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Mathematics Applied

T A S M A N I A N Q U A L I F I C A T I O N S

A U T H O R I T Y

MTA315114, TQA Level 3, Size Value = 15

THE COURSE DOCUMENT

This document contains the following sections:

RATIONALE 2

AIM 2

COURSE SIZE AND COMPLEXITY2

ACCESS 2

PATHWAYS 2

RESOURCES 2

LEARNING OUTCOMES 3

COURSE DELIVERY 3

COURSE CONTENT 3

Algebraic Modelling 4

Calculus 6

Applied Geometry 8

Data Analysis 10

Finance 12

ASSESSMENT 14

Quality Assurance Processes 14

Criteria 14

Standards 15

Qualifications Available 23

Award Requirements 23

EXPECTATIONS DEFINED BY NATIONAL STANDARDS 24

COURSE EVALUATION 27

ACCREDITATION 27

VERSION HISTORY 27

© Copyright for part(s) of this course may be held by organisations other than the TQA Period of Accreditation: 1/1/2014 – 31/12/2014Version 1 Date of Publishing: 6 May 2023

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Mathematics AppliedTQA Level 3

RATIONALEMathematics 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 used to describe much of the physical world. Mathematics also involves the study of ways of collecting and extracting information from data, and of methods of using that information to describe and make predictions about the behaviour of aspects of the real world, in the face of uncertainty. Mathematics provides a framework for thinking and a means of communication that is powerful, logical, concise and precise. It impacts upon the daily life of people everywhere and helps them to understand the world in which they live and work.

The study of Mathematics Applied offers a student the opportunity to apply learned mathematical concepts to real-world situations. The student will develop an appreciation of the direct relevance of mathematics to their present and future lives and to other learning areas.

AIMThe primary aim of this course is to develop students’ ability to apply mathematics to real world problems. As mathematics is being used intensively and increasingly throughout our society it is important that students learn the skills of applying mathematics in problem situations outside of mathematics. The focus of this course is on applications of mathematics rather than on continued skill development.

COURSE SIZE AND COMPLEXITYThis course has a complexity level of TQA level 3.

At TQA level 3, the student is expected to acquire a combination of theoretical and/or technical and factual knowledge and skills and use judgment when varying procedures to deal with unusual or unexpected aspects that may arise. Some skills in organising self and others are expected. TQA level 3 is a standard suitable to prepare students for further study at the tertiary level. VET competencies at this level are often those characteristic of an AQF Certificate III.

This course has a size value of 15.

ACCESSIt is recommended that students attempting this course will have successfully completed either Mathematics Applied – Foundation, TQA level 2, or have previously achieved at least a Grade 10 ‘B’ award in Australian Curriculum: Mathematics.

PATHWAYSMathematics Applied is designed for students who have a wide range of educational and employment aspirations, including continuing their studies at university or TAFE. While the successful completion of this course will gain entry into some post secondary courses, other courses may require the successful completion of Mathematics Methods, TQA level 3.

RESOURCESFor information regarding the use of a calculator when studying this course, refer to the current TQA Calculator Policy that applies to TQA level 3 courses. This policy is available at http://www.tqa.tas.gov.au/0021.

The use of computers is strongly recommended as an aid to student’s learning and mathematical development. A range of software packages are appropriate and, in particular, spreadsheets should be used.

Tasmanian Qualifications Authority Period of Accreditation: 1/1/2014 – 31/12/2014Version 1 Date of Publishing: 6 May 2023

http://www.tqa.tas.gov.au/0021

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LEARNING OUTCOMESOn successful completion of this course, students will:

understand the concepts and techniques introduced in algebraic modelling, calculus, data analysis, finance, trigonometry and world geometry

develop an understanding and appreciation of how mathematics can be used to model real world scenarios

apply reasoning skills and solve practical problems arising in algebraic modelling, calculus, data analysis, finance, trigonometry and world geometry

communicate their arguments and strategies, when solving problems, using appropriate mathematical language

interpret mathematical information, and ascertain the reasonableness of their solutions to problems

choose and use technology appropriately and efficiently.

COURSE DELIVERYApplications of mathematics can lend themselves to being studied in more informal ways than via the traditional methodology of mathematics classroom.

In order to develop students’ ability to apply mathematics to real world problems, each student is required to complete work of an investigative nature. This work is practical in nature and is focused upon the collection, analysis and interpretation of real-world data. While investigations may be planned, directed and initiated by the teacher, raw data should be sourced by the students, either by experimentation or through research.

This course has a design time of 150 hours. A minimum of 15 hours – not all of which need be class time – is to be spent on researching, conducting, analysing, interpreting, and reporting on investigations.

While many investigations will be short in nature and will reinforce understanding of the topics listed under the expansion of each of the content areas, there is clearly scope for students to extend investigative work beyond the course content or to complete an extended investigation that embraces more than one of the content areas.

COURSE CONTENTThere are five (5) content areas of study in this course:

Algebraic modelling

Calculus

Applied geometry

Data analysis

Finance

Each mathematics topic is compulsory, however the order of delivery is not prescribed. These topics relate directly to Criteria 4 – 8. Criteria 1 – 3 apply to all five topics of mathematics.

Algebraic Modelling


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The aim is to develop the capacity to select and use an appropriate algebra related problem solving strategy to represent a real world situation.

