Curriculum and Assessment Policy Statement
GRADES 7-9
MATHEMATICS
CAPS
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Mathematics · Grades 7, 8, 9 · English. CAPS document, 164 pages. Read online or download the PDF.
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MATHEMATICS GRADES 7-9
DISCLAIMER
In view of the stringent time requirements encountered by the Department of Basic Education to effect the necessary editorial changes and layout
to the Curriculum and Assessment Policy Statements and the supplementary policy documents, possible errors may occur in the said documents
placed on the official departmental websites.
There may also be vernacular inconsistencies in the language documents at Home-, First and Second Additional Language levels which have been
translated in the various African Languages. Please note that the content of the documents translated and versioned in the African Languages are
correct as they are based on the English generic language documents at all three language levels to be implemented in all four school phases.
If any editorial, layout or vernacular inconsistencies are detected, the user is kindly requested to bring this to the attention of the Department of
Basic Education.
E-mail: capslangcomments@dbe.gov.za or fax (012) 328 9828
Department of Basic Education
222 Struben Street
Private Bag X895
Pretoria 0001
South Africa
Tel: +27 12 357 3000
Fax: +27 12 323 0601
120 Plein Street Private Bag X9023
Cape Town 8000
South Africa
Tel: +27 21 465 1701
Fax: +27 21 461 8110
Website: http://www.education.gov.za
© 2011 Department of Basic Education
Isbn: 978-1-4315-0525-8
Design and Layout by: Ndabase Printing Solution
Printed by: Government Printing Works
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
MATHEMATICS GRADES 7-9
FOREWORD by the minister
Our national curriculum is the culmination of our efforts over a period of seventeen
years to transform the curriculum bequeathed to us by apartheid. From the start of
democracy we have built our curriculum on the values that inspired our Constitution
(Act 108 of 1996). The Preamble to the Constitution states that the aims of the
Constitution are to:
• heal the divisions of the past and establish a society based on democratic
values, social justice and fundamental human rights;
• improve the quality of life of all citizens and free the potential of each person;
• lay the foundations for a democratic and open society in which government is
based on the will of the people and every citizen is equally protected by law;
and
• build a united and democratic South Africa able to take its rightful place as a sovereign state in the family of
nations.
Education and the curriculum have an important role to play in realising these aims.
In 1997 we introduced outcomes-based education to overcome the curricular divisions of the past, but the experience
of implementation prompted a review in 2000. This led to the first curriculum revision: the Revised National Curriculum
Statement Grades R-9 and the National Curriculum Statement Grades 10-12 (2002).
Ongoing implementation challenges resulted in another review in 2009 and we revised the Revised National
Curriculum Statement (2002) and the National Curriculum Statement Grades 10-12 to produce this document.
From 2012 the two National Curriculum Statements, for Grades R-9 and Grades 10-12 respectively, are combined
in a single document and will simply be known as the National Curriculum Statement Grades R-12. The National
Curriculum Statement for Grades R-12 builds on the previous curriculum but also updates it and aims to provide
clearer specification of what is to be taught and learnt on a term-by-term basis.
The National Curriculum Statement Grades R-12 represents a policy statement for learning and teaching in South
African schools and comprises of the following:
(a) Curriculum and Assessment Policy Statements (CAPS) for all approved subjects listed in this document;
(b) National policy pertaining to the programme and promotion requirements of the National Curriculum Statement
Grades R-12; and
(c) National Protocol for Assessment Grades R-12.
MRS ANGIE MOTSHEKGA, MP
MINISTER OF BASIC EDUCATION
CAPS
MATHEMATICS GRADES 7-9
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
MATHEMATICS GRADES 7-9
TABLE OF CONTENTS
Section 1: INTRODUCTION AND BACKROUND ................................................................................ 3
1.1 Background.....................................................................................................................................................3
1.2 Overview ......................................................................................................................................................3
1.3 General aims of the South African curriculum. ...........................................................................................4
1.4 Time allocations..............................................................................................................................................6
1.4.1 Foundation Phase...................................................................................................................................6
1.4.2 Intermediate Phase.................................................................................................................................6
1.4.3 Senior Phase..........................................................................................................................................7
1.4.4 Grades 10-12..........................................................................................................................................7
SECTION 2: DEFINITION, AIMS, SKILLS AND CONTENT..................................................................... 8
2.1 Introduction.....................................................................................................................................................8
2.2 What is Mathematics?....................................................................................................................................8
2.3 Specific aims...................................................................................................................................................8
2.4 Specific skills..................................................................................................................................................8
2.5 Focus of content areas...................................................................................................................................9
Mathematics content knowledge.....................................................................................................................10
2.6 Weighting of content areas..........................................................................................................................11
2.7 Specification of content . .............................................................................................................................11
• Numbers, Operations and Relationships...................................................................................................12
• Patterns, Functions and Algebra................................................................................................................21
• Space and Shape (Geometry)...................................................................................................................27
• Measurement.............................................................................................................................................31
• Data Handling............................................................................................................................................33
SECTION 3: CLARIFICATION OF CONTENT........................................................................................ 37
3.1 Introduction...................................................................................................................................................37
3.2 Allocation of teaching time..........................................................................................................................37
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MATHEMATICS GRADES 7-9
3.3 Clarification notes with teaching guidelines..............................................................................................38
3.3.1 Clarification of content for Grade 7.......................................................................................................39
• Grade 7 term 1. ...............................................................................................................................39
• Grade 7 term 2. ...............................................................................................................................49
• Grade 7 term 3. ...............................................................................................................................58
• Grade 7 term 4. ...............................................................................................................................67
3.3.2 Clarification of content for Grade 8.......................................................................................................75
• Grade 8 term 1. ...............................................................................................................................75
• Grade 8 term 2. ...............................................................................................................................92
• Grade 8 term 3. .............................................................................................................................100
• Grade 8 term 4. .............................................................................................................................113
3.3.3 Clarification of content for Grade 9............................................................................................................... 119
• Grade 9 term 1. .............................................................................................................................119
• Grade 9 term 2. .............................................................................................................................134
• Grade 9 term 3. .............................................................................................................................141
• Grade 9 term 4. .............................................................................................................................147
SECTION 4: ASSESSMENT . ............................................................................................................... 154
4.1 Introduction ................................................................................................................................................154
4.2 Types of assessment..................................................................................................................................154
4.3 Informal or daily assessment. ...................................................................................................................155
4.4 Formal assessment. ...................................................................................................................................155
4.5 Recording and reporting............................................................................................................................157
4.6 Moderation of assessment.........................................................................................................................158
4.7 General ..................................................................................................................................................158
2 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
MATHEMATICS GRADES 7-9
Section 1: INTRODUCTION AND BACKGROUND
1.1 Background
The National Curriculum Statement Grades R-12 (NCS) stipulates policy on curriculum and assessment in the
schooling sector.
To improve implementation, the National Curriculum Statement was amended, with the amendments coming into
effect in January 2012. A single comprehensive Curriculum and Assessment Policy document was developed for
each subject to replace Subject Statements, Learning Programme Guidelines and Subject Assessment Guidelines
in Grades R-12.
1.2 Overview
(a) The National Curriculum Statement Grades R-12 (January 2012) represents a policy statement for learning
and teaching in South African schools and comprises the following:
(i) Curriculum and Assessment Policy Statements for each approved school subject;
(ii) The policy document, National policy pertaining to the programme and promotion requirements of the
National Curriculum Statement Grades R-12; and
(iii) The policy document, National Protocol for Assessment Grades R-12 (January 2012).
(b) The National Curriculum Statement Grades R-12 (January 2012) replaces the two current national curricula
statements, namely the
(i) Revised National Curriculum Statement Grades R-9, Government Gazette No. 23406 of 31 May 2002,
and
(ii) National Curriculum Statement Grades 10-12 Government Gazettes, No. 25545 of 6 October 2003 and
No. 27594 of 17 May 2005.
(c) The national curriculum statements contemplated in subparagraphs b(i) and (ii) comprise the following policy
documents which will be incrementally repealed by the National Curriculum Statement Grades R-12 (January
2012) during the period 2012-2014:
(i) The Learning Area/Subject Statements, Learning Programme Guidelines and Subject Assessment
Guidelines for Grades R-9 and Grades 10-12;
(ii) The policy document, National Policy on assessment and qualifications for schools in the General
Education and Training Band, promulgated in Government Notice No. 124 in Government Gazette No.
29626 of 12 February 2007;
(iii) The policy document, the National Senior Certificate: A qualification at Level 4 on the National
Qualifications Framework (NQF), promulgated in Government Gazette No.27819 of 20 July 2005;
CAPS 3
MATHEMATICS GRADES 7-9
(iv) The policy document, An addendum to the policy document, the National Senior Certificate: A
qualification at Level 4 on the National Qualifications Framework (NQF), regarding learners with special
needs, published in Government Gazette, No.29466 of 11 December 2006, is incorporated in the policy
document, National policy pertaining to the programme and promotion requirements of the National
Curriculum Statement Grades R-12; and
(v) The policy document, An addendum to the policy document, the National Senior Certificate: A
qualification at Level 4 on the National Qualifications Framework (NQF), regarding the National Protocol
for Assessment (Grades R-12), promulgated in Government Notice No.1267 in Government Gazette
No. 29467 of 11 December 2006.
(d) The policy document, National policy pertaining to the programme and promotion requirements of the
National Curriculum Statement Grades R-12, and the sections on the Curriculum and Assessment Policy as
contemplated in Chapters 2, 3 and 4 of this document constitute the norms and standards of the National
Curriculum Statement Grades R-12. It will therefore, in terms of section 6A of the South African Schools Act,
1996 (Act No. 84 of 1996,) form the basis for the Minister of Basic Education to determine minimum outcomes
and standards, as well as the processes and procedures for the assessment of learner achievement to be
applicable to public and independent schools.
1.3 General aims of the South African Curriculum
(a) The National Curriculum Statement Grades R-12 gives expression to the knowledge, skills and values worth
learning in South African schools. This curriculum aims to ensure that children acquire and apply knowledge
and skills in ways that are meaningful to their own lives. In this regard, the curriculum promotes knowledge in
local contexts, while being sensitive to global imperatives.
(b) The National Curriculum Statement Grades R-12 serves the purposes of:
• equipping learners, irrespective of their socio-economic background, race, gender, physical ability or
intellectual ability, with the knowledge, skills and values necessary for self-fulfilment, and meaningful
participation in society as citizens of a free country;
• providing access to higher education;
• facilitating the transition of learners from education institutions to the workplace; and
• providing employers with a sufficient profile of a learner’s competences.
(c) The National Curriculum Statement Grades R-12 is based on the following principles:
• Social transformation: ensuring that the educational imbalances of the past are redressed, and that equal
educational opportunities are provided for all sections of the population;
• Active and critical learning: encouraging an active and critical approach to learning, rather than rote and
uncritical learning of given truths;
• High knowledge and high skills: the minimum standards of knowledge and skills to be achieved at each
grade are specified and set high, achievable standards in all subjects;
• Progression: content and context of each grade shows progression from simple to complex;
4 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
MATHEMATICS GRADES 7-9
• Human rights, inclusivity, environmental and social justice: infusing the principles and practices of social and
environmental justice and human rights as defined in the Constitution of the Republic of South Africa. The
National Curriculum Statement Grades R-12 is sensitive to issues of diversity such as poverty, inequality,
race, gender, language, age, disability and other factors;
• Valuing indigenous knowledge systems: acknowledging the rich history and heritage of this country as
important contributors to nurturing the values contained in the Constitution; and
• Credibility, quality and efficiency: providing an education that is comparable in quality, breadth and depth to
those of other countries.
(d) The National Curriculum Statement Grades R-12 aims to produce learners that are able to:
• identify and solve problems and make decisions using critical and creative thinking;
• work effectively as individuals and with others as members of a team;
• organise and manage themselves and their activities responsibly and effectively;
• collect, analyse, organise and critically evaluate information;
• communicate effectively using visual, symbolic and/or language skills in various modes;
• use science and technology effectively and critically showing responsibility towards the environment and
the health of others; and
• demonstrate an understanding of the world as a set of related systems by recognising that problem solving
contexts do not exist in isolation.
(e) Inclusivity should become a central part of the organisation, planning and teaching at each school. This can
only happen if all teachers have a sound understanding of how to recognise and address barriers to learning,
and how to plan for diversity.
The key to managing inclusivity is ensuring that barriers are identified and addressed by all the relevant support
structures within the school community, including teachers, District-Based Support Teams, Institutional-Level
Support Teams, parents and Special Schools as Resource Centres. To address barriers in the classroom,
teachers should use various curriculum differentiation strategies such as those included in the Department of
Basic Education’s Guidelines for Inclusive Teaching and Learning (2010).
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MATHEMATICS GRADES 7-9
1.4 Time Allocation
1.4.1 Foundation Phase
(a) The instructional time in the Foundation Phase is as follows:
GRADE R GRADES 1-2 GRADE 3
SUBJECT
(HOURS) (HOURS) (HOURS)
Home Language 10 8/7 8/7
First Additional Language 2/3 3/4
Mathematics 7 7 7
Life Skills 6 6 7
• Beginning Knowledge (1) (1) (2)
• Creative Arts
(2) (2) (2)
• Physical Education
(2) (2) (2)
• Personal and Social Well-being
(1) (1) (1)
TOTAL 23 23 25
(b) Instructional time for Grades R, 1 and 2 is 23 hours and for Grade 3 is 25 hours.
(c) Ten hours are allocated for languages in Grades R-2 and 11 hours in Grade 3. A maximum of 8 hours and a
minimum of 7 hours are allocated for Home Language and a minimum of 2 hours and a maximum of 3 hours for
Additional Language in Grades 1-2. In Grade 3 a maximum of 8 hours and a minimum of 7 hours are allocated
for Home Language and a minimum of 3 hours and a maximum of 4 hours for First Additional Language.
(d) In Life Skills Beginning Knowledge is allocated 1 hour in Grades R – 2 and 2 hours as indicated by the hours
in brackets for Grade 3.
1.4.2 Intermediate Phase
(a) The instructional time in the Intermediate Phase is as follows:
SUBJECT HOURS
Home Language 6
First Additional Language 5
Mathematics 6
Natural Sciences and Technology 3,5
Social Sciences 3
Life Skills 4
• Creative Arts (1,5)
• Physical Education
(1)
• Personal and Social Well-being
(1,5)
TOTAL 27,5
6 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
MATHEMATICS GRADES 7-9
1.4.3 Senior Phase
(a) The instructional time in the Senior Phase is as follows:
SUBJECT HOURS
Home Language 5
First Additional Language 4
Mathematics 4,5
Natural Sciences 3
Social Sciences 3
Technology 2
Economic Management Sciences 2
Life Orientation 2
Creative Arts 2
TOTAL 27,5
1.4.4 Grades 10-12
(a) The instructional time in Grades 10-12 is as follows:
Subject Time allocation per week (hours)
Home Language 4.5
First Additional Language 4.5
Mathematics 4.5
Life Orientation 2
A minimum of any three subjects selected from Group B 12 (3x4h)
Annexure B, Tables B1-B8 of the policy document, National policy
pertaining to the programme and promotion requirements of
the National Curriculum Statement Grades R-12, subject to the
provisos stipulated in paragraph 28 of the said policy document.
TOTAL 27,5
The allocated time per week may be utilised only for the minimum required NCS subjects as specified above,
and may not be used for any additional subjects added to the list of minimum subjects. Should a learner wish
to offer additional subjects, additional time must be allocated for the offering of these subjects.
CAPS 7
MATHEMATICS GRADES 7-9
SECTION 2: DEFINITION, AIMS, SKILLS AND CONTENT
2.1 Introduction
In Section 2, the Senior Phase Mathematics Curriculum and Assessment Policy Statement (CAPS) provides teachers
with a definition of mathematics, specific aims, specific skills, focus of content areas, weighting of content areas and
content specification.
2.2 What is Mathematics?
Mathematics is a language that makes use of symbols and notations to describe numerical, geometric and graphical
relationships. It is a human activity that involves observing, representing and investigating patterns and quantitative
relationships in physical and social phenomena and between mathematical objects themselves. It helps to develop
mental processes that enhance logical and critical thinking, accuracy and problem-solving that will contribute in
decision-making.
2.3 Specific Aims
The teaching and learning of Mathematics aims to develop
• a critical awareness of how mathematical relationships are used in social, environmental, cultural and
economic relations
• confidence and competence to deal with any mathematical situation without being hindered by a fear of
Mathematics
• an appreciation for the beauty and elegance of Mathematics
• a spirit of curiosity and a love for Mathematics
• recognition that Mathematics is a creative part of human activity
• deep conceptual understandings in order to make sense of Mathematics
• acquisition of specific knowledge and skills necessary for:
-- the application of Mathematics to physical, social and mathematical problems
-- the study of related subject matter (e.g. other subjects)
-- further study in Mathematics.
2.4 Specific Skills
To develop essential mathematical skills the learner should
• develop the correct use of the language of Mathematics
• develop number vocabulary, number concept and calculation and application skills
8 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
MATHEMATICS GRADES 7-9
• learn to listen, communicate, think, reason logically and apply the mathematical knowledge gained
• learn to investigate, analyse, represent and interpret information
• learn to pose and solve problems
• build an awareness of the important role that Mathematics plays in real life situations including the personal
development of the learner.
2.5 Focus of Content Areas
Mathematics in the Senior Phase covers five main Content Areas.
• Numbers, Operations and Relationships;
• Patterns, Functions and Algebra;
• Space and Shape (Geometry);
• Measurement; and
• Data Handling.
Each content area contributes towards the acquisition of specific skills. The table below shows the general focus of
the content areas as well as the specific focus of the content areas for the Senior Phase.
CAPS 9
MATHEMATICS CONTENT KNOWLEDGE
Content area General content focus Senior Phase specific content focus
10
Development of number sense that includes: • Representation of numbers in a variety of ways and moving flexibly
• the meaning of different kinds of numbers between representations
• relationship between different kinds of numbers • Recognising and using properties of operations with different number
Numbers,
systems
Operations and • the relative size of different numbers
Relationships • Solving a variety of problems, using an increased range of numbers and the
• representation of numbers in various ways
ability to perform multiple operations correctly and fluently
• the effect of operating with numbers
• the ability to estimate and check solutions.
