal of the graph of $y = f(x)$.
• Displacement-Time Graphs: Construct and interpret displacement-time graphs, with velocity as the slope of the tangent.
• Velocity as Rate of Change: Recognize that velocity is the instantaneous rate of change of displacement with respect to time.
• Stationary Points: Use the first derivative of a function to determine and identify the nature of stationary points.
• Curve Sketching: Sketch curves associated with power functions and polynomials up to degree 4, finding stationary points and local/global maxima and minima, and examine behavior as $x \rightarrow \infty$ and $x \rightarrow -\infty$.

Topic 5: Further Differentiation

Sub-topic: Differentiation Rules (11 hours)

• Chain Rule: Use the chain rule, if $y = f(u)$ and $u = g(x)$ then $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$ to determine derivatives of composite functions involving power and polynomial functions.
• Product Rule: Use the product rule, $\frac{d(uv)}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$, to determine the derivative of products of functions involving power and polynomial functions.
• Quotient Rule: Use the quotient rule, $\frac{d(\frac{u}{v})}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$, to determine the derivative of quotients of functions involving power and polynomial functions.
• Problem Solving: Solve problems involving combinations of the chain, product, and quotient rules to differentiate functions involving power and polynomial functions, expressing derivatives in simplest and factorised form.

Unit 3: Further Calculus and Introduction to Statistics

• Topics:
  1. Differentiation of exponential and logarithmic functions
  2. Differentiation of trigonometric functions and differentiation rules
  3. Further applications of differentiation
  4. Introduction to integration
  5. Discrete random variables
• Overview: Derivatives of exponential, logarithmic, and trigonometric functions, differentiation techniques, optimization problems, graph sketching, integration, discrete random variables, modelling random processes.
• Unit Objectives:
  1. Recall mathematical knowledge.
  2. Use mathematical knowledge.
  3. Communicate mathematical knowledge.
  4. Evaluate the reasonableness of solutions.
  5. Justify procedures and decisions.
  6. Solve mathematical problems.

Topic 1: Differentiation of Exponential and Logarithmic Functions

Sub-topic: Calculus of Exponential Functions (6 hours)

• Limits: Estimate the limit of $\frac{a^h - 1}{h}$ as $h \rightarrow 0$ using technology, for various values of a > 0.
• Definition of e: Recognize that $e$ is the unique number $a$ for which the above limit is 1.
• Graph of $y = e^x$: Recognize and determine the qualitative features of the graph of $y = e^x$, including asymptote and intercept.
• Derivatives: Use the rules $\frac{d}{dx}e^x = e^x$ and $\frac{d}{dx}e^{f(x)} = f'(x)e^{f(x)}$.

Sub-topic: Calculus of Logarithmic Functions (8 hours)

• **Graph of $y = ln(x)$: **Recognize and determine the qualitative features of the graph of $y = \ln(x) = \log_e(x)$, including asymptote and intercept.
• Inverse Relationship: Recognize and use the inverse relationship of the functions $y = e^x$ and $y = \ln(x)$.
• Solving Equations: Solve equations involving exponential and logarithmic functions with base $e$.
• Derivatives: Use the rules $\frac{d}{dx}\ln(x) = \frac{1}{x}$ and $\frac{d}{dx}\ln(f(x)) = \frac{f'(x)}{f(x)}$.
• Modeling: Model and solve problems that involve derivatives of exponential and logarithmic functions.

Topic 2: Differentiation of Trigonometric Functions and Differentiation Rules

Sub-topic: Calculus of Trigonometric Functions (5 hours)

• Derivatives of Sine: Use the rules $\frac{d}{dx}\sin(x) = \cos(x)$ and $\frac{d}{dx}\sin(f(x)) = f'(x)\cos(f(x))$.
• Derivatives of Cosine: Use the rules $\frac{d}{dx}\cos(x) = -\sin(x)$ and $\frac{d}{dx}\cos(f(x)) = -f'(x)\sin(f(x))$.
• Modeling: Model and solve problems that involve derivatives of trigonometric functions.

Sub-topic: Differentiation Rules (5 hours)

• Chain Rule: Use the chain rule to determine the derivative of composite functions involving exponential, logarithmic, and trigonometric functions.
• Product Rule: Use the product rule to determine the derivative of exponential, logarithmic, and trigonometric functions.
• Quotient Rule: Use the quotient rule to determine the derivative of exponential, logarithmic, and trigonometric functions.
• Problem Solving: Solve problems that involve combinations of the chain, product, and quotient rule to differentiate exponential, logarithmic, and trigonometric functions.

Topic 3: Further Applications of Differentiation

Sub-topic: The Second Derivative and Applications of Differentiation (10 hours)

• Second Derivative: Understand the concept of the second derivative as the rate of change of the first derivative function.
• Acceleration: Recognize acceleration as the second derivative of displacement/position with respect to time.
• Concavity/Inflection Points: Understand concepts of concavity and inflection points and their relationship with the second derivative.
• Second Derivative Test: Understand and use the second derivative test for finding local maxima and minima.
• Graph Sketching: Sketch the graph of a function using first and second derivatives to locate stationary points and points of inflection.
• Optimization: Model and solve optimization problems using first and second derivatives.

