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WORK PROGRAMFOR

MATHEMATICS Bbased on the textbook:

MATHS QUEST: MATHS B

FOR QUEENSLANDYEAR 11NICK SIMPSONROB ROWLAND

This is intended as one example of a work program for the Mathematics B syllabus. It is not an accredited program but may be useful to teachers as a guide to producing their own school-based program. Thanks to Rachel Nooteboom (Dysart State High School) for her effort in producing this program based on the text.

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Contents

Rationale 3

Global Aims 4

General Objectives 4

Contexts 5

Technology 5

Language Statement 8

Educational Equity 8

Course Organisation 9

Learning Experiences 16

Assessment 22

Profiles 31

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1 RATIONALE

Mathematics is an integral part of a general education. It can enhance understanding of our world and the quality of our participation in a rapidly changing society. The range of career opportunities requiring an appropriate level of mathematical competence is rapidly expanding into such areas as health, environmental science, economics and management, while remaining crucial in such fields as the physical sciences, engineering, accounting, computer science and information technology areas. Mathematics is essential for widespread computational and scientific literacy, for the development of a more technologically skilled work force, for the development of problem solving skills and for the understanding and use of data and information to make well-considered decisions. It is valuable to people individually and collectively, providing important tools, which can be, used at personal, civic, professional and vocational levels.

At a personal level, the most obvious use of mathematics is to assist in making informed decisions in areas as diverse as buying and selling, home maintenance, interpreting media presentations and forward planning. The mathematics involved in these activities includes analysis, financial calculation, data description, inference, number, quantification and spatial measurement. The generic skills developed by mathematics are also constantly used at the personal level.

At the civic, professional and vocational levels, the generic skills, knowledge and application of mathematics underpin most of the significant activities in industry, trade and commerce, social and economic planning, and communication systems. In such areas, the concepts and application of functions, rates of change, total change and optimisation are very important. The knowledge and skills developed in Mathematics B are essential for further development in any quantitative areas. The demand for those who are skilled mathematically continues to rise, emphasising the need for schools to provide the opportunity for students to experience a thorough and well-rounded education in mathematical ideas, concepts, skills and processes.

Mathematics has provided a basis for the development of technology. In recent times, the uses of mathematics have increased substantially in response to changes in technology. The more technology is developed the greater 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 encourage students in understanding 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 real and life-like situations.

Mathematics B aims to provide the opportunity for students to participate more fully in life-long learning. It 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

Such development should occur in contexts. These contexts should range from purely mathematical through life-like to real, from simple through intermediate to complex, from basic to more advanced technology usage, and from routine rehearsed through to innovative. Of importance is the development of student thinking skills, as well as student recognition and use of mathematical patterns.

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The intent of Mathematics B is to encourage students to develop positive attitudes towards mathematics by approaches involving exploration, investigation, application of knowledge and skills, problem solving and communication. Students will be encouraged to mathematically model, to work systematically and logically, to conjecture and reflect, and to justify and communicate with and about mathematics. The subject is designed to raise the level of competence in the mathematics required for informed citizenship and life-long learning, to increase students’ confidence in using mathematics to solve problems, and especially to provide a basis for a wide range of further studies. Mathematics B 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.)

SCHOOL PROFILEInsert your school details here. AcknowledgementsThis Work Program was developed for the Year 11 cohort at Dysart State High School commencing Mathematics B in 2002. Rachel Nooteboom developed the content of this work program.

2 GLOBAL AIMS

Having completed the course of study, students of Mathematics B should: Have significantly broadened their mathematical knowledge and skills 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 aware of the uncertain nature of their world and be able to use mathematics to assist in

making informed decision in life-related situations Have experienced diverse applications of mathematics Have positive attitudes towards 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 use justification in and with mathematics 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 applications

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

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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 students be 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 experiences 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 graphics calculator.

Initiative

Learning experiences and the corresponding assessment must provide students with the opportunity to demonstrate their capability in 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.

3.3.1 Knowledge and procedures

The objectives of this category involve recalling and using 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 rules

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Access and apply rules and techniques Demonstrate number and spatial sense Demonstrate algebraic facility Demonstrate an ability to select and use technology such as calculators, measuring

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 that some tasks may be broken up into smaller components Transfer and apply mathematical procedures to similar situations

3.3.2 Modelling and Problem Solving

The objectives of this category involve the uses 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 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 a particular

situation Interpreting results from the mathematical model in terms of the given situation Exploring the strengths and limitations of mathematical modelsProblem 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 a 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

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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 this 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 every day 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 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 Using supporting arguments, when appropriate, to justify results obtained by calculator or

computer.

