Mathematical Methods (Version 8.4)

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Unit 1
Unit 2
Unit 3
Unit 4

Understand how Mathematical Methods works

Rationale

Mathematics is the study of order, relation and pattern. From its origins in counting and measuring it has evolved in highly sophisticated and elegant ways to become the language now used to describe much of the modern world.

Links to Foundation to Year 10

In Mathematical Methods, there is a strong emphasis on mutually reinforcing proficiencies in Understanding, Fluency, Problem solving and Reasoning.

Representation of General capabilities

The seven general capabilities of Literacy, Numeracy, Information and Communication technology (ICT) capability, Critical and creative thinking, Personal and social capability, Ethical understanding, and Intercultural understanding are identified where they offer opportunities to add depth and richness to student learning.

Structure of Mathematical Methods

Mathematical Methods is organised into four units. The topics broaden students’ mathematical experience and provide different scenarios for incorporating mathematical arguments and problem solving. The units provide a blending of algebraic and geometric thinking.

Glossary

Achievement standards

Unit 1

Unit 1 Description

This unit begins with a review of the basic algebraic concepts and techniques required for a successful introduction to the study of calculus. The basic trigonometric functions are then introduced. Simple relationships between variable quantities are reviewed, and these are used to introduce the key concepts of a function and its graph. The study of inferential statistics begins in this unit with a review of the fundamentals of probability and the introduction of the concepts of conditional probability and independence. Access to technology to support the computational aspects of these topics is assumed.

Unit 1 Learning Outcomes

By the end of this unit, students:

understand the concepts and techniques in algebra, functions, graphs, trigonometric functions and probability
solve problems using algebra, functions, graphs, trigonometric functions and probability
apply reasoning skills in the context of algebra, functions, graphs, trigonometric functions and probability
interpret and evaluate mathematical information and ascertain the reasonableness of solutions to problems
communicate their arguments and strategies when solving problems.

Unit 1 Content Descriptions

Topic 1: Functions and graphs

Lines and linear relationships:

determine the coordinates of the midpoint of two points (ACMMM001)

examine examples of direct proportion and linearly related variables (ACMMM002)

recognise features of the graph of $y=mx+c$, including its linear nature, its intercepts and its slope or gradient (ACMMM003)

find the equation of a straight line given sufficient information; parallel and perpendicular lines (ACMMM004)

solve linear equations. (ACMMM005)

Review of quadratic relationships:

examine examples of quadratically related variables (ACMMM006)

recognise features of the graphs of $y=x^2$, $y=a{(x-b)}^2+c$, and $y=a\left(x-b\right)\left(x-c\right)$ including their parabolic nature, turning points, axes of symmetry and intercepts (ACMMM007)

solve quadratic equations using the quadratic formula and by completing the square (ACMMM008)

find the equation of a quadratic given sufficient information (ACMMM009)

find turning points and zeros of quadratics and understand the role of the discriminant (ACMMM010)

recognise features of the graph of the general quadratic $y=ax^2+bx+c$ (ACMMM011)

Inverse proportion:

examine examples of inverse proportion (ACMMM012)

recognise features of the graphs of $y=\frac1x$ and $y=\frac a{x-b}$, including their hyperbolic shapes, and their asymptotes. (ACMMM013)

Powers and polynomials:

recognise features of the graphs of $y=x^n$ for $n\in\boldsymbol N,$ $n=-1$ and $n=½$, including shape, and behaviour as $x\rightarrow\infty$ and $x\rightarrow-\infty$ (ACMMM014)

identify the coefficients and the degree of a polynomial (ACMMM015)

expand quadratic and cubic polynomials from factors (ACMMM016)

recognise features of the graphs of $y=x^3$, $y=a{(x-b)}^3+c$ and $y=k(x-a)(x-b)(x-c)$, including shape, intercepts and behaviour as $x\rightarrow\infty$ and $x\rightarrow-\infty$ (ACMMM017)

factorise cubic polynomials in cases where a linear factor is easily obtained (ACMMM018)

solve cubic equations using technology, and algebraically in cases where a linear factor is easily obtained. (ACMMM019)

Graphs of relations:

recognise features of the graphs of $x^2+y^2=r^2$ and $\left(x-a\right)^2+\left(y-b\right)^2=r^2$, including their circular shapes, their centres and their radii (ACMMM020)

recognise features of the graph of $y^2=x$ including its parabolic shape and its axis of symmetry. (ACMMM021)

Functions:

understand the concept of a function as a mapping between sets, and as a rule or a formula that defines one variable quantity in terms of another (ACMMM022)

use function notation, domain and range, independent and dependent variables (ACMMM023)

understand the concept of the graph of a function (ACMMM024)

examine translations and the graphs of $y=f\left(x\right)+a$ and $y=f(x+b)$ (ACMMM025)

examine dilations and the graphs of $y=cf\left(x\right)$ and $y=f\left(kx\right)$ (ACMMM026)

recognise the distinction between functions and relations, and the vertical line test. (ACMMM027)