Students will have a range of previous experiences and skills, some of which may require some revision and consolidation. In particular, students will:

be able to sketch the graph of line ax+by+c = 0 by location of the x and y intercepts be aware of the significance of the gradient (a) and the y-intercept (b) when sketching the graph

of the line y = ax+b be able to estimate and calculate a and b from the graph of a line be able to solve algebraically and graphically linear equations in two unknowns

Two specific aspects of using algebra to model real situations are highlighted: linear algebraic models non-linear algebraic models

The range of modelling situations will include examples that require: possible elimination of data points that do not fit a pattern clearly established by other points interpolation (estimating y for given x and vice versa) extrapolation where there is emphasis on the limitations.

Linear Algebraic ModellingStudents will engage in modelling activities that present the basic principles, techniques and applications of:

linear relationships between variables using line of sight and the method of least squares (calculator method only)

the correlation coefficient (r) in terms of the variables under consideration(Note: the degree of correlation between two variables is described as weak, moderate or strong, depending on the magnitude of r (ignoring the sign) – weak if 0.25 – 0.49, moderate if 0.50 – 0.74, strong if 0.75 – 1.00)

the coefficient of determination (r2) in terms of the variables under consideration, e.g. a r2 value of 0.87 means that 87% of the variation in the y values can be associated with the variation of the x-values

graphs that are the combination of several linear segments (either continuous or discontinuous). Such graphs may be used to represent tax schedules, charging structures, production costs etc.

break-even analysis, limited to situations that may be modelled by, and involve locating the break-even point if:o the cost function and revenue function are each linear

o one function is linear and the other comprises segments of two linear functions (either continuous or discontinuous)

o the revenue function and the unknown cost function are each linear and two statements are made as to profitability.

Non-Linear Algebraic ModelsStudents will develop knowledge and skills to enable them to:


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recognise the general shape of graphs of type:o power equation y = axb including b < 0 and/or non-integral

o exponential equation y = abx , b > 0 but not necessarily integral (including 0 < b < 1) OR y= aebx (equivalent conditions)

choose the most appropriate of these models for the non linear data presented and find the values of the unknowns (graphics calculator only)

be able to use residual analysis to qualitatively assess the goodness of fit (graphics calculator methods only) for both linear and non-linear models.

Investigation Topics

The emphasis will be on the collection and analysis of real data. Possible topics include:

rate of burning of a candle the relationship between footprint size and height Newton’s law of cooling bounce height of a ball pendulum length and period light intensity Boyles’ Law.


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Calculus

The aims are: to introduce calculus to students using graphical and numerical methods to assist students to formulate algebraic and graphical models that relate to the rate of change of

a continuous variable to apply these models to the calculation of extreme values.

Students will develop a range of algebraic and graphical skills. In particular, students will be able to: factorise quadratic expressions sketch linear functions of the form y = ax + b plot and sketch graphs of the form y = ax2 + bx + c solve quadratic equations.

This module consists of two aspects: function study, in which there is an extension of basic algebraic and graphical techniques the derivative, in which the focus is on understanding the notion of the derivative, and on applying

the derivative to problem-solving situations.

Function StudyBasic algebraic and graphical skills required for differential calculus will be studied. The purpose is purely utilitarian. Students will:

factorise quadratics discuss the relationship between variables including the notion of dependent and independent

variables, and graphs of continuous functions discuss the notion of a function, its notation and its use to show relationships between variables sketch graphs of polynomial functions to the third degree presented in factorised form and be able

to justify the intercepts algebraically use polynomial modelling techniques (graphics calculator only to degree 3).

The DerivativeThe derivative will be explored in an investigative way and not in the manner of a theoretical Calculus course. The approach will be based upon geometric and numerical investigations into the gradient function of simple polynomial functions. Some excellent computer software that uses such an approach is available and graphics calculators provide similar opportunities. Calculus will be discussed as a special mathematical tool for solving certain algebraic modelling problems.Students will:

undertake graphical activities leading to an introduction to the rate of change of one quantity over another

develop an understanding of the derivative as a gradient function using graphical and numerical methods

determine stationary points of polynomial functions (of up to degree 3) using both pen and paper and graphics calculator methods

solve problems arising from situations that can be modelled using polynomial functions that involve finding maximum or minimum values in such context areas as:o profit, loss, revenue and cost

o areas, perimeters and volumes (limited to simple rectangular solids)

o speed as the rate of change of distance with time and acceleration as the rate of change of speed with time. The interpretation of distance/time and speed/time graphs.

Investigation Topics


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The emphasis will be on the collection and analysis of real data. Possible topics include: using ticker timer or photographic techniques to assist in modelling the motion of a moving object using a polynomial equation to model the production of a popcorn machine maximising the volume of a package constructed within constraints and comparison to the ‘real

life’ situation recording the height of a growing bean shoot over a period of several weeks and using

polynomial techniques to find a suitable equation to examine the rate of the shoot’s growth.


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Applied Geometry

The aims are: to develop skills in calculating heights, distances and angles to develop a working knowledge of the relationship between position on the Earth’s surface and

time, and calculations between such positions.

Students will have a range of calculation skills and previously acquired knowledge related to space and measurement. Students will:

possess knowledge of elementary results of Euclidean geometry involving angles, parallel lines, triangles, quadrilaterals, circles and the theorem of Pythagoras

possess a working knowledge of elementary trigonometry including the trigonometric ratios, the Pythagorean relations, solution of right angle triangles involving angles of elevation and depression and bearings, and the sine and cosine rules.

Two aspects of applied geometry are studied: applications of plane geometry spherical geometry.

Applications of Plane GeometryIn this section there are opportunities to develop further some of the pre-requisite skills and knowledge by considering applications of the basic principles in a variety of different contexts and practical situations. Students will:

calculate lengths and angles by applying trigonometrical techniques and a knowledge of bearings to two dimensional situations

calculate lengths and angles by applying right triangle trigonometric ratios and use bearings to solve triangles that model three dimensional situations. (Note that the base triangle need not be a right angle triangle).