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 Diverse and evolutionary nature of mathematics and the wide range of mathematics-based

vocations Contribution of mathematics to human culture and progress Power and beauty of mathematics

4 LANGUAGE STATEMENT

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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 B 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 B (such

as short responses, reports, multi-media presentations, seminars) Use the specialised vocabulary and terminology related to Mathematics B 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 B. Students should be involved in learning experiences that require them to comprehend and transform data in a variety of forms and, in doing so, 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 should use format and language that are familiar to students. They should be taught the language skills necessary to interpret questions accurately and develop coherent, logical and relevant responses. Attention to language education within Mathematics B should assist students to meet the language components of the exit criteria, especially the criterion Communication and justification.

This School will endeavour to support students in meeting the standards articulated in this language statement. 5 EDUCATIONAL EQUITY

This school will give due consideration to the issues associated with educational equity regarding fair treatment of all by providing:

Opportunities for all students to demonstrate what they know and what they can do

Equitable access to educational programs, human and material resources. This school makes available to students a wide range of resources to assist with their mathematics learning, including graphic and scientific calculators and electronic textbooks.

The needs of the following groups: 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; students from low socioeconomic backgrounds, independent students, students who are young parents and mature age students.

Suitable learning experiences to introduce and reinforce non-racist, non-sexist, culturally sensitive and unprejudiced attitudes and behavior.

The participation of students with disabilities and different learning styles, where applicable.

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Resource materials which recognise and value the contributions of both females and males to society and include the social experiences of both sexes

Resource materials which reflect the cultural diversity within the community and draw from the experiences of the range of cultural groups in the community

Assessment techniques which allow students of all backgrounds to demonstrate their knowledge and skills in the subject in relation to the criteria and standards stated in the syllabus, as outlined in ACACA (1996) Guidelines for Assessment Quality and Equity

6 COURSE ORGANISATION

6.1 DELIVERY OF THE COURSE

After considering the subject matter and the appropriate range of learning experiences to enable students to achieve the general objectives, a spiraling and integrated sequence will be offered which allows students to see a link between the different topics of mathematics rather than seeing them as discrete.

Although not all syllabus topics are covered in every semester, the concepts dealt with will be drawn upon for subsequent syllabus topics, and no syllabus subject matter will be studied before the relevant prerequisite material.

Throughout this course, time will be provided for the maintenance of basic knowledge and procedures required within each unit of work.

There will be a minimum of 55 hours per semester timetabled for this subject. In this school, each school week equates to approximately 3½ hours of timetabled time.

Students will be provided with personal TI-83+ graphics calculators for the duration of the course with additional access to computers and appropriate software to enhance student exploration and investigation of the concepts and processes of mathematics and enable students to develop the full range of skills required for successful problem solving during their course of study.

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6.2 COURSE SUMMARY

This course summary provides an overview of the delivery of the topics in this course.Hours

Semester 1Introduction to functions A 20Rates of Change A 15Periodic Functions and Applications A 20Total Hours 55

Semester 2Exponential and logarithmic functions and applications A

20

Applied Statistical Analysis A 10Introduction to functions B 15Rates of Change B 10Total Hours 55

Semester 3Periodic Functions and Applications B 20Exponential and logarithmic functions and applications B

20

Introduction to Integration A 15Total Hours 55

Semester 4Optimisation using derivatives 25Introduction to integration B 15Applied Statistical Analysis B 15Total Hours 55

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6.3 DETAILED COURSE SEQUENCEText: Simpson, N., and Rowland, R. (2001) Maths Quest: Maths B Year 11 for Queensland. John Wiley & Sons Australia, Brisbane.

Sem

este

r 1

Ref Unit title Time Subject matter Text Ref

1 Introduction to functions A

20hr Graphs as a representation of the points whose coordinates satisfy an equation Practical applications of linear functions including:

Direct variation Simple interest as an arithmetic progression Linear relationships between variables

algebraic and graphical solution of two simultaneous equations in two variables (to be applied only to linear and quadratic functions)

Concepts of function, domain and range Mappings, tables and graphs as representations of functions and relations Distinction between functions and relations Distinctions between continuous functions, discontinuous functions and

discrete functions

Ex 1AEx 1BEx 1CEx 1DEx 1E

Ex 1FEx 2A, B, CEx 2D

Ex 2E

2 Rates of Change A 15hr concept of a rate of change calculation of average rates of change in both practical and purely mathematical

situations interpretation of the average rate of change as the gradient of a secant Relating Gradient to original function intuitive understanding of a limitN.B. Calculations using limit theorems are not required.