Algebra is the language for investigating and communicating most of • Investigation of numerical and geometric patterns to establish the
Mathematics and can be extended to the study of functions and other relationships between variables
relationships between variables. A central part of this content area is for the
• Expressing rules governing patterns in algebraic language or symbols
learner to achieve efficient manipulative skills in the use of algebra. It also
MATHEMATICS GRADES 7-9
focuses on the: • Developing algebraic manipulative skills that recognize the equivalence
Patterns, Functions
• description of patterns and relationships through the use of symbolic between different representations of the same relationship
and Algebra
expressions, graphs and tables; and • Analysis of situations in a variety of contexts in order to make sense of
• identification and analysis of regularities and change in patterns, and them
relationships that enable learners to make predictions and solve problems. • Representation and description of situations in algebraic language,
formulae, expressions, equations and graphs
The study of Space and Shape improves understanding and appreciation • Drawing and constructing a wide range of geometric figures and solids
of the pattern, precision, achievement and beauty in natural and cultural using appropriate geometric instruments
forms. It focuses on the properties, relationships, orientations, positions and
• Developing an appreciation for the use of constructions to investigate the
transformations of two-dimensional shapes and three-dimensional objects.
Space and Shape properties of geometric figures and solids
(Geometry) • Developing clear and more precise descriptions and classification
categories of geometric figures and solids
• Solving a variety of geometric problems drawing on known properties of
geometric figures and solids
Measurement focuses on the selection and use of appropriate units, • Using formulae for measuring area, perimeter, surface area and volume of
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
instruments and formulae to quantify characteristics of events, shapes, objects geometric figures and solids
and the environment. It relates directly to the learner’s scientific, technological
Measurement • Selecting and converting between appropriate units of measurement
and economic worlds, enabling the learner to
• Using the Theorem of Pythagoras to solve problems involving right-angled
• make sensible estimates; and
triangles
• be alert to the reasonableness of measurements and results.
Data Handling involves asking questions and finding answers in order to • Posing of questions for investigation
describe events and the social, technological and economic environment. • Collecting, summarizing, representing and critically analysing data in order
Through the study of data handling, the learner develops the skills to collect, to interpret, report and make predictions about situations
Data Handling organize, represent, Interpret, analyse and report data.
• Probability of outcomes include both single and compound events and their
The study of probability enables the learner to develop skills and techniques relative frequency in simple experiments
for making informed predictions, and describing randomness and uncertainty.
MATHEMATICS GRADES 7-9
2.6 Weighting of content areas
The weighting of Mathematics content areas serves two primary purposes:
• guidance on the time needed to adequately address the content within each content area
• guidance on the spread of content in the examination (especially end-of-year summative assessment).
WEIGHTING OF CONTENT AREAS
Content Area Grade 7 Grade 8 Grade 9
Number, Operations and Relations 30% 25% 15%
Patterns, Functions and Algebra 25% 30% 35%
Space and Shape (Geometry) 25% 25% 30%
Measurement 10% 10% 10%
Data Handling 10% 10% 10%
100% 100% 100%
2.7 Specification of content
The Specification of Content in Section 2 shows progression in terms of concepts and skills from Grades 7 - 9 for
each Content Area. However, in certain topics the concepts and skills are similar in two or three successive grades.
The Clarification of Content in Section 3 provides guidelines on how progression should be addressed in these
cases. The Specification of Content in Section 2 should therefore be read in conjunction with the Clarification of
Content in Section 3.
CAPS 11
SPECIFICATION OF CONTENT (PHASE OVERVIEW)
Numbers, Operations and relationships
12
• Progression in Numbers, Operations and Relationships in the Senior Phase is achieved primarily by:
-- development of calculations using whole numbers to calculations using rational numbers, integers and numbers in exponential form
-- development of understanding of different number systems from natural and whole numbers to integers and rational numbers, as well as the recognition of irrational numbers
-- increasing use of properties of numbers to perform calculations
-- increasing complexity of contexts for solving problems
• Numbers, Operations and Relationships in the Senior Phase consolidates work done in the Intermediate Phase and is geared towards making learners competent and efficient in
performing calculations particularly with integers and rational numbers.
• Recognising and using the properties of operations for different numbers provides a critical foundation for work in algebra when learners work with variables in place of numbers and
manipulate algebraic expressions and solve algebraic equations.
MATHEMATICS GRADES 7-9
TOPICS GRADE 7 GRADE 8 GRADE 9
1.1 Mental calculations Mental calculations
Whole numbers Revise the following done in Grade 6: • Revise multiplication of whole numbers to at least
12 x 12
• Multiplication of whole numbers to at least 12 x 12
• Multiplication facts for:
-- units and tens by multiples of ten
-- units and tens by multiples of 100
-- units and tens by multiples of 1 000
-- units and tens by multiples of 10 000
• Inverse operation between multiplication and
division
Ordering and comparing whole numbers Ordering and comparing whole numbers
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
• Revise the following done in Grade 6: • Revise prime numbers to at least 100
-- order, compare and represent numbers to at
least 9-digit numbers
-- recognize and represent prime numbers to at
least 100
-- round off numbers to the nearest 5, 10, 100 or
1 000
TOPICS GRADE 7 GRADE 8 GRADE 9
1.1 Properties of whole numbers Properties of whole numbers Properties of numbers
Whole numbers • Revise the following done in Grade 6: • Revise: • Describe the real number system by recognising,
CAPS
defining and distinguishing properties of:
-- recognize and use the commutative; -- The commutative; associative; distributive
associative; distributive properties of whole properties of whole numbers -- natural numbers
numbers
-- 0 in terms of its additive property (identity -- whole numbers
-- recognize and use 0 in terms of its additive element for addition)
-- integers
property (identity element for addition)
-- 1 in terms of its multiplicative property (identity
-- rational numbers
-- recognize and use 1 in terms of its multiplicative element for multiplication)
property (identity element for multiplication) -- irrational numbers
• Recognize the division property of 0, whereby any
number divided by 0 is undefined
Calculations using whole numbers Calculations using whole numbers Calculations using whole numbers
• Revise the following done in Grade 6, without use • Revise calculations using all four operations on • Revise calculations using all four operations on
of calculators: whole numbers, estimating and using calculators whole numbers, estimating and using calculators
where appropriate where appropriate
-- Addition and subtraction of whole numbers to at
least 6-digit numbers
-- Multiplication of at least whole 4-digit by 2-digit
numbers
-- Division of at least whole 4-digit by 2-digit
numbers
-- Perform calculations using all four operations
on whole numbers, estimating and using
calculators where appropriate
Calculation techniques Calculation techniques Calculation techniques
• Use a range of strategies to perform and check • Use a range of techniques to perform and check • Use a range of techniques to perform and check
written and mental calculations of whole numbers written and mental calculations of whole numbers written and mental calculations of whole numbers
including: including: including:
-- long division -- long division -- long division
-- adding, subtracting and multiplying in columns -- adding, subtracting and multiplying in columns -- adding, subtracting and multiplying in columns
-- estimation -- estimation -- estimation
-- rounding off and compensating -- rounding off and compensating -- rounding off and compensating
-- using a calculator -- using a calculator -- using a calculator
13
MATHEMATICS GRADES 7-9
TOPICS GRADE 7 GRADE 8 GRADE 9
1.1 Multiples and factors Multiples and factors Multiples and factors
14
Whole numbers • Revise the following done in Grade 6: • Revise: • Use prime factorisation of numbers to find LCM
and HCF
-- multiples of 2-digit and 3-digit whole numbers -- Prime factors of numbers to at least 3-digit
whole numbers
-- factors of 2-digit and 3-digit whole numbers
-- LCM and HCF of numbers to at least 3-digit
-- prime factors of numbers to at least 100
whole numbers, by inspection or factorisation
• List prime factors of numbers to at least 3-digit
whole numbers
• Find the LCM and HCF of numbers to at
least 3-digit whole numbers, by inspection or
factorisation
Solving problems Solving problems Solving problems
MATHEMATICS GRADES 7-9
• Solve problems involving whole numbers, • Solve problems involving whole numbers, • Solve problems in contexts involving
including including
-- ratio and rate
-- comparing two or more quantities of the same -- comparing two or more quantities of the same
-- direct and indirect proportion
kind (ratio) kind (ratio)
-- comparing two quantities of different kinds (rate) -- comparing two quantities of different kinds (rate)
-- sharing in a given ratio where the whole is given -- sharing in a given ratio where the whole is given
-- increasing or decreasing of a number in a given
ratio
• Solve problems that involve whole numbers, • Solve problems that involve whole numbers, • Solve problems that involve whole numbers,
percentages and decimal fractions in financial percentages and decimal fractions in financial percentages and decimal fractions in financial
contexts such as: contexts such as: contexts such as:
-- profit, loss and discount -- profit, loss, discount and VAT -- profit, loss, discount and VAT
-- budgets -- budgets -- budgets
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
-- accounts -- accounts -- accounts
-- loans -- loans -- loans
-- simple interest -- simple interest -- Simple interest
-- hire purchase -- hire purchase
-- exchange rates -- exchange rates
-- commission
-- rentals
-- compound interest
TOPICS GRADE 7 GRADE 8 GRADE 9
1.2 Mental calculations Mental calculations
Exponents 2
• Determine squares to at least 12 and their • Revise:
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square roots
-- Squares to at least 122 and their square roots
3
• Determine cubes to at least 6 and cube roots
-- Cubes to at least 63 and their cube roots
Comparing and representing numbers in Comparing and representing numbers in Comparing and representing numbers in
exponential form exponential form exponential form
• Compare and represent whole numbers in • Revise compare and represent whole numbers in • Revise compare and represent integers in
exponential form: exponential form exponential form
ab = a x a x a x... for b number of factors
• Compare and represent integers in exponential -- compare and represent numbers in scientific
form notation
• Compare and represent numbers in scientific • Extend scientific notation to include negative
notation, limited to positive exponents exponents
Calculations using numbers in exponential form Calculations using numbers in exponential form Calculations using numbers in exponential form
• Recognize and use the appropriate laws of • Establish general laws of exponents, limited to: • Revise the following general laws of exponents:
operations with numbers involving exponents and
square and cube roots -- natural number exponents -- am x an = am + n
• Perform calculations involving all four operations -- am x an = am + n -- am ÷ an = am – n, if m>n
using numbers in exponential form, limited to -- am ÷ an = am – n, if m>n -- (am)n = am x n
exponents up to 5, and square and cube roots
-- (am)n = am x n -- (a x t)n = an x tn
-- (a x t)n = an x tn -- a0 = 1
-- a0 = 1
• Recognize and use the appropriate laws of • Extend the general laws of exponents to include:
operations using numbers involving exponents
-- integer exponents
and square and cube roots
• Perform calculations involving all four operations -- a–m= a1m
with numbers that involve the squares, cubes,
• Perform calculations involving all four operations
square roots and cube roots of integers
using numbers in exponential form, using the
• Calculate the squares, cubes, square roots and laws of exponents
cube roots of rational numbers
Solving problems Solving problems Solving problems
• Solve problems in contexts involving numbers in • Solve problems in contexts involving numbers in • Solve problems in contexts involving numbers in
15 exponential form. exponential form exponential form, including scientific notation
MATHEMATICS GRADES 7-9
TOPICS GRADE 7 GRADE 8 GRADE 9
1.3 Counting, ordering and comparing integers Counting, ordering and comparing integers
16
Integers • Count forwards and backwards in integers for any • Revise:
interval
-- counting forwards and backwards in integers
• Recognize, order and compare integers for any interval
-- recognizing, ordering and comparing integers
Calculations with integers Calculations with integers Calculations with integers
• Add and subtract with integers • Revise addition and subtraction with integers • Revise:
• Multiply and divide with integers -- perform calculations involving all four
• Perform calculations involving all four operations operations with integers
with integers -- perform calculations involving all four
• Perform calculations involving all four operations operations with numbers that involve the
MATHEMATICS GRADES 7-9
with numbers that involve the squares, cubes, squares, cubes, square roots and cube roots of
square roots and cube roots of integers integers
Properties of integers Properties of integers Properties of integers
• Recognise and use commutative and associative • Recognise and use commutative, associative • Revise:
properties of addition and multiplication for and distributive properties of addition and
multiplication for integers -- Commutative, associative and distributive
integers
properties of addition and multiplication for
• Recognize and use additive and multiplicative integers
inverses for integers
-- additive and multiplicative inverses for integers
Solving problems Solving problems Solving problems
Solve problems in contexts involving addition and Solve problems in contexts involving multiple Solve problems in contexts involving multiple
subtraction with integers operations with integers operations with integers
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
TOPICS GRADE 7 GRADE 8 GRADE 9
1.4 Ordering, comparing and simplifying fractions
Common • Revise the following done in Grade 6
CAPS
fractions
-- compare and order common fractions, including
specifically tenths and hundredths
• Extend to thousandths
Calculations with fractions Calculations with fractions Calculations with fractions
• Revise the following done in Grade 6: • Revise: • All four operations with common fractions and
-- addition and subtraction of common fractions, mixed numbers
-- addition and subtraction of common fractions,
including mixed numbers, limited to fractions including mixed numbers • All four operations, with numbers that involve the
with the same denominator or where one squares, cubes, square roots and cube roots of
denominator is a multiple of another -- finding fractions of whole numbers
common fractions
-- finding fractions of whole numbers -- multiplication of common fractions, including
mixed numbers
• Extend addition and subtraction to fractions
where one denominator is not a multiple of the • Divide whole numbers and common fractions by
other common fractions
• Multiplication of common fractions, including • Calculate the squares, cubes, square roots and
mixed numbers, not limited to fractions where one cube roots of common fractions
denominator is a multiple of another
Calculation techniques Calculation techniques Calculation techniques
• Convert mixed numbers to common fractions in • Revise: • Revise:
order to perform calculations with them
-- convert mixed numbers to common fractions in -- convert mixed numbers to common fractions in
• Use knowledge of multiples and factors to write order to perform calculations with them order to perform calculations with them
fractions in the simplest form before or after
calculations -- use knowledge of multiples and factors to write -- use knowledge of multiples and factors to write
fractions in the simplest form before or after fractions in the simplest form before or after
• Use knowledge of equivalent fractions to add and calculations calculations
subtract common fractions
-- use knowledge of equivalent fractions to add -- use knowledge of equivalent fractions to add
and subtract common fractions and subtract common fractions
• Use knowledge of reciprocal relationships to -- use knowledge of reciprocal relationships to
divide common fractions divide common fractions
Solving problems Solving problems Solving problems
• Solve problems in contexts involving common • Solve problems in contexts involving common • Solve problems in contexts involving common
fractions and mixed numbers, including grouping, fractions and mixed numbers, including grouping, fractions, mixed numbers and percentages
sharing and finding fractions of whole numbers sharing and finding fractions of whole numbers
17
MATHEMATICS GRADES 7-9
TOPICS GRADE 7 GRADE 8 GRADE 9
1.4 Percentages Percentages
18
Common • Revise the following done in Grade 6: • Revise:
Fractions
-- Finding percentages of whole numbers -- finding percentages of whole numbers
• Calculate the percentage of part of a whole -- calculating the percentage of part of a whole
• Calculate percentage increase or decrease of -- calculating percentage increase or decrease
whole numbers
• Calculate amounts if given percentage increase
• Solve problems in contexts involving percentages or decrease
• Solve problems in contexts involving percentages
Equivalent forms Equivalent forms Equivalent forms
Revise the following done in Grade 6: • Revise equivalent forms between: • Revise equivalent forms between:
MATHEMATICS GRADES 7-9
• recognize and use equivalent forms of common -- common fractions (fractions where one -- common fractions where one denominator is a
fractions with 1-digit or 2-digit denominators denominator is a multiple of the other) multiple of another
(fractions where one denominator is a multiple of
-- common fraction and decimal fraction forms of -- common fraction and decimal fraction forms of
the other)
the same number the same number
• recognize equivalence between common fraction
-- common fraction, decimal fraction and -- common fraction, decimal fraction and
and decimal fraction forms of the same number
percentage forms of the same number percentage forms of the same number
• recognize equivalence between common fraction,
decimal fraction and percentage forms of the
same number
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
TOPICS GRADE 7 GRADE 8 GRADE 9
1.5 Ordering and comparing decimal fractions Ordering and comparing decimal fractions Calculations with decimal fractions
Decimal • Revise the following done in Grade 6: • Revise: • Multiple operations with decimal fractions, using a
CAPS
fractions calculator where appropriate
-- count forwards and backwards in decimal -- ordering, comparing and place value of decimal
fractions to at least two decimal places fractions to at least 3 decimal places • Multiple operations with or without brackets, with
numbers that involve the squares, cubes, square
-- compare and order decimal fractions to at least -- rounding off decimal fractions to at least 2
roots and cube roots of decimal fractions
two decimal places decimal place
-- place value of digits to at least two decimal
places
-- rounding off decimal fractions to at least 1
decimal place
• Extend all of the above to decimal fractions to at
least three decimal places and rounding off to at
least 2 decimal places
Calculations with decimal fractions Calculations with decimal fractions
• Revise the following done in Grade 6: • Revise:
-- addition and subtraction of decimal fractions of -- addition, subtraction, multiplication and of
at least two decimal places decimal fractions to at least 3 decimal places
-- multiplication of decimal fractions by 10 and -- division of decimal fractions by whole numbers
100
• Extend multiplication to 'multiplication by decimal
• Extend addition and subtraction to decimal fractions' not limited to one decimal place
fractions of at least three decimal places
• Extend division to 'division of decimal fractions by
• Multiply decimal fractions to include: decimal fractions'
-- decimal fractions to at least 3 decimal places • Calculate the squares, cubes, square roots and
by whole numbers cube roots of decimal fractions
-- decimal fractions to at least 2 decimal places
by decimal fractions to at least 1 decimal place
• Divide decimal fractions to include decimal
fractions to at least 3 decimal places by whole
numbers
Calculation techniques Calculation techniques Calculation techniques
• Use knowledge of place value to estimate the • Use knowledge of place value to estimate the • Use knowledge of place value to estimate the
number of decimal places in the result before number of decimal places in the result before number of decimal places in the result before
performing calculations performing calculations performing calculations
19
• Use rounding off and a calculator to check results • Use rounding off and a calculator to check results • Use rounding off and a calculator to check results
where appropriate where appropriate where appropriate
MATHEMATICS GRADES 7-9
TOPICS GRADE 7 GRADE 8 GRADE 9
1.5 Solving problems Solving problems Solving problems
20
Decimal • Solve problems in context involving decimal • Solve problems in context involving decimal • Solve problems in context involving decimal
fractions fractions fractions fractions
Equivalent forms Equivalent forms Equivalent forms
• Revise the following done in Grade 6: • Revise equivalent forms between: Revise equivalent forms between:
-- recognize equivalence between common -- common fraction and decimal fraction forms of -- common fraction and decimal fraction forms of
fraction and decimal fraction forms of the same the same number the same number
number
-- common fraction, decimal fraction and -- common fraction, decimal fraction and
-- recognize equivalence between common percentage forms of the same number percentage forms of the same number
fraction, decimal fraction and percentage forms
of the same number
MATHEMATICS GRADES 7-9
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
sPECIFICATION OF CONTENT (PHASE OVERVIEW)
PATTERNS, FUNCTIONS AND ALGEBRA
• Progression in Patterns, Functions and Algebra is achieved primarily by
CAPS
-- increasing the range and complexity of:
♦♦ relationships between numbers in given patterns
♦♦ rules, formulae and equations for which input and output values can be found
♦♦ equations that can be solved
-- developing more sophisticated skills and techniques for:
♦♦ solving equations
♦♦ expanding and simplifying algebraic expressions
♦♦ drawing and interpreting graphs
-- developing the use of algebraic language and conventions.