Spherical GeomoetryIn this section the focus is on considering the spherical model for distances on the Earth’s surface as a more accurate model of the true situation. Distances on the Earth’s surface are found as well as calculations relating to time and time zones. Students will:

develop a working knowledge of, and perform calculations in relation to, great circles, small circles, latitude, longitude, angular distance, nautical miles and knots

use arc length and plane geometry to calculate distances along great and small circles associated with parallels of latitude and meridians of longitude

calculate great circle distances by performing arc length calculations in association with the use of the cosine rule formula for spherical triangles. The angular separation, between points P and Q on a great circle is given by:cos = sin(lat P).sin(lat Q) + cos(lat P).cos(lat Q).cos(longitude difference), where is the angle subtended at the centre of the great circle by the great circle arc between P and Q. If P and Q are in different hemispheres, northern latitude should be taken as positive, southern latitudes as negative

calculate the distance to the horizon along the line of sight from a position of altitude investigate zone time (standard time) at different meridians of longitude, and consider the

International Date Line. The time zone x hours ahead of GMT has as its centre 15x° E longitude and extends 7.5° either side of 15x° E. For the purpose of this course, the only exceptions will be for South Australia and the Northern Territory. Other regional time zone arrangements and daylight saving will not be considered. Australian time zones should be known.


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Investigation TopicsThe emphasis will be on the use of applied geometry (either plane or spherical) to analyse and solve real problems. Examples include:

analysing the relationship between great circle distance and air-fare between international destinations

developing a spreadsheet or computer program to find the zone time at different longitudes constructing a conversion wheel to find the zone time at different longitudes physically building a three dimensional model that describes a trigonometry problem.


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Data Analysis

The aims are: to consolidate skills in collecting, organising, displaying and interpreting numerical data, and

calculating summary statistics from this data to build upon and consolidate a student’s understanding of the statistical arguments (good and

bad) that they are confronted with daily.

In previous courses students will have been exposed to the notions of, and developed knowledge of and skills in:

collection and presentation of data using tables, charts and graphs sampling frequency distributions - discrete and grouped data terms such as cumulative frequency, range, mean, mode, median, and the effect of extreme

scores.

Data analysis will be as practical as possible with a major purpose being to assist students make sensible interpretations of collected and presented data. Techniques of exploratory data analysis will be employed. Such techniques involve choosing an appropriate way to classify and represent data, using visual representations, statistical measures and curve fitting. Such analysis enhances the quality of discussion of the significant features of a data set and the questions and issues being investigated.

In data analysis, slightly contrasting methods have become acceptable. In this course some recommendations have been made for the sake of uniformity. Students will:

plan, manage and appraise data collection and presentation, making decisions about the appropriateness to consider the population or a sample

calculate measures of central tendency including mean, mode, median for discrete and grouped data

understand the implication of approximations made when grouping data and the effects that it has on statistics calculated (for example: after grouping data the range may be calculated as the difference between the highest and lowest midpoints)

prepare histograms, stem and leaf plots, cumulative frequency tables and cumulative frequency polygons (ogives) to represent data

calculate percentiles, quartiles and interquartile range (a measure of spread) for discrete and grouped data (graphics calculator five figure summary is acceptable here)(Note: If the original readings are retained e.g. in stem and leaf diagrams, the median is found first as the middle score and then the lower and upper quartiles are determined as the middle scores of the lower and upper halves of the data remaining once the median has been removed. If the original readings are lost e.g. in frequency tables, these will be taken to be the 25th, 50th and 75th percentiles and may be estimated from the ogive. Calculation of these quartiles from the frequency table by applying interpolation is optional.)

construct box and whisker plots to represent the distribution of data. In this course, quartiles will be used instead of hinges to represent the end-points of the box

calculate the standard deviation of a set of readings (Graphics calculator method) understand the significance of the mean and standard deviation understand the difference between a sample and its population use the sample mean as a satisfactory estimate of the population mean, and find the best

estimate of the population standard deviation by using a calculator recognise different types of distributions from their histograms. (e.g. normal, skewed, clustered,

spread, bi-modal) solve normal distribution questions using graphics calculators use inverse normal processes (graphics calculator only) to work back from a known proportion (or

percentage) to the corresponding x-score (or range of x-scores).


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draw scatter diagrams and regression lines e.g. intuitive line of sight, method of least squares (calculator method only)

explore the notion of linear correlation with specific reference to Pearson’s (product) correlation coefficient (calculator method only)(Note: The degree of correlation between two variables is described as weak, moderate or strong, depending on the magnitude of r (ignoring the sign) – weak if 0.25 - 0.49, moderate if 0.50 - 0.74, strong if 0.75 - 1.00.)

undertake hypothesis testing using the t-test for:o testing the significance of the difference of a sample mean from a hypothesised population

meano testing the significance of the mean difference for paired data

o testing the significance of the difference between two sample means.

(Note: In this course t-testing will be limited to single tailed testing only and always at the 0.05 significance level.)

Investigation TopicsThe emphasis will be on the collection and analysis of real data. Possible topics include:

using ABS information to investigate Australian and Tasmanian data measuring reaction times - do boys have faster reactions than girls? the taste test – is ‘Coke’ better than home brand?