Ex 12A

Ex 12BEx 12C,DEx 12E, F, G

Ex 13A (B)

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Text: Simpson, N., and Rowland, R. (2001) Maths Quest: Maths B Year 11 for Queensland. John Wiley & Sons Australia, Brisbane.

Sem

este

r 1Ref Unit title Time Subject matter Text Ref

3 Periodic functions and applications A

20hr Trigonometry including the definition and practical applications of the sine,

cosine and tangent ratios Simple practical applications of the sine and cosine rules (the ambiguous case

is not essential) Definition of a periodic function, the period and amplitude Definition of a radian and its relationship with degrees Definition of the trigonometric functions sin, cos and tan of any angle in

degrees and radians Graphs of y = sin x, y = cos x and y = tan x Significance of the constants A, B, C and D on the graphs of y = A sin(Bx + C)

+D and y = A cos (Bx + C) + D Applications of periodic functions Pythagorean identity sin2x + cos2x = 1 Solution of simple trigonometric equations within a specified domain

Ex 4A, BEx 4C, DEx 4EEx 4F, GEx 5AEx 5BEx 5C, D

Ex 5E

Ex 5F

Ex 6A, B,C, D, E

Sem

este

r 2

4 Exponential and logarithmic functions A

20hr Index laws and definitions Solution of equations involving indices Graphs of exponential equations Logarithmic laws and definitions Use of logarithms to solve equations involving indices Applications of exponential and logarithmic functions Geometric sequences and series Growth and decay Compound interest and loan schedules

Ex 7A, BEx 7CEx 7DEx 7EEx 7FEx 7G

Ex 8A, BEx 8CEx 8D, E, (F)

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Text: Simpson, N., and Rowland, R. (2001) Maths Quest: Maths B Year 11 for Queensland. John Wiley & Sons Australia, Brisbane.

Sem

este

r 1Ref Unit title Time Subject matter Text Ref

5 Applied Statistical analysis A

10hr Identification of variables and types of data (continuous and discrete); practical

aspects of collection and entry of data Choice and use in context of appropriate graphical and tabular displays for

different types of data including pie charts, barcharts, tables, histograms, stem-and-leaf and box plots

Use of summary statistics including mean, median, standard deviation and interquartile distance as appropriate descriptors of features of data in context

Use of graphical displays and summary statistics in describing key features of data, particularly in comparing datasets and exploring possible relationships

Use of relative frequencies to estimate probabilities; the notion of probabilities of individual values for discrete variables and intervals for continuous variables

Ex 9A, B

Ex 9C, D, E

Ex 10A, B, CEx 10D, E, F

Ex 11A, B, C

6 Introduction to functions B

15hr practical applications of quadratic functions, the reciprocal function and inverse

variation relationships between the graph of f(x) and the graphs of f(x) + a, f(x + a),

af(x), f(ax) for both positive and negative values of the constant a general shapes of graphs of absolute value functions, the reciprocal function

and polynomial functions up to and including the fourth degree concept of the inverse of a function composition of two functions

Ex 3A, B

Ex 3C, D, E

7 Rates of Change B 10hr rules for differentiation Rules for differentiation Rates of change Maximum and minimum problems

Ex 13CEx 13DEx 13EEx 13F

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Sem

este

r 3Ref Unit title Time Subject matter

8 Periodic functions and Applications B

20hr derivatives of functions involving sin x and cos x applications of the derivatives of sin x and cos x in life-related situations

9 Exponential and Logarithmic Functions B

15hr definition of ex, logex development of algebraic models from data sets using logarithms derivatives of exponential and logarithmic functions for base e applications of exponential and logarithmic functions, and the derivative of exponential functions

10 Introduction to Integration A

15hr definition of the definite integral and its relationship to area under a curve the value of a limit of a sum as a definite integral definition of the indefinite integral rules for integration including:

, , indefinite integrals of simple polynomial functions, simple exponential functions,

sin (ax + b), cos (ax + b) and

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Sem

este

r 411 Optimisation using

derivatives25hr

positive and negative values of the derivative as an indication of the points at which the function is increasing or decreasing

zero values of the derivative as an indication of stationary points concept of relative maxima and minima and greatest and least value of functions relationships between the graph of a differentiable function, and the graph of the derivative of that

function methods of determining the nature of stationary points greatest and least values of a function in a given interval recognition of the problem to be optimized (maximized or minimized) identification of variables and construction of the function to be optimized applications of the derivative to optimisation in real-life situations interpretation of mathematical solutions and their communication in a form appropriate to the given

problem

12 Introduction to Integration B

15hr use of integration to find area practical applications of the integral trapezoidal rule for the approximation of a value of a definite integral numerically

13 Applied Statistical Analysis B

15hr probability distribution and expected value for a discrete variable identification of the binomial distribution and use of tables or technology or binomial probabilities introduction of the concept of hypothesis testing through assessing whether data are consistent with a

stated value of a proportion of life-related problems, using binomial probabilities concept of a probability distribution for a continuous random variable; notion of expected value and

median for a continuous variable the normal model and use of standard normal tables

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6.4 LEARNING EXPERIENCES

This school will provide a variety of learning experiences that are appropriate to the age group concerned and consistent with the objectives of this course.