Topic 2: Trigonometric functions

Cosine and sine rules:

review sine, cosine and tangent as ratios of side lengths in right-angled triangles (ACMMM028)

understand the unit circle definition of $\cos\theta,\;\sin\theta$ and $\tan\theta$ and periodicity using degrees (ACMMM029)

examine the relationship between the angle of inclination of a line and the gradient of that line (ACMMM030)

establish and use the sine and cosine rules and the formula $Area=\frac12bc\sin A$ for the area of a triangle. (ACMMM031)

Circular measure and radian measure:

define and use radian measure and understand its relationship with degree measure (ACMMM032)

calculate lengths of arcs and areas of sectors in circles. (ACMMM033)

Trigonometric functions:

understand the unit circle definition of $\cos\theta,\;\sin\theta$ and $\tan\theta$ and periodicity using radians (ACMMM034)

recognise the exact values of $\cos\theta,\;\sin\theta$ and $\tan\theta$ at integer multiples of $\frac\pi6$ and $\frac\pi4$ (ACMMM035)

recognise the graphs of $y=\sin x,\;y=\cos x,$ and $y=\tan x$ on extended domains (ACMMM036)

examine amplitude changes and the graphs of $y=a\sin x$ and $y=a\cos x$ (ACMMM037)

examine period changes and the graphs of $y=\sin bx,\;$ $y = \sin b x,$ y=\cos bx\), and $y=\tan bx$ (ACMMM038)

examine phase changes and the graphs of $y=\sin{(x+c)}$, $y=\cos{(x+c)}$ and $y=\tan{(x+c)}$ and the relationships $\sin\left(x+\frac\pi2\right)=\cos x$ and $\cos\left(x-\frac\pi2\right)=\sin x$ (ACMMM039)

prove and apply the angle sum and difference identities (ACMMM041)

identify contexts suitable for modelling by trigonometric functions and use them to solve practical problems (ACMMM042)

solve equations involving trigonometric functions using technology, and algebraically in simple cases. (ACMMM043)

Topic 3: Counting and probability

Combinations:

understand the notion of a combination as an unordered set of $r$ objects taken from a set of $n$ distinct objects (ACMMM044)

use the notation $\begin{pmatrix}n\\r\end{pmatrix}$ and the formula $\begin{pmatrix}n\\r\end{pmatrix}=\frac{n!}{r!\left(n-r\right)!}$ for the number of combinations of $r$ objects taken from a set of $n$ distinct objects (ACMMM045)

expand $\left(x+y\right)^n$ for small positive integers $n$ (ACMMM046)

recognise the numbers $\begin{pmatrix}n\\r\end{pmatrix}$ as binomial coefficients, (as coefficients in the expansion of $\left(x+y\right)^n)$ (ACMMM047)

use Pascal’s triangle and its properties. (ACMMM048)

Language of events and sets:

review the concepts and language of outcomes, sample spaces and events as sets of outcomes (ACMMM049)

use set language and notation for events, including $\overline A$ (or $A'$) for the complement of an event $A,$ $A?B$ for the intersection of events $A$ and $B$, and $A?B$ for the union, and recognise mutually exclusive events (ACMMM050)

use everyday occurrences to illustrate set descriptions and representations of events, and set operations. (ACMMM051)

Review of the fundamentals of probability:

review probability as a measure of ‘the likelihood of occurrence’ of an event (ACMMM052)

review the probability scale: $0\leq P(A)\leq1$ for each event $A,$ with $P\left(A\right)=0$ if $A$ is an impossibility and $P\left(A\right)=1$ if $A$ is a certaint (ACMMM053)

review the rules: $P\left(\overline A\right)=1-P\left(A\right)$ and $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$ (ACMMM054)

use relative frequencies obtained from data as point estimates of probabilities. (ACMMM055)

Conditional probability and independence:

understand the notion of a conditional probability and recognise and use language that indicates conditionality (ACMMM056)

use the notation $P(A\vert B)$ and the formula $P(A\vert B)=P(A\cap B) / P(B)$ (ACMMM057)

understand the notion of independence of an event $A$ from an event $B$, as defined by $P(A\vert B)=P(A)$ (ACMMM058)

establish and use the formula $P(A\cap B)=P(A)P(B)$ for independent events $A$ and $B$, and recognise the symmetry of independence (ACMMM059)

use relative frequencies obtained from data as point estimates of conditional probabilities and as indications of possible independence of events. (ACMMM060)