1 - Tripod
Mathematics C
Work Program
To accompany the Mathematics B textbook:
Mathematics for Queensland Years 11B and 12B
Important notes:
While this program has been submitted to the B.S.S.S.S. it has not yet been accredited.
For schools wishing to use numerical assessment in Modelling and Problem Solving an example of a marking rubric is given in the addendum of the Mathematics B work program.
It should be noted that marking rubrics, although common in the USA, have not been generally used in Queensland.
For updates and corrections, please visit mathematics-for-.
|TABLE OF CONTENTS |
| | |
| CONTENTS |PAGE |
| | |
|1 RATIONALE | 1 |
|2 GLOBAL AIMS | 2 |
|3 GENERAL OBJECTIVES | 2 |
|4 COURSE ORGANISATION | 6 |
|5 LEARNING EXPERIENCES | 16 |
|6 ASSESSMENT | 17 |
| 6.1 Assessment Techniques | 17 |
| 6.2 Assessment Outline | 19 |
| 6.3 Assigning Standards | 19 |
|7 STUDENT PROFILES | 27 |
|8 DETERMINING EXIT LEVELS OF ACHIEVEMENT | 30 |
|APPENDIX 1 Sample Sequence of Work | 31 |
|APPENDIX 2 Focus Statements and Learning Experiences | 32 |
|APPENDIX 3 Equity Statement | 41 |
1 RATIONALE
Mathematics is an integral part of a general education. It enhances both an understanding of the world and the quality of participation in a rapidly changing society.
Mathematics has been central to nearly all major scientific and technological advances. Many of the developments and decisions made in industry and commerce, in the provision of social and community services, and in government policy and planning, rely on the use of mathematics. Consequently, the range of career opportunities requiring and/or benefiting from an advanced level of mathematical expertise is rapidly expanding. For example, mathematics is increasingly important in health and life sciences, biotechnology, environmental science, economics, and business while remaining crucial in such fields as the physical sciences, engineering, accounting, computer science and the information technology areas.
Mathematics C aims to provide opportunities for students to participate more fully in life-long learning, to develop their mathematical potential, and to build upon and extend their mathematics. It is extremely valuable for students interested in mathematics. Students studying Mathematics C in addition to Mathematics B gain broader and deeper mathematical experiences that are very important for future studies in areas such as the physical sciences and engineering. They are also significantly advantaged in a wide range of areas such as finance, economics, accounting, information technology and all sciences. Mathematics C provides the opportunity for student development of:
← knowledge, procedures and skills in mathematics
← mathematical modelling and problem-solving strategies
← the capacity to justify and communicate in a variety of forms.
Mathematics students should recognise the dynamic nature of mathematics through the subject matter of Mathematics C which includes the concepts and application of matrices, vectors, complex numbers, structures and patterns, and the practical power of calculus. In the optional topics, students may also gain knowledge and skills in topics such as conics, dynamics, statistics, numerical methods, exponential and logarithmic functions, number theory, and recent developments in mathematics.
Mathematics has provided a basis for the development of technology. In recent times, the uses of mathematics have developed substantially in response to changes in technology. The more technology is developed, the greater is the level of mathematical skill required. Students must be given the opportunity to appreciate and experience the power which has been given to mathematics by this technology. Such technology should be used to help students understand mathematical concepts, allowing them to “see” relationships and graphical displays, to search for patterns and recurrence in mathematical situations, as well as to assist in the exploration and investigation of purely mathematical, real and life-like situations.
The intent of Mathematics C is to encourage students to develop positive attitudes towards mathematics by an approach involving exploration, investigation, problem solving and application in a variety of contexts. Of importance is the development of student thinking skills, as well as student recognition and use of mathematical structures and patterns. Students will be encouraged to model mathematically, to work systematically and logically, to conjecture and reflect, to prove and justify, and to communicate with and about mathematics.
The subject is designed to raise the level of competence and confidence in using mathematics, through aspects such as analysis, proof and justification, rigour, mathematical modelling and problem solving. Such activities will equip students well in more general situations, in the appreciation of the power and diversity of mathematics, and provide a very strong basis for a wide range of further mathematics studies.
Mathematics C provides opportunities for the development of the key competencies in situations that arise naturally from the general objectives and learning experiences of the subject. The seven key competencies are: collecting, analysing and organising information; communicating ideas and information; planning and organising activities; working with others and in teams; using mathematical ideas and techniques; solving problems; using technology. (Refer to Integrating the Key Competencies into the Assessment and Reporting of Student Achievement in Senior Secondary Schools in Queensland, published by QBSSSS in 1997.)
The Rockhampton Grammar School is a non denominational, coeducational, day and boarding school with a total school population of approximately 1000 students from Years 1 to 12. The school has a boarding population in excess of 400. The catchment area is the rural and mining areas to the west and north west of Rockhampton.
The school tends to be an academic school and offers Mathematics A, Mathematics B and Mathematics C. The S.A.S. Mathematics Courses are not offered by The Rockhampton Grammar School. Almost all students completing Mathematics B proceed to tertiary studies. Consequently the Mathematics C course offered by the school is designed to satisfy this outcome. Technologically, the school is well resourced with extensive on line facilities available to all students. In addition all students are required to own a Casio 9850 calculator.
2 GLOBAL AIMS
Having completed the course of study, students of Mathematics C should:
← be able to recognise when problems are suitable for mathematical analysis and solution, and be able to attempt such analysis or solution with confidence
← be able to visualise and represent spatial relationships in both two and three dimensions
← have experienced diverse applications of mathematics
← have positive attitudes to the learning and practice of mathematics
← comprehend mathematical information which is presented in a variety of forms
← communicate mathematical information in a variety of forms
← be able to benefit from the availability of a wide range of technologies
← be able to choose and use mathematical instruments appropriately
← be able to recognise functional relationships and dependent applications
← have significantly broadened their mathematical knowledge and skills
← have increased their understanding of mathematics and its structure through the depth and breadth of their study.
3 GENERAL OBJECTIVES
3.1 Introduction
The general objectives of this course are organised into four categories:
← Knowledge and procedures
← Modelling and problem solving
← Communication and justification
← Affective.
3.2 Contexts
The categories of Knowledge and procedures, Modelling and problem solving, and Communication and justification incorporate contexts of application, technology, initiative and complexity. Each of the contexts has a continuum for the particular aspect of mathematics it represents. Mathematics in a course of study developed from this syllabus must be taught, learned and assessed using a variety of contexts over the two years. It is expected that all students are provided with the opportunity to experience mathematics along the continuum within each of the contexts outlined below.
Application
Students must have the opportunity to recognise the usefulness of mathematics through its application, and the beauty and power of mathematics that comes from the capacity to abstract and generalise. Thus students’ learning experiences and assessment programs must include mathematical tasks that demonstrate a balance across the range from life-related through to pure abstraction.
Technology
A range of technological tools must be used in the learning and assessment experiences offered in this course. This ranges from pen and paper, measuring instruments and tables through to higher technologies such as graphing calculators and computers. The minimum level of higher technology appropriate for the teaching of this course is a graphing calculator.
Initiative
Learning experiences and the corresponding assessment must provide students with the opportunity to demonstrate their capability when dealing with tasks that range from routine and well rehearsed through to those that require demonstration of insight and creativity.
Complexity
Students must be provided with the opportunity to work on simple, single-step tasks through to tasks that are complex in nature. Complexity may derive from either the nature of the concepts involved or from the number of ideas or techniques that must be sequenced in order to produce an appropriate conclusion.
3.3 Objectives
The general objectives for each of the categories in section 3.1 are detailed below. These general objectives incorporate several key competencies. The first three categories of objectives, Knowledge and procedures, Modelling and problem solving, and Communication and justification, are linked to the exit criteria in section 7.3.
3.3.1 Knowledge and Procedures
The objectives of this category involve the recall and use of results and procedures within the contexts of application, technology, initiative and complexity. (see section 3.2)
By the conclusion of the course, students should be able to:
← recall definitions and results
← access and apply rules and techniques
← demonstrate number and spatial sense
← demonstrate algebraic facility
← demonstrate an ability to select and use appropriate technology such as calculators, measuring instruments, geometrical drawing instruments and tables
← demonstrate an ability to use graphing calculators and/or computers with selected software in working mathematically
← select and use appropriate mathematical procedures
← work accurately and manipulate formulae
← recognise some tasks may be broken up into smaller components
← transfer and apply mathematical procedures to similar situations
← understand the nature of proof.
3.3.2 Modelling and Problem Solving
The objectives of this category involve the use of mathematics in which the students will model mathematical situations and constructs, solve problems and investigate situations mathematically within the contexts of application, technology, initiative and complexity. (see section 3.2)
By the conclusion of the course, students should be able to demonstrate the category of modelling and problem solving through:
Modelling
← understanding that a mathematical model is a mathematical representation of a situation
← identifying the assumptions and variables of a simple mathematical model of a situation
← forming a mathematical model of a life-related situation
← deriving results from consideration of the mathematical model chosen for the particular situation
← interpreting results from the mathematical model in terms of the given situation
← exploring the strengths and limitations of a mathematical model and modifying the model as appropriate.
Problem solving
← interpreting, clarifying and analysing a problem
← using a range of problem solving strategies such as estimating, identifying patterns, guessing and checking, working backwards, using diagrams, considering similar problems and organising data
← understanding that there may be more than one way to solve a problem
← selecting appropriate mathematical procedures required to solve a problem
← developing a solution consistent with the problem
← developing procedures in problem solving.