The following sequence of work indicates the approach that this school will adopt regarding opportunities for students to achieve the general objectives within the contexts of the course.

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Topic 1: Introduction to Functions

Focus

Students are encouraged to develop an understanding and appreciation of relationships between variables, be conversant with the three methods of representation (algebraic, graphical and numerical) and interrelate these methods in a variety of modelling situations, ranging from life-related to abstract. Emphasis should be placed on the recognition of functions, sketching, investigating shapes and relationships, and the general forms of functions. The use of technology is expected to assist students in these processes.

Subject Matter

Concepts of function, domain and range Mappings, tables and graphs as representations of functions and relations Graphs as a representation of the points whose coordinates satisfy an equation Distinction between functions and relations Distinctions between continuous functions, discontinuous functions and discrete functions Practical applications of linear functions including:

Direct variation Simple interest as an arithmetic progression Linear relationships between variables

Algebraic and graphical solution of two simultaneous equations in two variables (to be applied only to linear and quadratic functions)

Suggested Learning Experiences

The following learning experiences may be developed as individual student work, or may be part of small-group or whole-class activities.

1. Find the domain and range of functions both in mathematical and life-related contexts, given data in a variety of forms such as graphs, tables of values, mathematical expressions or descriptions of situations

2. Use the vertical line test to determine if a relation is a function3. Draw graphs of step functions such as postal charges for packages against the weight of

packages, telephone charges per unit time against distance4. Examine the general shapes of polynomial functions of the type y = xn, n = 2, 3, 45. Using f(x ) = xn, for n = 1 to 4; investigate the relationship between the graph of f(x) and the

graphs of f(x + a), af(x) and f(ax), by means of a graphing calculator and algebraic methods.6. Investigate the shapes of common relations, which are not functions, for example circles and

ellipses.7. Locate the position algebraically, numerically and graphically of the highest point on a

projectile path defined by a quadratic function8. Derive the formula for the solution of general quadratic equation.9. Use a graphing calculator to investigate the shapes of different functions10. Investigate the difficulties encountered in using a graphing calculator or computer software to

draw graphs of relations which are not functions11. Solve simultaneous equations which model life-related situations12. Use a graphing calculator to investigate possible functions for data.

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13. Use linear functions in investigating break-even analysis14. Calculate the amount of simple interest generated over a given period using a graphing

calculator or a suitable software package; plot these discrete values and generate a function, which can be used to represent all values.

Suggested Resources

1. Simpson et al (2001) Maths Quest Maths B Year 11 for Queensland, John Wiley and Sons: Brisbane. Chapters 1 and 2.

2. Microsoft Excel3. Maths Helper Plus software4. Graphing calculators

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Topic 2: Rates of Change A

Focus

Students are encouraged to develop an understanding of average and instantaneous rates of change and of the derivative of a function. This understanding should be developed using both algebraic and graphical approaches. Students should be expected to apply the rules for differentiation and interpret the results. The use of technology is expected to help students in these processes.

Subject Matter

Concept of a rate of change Calculation of average rates of change in both practical and purely mathematical situations Interpretation of the average rate of change as the gradient of a secant Intuitive understanding of a limitN.B. Calculations using limit theorems are not required. Definition of the derivative of a function at a point Derivative of simple algebraic functions from first principles

Suggested Learning Experiences:


1. Determine average and instantaneous speeds from a distance-time graph2. Determine average and instantaneous accelerations from a velocity-time graph3. Use a numerical technique to estimate a limit or an average rate of change4. Graph a function and its gradient function; relate the features of each to the other

Suggested Resources:

1. Simpson et al (2001) Maths Quest Maths B Year 11 for Queensland, John Wiley and Sons: Brisbane. Chapter 12, 13(p542 –556 only)


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Topic 3: Periodic Functions and Applications A

Focus

Students are encouraged to develop an understanding and appreciation of periodic functions, be conversant with the three methods of representation (algebraic, graphical and numeric) and interrelate these methods in a variety of modeling situations, ranging from life-related to abstract. Emphasis should be placed on the recognition of periodic functions, sketching, investigating shapes and relationships, and the general forms of periodic functions. The use of technology is expected to assist students in these processes. Trigonometric identities need not be developed beyond the Pythagorean Identity.