• In Patterns, Functions and Algebra, learners’ conceptual development progresses from:
-- an understanding of number to an understanding of variables, where the variables are numbers of a given type (e.g. natural numbers, integers, rational numbers) in generalized form
-- the recognition of patterns and relationships to the recognition of functions, where functions have unique outputs values for specified input values
-- a view of Mathematics as memorized facts and separate topics to seeing Mathematics as interrelated concepts and ideas represented in a variety of equivalent forms (e.g. a number
pattern, an equation and a graph representing the same relationship)
• While techniques for solving equations are developed in Patterns, Functions and Algebra, learners also practise solving equations in Measurement and Space and Shape, when they
apply known formulae to solve problems.
TOPICS GRADE 7 GRADE 8 GRADE 9
2.1 Investigate and extend patterns Investigate and extend patterns Investigate and extend patterns
Numeric and • Investigate and extend numeric and geometric • Investigate and extend numeric and geometric • Investigate and extend numeric and geometric
geometric patterns looking for relationships between patterns looking for relationships between patterns looking for relationships between
patterns numbers, including patterns: numbers, including patterns: numbers, including patterns:
-- represented in physical or diagram form -- represented in physical or diagram form -- represented in physical or diagram form
-- not limited to sequences involving a constant -- not limited to sequences involving a constant -- not limited to sequences involving a constant
difference or ratio difference or ratio difference or ratio
-- of learner’s own creation -- of learner’s own creation -- of learner’s own creation
-- represented in tables -- represented in tables -- represented in tables
-- represented algebraically -- represented algebraically
21 • Describe and justify the general rules for • Describe and justify the general rules for • Describe and justify the general rules for
observed relationships between numbers in own observed relationships between numbers in own observed relationships between numbers in own
words words or in algebraic language words or in algebraic language
MATHEMATICS GRADES 7-9
TOPICS GRADE 7 GRADE 8 GRADE 9
22 2.2 Input and output values Input and output values Input and output values
Functions and • Determine input values, output values or rules for • Determine input values, output values or rules for • Determine input values, output values or rules for
relationships patterns and relationships using: patterns and relationships using: patterns and relationships using:
-- flow diagrams -- flow diagrams -- flow diagrams
-- tables -- tables -- tables
-- formulae -- formulae -- formulae
-- equations -- equations
Equivalent forms Equivalent forms Equivalent forms
• Determine, interpret and justify equivalence of • Determine, interpret and justify equivalence of • Determine, interpret and justify equivalence of
MATHEMATICS GRADES 7-9
different descriptions of the same relationship or different descriptions of the same relationship or different descriptions of the same relationship or
rule presented: rule presented: rule presented:
-- verbally -- verbally -- verbally
-- in flow diagrams -- in flow diagrams -- in flow diagrams
-- in tables -- in tables -- in tables
-- by formulae -- by formulae -- by formulae
-- by number sentences -- by equations -- by equations
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS) -- by graphs on a Cartesian plane
TOPICS GRADE 7 GRADE 8 GRADE 9
CAPS 2.3 Algebraic language Algebraic language Algebraic language
Algebraic • Recognize and interpret rules or relationships • Revise the following done in Grade 7:
expressions represented in symbolic form
-- recognize and interpret rules or relationships
• Identify variables and constants in given formulae represented in symbolic form
and/or equations
-- identify variables and constants in given
formulae and/or equations
• Recognize and identify conventions for writing • Revise the following done in Grade 8:
algebraic expressions
-- recognize and identify conventions for writing
• Identify and classify like and unlike terms in algebraic expressions
algebraic expressions
-- identify and classify like and unlike terms in
• Recognize and identify coefficients and algebraic expressions
exponents in algebraic expressions
-- recognize and identify coefficients and
exponents in algebraic expressions
• Recognize and differentiate between monomials,
binomials and trinomials
Expand and simplify algebraic expressions Expand and simplify algebraic expressions
Use commutative, associative and distributive laws • Revise the following done in Grade 8, using the
for rational numbers and laws of exponents to: commutative, associative and distributive laws for
rational numbers and laws of exponents to:
• add and subtract like terms in algebraic
expressions -- add and subtract like terms in algebraic
expressions
• multiply integers and monomials by:
-- multiply integers and monomials by:
-- monomials
♦♦ monomials
-- binomials
♦♦ binomials
-- trinomials
♦♦ trinomials
• divide the following by integers or monomials:
-- divide the following by integers or monomials:
-- Monomials
♦♦ monomials
-- Binomials
♦♦ binomials
-- trinomials
♦♦ trinomials
• simplify algebraic expressions involving the above
23 operations -- simplify algebraic expressions involving the
above operations
MATHEMATICS GRADES 7-9
TOPICS GRADE 7 GRADE 8 GRADE 9
24 2.3 • Determine the squares, cubes, square roots -- Determine the squares, cubes, square roots
and cube roots of single algebraic terms or like and cube roots of single algebraic terms or like
Algebraic
expressions algebraic terms algebraic terms
• Determine the numerical value of algebraic -- Determine the numerical value of algebraic
expressions by substitution expressions by substitution
• Extend the above algebraic manipulations to
include:
-- Multiply integers and monomials by polynomials
-- Divide polynomials by integers or monomials
-- The product of two binomials
-- The square of a binomial
MATHEMATICS GRADES 7-9
Factorize algebraic expressions
• Factorize algebraic expressions that involve:
-- common factors
-- difference of two squares
-- trinomials of the form:
♦♦ x2 + bx + c
♦♦ ax2 + bx + c, where a is a common factor.
• Simplify algebraic expressions that involve the
above factorisation processes.
• Simplify algebraic fractions using factorisation.
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
TOPICS GRADE 7 GRADE 8 GRADE 9
CAPS 2.4 Number sentences Equations Equations
Algebraic • Write number sentences to describe problem • Revise the following done in Grade 7: • Revise the following done in Grade 8:
equations situations
-- set up equations to describe problem situations -- set up equations to describe problem situations
• Analyse and interpret number sentences that
-- analyse and interpret equations that describe a -- analyse and interpret equations that describe a
describe a given situation
given situation given situation
• Solve and complete number sentences by:
-- solve equations by inspection -- solve equations by inspection
-- inspection
-- using additive and multiplicative inverses
-- trial and improvement
-- using laws of exponents
• Determine the numerical value of an expression -- determine the numerical value of an expression -- determine the numerical value of an expression
by substitution. by substitution. by substitution.
• Identify variables and constants in given formulae -- identify variables and constants in given
or equations formulae or equations
• Use substitution in equations to generate tables -- use substitution in equations to generate tables
of ordered pairs of ordered pairs
• Extend solving equations to include: • Extend solving equations to include:
-- using additive and multiplicative inverses -- using factorisation
-- using laws of exponents -- equations of the form: a product of factors = 0
25
MATHEMATICS GRADES 7-9
TOPICS GRADE 7 GRADE 8 GRADE 9
26 2.5 Interpreting graphs Interpreting graphs Interpreting graphs
Graphs • Analyse and interpret global graphs of problem • Revise the following done in Grade 7: • Revise the following done in Grade 8:
situations, with special focus on the following
-- analyse and interpret global graphs of problem -- analyse and interpret global graphs of problem
trends and features:
situations, with a special focus on the following situations, with a special focus on the following
-- linear or non-linear trends and features: trends and features:
-- constant, increasing or decreasing ♦♦ linear or non-linear ♦♦ linear or non-linear
♦♦ constant, increasing or decreasing ♦♦ constant, increasing or decreasing
♦♦ maximum or minimum
♦♦ discrete or continuous
• Extend the focus on features of graphs to include: • Extend the above with special focus on the
MATHEMATICS GRADES 7-9
following features of linear graphs:
-- maximum or minimum
-- x-intercept and y-intercept
-- discrete or continuous
-- gradient
Drawing graphs Drawing graphs Drawing graphs
• Draw global graphs from given descriptions of • Draw global graphs from given descriptions of • Revise the following done in Grade 8:
a problem situation, identifying features listed a problem situation, identifying features listed
-- draw global graphs from given descriptions of
above above
a problem situation, identifying features listed
• Use tables or ordered pairs to plot points and above.
draw graphs on the Cartesian plane
-- use tables of ordered pairs to plot points and
draw graphs on the Cartesian plane
• Extend the above with special focus on:
-- drawing linear graphs from given equations
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
-- determining equations from given linear graphs
sPECIFICATION OF CONTENT (PHASE OVERVIEW)
Space and Shape (Geometry)
• Progression in geometry in the Senior Phase is achieved primarily by:
CAPS
-- investigating new properties of shapes and objects
-- developing from informal descriptions of geometric figures to more formal definitions and classification of shapes and objects
-- solving more complex geometric problems using known properties of geometric figures
-- developing from inductive reasoning to deductive reasoning.
• The geometry topics are much more inter-related than in the Intermediate Phase, especially those relating to constructions and geometry of 2D shapes and straight lines, hence care
has to be taken regarding sequencing of topics through the terms.
• In the Senior Phase, transformation geometry develops from general descriptions of movement in space to more specific descriptions of movement in co-ordinate planes. This lays
the foundation for analytic geometry in the FET phase.
• Solving problems in geometry to find unknown angles or lengths provides a useful context to practise solving equations.
TOPICS GRADE 7 GRADE 8 GRADE 9
3.1 Classifying 2D shapes Classifying 2D shapes Classifying 2D shapes
Geometry of 2D • Describe, sort, name and compare triangles • Identify and write clear definitions of triangles in • Revise properties and definitions of triangles in
shapes according to their sides and angles, focusing on: terms of their sides and angles, distinguishing terms of their sides and angles, distinguishing
between: between:
-- equilateral triangles -- equilateral triangles -- equilateral triangles
-- isosceles triangles -- isosceles triangles -- isosceles triangles
-- right-angled triangles -- right-angled triangles -- right-angled triangles
• Describe, sort, name and compare quadrilaterals • Identify and write clear definitions of quadrilaterals • Revise and write clear definitions of quadrilaterals
in terms of: in terms of their sides and angles, distinguishing in terms of their sides, angles and diagonals,
between: distinguishing between:
-- length of sides
-- parallelogram -- parallelogram
-- parallel and perpendicular sides
-- rectangle -- rectangle
-- size of angles (right-angles or not)
-- square -- square
• Describe and name parts of a circle
-- rhombus -- rhombus
-- trapezium -- trapezium
-- kite -- kite
27
MATHEMATICS GRADES 7-9
TOPICS GRADE 7 GRADE 8 GRADE 9
3.1 Similar and congruent 2D shapes Similar and congruent 2D shapes Similar and congruent triangles
28
Geometry of 2D • Recognize and describe similar and congruent • Identify and describe the properties of congruent • Through investigation, establish the minimum
shapes figures by comparing: shapes conditions for congruent triangles
-- shape • Identify and describe the properties of similar • Through investigation, establish the minimum
shapes conditions for similar triangles
-- size
Solving problems Solving problems
Solving problems
• Solve geometric problems involving unknown • Solve geometric problems involving unknown
• Solve simple geometric problems involving
sides and angles in triangles and quadrilaterals, sides and angles in triangles and quadrilaterals,
unknown sides and angles in triangles and
using known properties and definitions. using known properties of triangles and
quadrilaterals, using known properties.
quadrilaterals, as well as properties of congruent
and similar triangles.
3.2 Classifying 3D objects Classifying 3D objects Classifying 3D objects
MATHEMATICS GRADES 7-9
Geometry of 3D • Describe, sort and compare polyhedra in terms • Describe, name and compare the 5 Platonic • Revise properties and definitions of the 5 Platonic
objects of: solids in terms of the shape and number of faces, solids in terms of the shape and number of faces,
the number of vertices and the number of edges the number of vertices and the number of edges
-- shape and number of faces
• Recognize and describe the properties of:
-- number of vertices
-- spheres
-- number of edges
-- cylinders
Building 3D models
Building 3D models Building 3D models
• Revise using nets to create models of geometric
solids, including: • Revise using nets to create models of geometric • Use nets to create models of geometric solids,
solids, including: including:
-- cubes
-- cubes -- cubes
-- prisms
-- prisms -- prisms
-- pyramids -- pyramids
-- cylinders
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
3.3 Define: Angle relationships Angle relationships
Geometry of • Line segment • Recognize and describe pairs of angles formed • Revise and write clear descriptions of the
straight lines by: relationship between angles formed by:
• Ray
-- perpendicular lines -- perpendicular lines
• Straight line
-- intersecting lines -- intersecting lines
• Parallel lines
-- parallel lines cut by a transversal -- parallel lines cut by a transversal
• Perpendicular lines
Solving problems Solving problems
• Solve geometric problems using the relationships • Solve geometric problems using the relationships
between pairs of angles described above between pairs of angles described above
TOPICS GRADE 7 GRADE 8 GRADE 9
3.4 Transformations Transformations Transformations
Transformation • Recognize, describe and perform translations, • Recognize, describe and perform transformations • Recognize, describe and perform transformations
CAPS
Geometry reflections and rotations with geometric figures with points on a co-ordinate plane, focusing on: with points, line segments and simple geometric
and shapes on squared paper figures on a co-ordinate plane, focusing on:
-- reflecting a point in the X-axis or Y-axis
• Identify and draw lines of symmetry in geometric -- reflection in the X-axis or Y-axis
-- translating a point within and across quadrants
figures
-- translation within and across quadrants
-- reflection in the line y = x
• Recognize, describe and perform transformations • Identify what the transformation of a point is, if
with triangles on a co-ordinate plane, focusing on given the co-ordinates of its image
the co-ordinates of the vertices when:
-- reflecting a triangle in the X-axis or Y-axis
-- translating a triangle within and across quadrants
-- rotating a triangle around the origin
Enlargements and reductions Enlargements and reductions Enlargements and reductions
• Draw enlargements and reductions of geometric • Use proportion to describe the effect of • Use proportion to describe the effect of
figures on squared paper and compare them in enlargement or reduction on area and perimeter enlargement or reduction on area and perimeter
terms of shape and size of geometric figures of geometric figures
• Investigate the co-ordinates of the vertices of
figures that have been enlarged or reduced by a
given scale factor
29
MATHEMATICS GRADES 7-9
TOPICS GRADE 7 GRADE 8 GRADE 9
3.5 Measuring angles
30
Construction • Accurately use a protractor to measure and
of geometric classify angles:
figures
-- < 90o (acute angles)
-- Right-angles
-- > 90o (obtuse angles)
-- Straight angles
-- > 180o (reflex angles)
Constructions Constructions Constructions
• Accurately construct geometric figures • Accurately construct geometric figures • Accurately construct geometric figures
appropriately using a compass, ruler and appropriately using a compass, ruler and appropriately using a compass, ruler and
MATHEMATICS GRADES 7-9
protractor, including: protractor, including: protractor, including bisecting angles of a triangle
-- angles, to one degree of accuracy -- bisecting lines and angles
-- circles -- perpendicular lines at a given point or from a
given point
-- parallel lines
-- triangles
-- perpendicular lines
-- quadrilaterals
• Construct angles of 30°, 45°, 60° and their • Construct angles of 30°, 45°, 60° and their
multiples without using a protractor multiples without using a protractor
Investigating properties of geometric figures Investigating properties of geometric figures
• By construction, investigate the angles in a • By construction, investigate the angles in a
triangle, focusing on: triangle, focusing on the relationship between the
exterior angle of a triangle and its interior angles
-- the sum of the interior angles of triangles
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
-- the size of angles in an equilateral triangle
-- the sides and base angles of an isosceles
triangle
• By construction, investigate sides and angles in • By construction, investigate sides, angles and
quadrilaterals, focusing on: diagonals in quadrilaterals, focusing on:
-- the sum of the interior angles of quadrilaterals -- the diagonals of rectangles, squares,
parallelograms, rhombi and kites
-- the sides and opposite angles of parallelograms
-- exploring the sum of the interior angles of
polygons
• By construction, explore the minimum conditions
for two triangles to be congruent
SPECIFICATION OF CONTENT (PHASE OVERVIEW)
MEASUREMENT
• Progression in Measurement is achieved by the selection of shapes and objects in each grade for which the formulae for finding area, perimeter, surface area and volume become
CAPS
more complex.
• The use of formulae in this phase provides a useful context to practise solving equations.
• The introduction of the Theorem of Pythagoras is a way of introducing a formula to calculate the lengths of sides in right-angled triangles. Hence, the Theorem of Pythagoras
becomes a useful tool when learners solve geometric problems involving right-angled triangles.
• Measurement disappears as a separate topic in the FET phase, and becomes part of the study of Geometry and Trigonometry.