Investigation
← identifying and/or posing a problem
← exploring the problem and from emerging patterns creating conjectures or theories
← reflecting on conjectures or theories making modifications if needed
← selecting and using problem-solving strategies to test and validate any conjectures or theories
← extending and generalising from problems
← developing strategies and procedures in investigations.
3.3.3 Communication and Justification
The objectives of this category involve presentation, communication (both mathematical and everyday language), logical arguments, interpretation and justification of mathematics within the contexts of application, technology, initiative and complexity. (see section 3.2)
Communication
By the conclusion of the course, students should be able to demonstrate communication through:
← organising and presenting information
← communicating ideas, information and results appropriately
← using mathematical terms and symbols accurately and appropriately
← using accepted spelling, punctuation and grammar in written communication
← understanding material presented in a variety of forms such as oral, written, symbolic, pictorial and graphical
← translating material from one form to another when appropriate
← presenting material for different audiences, in a variety of forms such as oral, written, symbolic, pictorial and graphical
← recognising necessary distinctions in the meanings of words and phrases according to whether they are used in a mathematical or non-mathematical situation.
Justification
By the conclusion of this course, the student should be able to demonstrate justification through:
← developing logical arguments expressed in everyday language, mathematical language or a combination of both, as required, to support conclusions, results and/or propositions
← evaluating the validity of arguments designed to convince others of the truth of propositions
← justifying procedures used
← recognising when and why derived results to a given problem are clearly improbable or unreasonable
← recognising that one counter example is sufficient to disprove a generalisation
← recognising the effect of assumptions on the conclusions that can be reached
← deciding whether it is valid to use a general result in a specific case
← recognising that a proof may require more than verification of a number of instances
← using supporting arguments, when appropriate, to justify results obtained by calculator or computer
← using different methods of proof.
3.3.4 Affective
Affective objectives refer to the attitudes, values and feelings which this subject aims at developing in students. Affective objectives are not assessed for the award of exit levels of achievement.
By the conclusion of the course, students should appreciate the:
← diverse applications of mathematics
← precise language and structure of mathematics
← uncertain nature of the world, and be able to use mathematics to assist in making informed decisions in life-related situations
← diverse and evolutionary nature of mathematics through an understanding of its history
← wide range of mathematics-based vocations
← contribution of mathematics to human culture and progress
← power and beauty of mathematics.
4 COURSE ORGANISATION
This school will continue to strive for educational equity by providing a curriculum which in subject matter, language, methodology, learning experiences and assessment instruments meets the educational needs and entitlements of all students. This program reflects school policy on equity in education (see Appendix 3), and teachers should implement the course with consideration of these issues.
4.1 Course Description
The course is intended to offer to students an integrated, spiralling curriculum. Although all topics are not covered in every semester, the concepts dealt with will be drawn upon in subsequent topics.
In allocating time to units, consideration has been given to the maintenance of basic skills and mathematical techniques as appropriate. The revision of basic mathematics should be done when needed and the maintenance of mathematical techniques should be ongoing throughout the course.
At the time of writing, Mathematics C has allocated 6 periods or about 230 minutes per week.
A brief summary of the integrated sequence of topics and a more detailed sequence are provided in the following pages. A summary of the focus statements for each topic, which should be referred to each time a topic is studied, is provided as Appendix 2. This sequence has been designed to:
• allow for the gradual development of the objectives over time
• ensure that pre-requisite material from Mathematics C has been covered at appropriate times
• allow for the use of technology wherever possible, particularly graphics calculators and computer software.
The School offers the two optional topics, Conics and Dynamics.
Although the sequence on the following pages does not show explicitly the integration of the syllabus topics, due to the difficulty of doing this in a limited space, teachers will be expected to integrate the topics wherever possible to ensure that students do not see mathematics as a series of discrete topics. Students will be encouraged to select from all their skills when problem solving.
While the sequence provided shows the order in which topics will generally be covered, the school reserves the right to modify this with specific cohorts to suit the specific conditions that year. The same work would, however, be covered for inclusion at Monitoring and Verification.
Focus Statement
When planning units of work, the following detailed sequence should be considered in conjunction with Appendix II which contains the focus statements, subject matter and learning experiences linked to the subject matter for each topic.
The topic sequence for Mathematics B and Mathematics C have been developed together, to ensure that significant pre-requisite material is taught in Mathematics B before being required in Mathematics C.
4.2 Detailed Sequence
SEMESTER 1
|Sequence |Topic |Time |Basic Skills and Maintenance |Subject Matter |
| |Real and Complex | |Calculation and estimation with & without |structure of the real number system including: |
|1 |Numbers |4 hours |instruments |rational numbers |
| | | |basic algebraic manipulations |irrational numbers (SLEs 2, 9, 10) |
| | | |absolute value |simple manipulation of surds |
| |Matrices and | |Basic algebraic manipulations |definition of a matrix as data storage and as a mathematical tool (SLEs 1B7) |
|2 |Applications I |12 hours | |dimension of a matrix |
| | | | |matrix operations |
| | | | |addition |
| | | | |transpose |
| | | | |inverse |
| | | | |multiplication by a scalar |
| | | | |multiplication by a matrix (SLEs 1B7, 13, 14, 15) |
| |Real and Complex | |Review Topic 1 (sem 1) |definition of complex numbers including standard and trigonometrical (modulus-argument) form |
|3 |Numbers II |12 hours |plotting points using cartesian co-ordinates |(SLEs 1, 2) |
| | | |the formula for zeros of a quadratic equation |algebraic representation of complex numbers in Cartesian, trigonometric and polar form (SLEs 3, |
| | | |identities, linear equations and inequations |4) |
| | | | |geometric representation of complex numbers—Argand diagrams (SLE 4) |
| | | | |operations with complex numbers including addition, subtraction, scalar multiplication, |
| | | | |multiplication of complex numbers, conjugation (SLEs 1B8, 12) |
SEMESTER 1 (CONT.)
|Sequence |Topic |Time |Basic Skills and Maintenance |Subject Matter |
|4 |Introduction to Groups |8 hours |Basic algebraic manipulations |Concepts of: |
| | | | |closure |
| | | | |associativity |
| | | | |identity |
| | | | |inverse (suggested learning experiences (SLEs) 1Β8) |
| | | | |definition of a group (SLEs 1Β8) |
|5 |Vectors and |10 hours |Review Trigonometry from Mathematics B |For vectors describing situations involving magnitude and direction: |
| |Applications I | |absolute value |definition of a vector (see appendix 2) |
| | | |basic algebraic manipulations |relationship between vectors and matrices (SLE 2) |
| | | | |two and three dimensional vectors and their algebraic and geometric representation (SLEs 3, 4, |
| | | | |6) |
| | | | |operations on vectors including: (SLEs 3Β5, 9, 10, 13) |
| | | | |addition and multiplication by a scalar |
| | | | |scalar product of two vectors (SLEs 2, 8) |
| | | | |unit vectors |
| | | | |resolution of vectors into components acting at right angles to each other (SLEs 3Β5, 11, 12) |
| | | | |calculation of the angle between two vectors |
| | | | |Applications of vectors in both life-related and purely mathematical situations(SLEs |
|6 |Structures and Patterns|12 hours |Calculation and estimation with & without |permutations and combinations and their use in purely mathematical & life-related situations |
| |I | |instruments |(SLEs 7, 8, 9, 14, 15) |
| | | |basic algebraic manipulations | |
SEMESTER 2
|Sequence |Topic |Time |Basic Skills and Maintenance |Subject Matter |
|1 |Vectors and |8 hours |Review Topic 5 (sem 1) |For vectors as a one-dimensional array: |
| |Applications II | |review Topic 2 (sem 1) |definition of a vector |
| | | | |relationship between vectors and matrices (SLE 1) |
| | | | |operations on vectors including: |
| | | | |addition |
| | | | |multiplication by a scalar |
| | | | |scalar product of two vectors (SLE 1) |
| | | | |simple life-related applications of vectors (SLEs 1, 3) |
|2 |Matrices and |14 hours |Review Topic 2 (sem 1) |definition & properties of the identity matrix (SLEs 1, 3, 15) |
| |Applications II | |linear equations |group properties of 2 x 2 matrices (SLE 5) |
| | | |formula for the zeros of a quadratic equation |determinant of a matrix (SLE 3) |
| | | | |singular and non-singular matrices (SLE 1) |
| | | | |solution of systems of homogeneous and non-homogeneous linear equations using matrices (SLEs 1, |
| | | | |6) |
|3 |Structures and Patterns|18 hours |Review AP and GP work from Maths B |sum to infinity of a geometric progression (SLEs 1, 2) |
| |II | |rates, percentages, ratio and proportion |purely mathematical and life-related applications of arithmetic and geometric progressions (SLEs|
| | | |basic algebraic manipulations |10, 11, 12) |
| | | |summation notation |sequences and series other than arithmetic and geometric (SLEs 3, 4, 16) |
| | | | |permutations and combinations and their use in purely mathematical & life-related situations |
| | | | |(SLEs 7, 8, 9, 14, 15) |
| | | | |recognition of patterns in well known structures including Pascal’s Triangle and Fibonacci |
| | | | |sequence (SLEs 5, 6, 13) |
| | | | |applications of patterns (SLEs 1B11, 17) |
| | | | |use of the method of finite differences (SLEs 4, 18) |
| | | | |proof by induction (SLE 4) |
SEMESTER 2 (CONT.)