Subject matter

Trigonometry including the definition and practical applications of the sine, cosine and tangent ratios Simple practical applications of the sine and cosine rules (the ambiguous case is not essential) Definition of a periodic function, the period and amplitude Definition of a radian and its relationship with degrees Definition of the trigonometric functions sin, cos and tan of any angle in degrees and radians Graphs of y = sin x, y = cos x and y = tan x Significance of the constants A, B, C and D on the graphs of y = A sin(Bx + C) +D and y = A cos (Bx +

C) + D Applications of periodic functions Pythagorean identity sin2x + cos2x = 1 Solution of simple trigonometric equations within a specified domain

Suggested Learning Experiences:


1. Use sine, cosine and tangent ratios to determine lengths/distances and magnitudes of angles in life-related situations such as: Guy ropes for tents or flag poles Distances across rivers and valleys

2. Use sine and cosine rules to solve triangles in two and three-dimensional contexts and determine lengths/distances and the magnitude of angles in life-related situations such as: The distance between two ships given the distances and bearings to a fixed point The height of a tower given the direction of angles of inclination from two fixed locations

of known distance apart3. Investigate the repetitive nature of daily temperature and human pulse4. Find the period, amplitude and frequency of periodic functions involving sine and cosine given

their graphs and/or equations5. Find the period, amplitude and frequency of trigonometric functions which are used to model

phenomena such as biorhythms, tide heights6. Using the concept of the unit circle:

Evaluate trigonometric functions sin, cos and tan of any angle in degrees and radians Explore the graphs of y = sin x, y = cos x and y = tan x

7. Investigate the periodic motion of a mass on the end of a spring; given the mathematical model of the displacement from a fixed position, find mathematical representations of its velocity and

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acceleration; solve these simple trigonometric equations to find displacement, velocity and acceleration at a given time during the periodic motion

8. Find solutions of trigonometric equations for -2π θ 2π such as sin θ = -0.7, 2 sin2 θ = cos θ9. Investigate the effect of the constants A, B, C and D on the graphs of y = A sin(Bx + C) +D and y

= A cos (Bx + C) + D.10. Sketch the graphs of y = sin x, y = cos x and y = tan x for any angle in degrees (-720 x

720) and in radians in the range of -4π x 4π11. Calculate the rate at which the water level is changing on a vertical marker given a sine

function as a model of tide height12. Investigate the periodic motion of a pendulum; mathematically model the motion of a

pendulum using trigonometric equations13. Plot the tide heights at a specified point over a 24 hour period14. Explore the period, amplitude and frequency of periodic (oscillatory) phenomena including

planetary motion, hormone cycles, ECG’s, Halley’s comet, reciprocating motion15. Investigate the path of a point on a moving bicycle wheel16. Graph the slopes of the tangent of a sine curve17. Investigate daily electrical energy consumption over a period of time18. Use computer software or a graphing calculator to investigate more complicate periodic

functions, for example, sin x + ½ sin 2x + sin 3x ….

Suggested Resources:

5. Simpson et al (2001) Maths Quest Maths B Year 11 for Queensland, John Wiley and Sons: Brisbane. Chapters 4, 5, 6


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7 ASSESSMENT

7.1 EXIT CRITERIA

Student achievement will be judged on the following three exit criteria: Knowledge and procedures Modeling and problem solving Communication and justification

7.2 THE ASSESSMENT PLAN

Dysart State High School will use a system of continuous assessment to obtain information on student achievement throughout the course.

The assessment plan will enable students to demonstrate their achievement across the full range of the general objectives in the categories of Knowledge and procedures, Modeling and problem solving, and Communication and justification, within each context and across topics. It is anticipated that most will allow students to demonstrate achievement in more than one criterion.

This school will ensure validity and reliability of assessment instruments, by using a variety of implementation conditions for assessment of student achievement such as formal tests, assignments undertaken in class time, assignments undertaken at home, and group projects. The school aims to use assessment practices that reflect the learning experiences of the students in this course of study and it is anticipated that the assessment items will allow students to demonstrate achievement in more than one criterion. The conditions appropriate to each assessment instruments will be supplied at monitoring and verification.

Assessment tasks other than tests will be included at least twice a year and will contribute significantly to the decision-making process in each criterion. The use of appropriate technology in the classroom will be reflected in assessment tasks where the school’s resources allow use of that technology.