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Finance

The aims are to develop a working knowledge of: the application of simple interest to various available short-term investment accounts the application of compound interest to growth, and a consideration of depreciation annuity formulae related to investments and loans available options that will reduce the time required to repay a credit-foncier housing loan procedures applied to the early discharge of loans construction and use of tables to calculate repayments, balance owing and number of repayments

on loans and investments.

Students will have knowledge and calculation skills in the areas of: percentages including profit, loss, commission and discount simple and compound interest calculations.

This study of finance is based on these skills and knowledge, and develops the relationship between them and the real data that exists in the publications of banks and other financial institutions, business houses etc. It is intended that activities be practical rather than theoretical. The focus is on the use of a wide variety of standard formulae (mathematical models) and spreadsheet methods of presenting data, that are used in financial calculations. These include applications of simple and compound interest formulae and annuity formulae. While extensive use of graphics calculators and spreadsheets is required, it is expected that students will still have some understanding of formulae use and algebraic solutions.

Simple InterestStudents will:

use the formula: I = PRT to find I, P, R or T apply this formula and spreadsheet techniques to cases where interest is earned or incurred, with

particular attention to calculations in which interest is determined for calculations in which the interest is determined for:o accounts allocating interest on a daily balance

o fixed term deposit accounts (e.g. $2000 at 10% for two years with interest paid quarterly).

Growth and DecayStudents will:

recognise circumstances when the compound growth formula A=P(1+i)n can be applied (for example, compound interest, inflation and appreciation)

use algebraic methods (substitution and transformation) to calculate A and P use calculator methods to find all variables (A, P, i, n) explore the ‘rule of 70’ as a checking procedure investigate the methods of calculation of depreciation:

o flat rate

o declining balance [using A = P(1 - i)n] for n years where n ≤ 6.

Comparison of Interest RatesStudents will:

distinguish between nominal rate per year, the effective rate per year (E), and rate per period (i) use the formula E=(1+i)n - 1 for effective interest rate per year to calculate the value of E, given i

(A calculator approach can be used to find the value of i given E) contrast the nominal rate per year with the effective rate per year explore the notion of real interest calculations using Real interest = Effective rate – Inflation rate.

Annuities in Advance - Investments


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Students will:

recognise when the formula can be applied to determine the

accumulated or future value (F) of an annuity in advance, where R is the regular repayment, i is the interest rate per period and n is the number of periods

use algebraic methods (substitution and transformation) to calculate F and R use calculator methods to find all variables (F, R, i, n) apply the formula to investments with equal periodic payments eg superannuation, insurance

bonds.

Annuities in Arrears - LoansIn this section, it is important for students to make calculations that show that the period of repayment may be shortened considerably by increasing the repayment amount, increasing the frequency of repayments or making extra payments (e.g. after receiving a ‘windfall’). Students will:

recognise circumstances when the formula can be applied to calculate P,

the amount owed, with n payments remaining

use algebraic methods (substitution and transformation) to calculate P and R use calculator methods to find all variables (P, R, i, n) apply the formula to any loans where repayments, calculated according to a given effective

interest rate, are made on a ‘reducing interest schedule’ calculate the lump-sum payment to discharge a loan after r repayments - r integral (Note: With ‘n’

repayments remaining the amount required is the principal outstanding).

Investigation TopicsPossible investigative topics using real financial information include:

using a web based home loan simulator for the comparison of different home loan scenarios designing a computer spreadsheet for modelling depreciation under different circumstances designing a computer spreadsheet for modelling car loan repayments investigation of credit cards and charge cards.


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ASSESSMENTCriterion-based assessment is a form of outcomes assessment that identifies the extent of student achievement at an appropriate end-point of study. Although assessment – as part of the learning program is continuous, much of it is formative, and is done to help students identify what they need to do to attain the maximum benefit from their study of the course. Therefore, assessment for summative reporting to the Tasmanian Qualifications Authority should focus on what both teacher and student understand to reflect end-point achievement.

The standard of achievement each student attains on each criterion is recorded as a rating ‘A’, ‘B’, or ‘C’, according to the outcomes specified in the standards section of the course.

A ‘t’ notation must be used where a student demonstrates any achievement against a criterion less than the standard specified for the ‘C’ rating.

A ‘z’ notation is to be used where a student provides no evidence of achievement at all.

Providers offering this course must participate in quality assurance processes specified by the Tasmanian Qualifications Authority to ensure provider validity and comparability of standards across all awards. Further information on quality assurance processes, as well as on assessment, is available in the TQA Senior Secondary Handbook or on the website at http://www.tqa.tas.gov.au.

Internal assessment of all criteria will be made by the provider. Providers will report the student’s rating for each criterion to the Tasmanian Qualifications Authority.

The Tasmanian Qualifications Authority will supervise the external assessment of designated criteria (*). The ratings obtained from the external assessments will be used in addition to those provided from the provider to determine the final award.

QUALITY ASSURANCE PROCESSES

The following processes will be facilitated by the TQA to ensure there is:

a match between the standards of achievement specified in the course and the skills and knowledge demonstrated by students

community confidence in the integrity and meaning of the qualifications.

Process - the Authority gives course providers feedback about any systematic differences in the relationship of their internal and external assessments and, where appropriate, seeks further evidence through audit and requires corrective action in the future.

CRITERIA

The assessment for Mathematics Applied, TQA level 3, will be based on the degree to which the student can:

1. communicate mathematical ideas and information

2. demonstrate mathematical reasoning, analysis and strategy in practical and problem solving situations

3. plan, organise and complete mathematical tasks

4. *use algebraic or graphical linear and non-linear models to solve problems

5. *use calculus to analyse and solve problems

6. *use geometrical methods to solve problems

7. *use data analysis techniques to analyse distributions

8. *use standard financial models to solve problems.