|Sequence |Topic |Time |Basic Skills and Maintenance |Subject Matter |
| | | | | |
|4 |Matrices and |10 hours |Review Topics 2, 5 (sem 1) |applications of matrices in both life-related and purely mathematical situations (SLEs 1B12) |
| |Applications III | | |relationship between matrices and vectors (SLEs 1, 6, 7, 12) |
| | | | | |
|5 |Real and Complex Number|10 hours |Review Topic 3 (sem 1) |roots of complex numbers (SLE 6) |
| |System III | | |use of complex numbers in proving trigonometric identities |
| | | | |powers of complex numbers including de Moivre’s Theorem |
| | | | |simple, purely mathematical applications of complex numbers (SLEs 6, 7, 8, 11, 12) |
| | | | |proof by mathematical induction (SLE 3) |
SEMESTER 3
|Sequence |Topic |Time |Basic Skills and Maintenance |Subject Matter |
|1 |Dynamics I |16 hours |Metric measurement (mass, length) |Newton’s laws of motion in vector form applied to objects of constant mass (SLEs 2B15) |
| | | |rates, percentages, ratio and proportion |application of the above to: |
| | | |linear and quadratic equations |straight line motion in a horizontal plane with variable force |
| | | | |vertical motion under gravity with and without air resistance |
| | | | |projectile motion without air resistance |
|2 |Conics |16 hours |Basic algebraic manipulations |concept of a locus, directrix and focal point (SLEs 1B17) |
| | | |absolute value |circle as a locus in: |
| | | |plotting points using Cartesian co-ordinates |Cartesian form [pic] |
| | | |review topic 3 (sem 1) |complex number form [pic](SLEs 1B6, 8, 11, 17) |
| | | | |definition of eccentricity e |
| | | | |ellipse as a locus in: |
| | | | |Cartesian form [pic] |
| | | | |parametric form x = a cos( , y = b sin( |
| | | | |complex number form [pic] where s >*∗ p - q∗* |
| | | | |hyperbola as a locus in: |
| | | | |Cartesian form [pic]; [pic] |
| | | | |complex number form[pic]; where 0 < s < ∗*p - q∗* |
| | | | |parabola as a locus in: |
| | | | |Cartesian form [pic] |
| | | | |In all cases above a, b, c, d, p, q and s are constants |
SEMESTER 3 (CONT.)
|Sequence |Topic |Time |Basic Skills and Maintenance |Subject Matter |
|3 |Vectors and |12 hours |Review Topic 5 (sem 1) |scalar product of two vectors (SLEs 2, 8) |
| |Applications III | | |vector product of two vectors (SLE 7) |
| | | | |unit vectors |
| | | | |resolution of vectors into components acting at right angles to each other (SLEs 3Β5, 11, 12) |
| | | | |calculation of the angle between two vectors |
| | | | |applications of vectors in both life-related and purely mathematical situations (SLEs 1–13) |
|4 |Conics |16 hours |Review Topic 2 (sem 3) |Polar co-ordinate form of conics |
| | | | |Parametric form of conics including: |
| | | | |x =a cosθ , y = a sin θ |
| | | | |x =a cosθ , y = b sin θ |
| | | | |[pic];[pic] |
| | | | |parametric form [pic] |
| | | | |where a, and b, are constants |
| | | | |simple applications of conics. |
SEMESTER 4
|Sequence |Topic |Time |Basic Skills and Maintenance |Subject Matter |
|1 |Calculus I |16 hours |Rates of change |approximating small changes in functions using derivatives (SLEs 1, 2) |
| | | |exponential and log functions |life-related applications of simple, linear, first order differential equations with constant |
| | | |differentiation |coefficients (SLEs 3, 8, 13, 14) |
| | | |integral as a sum (review all above from Maths |solution of simple, linear, first order differential equations with constant coefficients (SLEs 3,|
| | | |B topics) |6, 8, 10, 13, 14, 15) |
|2 |Dynamics II |16 hours |Review Topic 1 (sem 3) |derivatives and integrals of vectors (SLEs 1, 2, 3, 13) |
| | | |review related calculus topics from Maths B |Newton’s laws of motion in vector form applied to objects of constant mass (SLEs 2Β15) |
| | | | |application of the above to: |
| | | | |straight line motion in a horizontal plane with variable force |
| | | | |vertical motion under gravity with and without air resistance |
| | | | |projectile motion without air resistance |
| | | | |simple harmonic motion (derivation of the solutions to differential equations is not required) |
| | | | |circular motion with uniform angular velocity (SLEs 4Β12,14, 5) |
|3 |Calculus II |16 hours |Review all necessary previous calculus topics |integrals of the form: |
| | | |calculation and estimation with and without |simple integration by parts (SLE 4) |
| | | |instruments |development and use of Simpson’s rule (SLEs 5, 6, 7, 9,10,11,12, 15) |
SEMESTER 4 (CONT.)
|Sequence |Topic |Time |Basic Skills and Maintenance |Subject Matter |
|4 |Applications |7 hours | |Life-related and pure mathematical applications for the following topics |
| | | | |- complex numbers |
| | | | |- matrices |
| | | | |- vectors |
| | | | |- calculus |
| | | | |- dynamics and conics |
| | | | |- group theory. |
4.3 Time Allocation to Topics (in hours)
|TOPIC |SEMESTERS |Topic |Notional |
| | |Time |Time |
| |1 |2 |3 |4 | | |
| | | | | | | |
|Introduction to Groups |Unit 1 8 | | |Unit 2 7 |15 |7 |
| | | | | | | |
|Real and Com-plex Number |Unit 1 4 |Unit 3 10 | | |26 |25 |
|Systems |Unit 2 12 | | | | | |
| | | | | | | |
|Matrices and Applications |Unit 1 12 |Unit 2 14 | | |36 |30 |
| | |Unit 3 10 | | | | |
| | | | | | | |
|Vectors and Applications |Unit 1 10 |Unit 2 8 |Unit 3 12 | |30 |30 |
| | | | | | | |
|Calculus | | | |Unit 1 16 |32 |30 |
| | | | |Unit 2 16 | | |
| | | | | | | |
|Structures and Patterns |Unit 1 12 |Unit 2 18 | | |30 |30 |
| | | | | | | |
|Conics | | |Unit 1 16 | |32 |30 |
| | | |Unit 2 16 | | | |
| | | | | | | |
|Dynamics | | |Unit 1 16 |Unit 2 16 |32 |30 |
| | | | | | | |
|TOTAL |58 |60 |60 |55 |233 | |
4.4 Technology
Throughout the course of study, higher technologies such as graphics calculators and appropriate computer software will be used to reduce computational and algebraic drudgery and to aid in investigation and development of concepts. Such technology will also be used in assessment tasks where appropriate.
4.5 Language Education
Language is the means by which meaning is constructed and shared and communication is effected. It is the central means by which teachers and students learn. Mathematics C requires students to use language in a variety of ways – mathematical, spoken, written, graphical, symbolic. The responsibility for developing and monitoring students’ abilities to use effectively the forms of language demanded by this course rests with the teachers of mathematics. This responsibility includes developing students’ abilities to:
← select and sequence information
← manage the conventions related to the forms of communication used in Mathematics C (such as short responses, reports, multi-media presentations, seminars)
← use the specialised vocabulary and terminology related to Mathematics C
← use language conventions related to grammar, spelling, punctuation and layout.
The learning of language is a developmental process. When writing, reading, questioning, listening and talking about mathematics, teachers and students should use the specialised vocabulary related to Mathematics C. Students should be involved in learning experiences that require them to comprehend and transform data in a variety of forms and, in so doing, use the appropriate language conventions. Some language forms may need to be explicitly taught if students are to operate with a high degree of confidence within mathematics.
Assessment instruments will use format and language that are familiar to students. They will be taught the appropriate language skills to interpret questions accurately and to develop coherent, logical and relevant responses. Attention to language education within Mathematics C should assist students to meet the language components of the exit criteria, especially the Communication and justification criterion.
55 LEARNING EXPERIENCES
Throughout the course, students will be exposed to a variety of learning experiences to help them achieve the general objectives. These learning experiences will also incorporate the contexts of application, technology, initiative and complexity. Some of these are shown below in general terms:
• traditional methods of exposition, reinforcement, discussion
• investigations
• individual and group work requiring research, problem solving and modelling either as supervised school activities or as unsupervised out-of-class activities
• computer software and graphics calculators integrated into the course where appropriate.
Included in Appendix 2 are the suggested learning experiences from the syllabus linked to the subject matter. This information, combined with the focus statements should be used when designing a specific learning program and to give a guide to the depth of treatment required.
Included in Appendix 1 is a sample unit of work showing the range of learning experiences which might be included to develop the objectives of the course. While the actual learning experiences provided to students will vary from one topic to another and from one year to another, this school will provide a similar variety of learning experiences to ensure that all the objectives of the course are developed.
6 ASSESSMENT
Student achievement will be judged on the following three exit criteria:
← Knowledge and procedures
← Modelling and problem solving
← Communication and justification.
A system of continuous assessment is used by this school to gather data on student performance during the course. After each assessment instrument has been administered the students' results are recorded on an individual student profile.
This profiling of student performance allows both teachers and students to review student's performance and allows students to become aware of the criteria in which they need to improve.
Assessment instruments will be both formative and summative. Formative instruments will be administered in year 11. These instruments will be similar in style to those administered in year 12 to allow students to become familiar with different types of assessment. Year 12 instruments will provide the summative data for awarding exit levels of achievement for students who complete the course. Formative results will be referred to in the allocation of a level of achievement when a student exits before completion of the course and for awarding interim levels of achievement for reporting on student's progress and for Monitoring purposes at the end of year 11.