TOPICS GRADE 7 GRADE 8 GRADE 9
4.1 Area and perimeter Area and perimeter Area and perimeter
Area and • Calculate the perimeter of regular and irregular • Use appropriate formulae to calculate perimeter • Use appropriate formulae and conversions
perimeter of 2D polygons and area of: between SI units, to solve problems and calculate
shapes perimeter and area of:
• Use appropriate formulae to calculate perimeter -- squares
and area of: -- polygons
-- rectangles
-- squares -- circles
-- triangles
-- rectangles • Investigate how doubling any or all of the
-- circles
dimensions of a 2D figure affects its perimeter
-- triangles
• Calculate the areas of polygons, to at least and its area
2 decimal places, by decomposing them into
rectangles and/or triangles
• Use and describe the relationship between the
radius, diameter and circumference of a circle in
calculations
• Use and describe the relationship between the
radius and area of a circle in calculations
Calculations and solving problems Calculations and solving problems
• Solve problems involving perimeter and area of • Solve problems, with or without a calculator,
polygons involving perimeter and area of polygons and
circles
• Calculate to at least 1 decimal place
• Calculate to at least 2 decimal places
• Use and convert between appropriate SI units,
including: • Use and describe the meaning of the irrational
2 2 number Pi (π) in calculations involving circles
-- mm ↔ cm
• Use and convert between appropriate SI units,
-- cm2 ↔ m2
31 including: mm2 ↔ cm2 ↔ m2 ↔ km2
MATHEMATICS GRADES 7-9
TOPICS GRADE 7 GRADE 8 GRADE 9
4.2 Surface area and volume Surface area and volume Surface area and volume
32
Surface area and • Use appropriate formulae to calculate the surface • Use appropriate formulae to calculate the surface • Use appropriate formulae and conversions
volume of 3D area, volume and capacity of: area, volume and capacity of: between SI units to solve problems and calculate
objects the surface area, volume and capacity of:
-- cubes -- cubes
-- cubes
-- rectangular prisms -- rectangular prisms
-- rectangular prisms
-- triangular prisms
-- triangular prisms
-- cylinders
• Investigate how doubling any or all the
• Describe the interrelationship between surface • Describe the interrelationship between surface
dimensions of right prisms and cylinders affects
area and volume of the objects mentioned above area and volume of the objects mentioned above
their volume
Calculations and solving problems Calculations and solving problems
MATHEMATICS GRADES 7-9
• Solve problems involving surface area, volume • Solve problems, with or without a calculator,
and capacity involving surface area, volume and capacity
• Use and convert between appropriate SI units, • Use and convert between appropriate SI units,
including: including:
-- mm2 ↔ cm2 -- mm2 ↔ cm2 ↔ m2 ↔ km2
-- cm2 ↔ m2 -- mm3 ↔ cm3 ↔ m3
-- mm3 ↔ cm3 -- ml (cm3) ↔ l ↔ kl
-- cm3 ↔ m3
• Use equivalence between units when solving
problems:
-- 1 cm3 ↔ 1 ml
-- 1 m3 ↔ 1 kl
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
4.3 Develop and use the Theorem of Pythagoras Solve problems using the Theorem of
Pythagoras
The Theorem of • Investigate the relationship between the lengths
Pythagoras of the sides of a right-angled triangle to develop • Use the Theorem of Pythagoras to solve
the Theorem of Pythagoras problems involving unknown lengths in geometric
figures that contain right-angled triangles
• Determine whether a triangle is a right-angled
triangle or not if the length of the three sides of
the triangle are known
• Use the Theorem of Pythagoras to calculate a
missing length in a right-angled triangle, leaving
irrational answers in surd form
SPECIFICATION OF CONTENT (PHASE OVERVIEW)
Data Handling
• Progression in Data Handling is achieved primarily by:
CAPS
-- increasing complexity of data sets and contexts
-- reading, interpreting and drawing new types of data graphs
-- becoming more efficient at organizing and summarizing data
-- becoming more critical and aware of bias and manipulation in representing, analysing and reporting data
• Learners should work through at least 1 data cycle for the year – this involves collecting and organizing, representing, analysing, summarizing, interpreting and reporting data. The
data cycle provides the opportunity for doing projects.
• All of the above aspects of data handling should also be dealt with as discrete activities in order to consolidate concepts and practise skills. For example, learners need to practise
summarizing data presented in different forms, and summaries should be used when reporting data.
• Data handling contexts should be selected to build awareness of social, economic and environmental issues.
• Learners should become sensitized to bias in the collection of data, as well as misrepresentation of data through the use of different scales and different measures of central
tendency.
• The following resources provide interesting contexts for data comparison and analysis that can be used in this phase:
-- Census at School – for school based surveys
-- national surveys from Statistics South Africa (StatsSA) – for household and population surveys.
-- international surveys from United Nations (UN Data) – for international social, demographic and environmental surveys. Many other websites may be consulted, especially for health
and environmental data.
TOPICS GRADE 7 GRADE 8 GRADE 9
5.1 Collect data Collect data Collect data
Collect, organize • Pose questions relating to social, economic, and • Pose questions relating to social, economic, and • Pose questions relating to social, economic, and
and summarize environmental issues in own environment environmental issues environmental issues
data
• Select appropriate sources for the collection of • Select appropriate sources for the collection of • Select and justify appropriate sources for the
data (including peers, family, newspapers, books, data (including peers, family, newspapers, books, collection of data
magazines) magazines)
• Distinguish between samples and populations,
• Distinguish between samples and populations • Distinguish between samples and populations, and suggest appropriate samples for investigation
and suggest appropriate samples for investigation and suggest appropriate samples for investigation
• Select and justify appropriate methods for
• Design and use simple questionnaires to answer • Design and use simple questionnaires to answer collecting data
questions: questions with multiple choice responses
-- with yes/no type responses
33 -- with multiple choice responses
MATHEMATICS GRADES 7-9
TOPICS GRADE 7 GRADE 8 GRADE 9
5.1 Organize and summarize data Organize and summarize data Organize and summarize data
34
Collect, organize • Organize (including grouping where appropriate) • Organize (including grouping where appropriate) • Organize numerical data in different ways in order
and summarize and record data using and record data using to summarize by determining:
data
-- tally marks -- tally marks -- measures of central tendency
-- tables -- tables -- measures of dispersion, including extremes and
outliers
-- stem-and-leaf displays -- stem-and-leaf displays
• Organize data according to more than one criteria
• Group data into intervals • Group data into intervals
• Summarize and distinguishing between • Summarize data using measures of central
ungrouped numerical data by determining: tendency, including:
-- mean -- mean
-- median -- median
MATHEMATICS GRADES 7-9
-- mode -- mode
• Identify the largest and smallest scores in a data • Summarize data using measures of dispersion,
set and determine the difference between them in including:
order to determine the spread of the data (range)
-- range
-- extremes
5.2 Represent data Represent data Represent data
Represent data • Draw a variety of graphs by hand/technology • Draw a variety of graphs by hand/technology to • Draw a variety of graphs by hand/technology to
to display and interpret data (grouped and display and interpret data including: display and interpret data including:
ungrouped) including:
-- bar graphs and double bar graphs -- bar graphs and double bar graphs
-- bar graphs and double bar graphs
-- histograms with given and own intervals -- histograms with given and own intervals
-- histograms with given intervals
-- pie charts -- pie charts
-- pie charts
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
-- broken-line graphs -- broken-line graphs
-- scatter plots
TOPICS GRADE 7 GRADE 8 GRADE 9
5.3 Interpret data Interpret data Interpret data
Interpret, • Critically read and interpret data represented in: • Critically read and interpret data represented in: • Critically read and interpret data represented in a
CAPS
analyse, and variety of ways
report data -- words -- words
• Critically compare two sets of data related to the
-- bar graphs -- bar graphs
same issue
-- double bar graphs -- double bar graphs
-- pie charts -- pie charts
-- histograms -- histograms
-- broken-line graphs
Analyse data Analyse data Analyse data
• Critically analyse data by answering questions • Critically analyse data by answering questions • Critically analyse data by answering questions
related to: related to: related to:
-- data categories, including data intervals -- data categories, including data intervals -- data collection methods
-- data sources and contexts -- data sources and contexts -- summary of data
-- central tendencies (mean, mode, median) -- central tendencies (mean, mode, median) -- sources of error and bias in the data
-- scales used on graphs -- scales used on graphs
-- samples and populations
-- dispersion of data
-- error and bias in the data
Report data Report data Report data
• Summarize data in short paragraphs that include • Summarize data in short paragraphs that include • Summarize data in short paragraphs that include
-- drawing conclusions about the data -- drawing conclusions about the data -- drawing conclusions about the data
-- making predictions based on the data -- making predictions based on the data -- making predictions based on the data
-- identifying sources of error and bias in the data -- identifying sources of error and bias in the data -- making comparisons between two sets of data
-- choosing appropriate summary statistics for the -- choosing appropriate summary statistics for the -- identifying sources of error and bias in the data
data (mean, median, mode) data (mean, median, mode, range)
-- choosing appropriate summary statistics for the
-- the role of extremes in the data data (mean, median, mode, range)
-- the role of extremes and outliers in the data
35
MATHEMATICS GRADES 7-9
TOPICS GRADE 7 GRADE 8 GRADE 9
5.4 Probability Probability Probability
36
Probability • Perform simple experiments where the possible • Consider a simple situation (with equally likely • Consider situations with equally probable
outcomes are equally likely and: outcomes) that can be described using probability outcomes, and:
and:
-- list the possible outcomes based on the -- determine probabilities for compound events
conditions of the activity -- list all the possible outcomes using two-way tables and tree diagrams
-- determine the probability of each possible -- determine the probability of each possible -- determine the probabilities for outcomes of
outcome using the definiton of probability outcome using the definition of probability events and predict their relative frequency in
simple experiments
-- predict with reasons the relative frequency
of the possible outcomes for a series of trials -- compare relative frequency with probability and
based on probability explains possible differences
-- compare relative frequency with probability and
explains possible differences
MATHEMATICS GRADES 7-9
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
MATHEMATICS GRADES 7-9
Section 3: Content Clarification
3.1 Introduction
• In this chapter, content clarification includes:
-- teaching guidelines
-- suggested sequencing of topics per term
-- suggested pacing of topics over the year.
• Each Content Area has been broken down into topics. The sequencing of topics within terms gives an idea of
how content areas can be spread and re-visited throughout the year.
• Teachers may choose to sequence and pace the contents differently from the recommendations in this
section. However, cognisance should be taken of the relative weighting and notional hours of the Content
Areas for this phase.
3.2 Allocation of teaching time
Time has been allocated in the following way:
• 10 weeks per term, with 4,5 hours for Mathematics per week (10 x 4 x 4,5 hours = 180 hours per year)
• Between 6 and 12 hours have been allocated for revision and assessment per term
• Therefore, 150 hours of teaching have been distributed across the Content Areas
• The distribution of time per topic, has taken account of the weighting for the Content Area as specified for the
Senior Phase in Section 2.
• The weighting of Content Areas represents notional hours; therefore, the recommended distribution of hours
may vary slightly across grades.
CAPS 37
MATHEMATICS GRADES 7-9
3.3 Clarification notes with teaching guidelines
The tables below provide the teacher with:
• content areas and topics per grade per term;
• concepts and skills per term;
• clarification notes with teaching guidelines; and
• the duration of time allocated per topic in hours.
Time allocation per Topic: Grade 7
TERM 1 TERM 2 TERM 3 TERM 4
Topic Time Topic Time Topic Time Topic Time
Numeric and
9 9 6 9
Whole numbers Common fractions geometric Integers
hours hours hours hours
patterns
Numeric and
9 9 Functions and 3 3
Exponents Decimal fractions geometric
hours hours relationships hours hours
patterns
Construction of 10 Functions and 3 Algebraic 3 Functions and 3
Geometric figures hours relationships hours expressions hours relationships hours
Area and
Geometry of 2D 10 7 Algebraic 3 Algebraic 3
perimeter of 2D
shapes hours hours equations hours expressions hours
shapes
Geometry of 2 Surface area and
8 6 Algebraic 4
straight lines Volume of 3D Graphs
hours hours hours equations hours
objects
Transformation Collect, organize
geometry 9 4
and summarize
hours hours
data
Geometry of 3D 9 3
Represent data
objects hours hours
Interpret, analyse 3,5
and report data hours
4,5
Probability
hours
Revision/ 5 Revision/ 9 Revision/ 6 Revision/
8hours
assessment hours assessment hours assessment hours assessment
TOTAL: 45 hours TOTAL: 45 hours TOTAL: 45 hours TOTAL: 45 hours
38 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
3.3.1 Clarification of content for Grade 7
TERM 1 – Grade 7
CAPS
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
Numbers, 1.1 What is different to Grade 6? 9 hours
Operations and
Relationships Whole • Prime factors of 3-digit numbers
numbers
• LCM and HCF
• More complex financial contexts for solving problems
In Grade 7 learners consolidate number knowledge and calculation techniques for
whole numbers, developed in the Intermediate Phase.
Mental calculations Mental calculations
Revise the following done in Grade 6: Mental calculations should be used to practice concepts and skills developed
through the main lesson, sometimes with smaller number ranges. Learners should
• Multiplication of whole numbers to at
not be asked to do random calculations each day. Rather, mental calculations
least 12 x 12
should be used as an opportunity to consolidate four aspects of learners’ number
• Multiplication facts for: knowledge:
-- Units and tens by multiples of ten • number facts (number bonds and times tables)
-- Units and tens by multiples of 100 • calculation techniques ( doubling and halving, using multiplication to do division,
multiplying and dividing by 10, 100, 1 000
-- Units and tens by multiples of 1 000
• multiplying by multiples of 10, 100, 1 000,
-- Units and tens by multiples of
10 000 • building up and breaking down numbers, rounding off and compensating etc)
• Inverse operation between • number concept (counting, ordering and comparing, place value, odd and even
multiplication and division numbers, multiples and factors)
• properties of numbers (identity elements for addition and multiplication;
• commutative and associative property for addition and multiplication;
• inverse operation for multiplication and division; inverse operation for addition
and subtraction)
39
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
40 Numbers, 1.1 Ordering and comparing whole Ordering and comparing numbers
Operations and numbers
Relationships Whole Learners should be given a range of exercises such as:
numbers Revise the following done in Grade 6:
• Arrange given numbers from the smallest to the biggest: or biggest to smallest
• Order, compare and represent
• Fill in missing numbers in
numbers to at least 9-digit numbers
-- a sequence
• Recognise and represent prime
numbers to at least 100 -- on a number grid
• Rounding off numbers to the nearest -- on a number line e.g. which whole number is halfway between 471 340 and
5, 10, 100 or 1 000 471 350.
• Fill in <, = or >
Examples:
MATHEMATICS GRADES 7-9
a) 247 889 * 247 898
b) 784 109 * 785 190
Properties of whole numbers Properties of whole numbers
Revise the following done in Grade 6: • Revising the properties of whole numbers should be the starting point for work
with whole numbers. The properties of numbers should provide the motivation
• Recognise and use the commutative;
for why and how operations with numbers work.
associative; distributive properties
with whole numbers • When learners are introduced to new numbers, such as integers for example,
they can again explore how the properties of numbers work for the new set of
• Recognise and use 0 in terms of its
numbers.
additive property (identity element for
addition) • Learners also have to apply the properties of numbers in algebra, when they
work with variables in place of numbers.
• Recognise and use 1 in terms of
its multiplicative property (identity • Learners should know and be able to use the following properties:
element for multiplication)
-- The commutative property of addition and multiplication:
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
♦♦ a + b = b + a
♦♦ a x b = b x a
-- The associative (grouping) property of addition and multiplication:
♦♦ (a + b) + c = a + (b + a)
♦♦ (a x b) x c = a x (b x c)
-- The distributive property of multiplication over addition and subtraction:
♦♦ a(b + c) = (a x b) + (a x c)
♦♦ a(b – c) = (a x b) – (a x c)
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS Numbers, 1.1 -- Addition and subtraction as inverse operations
Operations and
Relationships Whole -- Multiplication and division as inverse operations
numbers
-- 0 is the identity element for addition: t + 0 = t
-- 1 is the identity element for multiplication: t x 1 = t
Illustrating the properties with whole numbers
a) 33 + 99 = 99 + 33 = 132
b) 51 + (19 + 46) = (51 + 19) + 46 = 116
c) 4(12 + 9) = (4 x 12) + (4 x 9) = 48 + 36 = 84
d) (9 x 64) + (9 x 36) = 9 x (64 + 36) = 9 x 100 = 900
e) If 33 + 99 = 132, then 132 – 99 = 33 and 132 – 33 = 99
f) If 20 x 5 = 110, then 110 ÷ 20 = 5 and 110 ÷ 5 = 20
Calculations with whole numbers Calculations with whole numbers
• Revise the following done in Grade 6, • Learners should do context free calculations and solve problems in contexts
without use of calculators:
• Learners should become more confident in and more independent at
-- Addition and subtraction of whole mathematics, if they have techniques
numbers to at least 6 –digit numbers
-- to check their solutions themselves, e.g. using inverse operations; using
-- Multiplication of at least whole calculators
4-digit by 2-digit numbers
-- to judge the reasonableness of their solutions e.g. estimate by rounding off;
-- Division of at least whole 4-digit by estimate by doubling or halving;
2-digit numbers
• Adding, subtracting and multiplying in columns, and long division, should
• Perform calculations using all four only be used to practice number facts and calculation techniques, and hence
operations on whole numbers, should be done with familiar and smaller number ranges. For big and unwieldy
estimating and using calculators calculations, learners should be encouraged to use a calculator.
where appropriate
Multiples and factors
Calculation techniques
• Practice with finding multiples and factors of whole numbers are especially
• Use a range of techniques to perform important when learners do calculations with fractions. They use this knowledge
and check written and mental to find the LCM when one denominator is a multiple of another, and also when
calculations of whole numbers they simplify fractions or have to find equivalent fractions.
including:
• Factorising whole numbers lays the foundation for factorisation of algebraic
-- estimation expressions.