* = internally and externally assessed criteria


http://www.tqa.tas.gov.au/

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STANDARDS

CRITERION 1 COMMUNICATE MATHEMATICAL IDEAS AND INFORMATION

Rating ‘C’ Rating ‘B’ Rating ‘A’

A student: A student: A student:

presents work that conveys a line of reasoning that has been followed between question and answer

presents work that conveys a logical line of reasoning that has been followed between question and answer

presents work that clearly conveys a logical line of reasoning between question and answer, including suitable justification and explanation of methods and processes used

shows an understanding of mathematical conventions and symbols. There may be some errors or omissions in doing so

generally presents work that follows mathematical conventions, and generally uses mathematical symbols correctly

consistently presents work that follows mathematical conventions, and consistently uses mathematical symbols correctly

presents work with the final answer apparent

presents work with the final answer clearly identified

presents work with the final answer clearly identified and articulated in terms of the question where necessary

generally presents the final answer with correct units when required

changes units appropriately and presents the final answer with correct units as required

changes units appropriately and presents the final answer with correct units as required

presents work that shows some attention to detail

presents work that shows attention to detail

presents work that shows a high level of attention to detail

presents graphs that convey meaning

presents detailed graphs presents graphs that are meticulous in detail

adds a diagram to a solution when prompted.

adds a diagram to supplement a solution.

adds a detailed diagram to supplement a solution.


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CRITERION 2 DEMONSTRATE MATHEMATICAL REASONING, ANALYSIS AND STRATEGY IN PRACTICAL AND PROBLEM SOLVING SITUATIONS



selects and follows an appropriate strategy to solve familiar problems

selects and follows an appropriate strategy to solve problems in unfamiliar contexts

selects and follows an appropriate strategy to solve complex problems in unfamiliar contexts

describes solutions to routine problems

describes solutions to routine and non routine problems

describes solutions to routine and non routine problems in a unfamiliar contexts

uses calculator techniques to solve familiar problems

selects calculator techniques appropriate to solving familiar problems

explores calculator techniques appropriate to solving unfamiliar problems

given written/oral instructions, sets up an experiment, and gathers data

given written instructions, sets up an experiment, and gathers accurate data

given an experiment’s aim, sets up an experiment, and gathers accurate and precise data

evaluates the reliability and validity of solutions to problems and experimental results

evaluates the reliability and validity of solutions to problems and experimental results, and recognises when an experiment is producing anomalous data

evaluates the reliability and validity of solutions to problems and experimental results, recognising when an experiment is producing anomalous data, and initiates steps to improve the experimental design

relates experimental findings to real-world phenomena, and draws conclusions using a template approach

relates experimental findings to real-world phenomena, noting differences between findings and what happens in the real world; draws conclusions with appropriate detail

relates experimental findings to real-world phenomena, noting differences between the findings and what happens in the real world; puts forward reasons for these differences, drawing conclusions showing detail, perception and insight

sources research data, and references it.

sources research data, and appropriately references it.

sources research data, appropriately references it, and evaluates its credibility and usefulness.


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CRITERION 3 PLAN, ORGANISE AND COMPLETE MATHEMATICAL TASKS



uses planning tools to achieve objectives within proposed times

selects and uses planning tools and strategies to achieve and manage activities within proposed times

evaluates, selects and uses planning tools and strategies to achieve and manage activities within proposed times

divides a task into sub-tasks as directed

divides a task into appropriate sub-tasks

assists others to divide a task into sub-tasks

selects from a range of strategies and formulae to successfully complete routine problems

selects from a range of strategies and formulae to successfully complete routine and complex problems

selects strategies and formulae to successfully complete routine and complex problems

monitors progress towards meeting goals and timelines

monitors and analyses progress towards meeting goals and timelines, and plans future actions

monitors and critically evaluates goals and timelines, and effectively plans future actions

meets specified timelines, and addresses most elements of the required task.

meets specified timelines, and addresses all required elements of the task, mostly with a high degree of accuracy.

meets specified timelines, and addresses all required elements of the task, with a high degree of accuracy.


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*CRITERION 4 USE ALGEBRAIC OR GRAPHICAL LINEAR AND NON-LINEAR MODELS TO SOLVE PROBLEMS



solves break-even problems graphically and algebraically where the linear cost and revenue equations are given

solves complex break-even problems which, for example, may involve setting up the linear cost and revenue equations

solves complex break-even problems graphically and algebraically, where one function is linear and the other comprises segments of two linear functions; interactively manipulates cost, profit, loss and revenue equations, where the cost function is unknown; (assuming two statements of profitability are made)

sketches combination graphs e.g. tax scales, commission scales

sketches combination graphs, and finds equations of segments from the graph and table of points

sketches combination graphs, and finds equations of segments from the graph and table of points, drawing conclusions from any given format

plots scatter graphs of real data in linear relationships, and finds the equation of the line of best fit

plots scatter graphs of real data, and finds a suggested suitable mathematical model in both linear and non-linear cases

plots scatter graphs of real data, and selects and finds a suitable mathematical model in both linear and non-linear cases

interpolates and extrapolates results both graphically and algebraically

interpolates and extrapolates results both graphically and algebraically, and discusses the reliability of results, using a templated approach

interpolates and extrapolates results both graphically and algebraically, and discusses the reliability of results, showing an understanding of why a particular indicator renders a result unreliable

uses a mathematical method to evaluate and justify the choice of a particular mathematical model

uses a variety of methods to evaluate and justify the choice of a particular mathematical model

uses a variety of methods including qualitative residual analysis to evaluate and justify the choice of a particular mathematical model

solves problems using non – linear equations (y=axb and y=abx or y=aebx) for y given x.

solves problems using non – linear equations (y=axb and y=abx or y=aebx) for y given x, and x given y, where interpolation is used.

solves problems using non – linear equations (y=axb and y=abx or y=aebx) for y given x, and x given y, where interpolation or extrapolation is used.