Students’ responses to at least one, and probably two, assessment tasks from semester IV will be included in the verification submission in October.
6.1 Assessment Techniques
Assessment techniques other than traditional written tests or examinations will be included in the assessment program at least twice each year.
A variety of assessment techniques will be used and might include those described below.
Extended modelling and problem-solving tasks
This form of assessment may require a response that involves mathematical language, graphs and diagrams, and could involve a significant amount of conventional English. It will typically be in written form, a combination of written and oral forms, or multimedia forms.
The activities leading to an extended modelling and problem-solving task could be done individually and/or in groups, and the extended modelling and problem-solving task could be prepared in class time and/or in students’ own time.
Reports
A report is typically an extended response to a task such as:
← an experiment in which data is collected, analysed and modelled
← a mathematical investigation
← a field activity
← a project.
A report could comprise such forms as:
← a scientific report
← a proposal to a company or organisation
← a feasibility study.
The activities leading to a report could be done individually and/or in groups and the report could be prepared in class time and/or in students’ own time. A report will typically be in written form, or a combination of written and oral multimedia forms.
The report will generally include an introduction, analysis of results and data, conclusions drawn, justification, and, when necessary, a bibliography, references and appendices.
Supervised Tests
Supervised tests commonly include tasks requiring quantitative and/or qualitative responses. Supervised tests could include a variety of items such as:
← multiple-choice questions
← questions requiring a short response:
▪ in mathematical language and symbols
▪ in conventional written English, ranging in length from a single word to a paragraph
← questions requiring a response including graphs, tables, diagrams and data
← questions requiring an extended answer where the response includes:
▪ mathematical language and symbols
▪ conventional written English, more than one paragraph in length
▪ a combination of the above.
The assessment instruments will reflect an appropriate emphasis according to time spent on a topic and the depth of treatment undertaken.
Conditions of Implementation
To ensure that the data gained from our assessment program is valid and reliable, we will use a variety of conditions of implementation which may include supervised tests, supervised open-book tests (Modelling and problem solving only), assignments done in class, assignments done at home, and group projects. When assignments are done entirely in the students' own time, mechanisms will be put in place to ensure ownership of the work presented. These mechanisms may include asking students to sign declarations of "ownership", acknowledging any help received or "random" oral questioning of students to check understanding of work presented. The importance of intellectual integrity and honesty will be stressed to students.
Where written tests are indicated these will generally be of the type:
a) A test assessing Knowledge and procedures and Communication and justification
b) A test of up to 4 questions assessing Modelling and problem solving and Communication and justification (this may need to be split into 2 parts to allow for sufficient thinking time).
(c) A test consisting of a number of items where part (a) of the item assesses specific knowledge and procedures followed by a part (b) which is modelling or problem solving which employs specifically part (a).
In all written tests, sufficient time will be allocated to ensure satisfactory coverage of topics studied, and to allow students adequate time to demonstrate their proficiency, thus ensuring time is not a hidden criterion.
The specific conditions appropriate to each instrument will be provided to the panel at Monitoring and Verification.
6.2 Assessment Outline
This school will have a minimum of 4 assessment instruments each semester. At least two of these tasks will be tests and at least one will be a task which is not a traditional written test.
All instruments will assess Communication and justification and all three criteria will be assessed on written tests (see note above). In any one semester, each criterion will be assessed on at least two instruments.
The sample profile (page 29) provides a possible assessment program that this school may offer. The actual program will be similar in terms of balance and variety.
6.3 Assigning Standards
(I) Knowledge and Procedures
The assessment of this criterion involves the recall and use of results and procedures within the contexts of application, technology and complexity. The assessment package will sample all topics and cover the continuum of the contexts.
The assessment package will enable students to demonstrate their ability in the general objectives of this category and the marking schemes/criteria sheets will provide appropriate data.
For each instrument assessing Knowledge and procedures either:
(a) Marks will be awarded and cut-offs applied to arrive at an A+ to E- grade; both the marks awarded and the grade will be recorded on the profile.
The actual cut-offs used will depend on the relative difficulty of the instrument and will be supplied, with the instrument, to the panel at Monitoring or Verification.
or
(b) Verbal standards will be used to arrive at an A to E result which will be recorded on the profile (again A+ to E- will generally be used).
Descriptors will be explicitly stated on the criteria sheet of the assessment item as they relate to the A,B,C,D or E standards.
This method may apply more appropriately to assessment of Practical Tests, Assignments, Projects or Oral presentations, but may be used throughout.
For tests assessing Modelling and problem solving:
A student’s responses to modelling and problem solving tests may provide some valuable data on the student’s ability to use procedures as they may have shown an ability to use some procedures that they may have not previously shown. This information will be used for borderline students and will be recorded on their profile in brackets. It will not be used in the regular decision making process as this may disadvantage students who are not able to start these problems and therefore do not know which procedures to use. Using this data in the regular process may also skew the level awarded as the procedures used in these tests may not cover the range of topics or contexts.
Awarding Summary Results
Summary results at the end of each semester or exit, are allocated according to the following guidelines describing minimum standards.
|Summary Result |Performance |
|A |Consistently A’s |
|B |Consistently B’s |
|C |Consistently C’s |
|D |Consistently D’s |
|E | All E-'s |
(i) "Latest and fullest" will be considered in awarding summary or Exit levels. For example, a gradual increase in student performance means that a student may be awarded a result reflecting later performance as long as the improvement is sustained.
(ii) Trade-offs might be applied to arrive at the summary result e.g. C,C,A,C,A,A,A,C could equate to a "B" under such a procedure.
(iii) Final A to E results must reflect the relevant descriptors as summarised below:
Knowledge and Procedures
|STANDARD |DESCRIPTOR |
| | |
|A |The overall quality of a student’s achievement across the full range within the contexts of application, technology and|
| |complexity, and across topics, consistently demonstrates: |
| |accurate recall, selection and use of definitions, results and rules |
| |appropriate use of technology |
| |appropriate selection and accurate and proficient use of procedures |
| |effective transfer and application of mathematical procedures |
|B |The overall quality of a student’s achievement across a range within the contexts of application, technology and |
| |complexity, and across topics, generally demonstrates: |
| |accurate recall, selection and use of definitions, results and rules |
| |appropriate use of technology |
| |appropriate selection and accurate use of procedures. |
|C |The overall quality of a student’s achievement in the contexts of application, technology and complexity, generally |
| |demonstrates: |
| |accurate recall and use of basic definitions, results and rules |
| |appropriate use of some technology |
| |generally accurate use of basic procedures |
|D |The overall quality of a student’s achievement in the contexts of application, technology and complexity, generally |
| |demonstrates: |
| |accurate recall and use of basic definitions, results and rules |
| |appropriate use of some technology |
|E |The overall quality of a student’s achievement rarely demonstrates knowledge and use of procedures |
(II) Modelling and Problem Solving
This criterion incorporates:
(a) Modelling
(b) Problem solving
(c) Investigations
The assessment package will enable students to demonstrate their ability in the general objectives of all three areas but this information will not be collected or recorded separately.
Assessment instruments for this criterion may contain closed items with a limited number of correct answers or open-ended items and the package will cover the continuum of all contexts.
Awarding Standards on Tests
Tests assessing performance on this criterion will contain a number of items ranging from the lower end of the complexity and initiative contexts through to the upper end and these items will usually be closed. In these situations, the following general descriptors will be used for rating each individual item on an instrument.
|RATING |DESCRIPTOR |
|A* |The problem is solved and the solution shows evidence of interpretation, analysis, identification of assumptions, selection,|
| |use and synthesis of strategies and procedures where appropriate. |
|A |An appropriate method is applied to yield a valid solution. Although a correct solution has not been achieved, the response|
| |shows evidence of appropriate interpretation, analysis, identification of assumptions, selection, use and synthesis of |
| |strategies and procedures |
|B |The student has made substantial progress towards a rational solution and the response shows evidence of appropriate |
| |interpretation, analysis, identification of assumptions and selection of strategies and procedures |
|C |The student has made a reasonable interpretation of the problem and has selected strategies and/or procedures appropriate to|
| |the problem. |
|D |The student has followed basic procedures and/or used strategies appropriate to the problem. |
|E |The student is unable to begin the problem or hands in work that is meaningless. |
| |OR The student has provided only an answer. |
When these general descriptors are not appropriate for a particular item, specific descriptors will be written which reflect the student's achievement of the objectives of Modelling and problem solving. This will generally be necessary for open ended assessment items. These descriptors will be stated on the assessment instruments.
Once each item has been rated, an overall grade will be awarded for the assessment instrument and this grade will be recorded on the student profile along with the rating for each item. The grade awarded for each instrument will reflect the requirement for the mathematical thinking to be displayed across the range within the contexts. A+ to E- will be used in recording this information.
The minimum requirements for A to E will differ for each instrument and will be included for the panel in all submissions. An example is included here by way of clarification.
Example:
A test contains four questions, ranging from a well-rehearsed modelling problem through to a question at the upper end of the complexity and/or initiative continuum
|RATING |MINIMUM REQUIREMENT |
|A* | 1 x a* plus 2 x a’s |
|A | 3 x a’s |
|B | 2 x b’s |
|C | 2 x c’s |
|D | 2 x d’s |
|E | All e’s |
*B would indicate that the student solved at least one question.