41 -- adding, subtracting and multiplying • Using the definition of prime numbers, emphasise that 1 is not classified as a
in columns prime number
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
42 Numbers, 1.1 -- long division Examples
Operations and
Relationships Whole -- rounding off and compensating a) The multiples of 6 are 6, 12, 18, 24,... or M6 = {6; 12; 18; 24;...}
numbers
-- using a calculator b) LCM of 6 and 18 is 18
LCM of 6 and 7 is 42
Multiples and factors
c) The factors of 24 are 1, 2, 3, 4, 6, 12 and 24 by inspection and, the prime
• Revise the following done in Grade 6:
factors of 24 are 2 and 3
-- Multiples of 2-digit and 3-digit
d) The factors of 140 are 1, 2, 5, 7, 10, 14, 28, 35, 70 and 140
whole numbers
e) Determine the HCF of 120; 300 and 900
-- Factors of 2-digit and 3-digit whole
numbers Learners do this by finding the prime factors of the numbers first.
-- Prime factors of numbers to at 120 = 5 x 3 x 23. Initially learners may write this as: 5 x 3 x 2 x 2 x 2
MATHEMATICS GRADES 7-9
least 100
300 = 52 x 3 x 22
• List prime factors of numbers to at
900 = 52 x 32 x 23
least 3-digit whole numbers
HCF = 5 x 3 x 22 = 60 (Multiply the common prime factors of the three numbers)
• Find the LCM and HCF of numbers
to at least 3-digit whole numbers, by
inspection or factorisation
Solving problems Solving problems
• Solve problems involving whole • Solving problems in contexts should take account of the number ranges learners
numbers, including: are familiar with.
-- Comparing two or more quantities • Contexts involving ratio and rate, should include speed, distance and time
of the same kind (ratio) problems.
-- Comparing two quantities of • In financial contexts, learners are not expected to use formulae for calculating
different kinds (rate) simple interest.
-- Sharing in a given ratio where the
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
whole is given
• Solve problems that involve whole
numbers, percentages and decimal
fractions in financial contexts such
as:
-- Profit, loss and discount
-- Budgets
-- Accounts
-- Loans
-- Simple interest
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS Numbers, 1.2 Mental calculations What is different to Grade 6? 9 hours
Operations and Exponents 2
Relationships • Determine squares to at least 12 • Although learners may have encountered square numbers in Grade 6, they
and their square roots were not expected to write these numbers in exponential form.
3
• Determine cubes to at least 6 and Comparing and representing numbers in exponential form
their cube roots • Learners need to understand that in the exponential form ab , the number is read
Comparing and representing as ‘a to the power b’, where a is called the base and b is called the exponent or
numbers in exponential form index. indicates the number of factors that are multiplied.
Example:
• Compare and represent whole
numbers in exponential form: a) a3 = a x a x a;
ab = a x a x a x... for b number of
b) a5 = a x a x a x a x a
factors
• Learners can represent any number in exponential form, without needing to
compute the value.
Example:
50 x 50 x 50 x 50 x 50 x 50 x 50 = 507
• Make sure learners understand that square roots and cube roots are the inverse
operations of squaring and cubing numbers.
Examples:
• Make sure learners understand that any number raised to the power 1 is equal
32 = 9 therefore √ 9 = 3
to the number.
Example m1 = m
• At this point, learners do not need to know the rule for raising a number to the
power 0. This will only be introduced in Grade 8 when they use other laws of
exponents in calculations.
• To avoid common misconceptions, emphasize the following with examples:
-- 122 = 12 x 12 and not 12 x 2
-- 13 means 1 x 1 x 1 and not 1 x 3
-- 1001 = 100
3
--
-- √ 81 = 9 because 92 = 81
-- The square of 9 = 81, whereas the square root of 9 = 3
√ 27 = 3 because 33 = 27
43 • Learners should use their knowledge of representing numbers in exponential
form when simplifying and expanding algebraic expressions and solving
algebraic equations.
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
44 Numbers, 1.2 Calculations using numbers in Calculations using numbers in exponential form
Operations and Exponents exponential form
Relationships • Knowing the rules of operations for calculations involving exponents, is
• Recognize and use the appropriate important.
laws of operations with numbers
Example:
involving exponents and square and
cube roots a) (7 – 4)3 = 33 and NOT 73 – 43
• Perform calculations involving all
four operations using numbers
b) √16 + 9 = √ 25 , and NOT √ 16 + √ 9
in exponential form, limited to
exponents up to 5, and square and
cube roots
Solving problems
MATHEMATICS GRADES 7-9
• Solve problems in contexts involving
numbers in exponential form
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS Shape and Space 3.5 Measuring angles What is different to Grade 6? 10 hours
(Geometry)
Construction • Accurately use a protractor to • Measure angles with a protractor
of geometric measure and classify angles:
• Geometric constructions using a compass, ruler and protractor
figures
-- < 90o (acute angles)
Measuring angles
-- Right-angles
• Learners have to be shown how to place the protractor on the arm of the angle
-- > 90o (obtuse angles) to be measured.
-- Straight angles • Learners also have to learn how to read the size of angles on a protractor.
-- > 180o (reflex angles) Constructions
Constructions • Constructions provide a useful context to explore or consolidate knowledge of
angles and shapes.
• Accurately construct geometric
figures appropriately using compass, • Learners have to be shown how to use a compass to draw circles, although they
ruler and protractor, including: might have done this in Grade 6.
-- angles, to one degree of accuracy • Learners should be aware that the centre of the circle is at the fixed point of the
compass and the radius of the circle is dependent on how wide the compass is
-- circles
opened up.
-- parallel lines
• Make sure learners understand that arcs are parts of the circles of a particular
-- perpendicular lines radius.
• Initially, learners have to be given careful instructions about how to do the
constructions of the various shapes
• Once they are comfortable with the apparatus and can do the constructions,
they can practise by drawing patterns, for example of circles or parallel lines.
45
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
46 Shape and Space 3.1 Classifying 2D shapes What is different to Grade 6? 10 hours
(Geometry)
Geometry of • Describe, sort, name and compare • Distinguishing and naming triangles in terms of their sides and angles
2D shapes triangles according to their sides and
• Distinguishing quadrilaterals in terms of parallel and perpendicular sides
angles, focusing on:
• Distinguishing similar and congruent figures
-- equilateral triangles
• Using known properties of shapes to solve geometric problems
-- isosceles triangles
Triangles
-- right-angled triangles
• Learners should be able to distinguish between an equilateral triangle (all the
• Describe, sort, name and compare
sides are equal), an isosceles triangle (at least two equal sides) and a right-
quadrilaterals in terms of:
angled triangle (one right-angle).
-- length of sides
Quadrilaterals
MATHEMATICS GRADES 7-9
-- parallel and perpendicular sides
• Learners should be able to sort and group quadrilaterals in the following ways:
-- size of angles (right-angles or not)
-- all sides equal (square and rhombus)
-- opposite sides equal (rectangle, parallelogram, square, rhombus)
-- at least one pair of adjacent sides equal (square, rhombus, kite)
-- all four angles right angles (square, rectangle)
-- perpendicular sides (square, rectangle)
-- two pairs of opposite sides parallel (rectangle, square, parallelogram)
-- only one pair of opposite sides parallel (trapezium)
• Describe and name parts of a circle Circles
• Parts of a circle learners should know:
-- radius
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
-- circumference
-- diameter
-- chord
-- segments
-- sectors
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS Shape and Space 3.1 Similar and congruent 2D shapes Similarity and congruency
(Geometry)
Geometry of • Recognize and describe similar and • Similarity and congruency can be explored with any 2D figures.
2D shapes congruent figures by comparing:
• Learners should recognize that two or more figures are congruent if they are
-- shape equal in all respects i.e. angles and sides are equal.
-- size • Learners should recognize that two or more figures are similar if they have
the same shape, but differ in size i.e. angles are the same, but sides are
proportionally longer or shorter. Similar figures are further explored when
Solving problems doing enlargements and reductions. Refer to “Clarification Notes” under 3.4
• Solve simple geometric problems Transformation Geometry.
involving unknown sides and angles Solving problems
in triangles and quadrilaterals, using
known properties. • At this stage learners can solve simple geometric problems to find unknown
sides in equilateral and isosceles triangles, and unknown sides and angles in
quadrilaterals.
• Learners should give reasons for their solutions.
Examples
a) If ∆ABC is an equilateral triangle, and side AB is 3 cm, what is the length
of BC? Here learners should answer: BC = 3 cm, because the sides of an
equilateral triangle are equal.
b) If ABCD is a kite, and AB = 2,5 cm and BC = 4,5 cm, what is the length of
AD and DC? Learners should use the property for kites, that adjacent pairs
of sides are equal, to find the unknown sides.
3.3 Define: • Line segment is a set of points with a definite starting-point and an end-point. 2 hours
Geometry of • Line segment • Ray is a set of points with a definite starting-point and no definite end-point.
straight lines
• Ray • Line is a set of points with no definite starting-point and end-point.
• Straight line • If two lines on the same plane are a constant distance apart, then the lines are
parallel. Example:
• Parallel lines
E > F
• Perpendicular lines
G > H This is written as EF║GH
• If vertical line AO meets or intersects with horizontal line BC at right angle, then
AO is perpendicular to BC.
Example:
A
47
B C This is written as AO┴BC
O
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
48 REVISION/ASSESSMENT: 5 hours
At this stage learners should have been assessed on:
• calculating and solving problems using whole numbers
• working with numbers in exponential form
• constructing geometric objects
• geometry of 2D shapes
MATHEMATICS GRADES 7-9
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
GRADE 7 – TERM 2
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
CAPS (in hours)
Numbers, 1.4 Ordering, comparing and simplifying What is different to Grade 6? 9 hours
Operations and fractions
Relationships Common • Compare and order thousandths
fractions • Revise the following done in Grade 6:
• Multiplication of common fractions
-- Compare and order common
• Percentage of part of a whole
fractions, including specifically
tenths and hundredths • Percentage increase or decrease
• Extend to thousandths In Grade 7 learners also consolidate number knowledge and calculation tech-
niques for common fractions, developed in the Intermediate Phase.
Calculations using fractions Calculations with fractions
• Revise the following done in Grade 6: • Learners should do context free calculations and solve problems in contexts.
-- addition and subtraction of • It is not expected that learners know rules for simplifying fractions or for
common fractions, including mixed converting between mixed numbers and fraction forms. Learners should know
numbers, limited to fractions with from working with equivalence, when a fraction is equal to or greater than 1.
the same denominator or where
• LCMs have to be found when adding and subtracting fractions of different
one denominator is a multiple of
denominators. Here learners use knowledge of common multiples to find the
another
LCM i.e. what number can both denominators be divided into.
-- finding fractions of whole numbers
• To simplify fractions, learners use knowledge of common factors i.e. what can
• Extend addition and subtraction to divide equally into the numerator and denominator of a fraction. Emphasize that
fractions where one denominator is when simplifying, the fractions must remain equivalent.
not a multiple of the other
Example
• Multiplication of common fractions,
3 2 4 3
including mixed numbers, not limited 4 x 5 = 20 = 10
to fractions where one denominator is or
a multiple of another
3 2 3
Calculation techniques 4 x 5 = 10
• Convert mixed numbers to common • Learners should recognize that finding a ‘fraction of a whole number’ or ‘finding
fractions in order to perform a fraction of a fraction’ means multiplying the fraction and the whole number or
calculations with them the fraction with the fraction.
• Use knowledge of multiples and • When learners find fractions of whole numbers, the examples can be chosen to
factors to write fractions in the result either in whole numbers or fractions or both.
simplest form before or after • Learners should also use the convention of writing the whole number as a
calculations fraction over when multiplying.
49 • Use knowledge of equivalent
fractions to add and subtract common
fractions
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
50 Numbers, 1.4 Solving problems Examples 9 hours
Operations and
Relationships Common • Solve problems in contexts involving a) Calculate
fractions common fractions and mixed 4
5 of 20
numbers, including grouping, sharing
and finding fractions of whole Answer:
numbers
4 4 20 4 4 4 4 20 80
5 of 20 = 5 x 1 = 1 x 1 = 16 OR 5 of 20 = 5 x 1 = 5 = 16
b) Calculate
2 5
3 of 6
Answer
2 5 2 5 1 5 5 2 5 2 5 1 10 5
MATHEMATICS GRADES 7-9
3 of 6 = 3 x 6 = 3 x 3 = 9 OR 3 of 6 = 3 x 6 = 3 x 18 = 9
Percentages Calculation using percentages
• Revise the following done in Grade 6: • Learners should do context free calculations and solve problems in contexts.
-- percentages of whole numbers • When doing calculations using percentages, learners have to use the equivalent
common fraction form, which is a fraction with denominator 100.
• Calculate the percentage of part of a
whole Learners should become familiar with the equivalent fraction and decimal forms
of common percentages like
• Calculate percentage increase or
decrease of whole numbers 1
a) 25% or 4 or 0,25;
• Solve problems in contexts involving
1
percentages b) 50% or 2 or 0,5;
Equivalent forms 3
c) 60% or 5 or 0,6.
Revise the following done in Grade 6:
• To calculate percentage of part of a whole, or percentage increase or decrease,
100
• Recognize and use equivalent forms learners have to learn the strategy of multiplying by 1 . It is useful for learners
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
of common fractions with 1-digit or to learn to use calculators for some of these calculations where the fractions are
2-digit denominators (fractions where not easily simplified.
one denominator is a multiple of the
other) • When using calculators, learners can also use the equivalent decimal fraction
form for percentages to do the calculations.
• Recognize equivalence between
common fraction and decimal fraction Examples:
forms of the same number a) Calculate 60% of R105
3
• Recognize equivalence between Amount = 5 X R105 = R63
common fraction, decimal fraction
and percentage forms of the same b) What percentage is 40c of R3,20?
number 40 100 100
Percentage = 320 x 1 = 8 = 12,5%
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS c) Calculate the percentage increase if the price of a bus ticket of R60 is
increased to R84.
Amount increased = R24.
24 100
Therefore percentage increase = 60 x 1 = 40%
d) Calculate the percentage decrease if the price of petrol goes down from 20
cents a litre to 18 cents a litre.
Amount decreased = 2 cents. Therefore percentage decrease
2 100
= 20 x 1 = 10%
Numbers, 1.5 What is different to Grade 6? 9 hours
Operations and
Relationships Decimal • Decimal fractions to at least decimal places
fractions
• Rounding off to at least decimal places
• Multiply and divide decimal fractions by whole numbers
• Multiply decimal fractions by decimal fractions
In Grade 7 learners consolidate number knowledge and calculation techniques for
decimal fractions, developed in the Intermediate Phase.
Ordering and comparing decimal Ordering, counting and comparing decimal fractions
fractions
• Counting should not only be thought of as verbal counting. Learners can count
• Revise the following done in Grade 6: in decimal intervals using:
-- count forwards and backwards in -- structured, semi-structured or empty number lines
decimal fractions to at least two
decimal places -- chain diagrams for counting
-- compare and order decimal • Learners should be given a range of exercises such as:
fractions to at least two decimal -- arrange given numbers from the smallest to the biggest: or biggest to smallest
places
-- fill in missing numbers in
-- place value of digits to at least two
decimal places ♦♦ a sequence
-- rounding off decimal fractions to at ♦♦ on a number grid
least 1 decimal place ♦♦ on a number line
• Extend all of the above to decimal ♦♦ fill in <, = or > Example: 0,4 * 0.04
fractions of at least three decimal
places and rounding off to at least 2 • Counting exercises in chain diagrams can be checked using calculators and
decimal places learners can explain any differences between their answers and those shown by
51 the calculator.
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
52 Numbers, 1.5 Calculations using decimal fractions Calculating with decimal fractions
Operations and
Relationships Decimal • Revise the following done in Grade 6: • Learners should do context free calculations and solve problems in contexts.
fractions
-- addition and subtraction of decimal • Learners should estimate their answers before calculating, especially
fractions of at least two decimal with multiplication by decimal fractions. They should be able to judge the
places reasonableness of answers relating to how many decimal places and also check
their own answers.
-- multiplication of decimal fractions
by 10 and 100 • Multiplication by decimal fractions should start with familiar numbers that
learners can calculate by inspection, so that learners get a sense of how
• Extend addition and subtraction to
decimal places are affected by multiplication.
decimal fractions of at least three
decimal places Examples:
• Multiply decimal fractions to include: a) 3x2=6
MATHEMATICS GRADES 7-9
-- decimal fractions to at least 3 0,3 x 2 = 0,6
decimal places by whole numbers
0,3 x 0,2 = 0,06
-- decimal fractions to at least 2
0,3 x 0,02 = 0,006
decimal places by decimal fractions
to at least 1 decimal place 0,03 x 0,002 = 0,0006 etc
• Divide decimal fractions to include b) 15 x 3 = 45
decimal fractions to at least 3 decimal
1,5 x 3 = 4,5
places by whole numbers
0,15 x 3 = 0,45
0,15 x 0,3 = 0,045
0,015 x 0,3 = 0,0045 etc
Calculation techniques Equivalence between common fractions and decimal fractions
• Use knowledge of place value to • Learners are not expected to be able to convert any common fraction into its
estimate the number of decimal decimal form, merely to see the relationship between tenths, hundredths and
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
places in the result before performing thousandths in their decimal forms.
calculations
• Learners should start by rewriting and converting tenths, hundredths and
• Use rounding off and a calculator to thousandths in common fraction form to decimal fractions. Where denominators
check results where appropriate of other fractions are factors of 10 e.g. 2,5 or factors of 100 e.g 2, 4, 20,
25 learners can convert these to hundredths using what they know about
Solving problems
equivalence.
• Solve problems in context involving
• It is useful to use calculators to help learners convert between common
decimal fractions
fractions and decimal fractions (here learners will use what they know about the
relationship between fractions and division).
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS Numbers, 1.5 Equivalent forms -- Dividing whole numbers by 10, 100, 1 000, etc. can help to build learners’
Operations and understanding of place value with decimals. This is also useful to do on the
Relationships Decimal • Revise the following done in Grade 6:
fractions calculator – learners can discuss the patterns they see when dividing.
-- recognize equivalence between
-- Similarly calculators can be useful tools for learners to learn about patterns
common fraction and decimal
when multiplying decimals by 10, 100 or,1 000 etc.
fraction forms of the same number
-- Recognize equivalence between
common fraction, decimal fraction
and percentage forms of the same
number
Patterns, 2.2 Input and output values What is different to the Intermediate Phase? 3 hours
functions and
algebra Functions • Determine input values, output • Finding input or output values using given formulae
and values or rules for patterns and
relationships • The rules and number patterns for which learners have to find input or output
relationships using:
values are extended to include patterns with integers, square numbers and
-- flow diagrams cubic numbers
-- tables Finding input and output values in flow diagrams, tables and formulae should be
done more than just once a year. It can be done after number work, to practise
-- formulae
properties and operations with numbers and after measurement or geometry to
Equivalent forms practise solving problems using formulae.