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*CRITERION 5 USE CALCULUS TO ANALYSE AND SOLVE PROBLEMS



uses the sketch method to prepare the graph of quadratic function that is presented in factorised form, identifying shape and all intercepts

uses the sketch method to prepare the graph of a quadratic function, factorising, if necessary, and identifying shape and all intercepts

uses the sketch method to prepare the graph of a quadratic function, factorising, if necessary, and identifying shape and all intercepts

sketches a factorised polynomial function of degree 3, identifying shape and intercepts

sketches a factorised polynomial function of degree 3, identifying shape and intercepts including double and triple zeros

sketches a factorised polynomial function of any degree, identifying shape and intercepts including multiple zeros

finds the derivative of a function (expanded and with integer coefficients)

finds the derivative of complex functions (expanded but with non-integer coefficients)

finds the derivatives of complex functions (that may be initially presented in factorised form)

locates the stationary points using the derivative for quadratic or cubic functions

locates and identifies the stationary points using the derivative for quadratic and cubic functions

locates and identifies stationary points using the derivative for quadratic and cubic functions and interprets their significance in context of a worded problem

uses calculus to solve a worded maximisation problem, where the function has been given

uses calculus to solve a worded maximisation problem, where the function must first be shown to be equivalent to a stated expression

uses calculus to solve a worded maximisation problem, where the function is unknown

prepares a screen plot of a polynomial function (up to degree 4) and determines relevant features from its graph, using a graphics calculator

prepares a screen plot of a polynomial function (up to degree 4), adjusts the calculators view window appropriately, and determines all relevant features from its graph, using a graphics calculator

prepares a screen plot of a polynomial function (up to degree 4), adjusts the calculators view window appropriately, and determines all relevant features from its graph, using a graphics calculator

uses the derivative as a rate of change to solve realistic problems

uses the derivative as a rate of change to solve realistic problems and gives the correct derived unit

uses the derivative as a rate of change to solve realistic problems, gives the correct derived unit and interprets the rate in words, relating it to the variables involved

uses polynomial modelling techniques to determine a polynomial equation (to degree 3) through a scatter plot.

uses polynomial modelling techniques to determine a polynomial equation (to degree 3) through a scatter plot and finds points of intersection with a linear horizontal constraint.

uses polynomial modelling techniques to determine a polynomial equation (to degree 3) through a scatter plot, finds points of intersection with a linear horizontal constraint, and interprets these values in terms of the variables involved.

*CRITERION 6 USE GEOMETRICAL METHODS TO SOLVE PROBLEMS


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given a 2D diagram, uses the trigonometrical ratios and the sine and cosine rule to calculate length and angle

given a 2D or a simple 3D diagram, uses the trigonometrical ratios and the sine and cosine rule to calculate length, angle and bearing

given a 2D or 3D diagram, uses the trigonometrical ratios and the sine and cosine rule to calculate length, angles and bearings

produces an accurate diagram to represent a 2D scenario, including angles of elevation/depression and bearing problems

produces an accurate diagram to represent a 2D scenario, including angles of elevation/depression and bearing problems and attempts to do the same with a 3D problem

produces an accurate diagram to represent a 2D or 3D scenario including angles of elevation/depression and bearing problems

uses a given formula to calculate the distance between two points on the earth’s surface (along same lines of longitude or latitude only) and expresses answer using correct units

uses a given formula to calculate the distance between two points on the earth’s surface (along same lines of longitude or latitude or on a great circle) and expresses answer using correct units

selects and uses an appropriate formula to calculate the distance between two points on the earth’s surface (along same lines of longitude or latitude or on a great circle) and expresses answer using correct units

determines the likely time zone of a place on the Earth’s surface given its longitude

solves time problems involving standard (zone) time

solves time problems involving standard (zone) time

applies both time and distance calculations to simple travel problems.

applies both time and distance calculations to more complex travel problems (e.g. more than one destination with a ‘stop over’), given prompting to the steps involved

applies both time and distance calculations to more complex travel problems (e.g more than one destination with a ‘stop over’), without prompting to the steps involved

solves distance to horizon problems.

solves complex distance to horizon problems.


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*CRITERION 7 USE DATA ANALYSIS TECHNIQUES TO ANALYSE DISTRIBUTIONS



calculates the mean, mode, median and range for discrete data

calculates the mean, mode, median and range for discrete and grouped data

calculates the mean, mode, median and range for discrete and grouped data, and understands the approximations that are being made when data is grouped

produces cumulative frequency histograms and ogives, and derives simple information from them, e.g. median and percentiles

produces cumulative frequency histograms and ogives, derives more complex information from them, e.g. median, percentiles and cut-off points and interprets the results in context

produces cumulative frequency histograms and ogives, derives more complex information from them, e.g. median, percentiles and cut-off points and interprets the results in context

constructs, stem and leaf diagrams and side by side boxplots, and compares statistics from each (Min, LQ, Median, UQ, Max)

constructs back to back stem and leaf diagrams, and side by side boxplots, and compares statistics from each (Min, LQ, Median, UQ, Max, range and interquartile range)

constructs back-to-back stem and leaf diagrams, and side by side boxplots, and makes a comprehensive comparative statement about the data

calculates probabilities using normal distribution

calculates probabilities using normal distribution, and converts the probability to reflect the number expected in a given sample

calculates probabilities using normal distribution, and converts the probability to reflect the number expected in a given sample. Given a proportion, uses inverse normal calculations to determine a suitable cut-off point

determines a linear correlation co-efficient

determines a linear correlation co-efficient, and interprets it in terms of the data variables

determines a linear correlation co-efficient, and interprets it in terms of the data variables

uses a suggested t-test. accurately uses a suggested t-test, and draws an appropriate conclusion.

selects and uses a suitable t-test, and draws an appropriate conclusion.