The school’s assessment program will be balanced over the course of study not necessarily over a semester or between semesters. Assessment will be formative in Year 11 and summative in Year 12. The formative assessment situations will be designed to develop students’ skills and lead naturally to the summative assessment situations. Formative assessment results will be selectively updated on the student’s profile by the summative assessment.

7.2.1 SPECIAL CONSIDERATION

The nature and appropriateness of special consideration and special arrangements for particular students will be in accordance with the responsibilities, principles and strategies outlined in the Board’s policy statement on special consideration: Special Consideration: Exemption and Special Arrangements in Senior Secondary School-Based Assessment.

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To enable special consideration to be effective for students so identified, this school will plan and implement strategies in the early stages of the assessment program and not at the point of deciding levels of achievement. The special consideration might involve alternative teaching approaches, assessment plans and learning experiences.

7.3 ASSESSMENT TECHNIQUES

This school will utilise a variety of assessment techniques that will allow students to demonstrate their abilities across a range of assessment items. The school will offer a balanced assessment package using a variety of assessment instruments, such as 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 modeled 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

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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.

Assessment tasks other than tests will be included at least twice each year and will contribute significantly to the decision making-process in each criterion.

7.3.1 Authorship of tasks

This school is mindful of the need to attest that the response to a task is genuinely that of the student. Depending on the circumstances, procedures such as the following may be utilised: The teacher may monitor the development of the task by seeing plans and a draft of the

student’s work The student may be required to produce and maintain documentation of the development of

the response The student may be required to acknowledge all resources used; this will include text and

source material and the type of assistance received The school may develop guidelines and procedures for students in relation to both print and

electronic source materials/resources, and to other types of assistance (including human) that have been sought.

7.4 Recording information

Relevant information collected on assessment instruments will be recorded on student profiles (see p 31) using grades in each criterion, although the school may record further information.

7.4.1 Knowledge and ProceduresThe assessment of this criterion involves the recall and use of results and procedures within the contexts of applications, technology and complexity. The assessment package will sample all topics and cover the continuum of contexts.The assessment package will enable students to demonstrate their ability in the general objectives of this category and marking schemes/ criteria sheets will allow appropriate data to be collected.For each instrument in 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 cut-offs will be applied according to the table below:

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NOTIONAL MINIMUM SCORE

STANDARD

>= 85% A>= 70% B>= 50% C>= 25% D< 25% E

The actual cut-offs used will depend on the relative difficulty of the assessment instrument and will be supplied, if necessary, with the instrument 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 (A to E will be used). Descriptors will be explicitly stated on the criteria sheet of the assessment. This method may apply more appropriately to assessment of Practical Tests, Assignments, Projects or Oral Presentations, but may be used throughout.

Where tests involve Modelling and problem solving, a student’s responses to these items may provide valuable data on the student’s ability to use some aspects of the knowledge of the subject matter. 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

At appropriate time (e.g. end of semester, monitoring, verification and exit) grades will be combined to arrive at a summary result using the table below. The overall performance required for a particular summary result may be adjusted depending on the complexity of the assessment provided, and the number and/or importance of the objectives addressed in the assessment instrument.

SUMMARY RESULT PERFORMANCE

A Consistently As

B Generally Bs, with the remainder generally Cs

C Generally Cs, with the remainder mostly DsD Generally Ds

E All Es

a) “Latest and fullest” will be considered in awarding summary or Exit levels. For example, a gradual upward trend in student performance means that a student may be awarded a result reflecting later performance as long as the performance is sustained.

b) 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.

c) Final A to E results must reflect the relevant descriptors summarised in the following table:

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7.4.2 Modelling and Problem SolvingThe assessment of this criterion involves modelling and problem solving. The assessment package will enable students to demonstrate their ability in the general objectives of both areas but this information will not be collected and recorded separately. Assessment instruments for this criterion may contain closed items with a limited number of correct solutions or open-ended items. The assessment package will cover the continuum of contexts.

Awarding Standards on Assessment ItemsFor each item assessing Modelling and problem solving a marking scheme using a grade from A to E will be used. For closed items and appropriate open-ended items, each item will be awarded a rating from A to E according to the table below.

GRADE DESCRIPTOR

A For problems: The student has solved problem correctly, without errorsFor open-ended tasks: The student has correctly extended and generalised the task, displaying significant insight and initiative.The student has recognised the strengths and limitations of the model or models (if required).

B The student had made substantial progress towards a rational solution and the response shows evidence of appropriate identification, clarification, analysis, selection of strategies and procedures.