• Determine, interpret and justify In Term 2 the focus of Functions and Relationships is on practising
equivalence of different descriptions operations with whole numbers as well as common fractions or decimal
of the same relationship or rule fractions as input values, or including common fractions and decimal
presented: fractions in the rules for finding output values. In Term 3 the focus of
Functions and Relationships is on using formulae and in Term 4, the focus is
-- verbally
on practising addition and subtraction of integers.
-- in flow diagrams
• In this phase, it is useful to begin to specify whether the input values are natural
-- in tables numbers, or integers or rational numbers. Hence, to find output values, learners
should be given the rule/formula as well as the input values.
-- by formulae
Flow diagrams are representations of functional relationships. Hence, when using
-- by number sentences flow diagrams, the correspondence between input and output values should be
clear in its representational form i.e. the first input produces the first output, the
second input produces the second output, etc.
53
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
54 Patterns, 2.2 Examples
functions and
algebra Functions a) Use the given rule to calculate the values of t for each value of p, where p is a
and natural number.
relationships
t=px3+1
MATHEMATICS GRADES 7-9
In this kind of flow diagram, learners can also be asked to find the value of p
for a given value t.
b) Find the rule for calculating the output value for every given input value in the
flow diagram below.
1 8
24
40
56
72
In flow diagrams such as these, more than one rule might be possible to
describe the relationship between input and output values. The rules are
acceptable if they match the given input values to the corresponding output
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
values.
c) If the rule for finding y in the table below is: y = 3x – 1, find y for the given x
values:
x 0 1 2 5 10 50 100
y
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS Patterns, 2.2 d) Describe the relationship between the numbers in the top row and bottom row
functions and in the table. Then write down the value of m and n.
algebra Functions
and
x 1 2 3 4 12 n
relationships
y 5 6 7 8 m 34
In tables such as these, more than one rule might be possible to describe the
relationship between x and y values. The rules are acceptable if they match the
given input values to the corresponding output values. For example, the rule
y = x + 4 describes the relationship between the given x and y values in the
table. To find m and n, you have to substitute the corresponding values for x or
y into this rule and solve the equation by inspection.
Measurement 4.1 Area and perimeter What is different to Grade 6? 7 hours
Area and • Calculate the perimeter of regular • In Grade 6 learners did not have to use formulae to calculate area and
perimeter of and irregular polygons perimeter.
2D shapes
• Use appropriate formulae to calculate • Formulae learners should know and use are:
perimeter and area of:
-- perimeter of a square = 4s
-- squares
-- perimeter of a rectangle = 2(l + b) or 2l + 2b
-- rectangles
-- area of a square = l2
-- triangles
-- area of a rectangle = l x b
Calculations and solving problems 1
-- area of a triangle = 2 (b x h)
• Solve problems involving perimeter
and area of polygons Solving equations using formulae
• Calculate to at least 1 decimal place • The use of formulae provides a context to practise solving equations by
inspection.
• Use and convert between appropriate
SI units, including: Example
-- mm2 ↔ cm2 1. If the perimeter of a square is 32 cm what is the length of each side?
Learners should write this as:
-- cm2 ↔ m2
4s = 32 and solve by inspection by asking: 4 times what will be 32?
2. If the area of a rectangle is 200 cm2, and its length is 50 cm what is its width?
Learners should write this as: 50 x b = 20 and solve by inspection by asking:
50 times what will be 200?
55
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
56 Measurement 4.1 Examples of calculations for area and perimeter
Area and Calculate:
perimeter of
a) Perimeter of a rectangle which is 24 cm long and 18 cm wide.
2D shapes
b) Perimeter of a regular octagon if the length of each side is 17 cm.
c) Area of ∆abc if bc = 12 cm and its height AT = 9 cm
d) Perimeter of a square if its area is 225 cm2
For areas of triangles:
• Make sure learners know that the height of a triangle is a line segment drawn
from any vertex perpendicular to the opposite side.
Example: aD is the height onto base bc of ∆abc.
MATHEMATICS GRADES 7-9
A A
C C
B D D B
• Point out that every triangle has 3 bases, each with a related height or altitude.
• For conversions, note:
-- if 1 cm = 10 mm then 1 cm2 = 100 mm2
-- if 1 m = 100 cm then 1 m2 = 10 000 cm2
Examples of solving problems involving perimeter and area.
a) Calculate the area of the shaded part in the diagram if ABCD is a rectangle,
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
ab = 18,6 cm, DC = 2TC and BC = 8 cm
b) The area of the floor of the dining room is 18,4 cm2. How many square tiles
with sides of 20 cm are needed to tile the floor?
c) The length of the side of a square is doubled. Will the area of the enlarged
square be double or four times that of the original square?
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS Measurement 4.2 Surface area and volume What is different to Grade 6? 8 hours
Surface area • Use appropriate formulae to calculate • In Grade 6 learners did not have to use formulae to calculate surface area and
and volume the surface area, volume and volume.
of 3D objects capacity of:
• Formulae learners should know and use:
-- cubes
-- the volume of a prism = the area of the base x the height
-- rectangular prisms
-- the surface area of a prism = the sum of the area of all its faces
• Describe the interrelationship
-- the volume of a cube = l3
between surface area and volume of
the objects mentioned above -- the volume of a rectangular prism = l x b x h
Calculations and solving problems • For conversions, note:
• Solve problems involving surface -- if 1 cm = 10 cm then 1 cm3 = 1 000 mm3 and
area, volume and capacity
-- if 1 m = 10 cm then 1 m3 = 1 000 000 mm3 or 1 000 000 or 106 cm3.
• Use and convert between appropriate
-- an object with a volume of 1 cm3 will displace exactly 1 ml of water; and
SI units, including:
-- an object with a volume of 1 m3 will displace exactly 1 kl of water.
-- mm2 ↔ cm2
• Emphasize that the amount of space inside a prism is called its capacity; and
-- cm2 ↔ m2
the amount of space occupied by a prism is called its volume.
-- mm3 ↔ cm3
• Investigate the nets of cubes and rectangular prisms in order to deduce
-- cm3 ↔ m3 formulae for calculating their surface areas.
• Use equivalence between units when
solving problems:
-- 1cm3 ↔ 1 ml
-- 1 m3 ↔ 1 kl
REVISION/ASSESSMENT: 9 hours
At this stage learners should be assessed on:
• calculating and solving problems with common fractions and decimal fractions
• using formulae to find area and perimeter of 2D shapes
• using formulae to find volume and surface area of 3D objects
57
MATHEMATICS GRADES 7-9
GRADE 7 – TERM 3
DURATION
58 CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
Patterns, 2.1 Investigate and extend patterns What is different to the Intermediate Phase? 6 hours
functions and
algebra Numeric and • Investigate and extend numeric • In the Senior Phase the emphasis is less on merely extending a pattern, and
geometric and geometric patterns looking for more on describing a general rule for the pattern or sequence and being able to
patterns relationships between numbers, predict unknown terms in a sequence using a general rule.
including patterns:
• Investigating number patterns is an opportunity to generalize – to give general
-- represented in physical or diagram algebraic descriptions of the relationship between terms and its position in a
form sequence and to justify solutions.
-- not limited to sequences involving a • The range of number patterns are extended to include patterns integers, square
constant difference or ratio numbers and cubic numbers
-- of learner’s own creation • As learners become used to describing patterns in their own words, their
MATHEMATICS GRADES 7-9
descriptions should become more precise and efficient with the use of algebraic
-- represented in tables
language to describe general rules of patterns.
• Describe and justify the general rules
• It is useful also to introduce the language of ‘term in a sequence’ in order to
for observed relationships between
distinguish the term from the position of a term in a sequence
numbers in own words
Numeric and geometric patterns are done again in Term 4, where patterns
can include integers. In this term patterns should be restricted to using
whole numbers, numbers in exponential form, common fractions and
decimal fractions.
Kinds of numeric patterns
• Provide a sequence of numbers, learners have to identify a pattern or
relationship between consecutive terms in order to extend the pattern.
Examples
Give a rule to describe the relationship between the numbers in the sequences
below. Use this rule to give the next three numbers in the sequence:
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
a) 3; 7; 11; 15;... ...
b) 120; 115; 110; 105;... ...
Here learners could identify the constant difference between consecutive terms
in order to extend the pattern. These patterns can be described in learners’
own words as (a) ‘adding 4’ or ‘counting in 4s’ or ‘add 4s to the previous
number in the pattern’ (b) ‘subtracting 5’ or ‘counting down in 5s’, or ‘subtract 5
from the previous number in the pattern’.
c) 2; 4; 8; 16;... ...
Here learners could identify the constant ratio between consecutive terms.
This pattern can be described in learners’ own words as ‘multiply the previous
number by 2’.
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS Patterns, 2.1 d) 1; 2; 4; 7; 11; 16;... ...
functions and
algebra Numeric and This pattern has neither a constant difference nor constant ratio. This pattern
geometric can be described in learners’ own words as ‘increase the difference between
patterns consecutive terms by 1 each time’ or ‘add 1 more than was added to get the
previous term’. Using this rule, the next 3 terms will be 22, 29, 37.
• Provide a sequence of numbers, learners have to identify a pattern or
relationship between the term and its position in the sequence. This
enables learners to predict a term in a sequence based on the position of that
term in the sequence. It is useful for learners to represent these sequences in
tables so that they can consider the position of the term.
Examples:
a) Provide a rule to describe the relationship between the numbers in this
sequence:1; 4; 9; 16;... ... Use the rule to find the 10th term in this sequence.
Firstly, learners have to understand that the ‘10th term’ refers to position 10 in
the number sequence. They have to find a rule in order to determine the 10th
term, rather than continuing the sequence to the tenth term.
This sequence can be represented in the following table:
Position in sequence 1 2 3 4 10
Term 1 4 9 16 ?
Learners should recognize that each term in the bottom row is obtained by
squaring the position number in the top row. Thus the 10th term will be '10
squared' or 102, which is 100. Using the same rule, learners can also be asked
what term number or position will 625 be? If the term is obtained by squaring
the position number of the term, then the position number can be obtained
by finding the square root of the term. Hence, 625 will be the 25th term in the
sequence since √625 = 25
59
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
60 b) Provide a rule to describe the relationship between the numbers in this
sequence: 4; 7; 10; 13; …... Use the rule to find the 20th term in the sequence.
If learners consider only the relationship between consecutive terms, then they
can continue the pattern (‘add 3 to previous number’) to the 20th term to find
the answer. However, if they look for a relationship or rule between the term
and the position of the term, they can predict the answer without continuing the
pattern. Using number sentences can be useful to find the rule:
1st term: 4 = 3 (1) + 1
2nd term: 7 = 3 (2) + 1
3rd term: 10 = 3 (3) + 1
4th term: 13 = 3 (4) + 1
The number in the brackets corresponds to the position of the term. Hence, the
MATHEMATICS GRADES 7-9
20th term will be: 3 (20) + 1 = 61
The rule in learners’ own words can be written as ‘3 x the position of the
term + 1
• These types of numeric patterns develop an understanding of functional
relationships, in which you have a dependent variable (position of the term) and
independent variable (the term itself), and where you have a unique output for
any given input value.
Kinds of geometric patterns
• Geometric patterns are number patterns represented diagrammatically. The
diagrammatic representation reveals the structure of the number pattern.
• Hence, representing the number patterns in tables, makes it easier for learners
to describe the general rule for the pattern.
Example
Consider this pattern for building hexagons with matchsticks. How many
matchsticks will be used to build the 10th hexagon?
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
The rule for the pattern is contained in the structure (construction) of the
successive hexagonal shapes:
(1) add 1 on matchstick per side
(2) there are 6 sides, so
(3) add on 6 matchsticks per hexagon as you proceed from a given hexagon to
the next one.
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS Patterns, 2.1 For the 2nd hexagon, you have 2 x 6 matches; for the 3rd hexagon you have 3 x 6
functions and matches; Using this pattern for building hexagons, the 10th hexagon will have
algebra Numeric and 10 x 6 matches.
geometric
patterns Learners can also use a table to record the number of matches used for each
hexagon. This way they can look at the number pattern related to the number of
matches used for each new hexagon.
Position of hexagon in pattern 1 2 3 4 5 6 10
Number of matches 6 12 18
Describing patterns
• It does not matter if learners are already familiar with a particular pattern. Their
descriptions of the same pattern can be different when they encounter it at
different stages of their mathematical development.
Example
The rule for the sequence: 4; 7; 10; 13 can be described in the following ways:
a) add three to the previous term
b) 3 times the position of the term + 1 or 3 x the position of the term + 1
c) 3(n) + 1, where n is the position of the term
d) 3(n) + 1, where n is a Natural number.
61
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
62 2.2 Input and output values Functions and relationships were also done in Term 2, and will be done again 3 hours
in Term 4, focusing on integers. In this term, the focus is on finding output
Functions • Determine input values, output values for given formulae and input values.
and values or rules for patterns and
relationships relationships using: See additional notes and examples in Term 2.
-- flow diagrams Note, when learners find input or output values for given rules or formulae, they
are actually finding the numerical value of algebraic expressions using substitu-
-- tables tion.
-- formulae Examples
Equivalent forms Use the formula for the area of a rectangle: A = l x b to calculate the following:
• Determine, interpret and justify a) The area, if the length is 4 cm and the width is 2 cm
equivalence of different descriptions
of the same relationship or rule b) The length, if the area is 20 cm2 and the width is 4 cm
MATHEMATICS GRADES 7-9
presented: c) The width, if the area is 24 cm2 and the length is 8 cm
-- verbally Learners can write these as number sentences, and solve by inspection.
-- in flow diagrams
-- in tables
-- by formulae
-- by number sentences
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS 2.3 Algebraic language What is different to the Intermediate Phase? 3 hours
Algebraic • Recognize and interpret rules or This is an introduction to formal algebraic language and is new in the Senior
expressions relationships represented in symbolic Phase. The use of symbolic language helps to develop an understanding of
form variables.
• Identify variables and constants in Algebraic expressions are done again in Term 4, where rules and
given formulae and equations relationships can include integers.
Learners have opportunities to write and interpret algebraic expressions when they
write general rules to describe relationships between numbers in number patterns,
and when they find input and output values for given rules in flow diagrams, tables
and formulae.
Examples
a) What does the rule 2 x n –1 mean for the following number sequence: 1; 3; 5;
7; 9;....
Here learners should recognize that 2 x n –1 represents the general term in this
sequence, where n represents the position of the term in the sequence. Thus it
is the rule that can be used to find any term in the given sequence.
b) The relationship between a boy’s age (x yrs old) and his mother’s age is given
as 25 + x. How can this relationship be used to find the mother’s age when the
boy is 11 years old?
Here learners should recognize that to find the mother’s age, they should
substitute the boy’s given age into the rule 25 + x. They should also recognize
that the given rule means the mother is 25 years older than the boy.
See further examples given for functions and relationships.
63
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
64 2.4 Number sentences What is different to the Intermediate Phase? 3 hours
Algebraic • Write number sentences to describe The number sentences that learners can solve are extended to include number
equations problem situations sentences with integers, square numbers and cubic numbers.
• Analyse and interpret number Number sentences are used here as a more familiar term for Grade 7 learners than
sentences that describe a given equations.
situation However, the term equation will be used instead of number sentences in later
grades.
• Solve and complete number
sentences by: Algebraic equations are done again in Term 4, where number sentences can
include integers.
-- inspection
Learners have opportunities to write, solve and complete number sentences when
-- trial and improvement they write general rules to describe relationships between numbers in number
patterns, and when they find input and output values for given rules in flow diagrams,
• Identify variables and constants in
MATHEMATICS GRADES 7-9
tables and formulae.
given formulae or equations
Rather than use formal algebraic processes, learners solve number sentences by
• Determine the numerical value of an inspection or determine the numerical value of expressions by substitution.
expression by substitution.
In this phase, it is useful when solving equations to begin to specify whether x is a
natural number, integer or rational number. This builds learners’ awareness of the
domain of x.
Examples
a) Solve x if x + 4 = 7, where x is a natural number. (What must be added to 4 to
give 7?)
b) Solve x if x + 4 = –7, where x is an integer. (What must be added to 4 to give
–7?)
c) Solve x if 2x = 30, where x is a natural number. (What must be multiplied by 2
to give 30?)
d) Write a number sentence to find the area of a rectangle with length 4,5 cm and
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
breadth 2 cm.
e) If y = x2 + 1, calculate y when x = 3
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS 2.5 Interpreting graphs What is different to the Intermediate Phase? 6 hours
Graphs • Analyse and interpret global graphs In the Intermediate Phase learners encountered graphs in the form of data bar
of problem situations, with special graphs and pie charts. This means they do have some experience reading and
focus on the following trends and interpreting graphs. However, in the Senior Phase, they are introduced to line
features: graphs that show functional relationships described in terms of dependent and
independent variables.
-- linear or non-linear
In Grade 7, the focus is on drawing, analysing and interpreting global graphs only.
-- constant, increasing or decreasing
That is, learners do not have to plot points to draw graphs and they focus on the
Drawing graphs features of the global relationship shown in the graph.
• Draw global graphs from given Examples of contexts for global graphs include:
descriptions of a problem situation,
• the relationship between time and distance travelled
identifying features listed above
• the relationship between temperature and time over which it is measured
• the relationship between rainfall and time over which it is measured, etc.
Space and Shape 3.4 Transformations What is different to Grade 6 9 hours
(geometry)
Transforma- • Recognize, describe and perform Learners in Grade 7 have to do transformations on squared paper.
tion translations, reflections and rotations
Focus of transformations
Geometry with geometric figures and shapes on
squared paper • Using squared paper for transformations allows learners to more accurately
perform transformations and to compare the shape and size of figures.
• Identify and draw lines of symmetry
in geometric figures • Learners should recognize that translations, reflections and rotations only
change the position of the figure, and not its shape or size.