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*CRITERION 8 USE STANDARD FINANCIAL MODELS TO SOLVE PROBLEMS



selects the appropriate formula for solving financial maths problems, as guided

selects the appropriate formula for solving financial maths problems

consistently selects the appropriate formula for solving financial maths problems

accurately applies the simple interest formula in straight forward settings (e.g. single application)

accurately applies the simple interest formula in more complex settings (e.g. daily balance bank account where several transactions have been made)

accurately applies the simple interest formula in more complex settings (e.g. daily balance bank account where several transactions have been made)

applies the growth and decay formulae to straight forward compound interest, depreciation and inflation situations

applies the growth and decay formulae to more complex compound interest, depreciation and inflation situations

applies the growth and decay formulae to complex compound interest, depreciation and inflation situations and interprets results or makes recommendations

calculates effective interest and uses it appropriately when prompted

calculates effective interest, realising when its use is necessary and uses it appropriately without prompting

understands and explains the difference between effective interest and apparent interest. Calculates effective interest, realising when its use is necessary, and uses it appropriately without prompting

uses a suggested annuities formula to calculate the subject of its formula.

selects the appropriate annuities formula and calculates any variable (using graphics calculator if necessary) in the context of a straight forward scenario

selects the appropriate annuities formula and calculates any variable (using graphics calculator if necessary) in the context of a more complex scenario (e.g. extended question)

as guided, investigates scenarios using the annuities formulae to determine the effect of a range of hypothetical changes e.g. increasing repayments on a loan, increasing super contributions.

investigates scenarios using the annuities formulae to determine the effects of a range of hypothetical changes e.g. increasing repayments on a loan, increasing super contributions.


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QUALIFICATIONS AVAILABLE

Mathematics Applied, TQA level 3 (with the award of):

EXCEPTIONAL ACHIEVEMENT

HIGH ACHIEVEMENT

COMMENDABLE ACHIEVEMENT

SATISFACTORY ACHIEVEMENT

PRELIMINARY ACHIEVEMENT

AWARD REQUIREMENTS

The final award will be determined by the Tasmanian Qualifications Authority from the 13 rating (8 ratings from the internal assessment and 5 ratings from the external assessment).

The minimum requirements for an award in Mathematics Applied, TQA level 3, are as follows:

EXCEPTIONAL ACHIEVEMENT (EA)11 ‘A’ ratings, 2 ‘B’ ratings (4 ‘A’ ratings, 1 ‘B’ rating from external assessment)

HIGH ACHIEVEMENT (HA)5 ‘A’ ratings, 5 ‘B’ ratings, 3 ‘C’ ratings (2 ‘A’ ratings, 2 ‘B’ ratings, 1 ‘C’ rating from external assessment)

COMMENDABLE ACHIEVEMENT (CA)7 ‘B’ ratings, 5 ‘C’ ratings (2 ‘B’ ratings, 2 ‘C’ ratings from external assessment)

SATISFACTORY ACHIEVEMENT (SA)11 ‘C’ ratings (3 ‘C’ ratings from external assessment)

PRELIMINARY ACHIEVEMENT (PA)6 ‘C’ ratings

A student who otherwise achieves the ratings for a CA (Commendable Achievement) or SA (Satisfactory Achievement) award but who fails to show any evidence of achievement in one or more criteria (‘z’ notation) will be issued with a PA (Preliminary Achievement) award.


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EXPECTATIONS DEFINED BY NATIONAL STANDARDS IN CONTENT STATEMENTS DEVELOPED BY ACARAThe statements in this section, taken from the Australian Senior Secondary Curriculum: General Mathematics endorsed by Education Ministers as the agreed and common base for course development, are to be used to define expectations for the meaning (nature, scope and level of demand) of relevant aspects of the sections in this document setting out course requirements, learning outcomes, the course content and standards in the assessment.

Unit 1 – Topic 2: Algebra and Matrices

Linear and non-linear expressions: substitute numerical values into linear algebraic and simple non-linear algebraic expressions, and

evaluate (ACMGM010)

find the value of the subject of the formula, given the values of the other pronumerals in the formula (ACMGM011)

use a spreadsheet or an equivalent technology to construct a table of values from a formula, including two-by-two tables for formulas with two variable quantities; for example, a table displaying the body mass index (BMI) of people of different weights and heights. (ACMGM012)

Unit 1 – Topic 3: Shape and Measurement

Pythagoras’ Theorem: review Pythagoras’ Theorem and use it to solve practical problems in two dimensions and for

simple applications in three dimensions. (ACMGM017)

Unit 2 – Topic 2: Applications of Trigonometry:

Applications of Trigonometry review the use of the trigonometric ratios to find the length of an unknown side or the size of an

unknown angle in a right-angled triangle (ACMGM034)

determine the area of a triangle given two sides and an included angle by using the rule Area=12absinC, or given three sides by using Heron’s rule, and solve related practical problems (ACMGM035)

solve problems involving non-right-angled triangles using the sine rule (ambiguous case excluded) and the cosine rule (ACMGM036)

solve practical problems involving the trigonometry of right-angled and non-right-angled triangles, including problems involving angles of elevation and depression and the use of bearings in navigation. (ACMGM037)


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Unit 2 – Topic 3: Linear Equations and their Graphs

Linear equations: identify and solve linear equations (ACMGM038)

develop a linear formula from a word description. (ACMGM039)

Straight-line graphs and their applications: construct straight-line graphs both with and without the aid of technology (ACMGM040)

determine the slope and intercepts of a straight-line graph from both its equation and its plot (ACMGM041)

interpret, in context, the slope and intercept of a straight-line graph used to model and analyse a practical situation (ACMGM042)

construct and analyse a straight-line graph to model a given linear relationship; for example, modelling the cost of filling a fuel tank of a car against the number of litres of petrol required. (ACMGM043)

Unit 3 – Topic 1: Bivariate Data Analysis

The statistical investigation process: review the statistical investigation process; for example, identifying a problem and posing a

statistical question, collecting or obtaining data, analysing the data, interpreting and communicating the results. (ACMGM048)

Identifying and describing associations between two categorical variables: construct two-way frequency tables and determine the associated row and column sums and

percentages (ACMGM049)

use an appropriately percentaged two-way frequency table to identify patterns that suggest the presence of an association (ACMGM050)

describe an association in terms of differences observed in percentages across categories in a systematic and concise manner, and interpret this in the context of the data. (ACMGM051)

Identifying and describing associations between two numerical variables: construct a scatterplot to identify patterns in the data suggesting the presence of an association

(ACMGM052)

describe an association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak) (ACMGM053)

calculate and interpret the correlation coefficient (r) to quantify the strength of a linear association. (ACMGM054)


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Fitting a linear model to numerical data: identify the response variable and the explanatory variable (ACMGM055)

use a scatterplot to identify the nature of the relationship between variables (ACMGM056)

model a linear relationship by fitting a least-squares line to the data (ACMGM057)

use a residual plot to assess the appropriateness of fitting a linear model to the data (ACMGM058)

interpret the intercept and slope of the fitted line (ACMGM059)

use the coefficient of determination to assess the strength of a linear association in terms of the explained variation (ACMGM060)

use the equation of a fitted line to make predictions (ACMGM061)

distinguish between interpolation and extrapolation when using the fitted line to make predictions, recognising the potential dangers of extrapolation (ACMGM062)

write up the results of the above analysis in a systematic and concise manner. (ACMGM063)

Association and causation: recognise that an observed association between two variables does not necessarily mean that

there is a causal relationship between them (ACMGM064)

identify possible non-causal explanations for an association, including coincidence and confounding due to a common response to another variable, and communicate these explanations in a systematic and concise manner. (ACMGM065)

The data investigation process: implement the statistical investigation process to answer questions that involve identifying,

analysing and describing associations between two categorical variables or between two numerical variables; for example, is there an association between attitude to capital punishment (agree with, no opinion, disagree with) and sex (male, female)? Is there an association between height and foot length? (ACMGM066)

Unit 4 – Topic 2: Loans, Investments and Annuities

Compound interest loans and investments: use a recurrence relation to model a compound interest loan or investment, and investigate

(numerically or graphically) the effect of the interest rate and the number of compounding periods on the future value of the loan or investment (ACMGM094)

calculate the effective annual rate of interest and use the results to compare investment returns and cost of loans when interest is paid or charged daily, monthly, quarterly or six-monthly (ACMGM095)

with the aid of a calculator or computer-based financial software, solve problems involving compound interest loans or investments; for example, determining the future value of a loan, the number of compounding periods for an investment to exceed a given value, the interest rate needed for an investment to exceed a given value. (ACMGM096)


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Reducing balance loans (compound interest loans with periodic repayments): use a recurrence relation to model a reducing balance loan and investigate (numerically or

graphically) the effect of the interest rate and repayment amount on the time taken to repay the loan (ACMGM097)

with the aid of a financial calculator or computer-based financial software, solve problems involving reducing balance loans; for example, determining the monthly repayments required to pay off a housing loan. (ACMGM098)

Annuities and perpetuities (compound interest investments with periodic payments made from the investment):

use a recurrence relation to model an annuity, and investigate (numerically or graphically) the effect of the amount invested, the interest rate, and the payment amount on the duration of the annuity (ACMGM099)

with the aid of a financial calculator or computer-based financial software, solve problems involving annuities (including perpetuities as a special case); for example, determining the amount to be invested in an annuity to provide a regular monthly income of a certain amount. (ACMGM100)

COURSE EVALUATIONCourses are accredited for a specific period of time (up to five years) and they are evaluated in the year prior to the expiry of accreditation.

As well, anyone may request a review of a particular aspect of an accredited course throughout the period of accreditation. Such requests for amendment will be considered in terms of the likely improvements to the outcomes for students and the possible consequences for delivery of the course.

The TQA can evaluate the need and appropriateness of an accredited course at any point throughout the period of accreditation.

ACCREDITATIONThe accreditation period for this course is from 1st January 2014 – 31st December 2014.

VERSION HISTORYVersion 1 – Accredited 23 July 2013 for use in 2014. This course replaces Mathematics Applied

(MTA315109) that expired on 31 December 2013.

VERSION CONTROLThis document is a Word version of the course. It is not a TQA controlled version. The current PDF version of the course on the TQA website is the definitive one.


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