NOTE: Tests will be set so that the questions are in order of increasing difficulty. It is acknowledged that the level of complexity is a continuum and within the questions considered complex, there should be a range of difficulty from those that a low HA student should be able to solve to those which would challenge VHA students. In addition, a hash (#) appearing below an item rating indicates that the item is regarded to be a more substantial item than others in an assessment package. For example a “#b” could contribute more towards the student’s final grade than (say) an “a”.
Awarding Standards on Assessment other than Tests
As these tasks will generally be open-ended, task-specific criteria will be written which allow data to be gathered on the achievement of the general objectives of Modelling and problem solving and a single level recorded on the profile. In general, these tasks will give the students the opportunity to explore the strengths and limitations of the model and to refine the model.
Awarding Summary Results
Summary results at the end of each semester or exit, are allocated according to the following guidelines describing minimum standards.
|Summary Result |Performance |
|A |Generally A’s (with some *’s) |
|B |Generally B’s (with synthesis in some items) |
|C |Generally C’s |
|D |Generally D’s |
|E |All E’s |
Note: This mechanism shows us students who may have achieved the syllabus requirements for this criterion but the final decision must be made on the overall quality of the work.
Final A to E results must reflect the relevant descriptors as summarized below:
Modelling and Problem Solving
|STANDARD |DESCRIPTOR |
|A |The overall quality of a student’s achievement across the full range within each context, and across topics generally|
| |demonstrates mathematical thinking which includes: |
| |interpreting, clarifying and analysing a range of situations, identifying assumptions and variables |
| |selecting and using effective strategies |
| |selecting appropriate procedures required to solve a wide range of problems |
| |appropriate synthesis of procedures and strategies; |
| | |
| |and in some contexts and topics demonstrates mathematical thinking which includes: |
| |synthesis of procedures and strategies to solve problems |
| |initiative and insight in exploring the problem |
| |exploring strengths and limitations of models |
| |refining a model |
| |extending and generalising from solutions. |
|B |The overall quality of a student’s achievement across a range within each context, across topics, generally |
| |demonstrates mathematical thinking which includes: |
| |interpreting, clarifying and analysing a range of situations, identifying assumptions and variables |
| |selecting and using effective strategies |
| |selecting appropriate procedures required to solve a range of problems; |
| | |
| |and in some contexts and topics demonstrates mathematical thinking which includes appropriate synthesis of procedures|
| |and strategies. |
|C |The overall quality of a student’s achievement in all contexts generally demonstrates mathematical thinking which |
| |includes: |
| |interpreting and clarifying a range of situations |
| |selecting strategies and/or procedures required to solve problems. |
|D |The overall quality of a student’s achievement sometimes demonstrates mathematical thinking which includes following |
| |basic procedures and/or using strategies required to solve problems. |
|E |The overall quality of a student’s achievement rarely demonstrates mathematical thinking which includes following |
| |basic procedures and/or using strategies required to solve problems. |
(III) Communication and Justification
Information on student achievement in this criterion will be collected by a global consideration of the communication and justification skills evident in responding to tasks used to assess student performance in Knowledge and procedures and Modelling and problem solving.
(i) A student's communication on each assessment task will be considered globally and rated as A, B, C, D, E or NR reflecting the standards detailed in the table below:
|STANDARD |DESCRIPTOR |
|A |The student consistently demonstrates: |
| |Correct use of mathematical terms and symbols |
| |Working shown and legible (as appropriate to the assessment conditions |
| |Appropriate presentation of relevant information |
| |Arguments which are logically developed to support a logical conclusion |
| |Recognition of the effects of assumptions used |
| |Justification of procedures |
| |Evaluation of the validity of arguments |
|B |The student generally demonstrates: |
| |Correct use of mathematical terms and symbols |
| |Working shown and legible (as appropriate to the assessment conditions |
| |Reasonable presentation of relevant information |
| |Simple arguments are developed to support a conclusion |
| |Justification of procedures |
|C |The student generally demonstrates: |
| |Correct use of basic mathematical terms and symbols |
| |Some working shown and legible (as appropriate to the assessment conditions |
| |Some simple arguments are developed to support arguments |
|D |The student sometimes demonstrates evidence of the use of the basic conventions of language and mathematics |
|E |The student rarely demonstrates evidence of the use of the basic conventions of language and mathematics |
|NR |Insufficient data provided to make a valid judgement. |
(i) The student's result will be recorded on the student profile.
(ii) For some assessment instruments, it may be desirable to write specific descriptors for Communication. These would be provided for students on the task sheet.
Awarding Summary Results
Summary results at the end of each semester or on exit, are allocated according to the following guidelines describing minimum standards.
|Summary Result |Performance |
|A |Consistently A’s |
|B |Consistently B’s |
|C |Generally C’s |
|D |Generally D’s |
|E |All U's |
Final A to E results must reflect the relevant descriptors from the syllabus as summarized in the table below.
Communication and Justification
|STANDARD |DESCRIPTOR |
| | |
|A |The overall quality of a student’s achievement across the full range within each context consistently demonstrates: |
| |accurate and appropriate use of mathematical terms and symbols |
| |accurate and appropriate use of language |
| |collection and organisation of information into various forms of presentation suitable for a given use or audience |
| |use of mathematical reasoning and proof to develop logical arguments in support of conclusions, results and/or |
| |propositions |
| |recognition of the effects of assumptions used |
| |evaluation of the validity of arguments |
| |justification of procedures. |
| | |
|B |The overall quality of a student’s achievement across a range within each context generally demonstrates: |
| |accurate and appropriate use of mathematical terms and symbols |
| |accurate and appropriate use of language |
| |collection and organisation of information into various forms of presentation suitable for a given use or audience |
| |use of mathematical reasoning and proof to develop simple logical arguments in support of conclusions, results and/or|
| |propositions |
| |justification of procedures. |
| | |
|C |The overall quality of a student’s achievement in some contexts generally demonstrates: |
| |accurate and appropriate use of basic mathematical terms and symbols |
| |accurate and appropriate use of basic language |
| |collection and organisation of information into various forms of presentation |
| |use of some mathematical reasoning to develop simple logical arguments in support of conclusions, results and/or |
| |propositions. |
| | |
|D |The overall quality of a student’s achievement sometimes demonstrates evidence of the use of the basic conventions of|
| |language and mathematics. |
| | |
|E |The overall quality of a student’s achievement rarely demonstrates use of the basic conventions of language or |
| |mathematics. |
7 STUDENT PROFILES
As described in the previous section, data gathered on a student's performance on each of the criteria will be recorded on an individual student profile.
N.B. All summary results for each criterion are based on assessment items to date for the whole of the current year (see sample profiles). For example, the summary results for Year 11 incorporate the data for both semesters 1 and 2.
The rules for awarding summary grades described in the previous section are designed to identify students whose work may match the syllabus standards. The work of any students who are identified as borderline, both above and below the minimum, will be compared with the syllabus standards before an exit standard is awarded.
An example of a completed profile is provided (page 29). Pluses and minuses have not been used in the example but may be used if required.
|THE ROCKHAMPTON GRAMMAR SCHOOL | Name: |
|Student Profile |2002 |
|Mathematics C |2003 |
|SEMESTER | |K & P |M & PS |C & J |L OF A |
| |ASSESSMENT | | | | |
| |INSTRUMENT | | | | |
| | |MARKS |RESULT |QUESTION RATING |RESULT |RESULT | |
| |1.2 Mid Sem. Test | | | | | | |
| |1.3 End Sem. Test | | | | | | |
| |1.4 | | | | | | |
| |1.5 | | | | | | |
|Sem. 1 | | | | | | | |
|Result | | | | | | | |
| |2.2 Mid Sem. Test | | | | | | |
| |2.3 End Sem. Test | | | | | | |
| |2.4 | | | | | | |
| |2.5 | | | | | | |
|Sem. 2 | | | | | | | |
|Result | | | | | | | |
| |3.2 Mid Sem. Test | | | | | | |
| |3.3 End Sem. Test | | | | | | |
| |3.4 | | | | | | |
| |3.5 | | | | | | |
|Sem. 3 | | | | | | | |
|Result | | | | | | | |
| |4.2 Mid Sem. Test | | | | | | |
| |4.3 End Sem. Test | | | | | | |
| |4.4 | | | | | | |
| |4.5 | | | | | | |
|Sem. 4 | | | | | | |
|Result | | | | | | |
|VERIFICATION | | | | | | |
|EXIT | | | | | | |
|RANK (End Sem 3) | | | | | | |
|THE ROCKHAMPTON GRAMMAR SCHOOL | Name: |
|Student Profile |2002 |
|Mathematics C |2003 |
|SEMESTER | |K & P |M & PS |C & J |L OF A |
| |ASSESSMENT | | | | |
| |INSTRUMENT | | | | |
| | |MARKS |RESULT |QUESTION RATING |RESULT |RESULT | |
| |1.2 Mid Sem. Test |45/50 |A |a, a*, c, a |*A |B | |
| |1.3 End Sem. Test |43/50 |A |a, a*, #b, b |A |B | |
| |1.4 | | | | | | |
| |1.5 | | | | | | |
|Sem. 1 | | | | | | | |
|Result | | |A | |A |B |VHA |
| |2.2 Mid Sem. Test |40/50 |B |#a, b, a, b |A |A | |
| |2.3 End Sem. Test |44/50 |A |a, a, a, #b |A |B | |
| |2.4 | | | | | | |
| |2.5 | | | | | | |
|Sem. 2 | | | | | | | |
|Result | | |A | |A |A |VHA |
| |3.2 Mid Sem. Test |27/30 |A |a*, a*, b, a |A |A | |
| |3.3 End Sem. Test |63/70 |A |a, b, #b, a |A |A | |
| |3.4 | | | | | | |
| |3.5 | | | | | | |
|Sem. 3 | | | | | | | |
|Result | | |A | |A |A |VHA |
| |4.2 Mid Sem. Test |40/50 |B |a, a, #a* |A |A | |
| |4.3 End Sem. Test |39/50 |B |a, a, a |A |A | |
| |4.4 | | | | | | |
| |4.5 | | | | | | |
|Sem. 4 | | | | | | |
|Result | | |B | |A |A |
|VERIFICATION | |A | |A |A |VHA |
|EXIT | |A | |A |A |VHA |
|RANK (End Sem 3) | |A | |A |A |VHA |
8 DETERMINING EXIT LEVELS OF ACHIEVEMENT
On completion of the course of study, each student will be awarded a level of achievement, for each of the three criteria, using the tables in the previous section. These levels are then used to determine each individual student's exit level of achievement.