C The student has made a reasonable interpretation of the problem and has selected some strategies or procedures appropriate to the problem.

D The student followed basic procedures or used strategies or procedures appropriate to the problem.

E The student is unable to begin the problem, submits work that is meaninglessOR the student has provided only the answer.

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GRADE 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 analyzing 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 Extending and generalizing from solutions

B The overall quality of a student’s achievement across a range within each context, and across topics generally demonstrates mathematical thinking which includes:

Interpreting, clarifying and analyzing a range of situations, identifying assumptions and variables

Selecting and using effective strategies Selecting appropriate procedures required to solve a wide 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, and across topics generally demonstrates mathematical thinking which includes:

Interpreting and clarifying a range of situations Selecting strategies and/or procedures appropriate to problems

D The overall quality of a student’s achievement sometimes demonstrates mathematical thinking which includes following basic procedures and/or using basic strategies

E The overall quality of a student’s achievement rarely demonstrates mathematical thinking which includes following basic procedures and/or using basic strategies

When these descriptors are not appropriate for a particular item, specific descriptors will be written which reflect the student’s achievement of objectives of Modelling and problem solving. This will generally be necessary for open-ended assessment items.For each instrument involving Modelling and problem solving all A to E grades obtained will be recorded on the student profile, although further information might be recorded.

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Awarding of Summary Grades

At appropriate time (e.g. end of semester, monitoring, verification and exit) grades will be combined to arrive at a summary result using the table below. The overall performance required for a particular summary result may be adjusted depending on the complexity of the assessment provided, and the number and/or importance of the objectives addressed in the assessment instrument.


A Generally As

B Generally Bs

C Generally CsD Generally Ds

E All Es

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Final A to E results must reflect the relevant descriptors as summarised in the following table:

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, and

identifying variables Selecting and using effective strategies Informed decision making…And in some contexts and topics, demonstrates mathematical thinking

which includes: Selecting and using appropriate procedures to solve a wide range of

problems Initiative in exploring the problem recognising strengths and limitations of models.

B The overall quality of a student’s achievement across a range within each context, and across topics, generally demonstrates mathematical thinking which includes: Interpreting, clarifying and analysing a range of situations, and

identifying variables Selecting and using strategies…And in some contexts and topics demonstrates mathematical thinking

which includes: Selecting and using procedures to solve a range of problems Informed decision-making.

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 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 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.

7.4.3 Communication and JustificationInformation on 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.a) A criteria sheet will accompany each task and used as a guide to collect information

reflecting the standards belowb) A student’s communication on each assessment task will be considered globally and rated

A, B, C, D or E reflecting standards detailed in the following table below:

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STANDARD DESCRIPTOR

A The student consistently demonstrates: Correct use of mathematical terms and symbols Evident mathematical and legible working as appropriate to the

assessment conditions Appropriate presentation of relevant information Arguments which are logically developed to support a logical

conclusion Justification of procedures

B The student generally demonstrates: Correct use of mathematical terms and symbols Evident mathematical and legible working 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 evident mathematical and legible working as appropriate to the

assessment conditions Simple arguments are developed to support a conclusion

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 basic conventions of language and mathematics

The student’s result will be recorded on the profile sheetFor some assessment items, it may be desirable to write specific descriptors for Communication and justification. These would be provided for students on the task sheet.

Awarding of Summary GradesAt appropriate time (e.g. end of semester, monitoring, verification and exit) grades will be combined to arrive at a summary result using the table below. The overall performance required for a particular summary result may be adjusted depending on the complexity of the assessment provided, and the number and/or importance of the objectives addressed in the assessment instrument.


A Consistently As

B Consistently Bs

C Consistently CsD Consistently Ds

E All Es

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Final A to E results must reflect the relevant descriptors as summarised below:

STANDARD DESCRIPTORA 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 to develop logical arguments in

support of conclusions, results and/or decisions Justification of procedures

B The overall quality of a student’s achievement in the range of contexts 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 to develop simple logical arguments in

support of conclusions, results and/or decisions.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. 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.

7.5 Student ProfilesAs described in the previous section, data gathered on a student’s performance on each of the criteria will be recorded on the student profile.The procedures 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.At verification all available summative assessment will be included in the verification folios for submission for review, including all completed assessment from semester 4.