Enlargements and reductions
• They should recognize that the above transformations produce congruent
• Draw enlargements and reductions of
figures.
geometric figures on squared paper
and compare them in terms of shape • Learners should recognize that enlargements and reductions change the size of
and size figures by increasing or decreasing the length of sides, but keeping the angles
the same, producing similar rather than congruent figures.
• Learners should also be able to work out the factor of enlargement or reduction.
65
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
66 3.2 What is different to Grade 6? 9 hours
Geometry of • Most of this work consolidates what has been done in Grade 6.
3D objects
Classifying 3D objects Polyhedra
• Describe, sort and compare Examples of sorting or grouping categories:
polyhedra in terms of
• cubes (only square faces)
-- shape and number of faces
• rectangular prisms (only rectangular faces)
-- number of vertices
• triangular prisms (only triangular and rectangular faces)
-- number of edges
• pyramids (square and triangular faces)
• cylinders (circular and rectangular faces).
MATHEMATICS GRADES 7-9
Building 3D model Using and constructing nets
• Revise using nets to create models of • Using and constructing nets are useful contexts for exploring or consolidating
geometric solids, including: properties of polyhedra.
-- cubes • Learners should recognize the nets of different solids.
-- prisms • Learners should draw sketches of the nets using their knowledge of shape and
number of faces of the solids, before drawing and cutting out the nets to build
models.
• The construction of nets is based on the number and shape of faces of the
solids, and do not require measuring of internal angles of polygons.
• Learners have to work out the relative position of the faces of the nets, and
use trial and error to match up the edges and vertices, in order to build the 3D
object.
REVISION/ASSESSMENT: 6 hours
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
At this stage learners should have been assessed on:
• numeric and geometric patterns
• functions and relationships
• algebraic expressions
• algebraic equations
• graphs
• transformation geometry
• geometry of 3D objects
GRADE 7 – TERM 4
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
CAPS (in hours)
Numbers, 1.3 Counting, ordering and comparing What is different to Grade 6? 9 hours
Operations and integers
Relationships Integers Integers are new numbers introduced in Grade 7.
• Count forwards and backwards in
Counting, ordering and comparing integers
integers for any interval
• Counting should not only be thought of as verbal counting. Learners can count
• Recognize, order and compare
using:
integers
-- structured, semi-structured or empty number lines
-- chain diagrams for counting
• Learners should be given a range of exercises such as:
-- arrange given numbers from the smallest to the biggest: or biggest to smallest
-- fill in missing numbers in
♦♦ a sequence
♦♦ on a number grid
♦♦ on a number line
♦♦ fill in <, = or > Example: – 425 * – 450
Calculations with integers Calculations using integers
• Add and subtract with integers • Start calculations using integers in small number ranges.
• Develop an understanding that subtracting an integer is the same as adding its
additive inverse.
• Example:
7 – 4 = 7 + (– 4) = 3 OR –7 – 4 = –7 + (– 4) = –11
So too, 7 – (– 4) = 7 + (+4) = 11 OR –7 – (– 4) = –7 + (+4) = –3
Here the use of brackets around the integers are useful.
Properties of integers Properties of integers
• Recognize and use commutative and • Learners should investigate the properties for operations using whole numbers
associative properties of addition and on the set of integers.
multiplication for integers • These properties should serve as motivation for the operations they can perform
using integers.
67 Solving problems • Learners should see that the commutative property for addition holds for
integers e.g 8 + (–3) = (–3) + 8 = 5
• Solve problems in contexts involving
• Learners should see that they can still use subtraction to check addition or vice
MATHEMATICS GRADES 7-9
addition and subtraction of integers
versa. Example: If 8 + (–3) = 5, then 5 – 8 = – 3 and 5 –(–3) = 8
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
68 Numbers, 1.3 • Learners should see that the associative property for addition holds for integers.
Operations and Example: [(–6) + 4] + (–1) = (–6) + [4 + (–1)] = –3
Integers
Relationships
• Learners should only explore the distributive property once they can multiply
with integers
Patterns, 2.1 Investigate and extend patterns The focus of numeric patterns in this term should be on practising operations with 3 hours
functions and integers.
Numeric and • Investigate and extend numeric
algebra
geometric and geometric patterns looking for See additional notes in Term 3.
patterns relationships between numbers,
including patterns:
-- represented in physical or diagram
form
-- not limited to sequences involving a
MATHEMATICS GRADES 7-9
constant difference or ratio
-- of learner’s own creation
-- represented in tables
• Describe and justify the general rules
for observed relationships between
numbers in own words
2.2 Input and output values Functions and relationships in this term should include integers as input or output 3 hours
Functions values, as well as using integers in the rules for patterns and relationships.
• Determine input values, output
and values or rules for patterns and See additional notes in Terms 2 & 3.
relationships relationships using:
-- flow diagrams
-- tables
-- formulae
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
Equivalent forms
• Determine, interpret and justify
equivalence of different descriptions
of the same relationship or rule
presented:
-- verbally
-- in flow diagrams
-- in tables
-- by formulae
-- by number sentences
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS 2.3 Algebraic language Algebraic expressions should include integers in the rules or relationships repre- 3 hours
sented in symbolic form.
Algebraic • Recognize and interpret rules or
expressions relationships represented in symbolic See additional notes in Term 3.
form
• Identify variables and constants in
formulae and equations
2.4 Number sentences Number sentences should include integers. 4 hours
Algebraic • Write number sentences to describe See additional notes in Term 3.
equations problem situations
• Analyse and interpret number
sentences that describe a given
situation
• Solve and complete number
sentences by:
-- inspection
-- trial and improvement
• Identify variables and constants in
given formulae or equations
• Determine the numerical value of an
expression by substitution.
69
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
70 DATA HANDLING 5.1 What is different to Grade 6? Time for
collecting
Collect, The following are new in Grade 7 and
organize and organizing
• samples and populations
summarize data: 4 hours
data • multiple choice questionnaires
• stem-and-leaf displays
• grouping data in intervals
• mean
• range
• histograms
MATHEMATICS GRADES 7-9
• scales on graphs
Collect data Data sets and contexts
• Pose questions relating to social, Learners should be exposed to a variety of contexts that deal with social and
economic, and environmental issues environmental issues, and should work with given data sets, represented in
in own environment a variety of ways, that include big number ranges, percentages and decimal
fractions. Learners should then practise organizing and summarizing this data,
• Select appropriate sources for
analysing and interpreting the data, and writing a report about the data.
the collection of data (including
peers, family, newspapers, books,
magazines)
• Distinguish between samples and
populations
• Design and use simple
questionnaires to answer questions:
-- with yes/no type responses
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
-- with multiple choice responses.
Organize and summarize data Complete a data cycle
• Organize (including grouping where Learners should complete at least one data cycle for the year, starting with
appropriate) and record data using posing their own questions, selecting the sources and method for collecting data,
recording the data, organizing the data, representing the data, then analysing,
-- tally marks
summarizing, interpreting and reporting the data. Challenge learners to think about
-- tables what kinds of questions and data need to be collected to be represented in a
histogram, pie chart, or bar graph.
-- stem-and-leaf displays
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS DATA HANDLING 5.1 • Group data into intervals Representing data Time for
representing
Collect, • Summarize and distinguish between • Drawing pie charts to represent data do not have to be accurately drawn with a data: 3 hours
organize and ungrouped numerical data by compass and protractor, etc. Learners can use any round object to draw a circle,
summarize determining: then divide the circle into halves and quarters and eighths if needed, as a guide
data to estimate the proportions of the circle that need to be shown to represent the
-- mean
data. What is important is that the values or percentages associated with the
-- median data, are shown proportionally on the pie chart.
-- mode • Drawing, reading and interpreting pie charts is a useful context to re-visit
equivalence between fractions and percentages, e.g. 25% of the data is
• Identify the largest and smallest 1
represented by a 4 sector of the circle.
scores in a data set and determine
the difference between them in order • It is also a context in which learners can find percentages of whole numbers e.g.
to determine the spread of the data if 25% of 300 learners like rugby, how many (actual number) learners like rugby?
(range).
• Histograms are used to represent grouped data shown in intervals on the
horizontal axis of the graph. Point out the differences between histograms and
bar graphs, in particular bar graphs that represent discrete data (e.g. favourite
sports) compared to histograms that show categories in consecutive, non-
overlapping intervals, (e.g. test scores out of 100 shown in intervals of 10). The
bars on bar graphs do not have to touch each other, while in a histogram they
have to touch since they show consecutive intervals.
Developing critical analysis skills
• Learners should compare the same data represented in different ways e.g. in a
pie chart or a bar graph or a table, and discuss what information is shown and
what is hidden; they should evaluate what form of representation works best for
the given data.
• Learners should compare graphs on the same topic but where data has been
collected from different groups of people, at different times, in different places
or in different ways. Here learners should discuss differences between the data
with an awareness of bias related to the impact of data sources and methods of
data collection on the interpretation of the data.
• Learners should compare different ways of summarising the same data sets,
developing an awareness of how data reporting can be manipulated; evaluating
which summary statistics best represent the data.
• Learners should compare graphs of the same data, where the scales of Time for
analysing,
the graphs are different. Here learners should discuss differences with an
interpreting
awareness of how representation of data can be manipulated; they should and reporting
evaluate which form of representation works best for the given data. data: 3,5
71 hours
• Learners should write reports on the data in short paragraphs.
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
72 5.2 Represent data
Representing • Draw a variety of graphs by hand/
data technology to display and interpret
data (grouped and ungrouped)
including:
-- bar graphs and double bar graphs
-- histograms with given intervals
-- pie charts
5.3 Interpret data
Interpret, • Critically read and interpret data
analyse and represented in:
MATHEMATICS GRADES 7-9
report data -- words
-- bar graphs
-- double bar graphs
-- pie charts
-- histograms
Analyse data
• Critically analyse data by answering
questions related to:
-- data categories, including data
intervals
-- data sources and contexts
-- central tendencies (mean, mode,
median)
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
-- scales used on graphs
Report data
• Summarize data in short paragraphs
that include
-- drawing conclusions about the data
-- making predictions based on the
data
-- identifying sources of error and
bias in the data
-- choosing appropriate summary
statistics for the data (mean,
median, mode)
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS 5.4 Probability Probability experiments 4,5 hours
Probability Perform simple experiments where the In the Intermediate Phase learners did experiments with coins, dice and spinners.
possible outcomes are equally likely In this grade experiments can be done with other objects, like, different coloured
and buttons in a bag; choosing specific cards from a deck of cards, etc.
• list the possible outcomes based on Example
the conditions of the activity If you toss a coin there are two possible outcomes (head or tail). The probability
1
• determine the probability of each of a head is 2 which is equivalent to 50% (since it is one out of two possible
possible outcome, using the definition outcomes)
of probability
REVISION/ASSESSMENT: Total time for
revision/
At this stage learners should have been assessed on:
assessment
• adding and subtracting with integers for the term
• collecting, organizing, representing, analysing, summarizing, interpreting and reporting data 8 hours
• probability
73
MATHEMATICS GRADES 7-9
MATHEMATICS GRADES 7-9
Time allocation per Term: Grade 8
TERM 1 TERM 2 TERM 3 TERM 4
Topic Time Topic Time Topic Time Topic Time
6 Algebraic 9 7 Functions and 6
Whole numbers Common fractions
hours expressions hours hours relationships hours
9 Algebraic 3 6 Algebraic 3
Integers Decimal fractions
hours equations hours hours equations hours
9 Construction of 8 Theorem of 5 9
Exponents Graphs
hours Geometric figures hours Pythagoras hours hours
Numeric and 4,5 Geometry of 2D 8 Area and 5 6
Transformation
geometric hours shapes hours perimeter of 2D hours hours
geometry
patterns shapes
3 Geometry of 9 Surface area and 5 7
Functions and Geometry of 3D
hours straight lines hours volume of 3D hours hours
relationships objects
objects
4,5 Collect, organize 4 4,5
Algebraic
hours and summarize hours Probability hours
expressions
data
Algebraic 3 3
Represent data
equations hours hours
Interpret, analyse 3,5
and report data hours
Revision/ 6 Revision/ 8 Revision/ 6,5 Revision/ 9,5
hours hours hours hours
Assessment Assessment Assessment Assessment
TOTAL: 45 hours TOTAL: 45 hours TOTAL: 45 hours TOTAL: 45 hours
74 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
3.3.2 Clarification of content for Grade 8
CAPS GRADE 8 – TERM 1
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
Numbers, 1.1 Mental calculations What is different to Grade 7? 6 hours
Operations and
Relationships Whole Revise: In Grade 8 learners consolidate number knowledge and calculation techniques for
numbers whole numbers, developed in the Intermediate Phase and Grade 7
• Multiplication of whole numbers to at
least 12 x 12
Ordering and comparing whole
numbers
Revise Prime numbers to at least 100
Properties of whole numbers Properties of whole numbers
• Revise: • Revising the properties of whole numbers should be the starting point for work
with whole numbers. The properties of numbers should provide the motivation for
-- The commutative; associative; why and how operations with numbers work.
distributive properties of whole
numbers • When learners are introduced to new numbers, such as integers for example,
they can again explore how the properties of numbers work for the new set of
-- 0 in terms of its additive property numbers.
(identity element for addition)
• Learners also have to apply the properties of numbers in algebra, when they
-- 1 in terms of its multiplicative work with variables in place of numbers.
property (identify element for
multiplication) • Learners should know and be able to use the following properties:
-- Recognise the division property of -- The commutative property of addition and multiplication:
0, whereby any number divided by ♦♦ a + b = b + a
0 is undefined
♦♦ a x b = b x a
-- The associative (grouping) property of addition and multiplication:
♦♦ (a + b) + c = a + (b + c)
♦♦ (a x b) x c = a x (b x c)
-- The distributive property of multiplication over addition and subtraction:
♦♦ a(b + c) = (a x b) + (a x c)
♦♦ a(b – c) = (a x b) – (a x c)
-- Addition and subtraction as inverse operations
75 -- Multiplication and division as inverse operations
-- 0 is the identity element for addition: t + 0 = t
MATHEMATICS GRADES 7-9
-- 1 is the identity element for multiplication: t x 1 = t
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
76 Numbers, 1.1 Illustrating the properties with whole numbers
Operations and
Relationships Whole a) 33 + 99 = 99 + 33 = 132
numbers
b) 51 + (19 + 46) = (51 + 19) + 46 = 116
c) 4(12 + 9) = (4 x 12) + (4 x 9) = 48 + 36 = 84
d) (9 x 64) + (9 x 36) = 9(64 + 36) = 9 x 100 = 900
e) if 33 + 99 = 132, then 132 – 99 = 33 ad 132 – 33 = 99
f) if 20 x 5 = 110, then 110 ÷ 20 = 5 and 110 ÷ 5 =20
Calculations using whole numbers Calculations with whole numbers
Revise: • Learners should continue to do context free calculations and solve problems in
contexts using whole numbers, integers and fractions
MATHEMATICS GRADES 7-9
• Calculations using all four operations
on whole numbers, estimating and • Learners should become more confident in and more independent at
using calculators where appropriate mathematics, if they have techniques
Calculation techniques -- to check their solutions themselves, e.g. using inverse operations; using
calculators
• Use a range of strategies to perform
and check written and mental -- to judge the reasonableness of their solutions e.g. estimate by rounding off;
calculations with whole numbers estimate by doubling or halving;
including:
• Adding, subtracting and multiplying in columns, and long division, should only be
-- estimation used to practice number facts and calculation techniques, and hence should be
done with familiar and smaller number ranges. For big and unwieldy calculations,
-- adding, subtracting and multiplying
learners should be encouraged to use a calculator.
in columns
Multiples and factors
-- long division
• Learners should continue practising finding multiples and factors of whole
-- rounding off and compensating
numbers. This is especially important when learners do calculations with
-- using a calculator fractions. They use this knowledge to find the LCM for denominators that are
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
different from each other, and also when they simplify fractions or have to find
Multiples and factors
equivalent fractions.
Revise:
• Factorising whole numbers lays the foundation for factorisation of algebraic
-- Prime factors of numbers to at least expressions.
3-digit whole numbers
• Using the definition of prime numbers, emphasise that 1 is not classified as a
-- LCM and HCF of numbers to at prime number
least 3-digit whole numbers, by
inspection or factorisation
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS Numbers, 1.1 Examples
Operations and
Whole a) LCM of 6 and 18 is 18
Relationships
numbers
LCM of 6 and 7 is 42
b) The factors of 24 are 1, 2, 3, 4, 6,12 and 24 by inspection.
And, the prime factors of 24 are 2 and 3
c) The factors of 140 are 1, 2, 5, 7, 10, 14, 28, 35, 70 and 140
d) Determine the HCF of 120; 300 and 900.
Learners do this by finding the prime factors of the numbers first.
120 = 5 x 3 x 23 Initially learners may write this as: 5 x 3 x 2 x 2 x 2
300 = 52 x 3 x 22
900 = 52 x 32 x 23
HCF = 5 x 3 x 22 = 60 (Multiply the common prime factors of the three numbers)
Solving problems
Solving problems
• Solve problems involving whole
• Solving problems in contexts should take account of the number ranges learners
numbers, including:
are familiar with.
-- Comparing two or more quantities
• Contexts involving ratio and rate should include speed, distance and time
of the same kind (ratio)
problems.
-- Comparing two quantities of
• In financial contexts, learners are not expected to use formulae for calculating
different kinds (rate)
simple interest.
-- Sharing in a given ratio where the
whole is given
-- Increasing or decreasing of a
number in a given ratio
• Solve problems that involve whole
numbers, percentages and decimal
fractions in financial contexts such as:
-- Profit, loss, discount and VAT
-- Budgets
-- Accounts
-- Loans
-- Simple interest
-- Hire Purchase
77
-- Exchange rates
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
78 1.3 What is different to Grade 7? Total time for
Integers:
Integers • Multiply and divide with integers
• All four operations with integers
9 hours
• All four operations with squares, cubes, square and cube roots of integers
In Grade 8 learners consolidate number knowledge and calculation techniques for
integers, developed in Grade 7.
Counting, ordering and comparing Counting, ordering and comparing integers
integers • Learners should continue practising counting, ordering and comparing integers.