N.B. The minimum standards table includes allowable trade-offs within criteria I-III, therefore NO TRADING-OFF is permitted BETWEEN the criteria when determining levels of achievement.
The following table gives the MINIMUM performance composite of standards for each level of achievement.
Minimum requirements for exit levels
|VHA |Standard A in any two exit criteria and no less than a B in the remaining criterion |
|HA |Standard B in any two exit criteria and no less than a C in the remaining criterion |
|SA |Standard C in any two exit criteria, one of which must be the Knowledge and procedures criterion, and no less than a D in the |
| |remaining criterion |
|LA |Standard D in any two exit criteria, one of which must be the Knowledge and procedures criterion |
|VLA |Does not meet the requirements for Limited Achievement |
Note: The minimum standards table applies when placing students into a band, it does not apply when ranking students within the band. For example, a student who has the following results:-
K & P - A; M & PS - B; C & J - B
would be an HA overall but would not be placed on the bottom rung of the HA band.
APPENDIX 1
Sample Sequence of Work
|Unit title: Calculus |
|Time: 30 hours |
|Subject matter |Suggested learning experiences |Contexts addressed |
| | |A |C |I |T |
|Integrals of the form |Use integration by parts to evaluate expressions such as: [pic] | |( |( |( |
|[pic] | | | | | |
| | | | | | |
|[pic] | | | | | |
| | | | | | |
|Simple integration by parts | | | | | |
| | | | | | |
|Development and use of Simpson’s rule | | | | | |
| | | | | | |
|Approximating small changes in functions | | | | | |
|using derivatives | | | | | |
| | | | | | |
|Life-related applications of simple, | | | | | |
|linear, first order differential | | | | | |
|equations with constant coefficients | | | | | |
| | | | | | |
|Solution of simple, linear, first order | | | | | |
|differential equations with constant | | | | | |
|coefficients | | | | | |
| |Use Simpson’s rule to evaluate definite integrals where the indefinite integral | | |( |( |
| |cannot be found such as [pic] | | | | |
| | | | | | |
| |Compare the values of areas of simple shapes determined from known rules with | | | | |
| |approximations determined from Simpson’s rule such as the area of a quadrant of a | | | | |
| |circle of radius 4 compared with the area represented by [pic] evaluated using| |( |( |( |
| |Simpson’s rule |( | | | |
| | | | | | |
| |Use Simpson’s rule with discrete data in situations such as the volume of fill to | | | | |
| |be removed in the construction of a road cutting | | | | |
| | | | | | |
| | | | | |( |
| | |( | | | |
| |Investigate life related situations where small changes in calculated quantities |( |( |( |( |
| |due to small errors in measurements can be approximated using derivatives, such as| | | | |
| |the tolerance in the volume of a soft drink can, produced by small errors in the | | | | |
| |diameter or height | | | | |
| | | | | | |
| |Find approximate solutions to equations by using small changes from equations with| |( | | |
| |exact solutions such as an approximate value of [pic] | | | | |
| |Investigate life-related situations that can be modelled by simple differential |( |( |( |( |
| |equations such as growth of bacteria, cooling of a substance | | | | |
| | | | | | |
| |Investigate the motion of falling objects, where resistance is proportional to the| | | | |
| |velocity, by considering the differential equation [pic] |( |( | |( |
| | | | | | |
| |Find an expression for the amount of the desired product Pu239 present, as a | | | | |
| |function of time after start up in a breeder reactor where U238 is converted to | | | | |
| |Pu239 at a constant rate and Pu239 is converted to Pu240 at a rate proportional to| | | | |
| |the amount of Pu239 present | | | | |
| | |( |( | |( |
| |Compare the values of areas of simple shapes determined from known rules with |( | |( |( |
| |approximations determined from Simpson’s rule such as the area of a quadrant of a | | | | |
| |circle of radius 4 compared with the area represented by [pic] | | | | |
| | | | | | |
| |Verify integrals in integral tables by differentiation of the result | | | | |
| | | | | | |
| |Use tables of integrals or computer software to evaluate a given integral |( | | |( |
| | | | | | |
| | |( | | |( |
APPENDIX 2
Focus Statements and Learning Experiences
Introduction to Groups (notional time 7 hours)
Focus
Students are encouraged to investigate the structures and properties of groups. It is intended that this introduction to groups should provide a basis for identifying the common features which are found in systems such as real and complex numbers, matrices and vectors.
Subject matter
Concepts of:
← closure
← associativity
← identity
← inverse (suggested learning experiences (SLEs) 1B8)
← definition of a group (SLEs 1B8)
Suggested learning experiences
The following suggested learning experiences may be developed as individual student work, or may be part of small group or whole class activities.
1. determine if the elements of a set form a group under a binary operation
2. determine the identity element and inverses in a group table
3. use a small Cayley table to determine whether a set of elements under a binary operation forms a group
4. investigate when the integers modulo n form groups under addition or multiplication
5. investigate groups formed by geometric transformations such as the reflections of a rectangle in its axes of symmetry and rotations of an equilateral triangle
6. construct a Cayley table and use it to identify subgroups (if any) such as the rotations of a square about its centre
7. find the element(s) which generate(s) the group in a group table
8. investigate the group structure of friezes, wall-papers or simple crystals by studying their symmetries under translations, rotations and reflections
Real and Complex Number Systems (notional time 25 hours)
Focus
Students are encouraged to extend their knowledge of the real number system and to develop an understanding of the complex number system. Students should see the group structure within these systems as a link between the unfamiliar complex numbers and the familiar real numbers.
Subject matter
← structure of the real number system including:
– rational numbers
– irrational numbers (SLEs 2, 9, 10)
← simple manipulation of surds
← definition of complex numbers including standard and trigonometrical (modulus-argument) form (SLEs 1, 2)
← algebraic representation of complex numbers in Cartesian, trigonometric and polar form (SLEs 3, 4)
← geometric representation of complex numbers—Argand diagrams (SLE 4)
← operations with complex numbers including addition, subtraction, scalar multiplication, multiplication of complex numbers, conjugation (SLEs 1B8, 12)
← roots of complex numbers (SLE 6)
← use of complex numbers in proving trigonometric identities
← powers of complex numbers including de Moivre’s Theorem
← simple, purely mathematical applications of complex numbers (SLEs 6, 7, 8, 11, 12)
← proof by mathematical induction (SLE 3)
Suggested learning experiences
The following suggested learning experiences may be developed as individual student work, or may be part of small group or whole class activities.
1. solve quadratic equations whose discriminant is negative
2. solve simple inequality statements such as ∗ *z - a* ∗ > b in both the real and complex systems, and be able to give a verbal description of the meaning of the mathematical symbolism
3. use mathematical induction to prove De Moivre’s Theorem
4. use polar forms to demonstrate multiplication and division of complex numbers
5. use geometry to demonstrate the effect of addition, subtraction and multiplication of complex numbers
6. solve simple equations of the form zn = w where n is an integer and w is a complex number
7. solve polynomial equations with real and complex coefficients (degree # 3)
8. investigate the use of complex conjugates in the solution of polynomial equations with real coefficients
9. use a proof by contradiction to show that [pic] is irrational
10. investigate some of the approximations to ( which have been used
11. research areas in which complex numbers are used in life-related applications such as electric circuit theory, vibrating systems and aerofoil designs
12. investigate the group properties of matrices of the form
under both addition and multiplication; find interesting subsets of this class of matrices (known as quaternions); in particular, show that the eight matrices:
form a group under multiplication
Matrices and Applications (notional time 30 hours)
Focus
Students are encouraged to develop an understanding of the algebraic structure of matrices including situations where they form groups. Students should apply matrices in a variety of situations and use technology to facilitate the solution of problems involving matrices.
Subject matter
← definition of a matrix as data storage and as a mathematical tool (SLEs 1B7)
← dimension of a matrix
← matrix operations
– addition
– transpose
– inverse
– multiplication by a scalar
– multiplication by a matrix (SLEs 1B7, 13, 14, 15)
• definition and properties of the identity matrix (SLEs 1, 3, 15)
← group properties of 2 H 2 matrices (SLE 5)
← determinant of a matrix (SLE 3)
← singular and non-singular matrices (SLE 1)
← solution of systems of homogeneous and non-homogeneous linear equations using matrices (SLEs 1, 6)
← applications of matrices in both life-related and purely mathematical situations (SLEs 1B12)
← relationship between matrices and vectors (SLEs 1, 6, 7, 12)
Suggested learning experiences
NB Many learning experiences in this topic are enhanced by the use of a calculator with matrix operations.