Topic 4: Introduction to Integration

Sub-topic: Anti-differentiation (9 hours)

• Anti-Differentiation: Recognize anti-differentiation as the reverse of differentiation.
• Notation: Use the notation $\int f(x) dx$ for anti-derivatives or indefinite integrals.
• Power Rule for Integration: Use the formula $\int x^n dx = \frac{x^{n+1}}{n+1} + c$ for $n \neq -1$.
• Integral of $e^x$: Use the formula $\int e^x dx = e^x + c$.
• Integral of $1/x$: Use the formula $\int \frac{1}{x} dx = \ln(x) + c$, for x > 0.
• Integrals of Sine and Cosine: Use the formulas $\int \sin(x) dx = -\cos(x) + c$ and $\int \cos(x) dx = \sin(x) + c$.
• Linearity of Integration: Use the formulas $\int (f(x) + g(x))dx = \int f(x) dx + \int g(x) dx$ and $\int k f(x) dx = k \int f(x) dx$.
• Integrals of Linear Functions: Determine indefinite integrals of the form $\int f(ax + b) dx$.
• Determining $f(x)$: Determine $f(x)$ given $f'(x)$ and an initial condition $f(a) = b$.
• Determining Displacement: Determine displacement given velocity and the initial value of displacement.
• Determining Displacement (Acceleration): Determine displacement given acceleration and initial values of displacement and velocity.
• Modeling: Model and solve problems that involve indefinite integrals.

Topic 5: Discrete Random Variables

Sub-topic: General Discrete Random Variables (5 hours)

• Discrete Random Variable: Understand the concepts of a discrete random variable and its associated probability function.
• Point Estimates: Use relative frequencies obtained from data to determine point estimates of probabilities.
• Uniform Discrete RVs: Recognize uniform discrete random variables and model phenomena with equally likely outcomes.
• Non-Uniform Discrete RVs: Recognize non-uniform discrete random variables and use them to model random phenomena.
• Mean (Expected Value): Determine and use the mean (expected value) of a discrete random variable: $E(X) = \mu = \sum p<em>i x</em>i$, where $p<em>i$ is the probability of outcome $x</em>i$.
• Variance: Determine and use the variance of a discrete random variable: $Var(X) = \sum p<em>i (x</em>i - \mu)^2$, where $p<em>i$ is the probability of outcome $x</em>i$, and $\mu$ is the mean.
• Standard Deviation: Determine and use the standard deviation of a discrete random variable, $\sqrt{Var(X)}$.
• Modeling: Model and solve problems that involve discrete random variables and associated probabilities.

Sub-topic: Bernoulli Distributions (2 hours)

• Bernoulli Model: Use a Bernoulli random variable as a model for two-outcome situations.
• Context Identification: Identify contexts suitable for modelling by Bernoulli random variables.
• Mean and Variance: Recognize and determine the mean $p$ and variance $p(1 - p)$ of the Bernoulli distribution with parameter $p$.
• Modeling: Model and solve problems that involve Bernoulli random variables and associated probabilities.

Sub-topic: Binomial Distributions (5 hours)

• Bernoulli Trials: Understand the concepts of Bernoulli trials and a binomial RV as the number of 'successes' $r$, in $n$ independent Bernoulli trials, with success probability $p$.
• Context Identification: Identify contexts suitable for modelling by binomial random variables.
• Probabilities: Determine and use probabilities $P(X = r) = {n \choose r} p^r (1 - p)^{n-r}$ associated with the binomial distribution with parameters $n$ and $p$.
• Mean and Variance: Calculate the mean $np$ and variance $np(1 - p)$ of a binomial distribution.
• Probability Language: Use probability language, including at most, at least, no more than, no less than, inclusive and between.
• Modeling: Model and solve problems that involve binomial distributions.

Unit 4: Further Calculus, Trigonometry and Statistics

• Topics:
  1. Further integration
  2. Trigonometry
  3. Continuous random variables and the normal distribution
  4. Sampling and proportions
  5. Interval estimates for proportions
• Overview: Integral calculus, fundamental theorem, areas under/between curves, cosine and sine rules, continuous random variables, normal distribution, sample and population proportions, statistical inference.
• Unit Objectives:
  1. Recall mathematical knowledge.
  2. Use mathematical knowledge.
  3. Communicate mathematical knowledge.
  4. Evaluate the reasonableness of solutions.
  5. Justify procedures and decisions.
  6. Solve mathematical problems.

Topic 1: Further Integration

Sub-topic: Fundamental Theorem of Calculus and Definite Integrals (3 hours)

• Area Estimation: Use sums of the form $\sum f(x^*<em>i) \delta x</em>i$ to estimate the area under the curve $y = f(x)$.
• Definite Integral: Recognize the definite integral $\int<em>a^b f(x) dx$ as a limit of sums of the form $\sum f(x^*</em>i) \delta x_i$.
• Fundamental Theorem: Understand and use the fundamental theorem of calculus: $\int_a^b f(x) dx = F(b) - F(a)$.
• Area Calculation: Use the definite integral to determine the area under the curve $y = f(x)$ between $x = a$ and $x = b$ if f(x) > 0 over this interval.

Sub-topic: Applications of Integration (8 hours)

• Area Calculation: Calculate the area enclosed by a curve and the x-axis over a given domain.
• Area Between Curves: Calculate the area between curves.
• Trapezoidal Rule: Use the trapezoidal rule, $\int<em>a^b f(x) dx \approx \frac{w}{2} [f(x</em>0) + 2(f(x<em>1) + f(x</em>2) + f(x<em>3) + … f(x</em>{n-1})) + f(x_n)]$, where $w = \frac{b-a}{n}$, to approximate an area and the value of a definite integral.
• Total Change: Calculate total change by integrating instantaneous or marginal rates of change.
• Modeling: Model and solve problems that involve definite integrals, including motion problems.

Topic 2: Trigonometry

Sub-topic: Cosine and Sine Rules (10 hours)

• Sine Rule: Use the sine rule (ambiguous case is required):$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$, where $a$, $b$, and $c$ are the side lengths, and $A$, $B$, and $C$ are opposite angles.
• Cosine Rule: Use the cosine rule: $$c^2 = a