Page 31

QUEENSLAND STATE HIGH SCHOOL

Mathematics Department BSSSS R4Mathematics B Year 11-

12Student:_______ Level of Achievement:________

Name:Semester

Assessment Knowledge & Procedures

Modelling &Problem Solving

Communication & Justification

Level of AchievementScore Grade

Semester 1

1.Assignment2. Test

3. Test

Semester 1 Summary

Semester 2

4.Assignment5. Test

6. Test

Semester 2 Summary

Semester 3

1. Assignment2. Test

3. Test

Semester 3 Summary

Semester 4

4. Assignment5. Test

Verification6. Test

Semester 4 Summary

Exit Summary

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QUEENSLAND STATE HIGH SCHOOL

Mathematics Department BSSSS R4Mathematics A Year 11-

12Student:_______ Level of Achievement:________

Name:Semester

Assessment Knowledge & Procedures

Modelling &Problem Solving

Communication & Justification

Level of AchievementScore Grade

Semester 1

1.Assignment2. Test

3. Test

Semester 1 Summary

Semester 2

4.Assignment

27.5/30 A A, A A

5. Test 25.5/30 B B A A A A

6. Test 17.5/30 C C A D E B

Semester 2 Summary B B A HA

Semester 3

1. Assignment

29/30 A B B A

2. Test 27.5/30 A A E D E B

3. Test

Semester 3 Summary A B A VH

Semester 4

4. Assignment

27/30 A A B D E

5. Test 28/30 A A B D A A

Verification A B A VH

6. Test 29.5/30 A A A A

Semester 4 Summary A B A VH

Exit Summary A B A VH

7.6 Determining exit levels of achievementThe school will award an exit standard for each of the three criteria (Knowledge and procedures, Modelling and problem solving, Communication and justification), based on the principles of assessment. When teachers are determining a standard for each criterion, the extent to which the qualities of the work match the descriptors overall will strongly influence the standard awarded.The exit standards will be applied to the summative body of work selected for exit.

Page 33

When standards have been determined in each of the three criteria, Table 1 will be used to determine the exit level of achievement.On completion of the course of study, the school will award each student an exit level of achievement from:Very High Achievement (VHA)High Achievement (HA)Sound Achievement (SA)Limited Achievement (LA)Very Limited Achievement (VLA).

Table 1: 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

NB For a student to be awarded an HA or a VHA, an elective topic will have been studied and contributed to summative assessment.

Page 34

MATHEMATICS B Senior Syllabus

Table 2: Minimum standards associated with exit criteria

Standard A Standard B Standard C Standard D Standard E

Cri

teri

on: K

now

ledg

e an

d pr

oced

ures

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 and rules Appropriate use of technology Appropriate recall and

selection of procedures, and their accurate and proficient use.

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 and rules

Appropriate use of technology

Appropriate recall and selection of procedures, and their accurate use.

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 and rules

Appropriate use of some technology

Accurate use of basic procedures.

The overall quality of a student’s achievement in the contexts of application, technology and complexity, sometimes demonstrates: Accurate recall

and use of some definitions and rules

Appropriate use of some technology.

The overall quality of a student’s achievement rarely demonstrates knowledge and use of procedures.

35



Cri

teri

on: M

odel

ing

and

prob

lem

solv

ing

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, and identifying variables

Selecting and using effective strategies

Informed decision making

…And in some contexts and topics, demonstrates mathematical thinking which includes: Selecting and using

appropriate procedures to solve a wide range of problems

Initiative in exploring the problem

Recognising strengths and limitations of models.

The overall quality of a student’s achievement across a range within each context, and across topics, generally demonstrates mathematical thinking which includes: Interpreting, clarifying

and analysing a range of situations, and identifying variables

Selecting and using strategies

…And in some contexts and topics demonstrates mathematical thinking which includes: Selecting and using

procedures to solve a range of problems

Informed decision-making.

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 to solve problems.

The overall quality of a student’s achievement sometimes demonstrates mathematical thinking which includes following basic procedures and/or using strategies to solve problems.

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.

36



Cri

teri

on: C

omm

unic

atio

n an

d ju

stifi

catio

n

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 to develop logical arguments in support of conclusions, results and/or decisions

Justification of procedures.

The overall quality of a student’s achievement in the range of contexts 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 to develop simple logical arguments in support of conclusions, results and/or decisions.

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.

The overall quality of a student’s achievement sometimes demonstrates evidence of the use of the basic conventions of language and mathematics.

The overall quality of a student’s achievement rarely demonstrates use of the basic conventions of language or mathematics.

Contexts are explained in section 3.2.

37


Dysart State High School – Mathematics A Work Program 20021

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Secondary Teaching Resources & Textbooks | Jacaranda Australia€¦ · Web viewnotional minimum score standard >= 85% A >= 70% B >= 50% C >= 25% D < 25% E The actual