The following suggested learning experiences may be developed as individual student work, or may be part of small group or whole class activities.
1. solve linear equations by using matrices and Gaussian elimination; solution of equations involving more than three variables will involve the use of graphing calculators.
2. investigate transition probability matrices in situations such as, recording over a period of time, the changes in major weather conditions as stated by newspaper or TV weather reports (fine, showers, cloudy, clearing); construct the matrix describing the probabilities that one condition will be followed by each different condition; given today’s weather find the most probable sequence of weather conditions in the near future
3. use matrices to encode and decode messages
4. demonstrate the use of the transformation matrices (rotation, reflection, dilation) as an application of 2 H 2 matrices to geometric transformations in the plane
5. consider subsets of 2 H 2 matrices forming a group under addition or multiplication.
6. show that the change of frame of reference used in Newtonian mechanics,
[pic], can be written using matrices
7. use input-output (Leontief) matrices in economics
8. investigate the use of matrices in dietary problems in health
9. research the use of Leslie matrices in ecology
10. investigate the use of matrices in dominance problems such as in predicting the next round results (rankings) for the national netball competition
11. investigate the use of matrices in game strategies
12. investigate the application of matrices in formulating a mathematical model for a closed economic system
13. research nilpotent matrices where the matrix, A, is nilpotent if it has the property A2 =O, (O is the zero matrix)
14. research idempotent matrices where the matrix, A, is idempotent if it has the property A2 = A
15. investigate the group properties of matrices of the form
under both addition and multiplication; find interesting subsets of this class of matrices (known as quaternions); in particular, show that the eight matrices:
form a group under multiplication
Vectors and Applications (notional time 30 hours)
Focus
Students are encouraged to develop an understanding of vectors as entities which can be used to describe naturally occurring systems. They should also understand that the meaning of a vector comes from the situation and the model being considered. Students should become aware of the links between vectors and matrices. The emphasis should be on those vectors which describe situations involving magnitude and direction.
Subject matter
For vectors as a one-dimensional array
← definition of a vector
← relationship between vectors and matrices (SLE 1)
← operations on vectors including:
– addition
– multiplication by a scalar
← scalar product of two vectors (SLE 1)
← simple life-related applications of vectors (SLEs 1, 3)
(b) For vectors describing situations involving magnitude and direction
← definition of a vector (see Appendix 2)
← relationship between vectors and matrices (SLE 2)
← two and three dimensional vectors and their algebraic and geometric representation (SLEs 3, 4, 6)
← operations on vectors including: (SLEs 3B5, 9, 10, 13)
– addition
– multiplication by a scalar
← scalar product of two vectors (SLEs 2, 8)
← vector product of two vectors (SLE 7)
← unit vectors
← resolution of vectors into components acting at right angles to each other (SLEs 3B5, 11, 12)
← calculation of the angle between two vectors
← applications of vectors in both life-related and purely mathematical situations(SLEs 1–13)
Suggested learning experiences
The following suggested learning experiences may be developed as individual student work, or may be part of small group or whole class activities.
1. show that the cost of weekly shopping is the scalar product of the shopping list vector and the unit cost vector
2. show that if x is a column vector of order n
xTx = [pic]
3. use addition and subtraction in life-related situations such as the effect of current flow on a boat; consider the concept of relative velocity
4. use resolution of vectors to consider the equilibrium of a body subject to a number of coplanar forces acting at a point
5. use vectors in three dimensions such as the placement of a TV aerial mast and its wire supports on a roof of a building
6. solve problems from geometry using vectors such as proving the concurrency of (a) the medians and (b) the bisectors of the internal angles of a triangle
7. use the vector product to calculate areas in situations such as the calculation of the area of a suspended triangular shade canopy
8. use scalar product to solve problems in situations such as the evaluation of work as a product of force and displacement
9. investigate the use of vectors in surveying
10. investigate the effect of wind on wind propelled craft
11. investigate the way medical staff use vectors to put a broken bone in suitable traction; consider the weights and angles of the ropes that are needed
12. investigate the forces exerted by the hip, knee and ankle joints in pushing and pulling a bicycle pedal
13. use addition of vectors to see how the apparent motion of planets in the solar system depends on the frame of reference chosen
Calculus (notional time 30 hours)
Focus
Students are encouraged to extend their knowledge of analytical and numerical techniques of integration. Students should also gain further experience in applying differentiation and integration to both life-related and purely mathematical situations. They should appreciate the importance of differential equations in representing problems involving rates of change.
Subject matter
← integrals of the form
← simple integration by parts (SLE 4)
← development and use of Simpson’s rule (SLEs 5, 6, 7, 9, 10, 11, 12, 15)
← approximating small changes in functions using derivatives (SLEs 1, 2)
← life-related applications of simple, linear, first order differential equations with constant coefficients (SLEs 3, 8, 13, 14)
← solution of simple, linear, first order differential equations with constant coefficients (SLEs 3, 6, 8, 10, 13, 14, 15)
Suggested learning experiences
The following suggested learning experiences may be developed as individual student work, or may be part of small group or whole class activities.
1. investigate life related situations where small changes in calculated quantities due to small errors in measurements can be approximated using derivatives, such as the tolerance in the volume of a soft drink can, produced by small errors in the diameter or height
2. find approximate solutions to equations by using small changes from equations with exact solutions such as an approximate value of [pic]
3. investigate life-related situations that can be modelled by simple differential equations such as growth of bacteria, cooling of a substance
4. use integration by parts to evaluate expressions such as
[pic]
5. use Simpson’s rule to evaluate definite integrals where the indefinite integral cannot be found such as
[pic]
6. compare the values of areas of simple shapes determined from known rules with approximations determined from Simpson’s rule such as the area of a quadrant of a circle of radius 4 compared with the area represented by
[pic]
evaluated using Simpson’s rule
7. use Simpson’s rule with discrete data in situations such as the volume of fill to be removed in the construction of a road cutting
8. investigate the motion of falling objects, where resistance is proportional to the velocity, by considering the differential equation
[pic]
9. compare the accuracy of numerical techniques with analytical results for selected integrals
10. verify integrals in integral tables by differentiation of the result
11. show that Simpson’s Rule is exact for polynomials of degree three or less
12. investigate the varying volumes for the earth obtained when its shape is assumed to be (a) a sphere, and (b) an ellipsoid
13. find an expression for the pressure, P, as a function of altitude in an isothermal atmosphere where the rate of decrease of atmospheric pressure with increasing altitude is proportional to the density of the air, (, pressure, density and temperature, T, are related by P = R(T, and R is a constant
14. find an expression for the amount of the desired product Pu239 present, as a function of time after start up in a breeder reactor where U238 is converted to Pu239 at a constant rate and Pu239 is converted to Pu240 at a rate proportional to the amount of Pu239 present
15. use tables of integrals or computer software to evaluate a given integral
Structures and Patterns (notional time 30 hours)
Focus
Students are encouraged to develop their ability to recognise and use structures and patterns in a wide variety of situations. They should appreciate the value of symmetries and patterns in making generalisations to explain, simplify or extend their mathematical understanding. Justification of results is important and, where appropriate, results should be validated inductively or deductively. It is not intended that a great emphasis be placed on repetitive calculations in arithmetic progressions, geometric progressions, permutations or combinations.
Subject matter
← sum to infinity of a geometric progression (SLEs 1, 2)
← purely mathematical and life-related applications of arithmetic and geometric progressions (SLEs 10, 11, 12)
← sequences and series other than arithmetic and geometric (SLEs 3, 4, 16)
← permutations and combinations and their use in purely mathematical and life-related situations (SLEs 7, 8, 9, 14, 15)
← recognition of patterns in well known structures including Pascal’s Triangle and Fibonacci sequence (SLEs 5, 6, 13)
← applications of patterns (SLEs 1B11, 17)
← use of the method of finite differences (SLEs 4, 18)
← proof by induction (SLE 4)
Suggested learning experiences
The following suggested learning experiences may be developed as individual student work, or may be part of small group or whole class activities.
1. establish the formula for the sum to n terms of a geometric progression, and hence the formula for the sum to infinity of a geometric progression; verify the formula by mathematical induction
2. recognise geometric progressions in many different algebraic forms such as 2, 4p2, 8p4, 16p6 ...; determine the general term, sum of n terms, sum to infinity of such sequences
3. use finite differences in determining polynomial coefficients for polynomials of degree ## 3
4. use finite difference methods to establish the formula in situations such as the sum of the first n positive integers, the sum of the first n squares, the sum of the first n cubes; use the principle of mathematical induction to prove the formula obtained
5. search for patterns in Pascal’s Triangle and verify any claims algebraically or otherwise, such as
nCr + nCr+1 = n+1Cr+1
6. investigate patterns in Fibonacci numbers such as:
f1 + f2 + f3 + ... + fn = fn+2 - 1
f1 + f3 + f5 + ... + f2n-1 = f2n - 1
7. investigate permutations and combinations which arise in games of chance such as, in poker, the number of hands containing two aces, the number of hands that are a full house
8. apply counting techniques to investigate problems in situations such as:
• the amount of wool eaten by the offspring of one female moth who lays 300 eggs if each larva eats 15 milligrams of wool, two-thirds of eggs die, fifty per cent of remaining eggs are female and there are five generations per year
• the number of different sequences possible in 4 months, when a patient must receive H-insulin for 2 months and P-insulin for the other 2 months in a medical study
9. investigate the use of the inclusion-exclusion principle in counting cases in situations such as determine how many individuals were interviewed in a survey of the eating habits of teenagers if 10 ate pizzas, 5 ate pies, 2 ate both and 7 ate neither; consider the effect of adding hamburgers as a third option
10. investigate the use of an arithmetic progression in situations such as:
• the calculation of the length of batten material required for tiling a hip roof
• the calculation of the total number of potatoes required for a “potato race” over a given distance if the distance between potatoes is a specified constant
11. use geometric progressions in situations involving the sum to infinity
12. investigate logarithmic spirals and polar curves which occur in nature such as in nautilus shells
13. investigate the occurrence of Fibonacci numbers in nature such as in spirals in sunflowers, pineapples and other plants
14. given a cube and six different colours, determine how many different ways the cube can be painted so that each face is a different colour; extend to other regular solids
15. apply the pigeonhole principle to solve problems in situations such as showing there are at least 2 in a group of 8 people whose birthdays fall on the same day of the week in any given year
16. use series expansions for ex, sin x and cos x to illustrate :
• derivatives of ex, sin x and cos x
• eix = cos x + isin x
17. use symmetries to find the order of groups of rotations such as rotations of squares, equilateral triangles
18. verify the outcomes of finite differences by using graphing calculators
Conics (notional time 30 hours)
Focus
Students are encouraged to extend their knowledge of coordinate geometry in two dimensions. They are encouraged to appreciate the interrelationships that exist between areas of mathematics. These relationships should be illustrated by applying coordinate geometry and complex numbers to conics.
Subject matter
← concept of a locus, directrix and focal point (SLEs 1B17)
← circle as a locus in:
– Cartesian form [pic]
– parametric form x = a cos ( , y = a sin (
– complex number form [pic](SLEs 1B6, 8, 11, 17)
← definition of eccentricity e
← ellipse as a locus in:
– Cartesian form [pic]
– parametric form x = a cos( , y = b sin(
– complex number form [pic] where s > *∗ p - q*∗
– polar form [pic] (SLEs 3, 5 ,7 ,9 , 10, 11, 14, 16, 17)
← hyperbola as a locus in:
– Cartesian form [pic]; [pic]
– parametric form [pic];[pic]
– complex number form[pic]; where 0 < s < ∗*p - q∗*
– polar form [pic] (SLEs 3, 5, 10, 11, 15, 17)
← parabola as a locus in:
– Cartesian form [pic]
– parametric form [pic]
– polar form [pic] (SLEs 1, 2, 3, 5, 8, 10, 11, 12, 13, 17)
in all cases above a, b, c, d, p, q and s are constants
← simple applications of conics (SLEs 1B17)
Suggested learning experiences
The following suggested learning experiences may be developed as individual student work, or may be part of small-group or whole-class activities.
1. introduce the concept of locus using situations such as the path followed by:
• the handle on an opening door
• a boat sailing equidistant from two fixed lights
• a netballer moving to remain equidistant from a goal post and a sideline
2. derive the Cartesian forms for the circle and parabola
3. find the Cartesian equations of conics after a given translation has been applied
4. find the centre and radius of a circle given its Cartesian equation
5. sketch conics given the Cartesian form showing directrices, focuses, asymptotes, and axes of symmetry as appropriate
6. convert the complex number form for a circle to the corresponding Cartesian form using both algebraic and geometric methods
7. convert the complex number form for an ellipse to the corresponding Cartesian form using both algebraic and geometric methods; situations should be limited to those where both focuses lie on the same coordinate axis
8. find the equation of a tangent to a circle or a parabola given its parametric form
9. investigate how to construct the elliptical hole required to be cut out of a sloping roof to fit a vertical cylindrical vent
10. use a graphing calculator to plot a conic whose form is given parametrically
11. find the equation of tangents, chords, and normals to conics whose equations are given in Cartesian form
12. investigate why parabolic reflectors are used in astronomical telescopes, hand-held torches and microwave repeater stations
13. investigate the shape on the ground of the leading edge of the sonic boom produced by a supersonic aircraft flying at high altitudes
14. research the use of elliptic reflectors in the treatment for kidney stones
15. research how hyperbolas were used in the Omega navigation system
16. investigate how the properties of ellipses are used in whispering rooms
17. use a dynamic geometry package to investigate locus behaviour
Dynamics (notional time 30 hours)
Focus
Students are encouraged to develop an understanding of the motion of objects which are subjected to forces. The approach used throughout this topic should bring together concepts from both vectors and calculus.
Subject matter
← derivatives and integrals of vectors (SLEs 1, 2, 3, 13)
← Newton’s laws of motion in vector form applied to objects of constant mass (SLEs 2B15)
← application of the above to:
– straight line motion in a horizontal plane with variable force
– vertical motion under gravity with and without air resistance
– projectile motion without air resistance
– simple harmonic motion (derivation of the solutions to differential equations is not required)
– circular motion with uniform angular velocity
(SLEs 4B12, 14, 15)
Suggested learning experiences
The following suggested learning experiences may be developed as individual student work, or may be part of small-group or whole-class activities.
1. given the position vector of a point as a function of time such as r (t) = t i + t2 j + sin t k determine the velocity and acceleration vectors
2. given the displacement vector of an object as a function of time, by the processes of differentiation, find the force which gives this motion
3. given the force on an object as a function of time and suitable prescribed conditions, such as velocity and displacement at certain times, use integration to find the position vector of the object
4. investigate the motion of falling objects such as in situations in which:
• resistance is proportional to the velocity, by considering the differential equation
[pic]
• resistance is proportional to the square of velocity, by considering the differential equation
[pic]
where k is a positive constant
5. model vertical motion under gravity alone; investigate the effects of the inclusion of drag on the motion
6. develop the equations of motion under Hooke’s law; verify the solutions for displacement by substitution and differentiation; relate the solutions to simple harmonic motion and circular motion with uniform angular velocity
7. from a table of vehicle stopping distances from various speeds, calculate (a) the reaction time of the driver and (b) the deceleration of the vehicle, which were assumed in the calculation of the table
8. model the path of a projectile without air resistance, using the vector form of the equations of motion starting with a = -g j where upwards is positive
9. use the parametric facility of a graphing calculator to model the flight of a projectile
10. investigate the flow of water from a hose held at varying angles, and model the path of the water
11. investigate the motion of a simple pendulum with varying amplitudes
12. investigate the angle of lean required by a motorcycle rider to negotiate a corner at various speeds
13. use the chain rule to show that the acceleration can be written as
[pic]
if the velocity, v, of a particle moving in a straight line is given as a function of the distance, x
14. investigate the speed required for a projectile launched vertically to escape from the earth’s gravitational field, ignoring air resistance but including the variation of gravitational attraction with distance
15. use motion detectors to investigate problems e.g. rolling a ball down a plank
APPENDIX 3
Equity Statement
Equity means fair treatment of all. In developing work programs from this syllabus, schools are urged to consider the most appropriate means of incorporating the following notions of equity.
Schools need to provide opportunities for all students to demonstrate what they know and what they can do. All students, therefore, should have equitable access to educational programs and human and material resources. Teachers should ensure that the particular needs of the following groups of students are met: female students; male students; Aboriginal students; Torres Strait Islander students; students from non–English-speaking backgrounds; students with disabilities; students with gifts and talents; geographically isolated students; and students from low socioeconomic backgrounds.
The subject matter chosen should include, where appropriate, the contributions and experiences of all groups of people. Learning contexts and community needs and aspirations should also be considered when selecting subject matter. In choosing suitable learning experiences teachers should introduce and reinforce non-racist, non-sexist, culturally sensitive and unprejudiced attitudes and behaviour. Learning experiences should encourage the participation of students with disabilities and accommodate different learning styles.
It is desirable that the resource materials chosen recognise and value the contributions of both females and males to society and include the social experiences of both sexes. Resource materials should also reflect the cultural diversity within the community and draw from the experiences of the range of cultural groups in the community.
Efforts should be made to identify, investigate and remove barriers to equal opportunity to demonstrate achievement. This may involve being proactive in finding out about the best ways to meet the special needs, in terms of learning and assessment, of particular students. The variety of assessment techniques in the work program should allow students of all backgrounds to demonstrate their knowledge and skills in a subject in relation to the criteria and standards stated in this syllabus. The syllabus criteria and standards should be applied in the same way to all students.
Teachers may find the following useful for devising an inclusive work program.
➢ Australian Curriculum, Assessment and Certification Authorities 1996, Guidelines for Assessment Quality and Equity 1996, Australian Curriculum, Assessment and Certification Authorities, available through QBSSSS, Brisbane.
➢ Department of Education, Queensland 1991, A Fair Deal: Equity guidelines for developing and reviewing educational resources, Department of Education, Brisbane.
➢ Department of Training and Industrial Relations 1998, Access and Equity Policy for the Vocational Education and Training System, DTIR, Brisbane.
➢ [Queensland] Board of Senior Secondary School Studies 1994, Policy Statement on Special Consideration, QBSSSS, Brisbane.
➢ [Queensland] Board of Senior Secondary School Studies 1995, Language and Equity: A discussion paper for writers of school-based assessment instruments, QBSSSS, Brisbane.
➢ [Queensland] Board of Senior Secondary School Studies 1995, Studying Assessment Practices: A resource for teachers in schools, QBSSSS, Brisbane.
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