• Revise: Counting should not only be thought of as verbal counting. Learners can count
using:
-- counting forwards and backwards
MATHEMATICS GRADES 7-9
in integers for any interval -- structured, semi-structured or empty number lines
-- recognising, ordering and -- chain diagrams for counting
comparing integers • Learners should be given a range of exercises
• Arrange given numbers from the smallest to the biggest: or biggest to smallest
• Fill in missing numbers in
-- a sequence
-- on a number grid
-- on a number line
-- fill in <, = or > e.g. – 425 * – 450;
Calculations with integers Calculations using integers
• Revise addition and subtraction with • Start calculations with integers using small number ranges.
integers
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
• Develop an understanding that subtracting an integer is the same as adding its
• Multiply and divide with integers additive inverse.
• Perform calculations involving all four Example:
operations with integers 7 – 4 = 7 + (– 4) = 3 OR –7 – 4 = –7 + (–4) = –11
• Perform calculations involving all four So too, 7 – (– 4) = 7 + (+4) = 11 OR –7 – (– 4) = –7 + (+ 4) = – 3. Here the use of
operations with numbers that involve brackets around the integers are useful.
the squares, cubes, square roots and
cube roots of integers • A useful strategy is to use repeated addition and number patterns to show
learners the reasonableness of rules for the resultant sign for multiplication with
integers.
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS 1.3 Example:
Integers a) Repeated addition of (–3): (–3) + (–3) + (–3) = –9 = 3 x (–3)
b) Repeated addition of (–2): (–2) + (–2) + (–2) + (–2) = –8 = 4 x (–2)
c) Counting down in intervals of 3 from 9:
3x3=9
3x2=6
3x1=3
3x0=0
3 x –1 = –3
3 x –2 = ?
3 x –3 = ?
Hence the rule: a positive integer x a negative integer = a negative integer
d) Using the rule that a positive integer x a negative integer = a negative integer,
established from examples above, the following pattern can be used:
–1 x 3 = –3
–1 x 2 = –2
–1 x 1 = –1
–1 x 0 = 0
–1 x –1 = 1
–1 x –2 = ?
–1 x –3 = ?
Hence the rule: a negative integer x a negative integer = a positive integer
• Use the inverse operation for multiplication and division to develop a rule for the
resultant sign for division with integers.
Example:
a) If 4 x (–2) = –8, then –8 ÷ 4 = –2 and –8 ÷ (–2) = 4
b) If (–1) x (–3) = 3, then 3 ÷ (–1) = –3 and 3 ÷ (–3) = –1
Hence the rules: division of a positive and negative integer equals a negative
79
integer and division of two negative integers equal a positive integer.
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
80 1.3 • Finding the squares, cubes, square roots and cube roots of integers are also
opportunities to check that learners know the rules for resultant signs when
Integers
multiplying integers.
• Therefore, make sure that learners understand why you cannot find the square
root of a negative integer, and that the square of a negative integer is always
positive.
Example:
a) (–5)2 = (–5) x (–5) = 25
b) (–4)3 = (–4) x (–4) x (–4) = –64
3
c)
Properties of integers
MATHEMATICS GRADES 7-9
√ –27 = –3 because –3 x –3 x–3 = –27
Properties of integers • Learners should investigate the properties for operations with whole numbers on
the set of integers.
• Recognize and use commutative,
associative and distributive properties • These properties should serve as motivation for the operations they can perform
of addition and multiplication for with integers.
integers
• Learners should see that the commutative property for addition and multiplication
• Recognize and use additive and holds for integers, e.g.
multiplicative inverses for integers 8 + (–3) = (–3) + 8 = 5; 8 x (–3) = (–3) x 8 = –24
Solving problems • Learners should see that they can still use subtraction to check addition or vice
versa, e.g. if 8 + (–3) = 5, then 5 – 8 = –3 and 5 –(–3) = 8
• Solve problems in contexts involving
multiple operations with integers • Learners should see that the associative property for addition holds for integers,
e.g. [(–6) + 4] + (–1) = (–6) + [4 + (–1)] = –3
• Learners should see that the inverse operation for multiplication and division
holds for integers, e.g. if 5 x (–6) = –30, then –30 ÷ 5 = –6 and –30 ÷ (–6) = 5
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
• Learners should develop the rules, through patterning, for resultant signs when
multiplying and dividing integers:
(+5) x (+5) = (+25);
(–5) x (–5) = (+25);
(–5) x (+5) = (–25);
(+25) ÷ (+5) = (+5);
(–25) ÷ (–5) = (+5);
(–25) ÷ (+5) = (–5);
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS 1.2 Mental calculations What is different to Grade 7? Total time for
Exponents exponents:
• Revise: • Integers and rational numbers in exponential form
2
-- Squares to at least 12 and their • Scientific notation of numbers
square roots 9 hours
3
-- Cubes to at least 6 and their cube
roots
Comparing and representing Comparing and representing numbers in exponential form
numbers in exponential form
• Learners need to understand that in the exponential form ab, the number is read
• Revise compare and represent whole as ‘a to the power of b, where a is called the base and b is called the exponent or
numbers in exponential form index and b indicates the number of factors that are multiplied.
• Compare and represent integers in Example: a3 = a x a x a; a5 = a x a x a x a x a
exponential form Learners can represent any number in exponential form, without needing to
• Compare and represent numbers in compute the value. Example: 50 x 50 x 50 x 50 x 50 x 50 x 50 = 507
scientific notation, limited to positive • Make sure learners understand that square roots and cube roots are the inverse
exponents operations of squaring and cubing numbers. Example: 32 = 9 therefore √9 = 3
• Make sure learners understand that any number raised to the power 1 is equal to
that particular number
Example: m1 = m
• Using patterns and their knowledge of multiplication with integers, learners
should anticipate the resultant sign of an integer raised to an odd or even power
e.g. (–15)4 will be positive, while (–15)3 will be negative
• To avoid common misconceptions, emphasize the following
-- 122 = 12 x 12 and not 12 x 2
-- 13 means 1 x 1 x 1 and not 1 x 3
-- 1001 = 100
3
--
-- √ 81 = 9 because 92 = 81
-- the square of 9 = 81, whereas the square root of 9 = 3
√ 27 = 3 because 33 = 27
• Learners should use their knowledge of representing numbers in exponential
form when simplifying and expanding algebraic expressions and solving
algebraic equations.
81
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
82 1.2 Calculations using numbers in Laws of exponents
Exponents exponential form
• The laws of exponents should be introduced through a range of numeric
• Establish general laws of exponents, examples first, then variables can be used. In other words, the numbers are
limited to: replaced with letters, but the rules work the same.
-- natural number exponents • The following laws of exponents should be introduced, where m and n are natural
m n m+n
numbers and a and t are not equal to 0:
♦♦ a x a = a
♦♦ am ÷ an = am–n, if m > n am x an = am+n
♦♦ (am)n = am x n Example
♦♦ (a x t)n = an x tn a) 23 x 24 = 23+4 = 27
♦♦ a0 = 1 b) x3 x x4 = x3+4 = x7
MATHEMATICS GRADES 7-9
• Recognize and use the appropriate
laws of operations using numbers am ÷ an = am–n if m >n
involving exponents and square and Example:
cube roots
a) 35 ÷ 32 = 33 = 27
• Perform calculations involving all four
operations with numbers that involve b) x5 ÷ x3 = x2
squares, cubes, square roots and
cube roots of integers (am)n = amn
• Calculate the squares, cubes, square Example:
roots and cube roots of rational
a) (23)2 = 26 = 64
numbers
b) (x3)2 = x6
Solving problems
• Solve problems in contexts involving (a x t)n = an x tn
numbers in exponential form
Example: (3x2)3 = 33. x6 = 27x6
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
a0 = 1
Examples: (37)0 = 1; (4x2)0 = 1;
• Make sure learners understand these laws reading from both sides of the equal
sign i.e. if the LHS = RHS, then the RHS = LHS.
• The law a0 = 1 can be derived by using the law of exponents for division in a few
axaxaxa
examples. a4 ÷ a4 = a x a x a x a = 1 therefore a4–4 = a0 = 1
• Learners should be able to use the laws of exponents in calculations and
for solving simple exponential equations as well as expanding or simplifying
algebraic expressions.
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS 1.2 • Look out for the following common misconceptions where:
Exponents
-- learners multiply unlike bases and add the exponent.
Example:
xm x yn = (xy)m+n
-- learners multiply like bases and add the exponents
Example:
25 x 27 = 412 instead of the correct answer 212.
-- learners forget, for example, that when squaring a binomial there is a middle
term
Example:
m m m
(x + y) = x + y
-- learners confuse adding the exponents and adding the terms
Example:
m n m+n mn
x +x =x or x
Calculations using numbers in exponential form
• Knowing the rules of operations for calculations involving exponents are
important, e.g.
a) (7 – 4)3 = 33 and NOT 73 – 43
b)
• Learners can also do simple calculations where the numerator and denominator
3
√16 + 9 = √ 25 and NOT the √ 16 + √ 9
2 2x2x2 8
2
of a fraction are written in exponential form, e.g. 2 = 2 x 2 = 4 = 2
• Learners can also find squares, cubes, square roots and cube roots of decimal
and common fractions by inspection.
Examples
a) (0,7)2 = 0,49
b) (0,1)3 = 0,001
c) = 0,3
2
9
d) ( 34 )2 = 342 = 16
√0,09
83
e)
√9
MATHEMATICS GRADES 7-9
√259 = √ 25 = 35
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
84 1.2 Scientific notation
Exponents
• When writing numbers in scientific notation, learners have to understand the
relationship between the number of decimal places and the index of 10
Examples:
a) 25 = 2,5 x 101
b) 250 = 2,5 x 102
c) 2 500 = 2,5 x 103
• Scientific notation limited to positive exponents, includes writing very large
numbers in scientific notation.
Example: 25 million = 2,5 x 107
MATHEMATICS GRADES 7-9
• Learners practise writing large numbers in scientific notation, they will realize
they have encountered these in Natural Science. It is useful to refer to these
contexts when talking about scientific notation.
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS Patterns, 2.1 Investigate and extend patterns What is different to Grade 7? Total time for
functions and Numeric and
algebra Numeric and • Investigate and extend numeric • The range of number patterns are extended to include patterns with multiplication geometric
geometric and geometric patterns looking for and division with integers, numbers in exponential form patterns:
patterns relationships between numbers,
• As learners become used to describing patterns in their own words, their
including patterns:
descriptions should become more precise and efficient with the use of algebraic
4,5 hours
-- represented in physical or diagram language to describe general rules of patterns
form
• It is useful also to introduce the language of ‘term in a sequence’ in order to
-- not limited to sequences involving a distinguish the term from the position of a term in a sequence
constant difference or ratio
• Investigating number patterns is an opportunity to generalize – to give general
-- of learner’s own creation algebraic descriptions of the relationship between terms and their position in a
sequence and to justify solutions.
-- represented in tables
Kinds of numeric patterns
-- represented algebraically
• Given a sequence of numbers, learners have to identify a pattern or relationship
• Describe and justify the general rules
between consecutive terms in order to extend the pattern.
for observed relationships between
numbers in own words or in algebraic Examples
language
Provide a rule to describe the relationship between the numbers in the
sequences below. Use this rule to provide the next three numbers in the
sequence:
a) –3; –7; –11; –15; ... ...
Here learners should identify the constant difference between consecutive
terms in order to extend the pattern. This pattern can be described in learners’
own words as ‘adding –4’ or ‘counting in –4s’ or ‘add –4 to the previous
number in the pattern’.
b) 2; –4; 8; –16; 32 ... ...
Here learners should identify the constant ratio between consecutive terms.
This pattern can be described in learners’ own words as ‘multiply the previous
number by –2’.
c) 1; 2; 4; 7; 11; 16 ... ...
This pattern has neither a constant difference nor constant ratio. This pattern
can be described in learners’ own words as ‘increase the difference between
consecutive terms by 1 each time’ or ‘add 1 more than was added to get the
previous term’. Using this rule, the next 3 terms will be 22; 29; 37.
• Given a sequence of numbers, learners have to identify a pattern or relationship
85 between the term and its position in the sequence. This enables learners to
predict a term in a sequence based on the position of that term in the sequence.
It is useful for learners to represent these sequences in tables so that they can
MATHEMATICS GRADES 7-9
consider the position of the term.
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
86 Patterns, 2.1 Examples
functions and
algebra Numeric and a) Provide a rule to describe the relationship between the numbers in this
geometric sequence: 1; 8; 27; 64; ... .. Use this rule to find the 10th term in this sequence.
patterns
Firstly, learners have to understand that the ‘10th term’ refers to position in the
number sequence. They have to find a rule in order to determine the 10th term,
rather than continuing the sequence to the tenth term.
Example
b) This sequence can be re-presented in the following table:
Position in sequence 1 2 3 4 10
Term 1 8 27 64 ?
MATHEMATICS GRADES 7-9
Learners have to recognize that each term in the bottom row is obtained by cubing
the position number in the top row. Thus the 10th term will be 10 cubed or 103,
which is. Using the same rule, learners can also be asked what term number or
position will 512 be? If the term is obtained by cubing the position number of the
term, then the position number can be obtained by finding the cube root of the
c) Provide a rule to describe the relationship between the numbers in this
term. Hence, 512 will be the 8th term in the sequence since 3√512 = 8
sequence: 4; 7; 10; 13; ... Use this rule to find the 20th term in this sequence.
If learners consider only the relationship between consecutive terms, then they
have to continue the pattern (‘add 3 to previous number’) to the 20th term to
find the answer. However, if they look for a relationship or rule between the term
and the position of the term, they can predict the answer without continuing the
pattern. Using number sentences can be useful to find the rule:
1st term: 4 = 3(1) + 1
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
2nd term: 7 = 3(2) + 1
3rd term: 10 = 3(3) + 1
4th term: 13 = 3(4) + 1
The number in the brackets corresponds to the position of the term. Hence, the
20th term will be: 3(20) + 1 = 61
The rule in learners’ own words can be written as ‘3 times the position of the term
+1’ or 3n + 1 where n is the position of the term.
• These types of numeric patterns develop an understanding of functional
relationships, in which you have a dependent variable (position of the term) and
independent variable (the term itself), and where you have a unique output for
any given input value.
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS Patterns, 2.1 Kinds of geometric patterns
functions and
algebra Numeric and • Geometric patterns are number patterns represented diagrammatically. The
geometric diagrammatic representation reveals the structure of the number pattern.
patterns
• Hence, representing the number patterns in tables makes it easier for learners to
describe the general rule for the pattern.
Example
Consider this pattern for building hexagons with matchsticks. How many
matchsticks will be used to build the 10th hexagon?
The rule for the pattern is contained in the structure (construction) of the
successive hexagonal shapes:
(1) add 1 matchstick per side
(2) there are 6 sides
(3) add 6 matchsticks per hexagon as you proceed from a given hexagon to the
next one.
So, for the 2nd hexagon, you have 2 x 6 matches; for the 3rd hexagon you have
3 x 6 matches: Using this pattern for building hexagons, the 10th hexagon will have
10 x 6 matches.
Learners can also use a table to record the number of matches used for each
hexagon. This way they can look at the number pattern related to the number of
matches used for each new hexagon.
Position of hexagon in pattern 1 2 3 4 5 6 10
Number of matches 6 12 18
Describing patterns
• It does not matter if learners are already familiar with a particular pattern. Their
descriptions of the same pattern can be different when they encounter it at
87 different stages of their mathematical development.
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
88 Patterns, 2.1 Example
functions and Numeric and The rule for the sequence: can be described in the following ways:
algebra geometric
patterns a) add 3 to the previous term
b) 3 x the position of the term +1
c) 3(n) + 1, where n is the position of the term
d) 3(n) + 1, where n where is a natural number
2.2 Input and output values What is different to Grade 7? Time for
Functions Functions
• Determine input values, output values • Finding input or output values using given equations
and and Rela-
or rules for patterns and relationships • The rules and number patterns for which learners have to find input and output tionships in
relationships using: values are extended to include patterns with multiplication and division of this term:
MATHEMATICS GRADES 7-9
-- flow diagrams integers and numbers in exponential form
-- tables • In this phase, it is useful to begin to specify whether the input values are natural 3 hours
-- formulae numbers, or integers or rational numbers. This builds learners’ awareness of the
domain of input values. Hence, to find output values, learners should be given
-- equations the rule/formula as well as the domain of the input values.
Equivalent forms Functions and relationships will be done again in Term 4. In Term 1 the focus
• Determine, interpret and justify of Functions and relationships is on practising operations with integers, or
equivalence of different descriptions including integers in the rules for finding output values.
of the same relationship or rule Note that when learners find input or output values for given rules, they are actually
presented: finding the numerical value of algebraic expressions using substitution.
-- verbally Flow diagrams are representations of functional relationships. Hence, when using
-- in flow diagrams flow diagrams, the correspondence between input and output values should be
clear in its representational form i.e. the first input produces the first output, the
-- in tables
second input produces the second output, etc.
-- by formulae
Examples
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
-- by equations
a) Use the given rule to calculate the values of t for each value of p, where p is a
natural number.
p t
0
Rule
-3
-5 t = p x 3 + 1
-7
-9
In this kind of flow diagram, learners can also be asked to find the value of p for
a given t value.
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS 2.2 b) Find the rule for calculating the output value for every given input value in the
flow diagram below.
Functions
and p t
relationships
0 -3
Rule
3 -9
5
-15
7 -21
9 -27
In flow diagrams such as these, more than one rule might be possible to describe the
relationship between input and output values. The rules are acceptable if they match
the given input values to the corresponding output values.
c) If the rule for finding y in the table below is: y = –3x – 1, find y for the given x
values:
x 0 1 2 5 10 50 100
y
b) Describe the relationship between the numbers in the top row and bottom row
in the table. Then write down the value of and
x –2 –1 0 1 2 12 n
y –5 –4 –3 –2 –1 m 34
In tables such as these, more than one rule might be possible to describe the
relationship between x and y values. The rules are acceptable if they match the
given input values to the corresponding output values. For example, the rule
y = x –3 describes the relationship between the x values and given y values. To find
m and n, learners have to substitute the corresponding values for x or y in the rule
and solve the equation by inspection.
89
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
90 2.3 Algebraic language What is different to Grade 7? Time for
algebraic
Algebraic • Revise the following done in Grade 7: • Introduction to conventions of algebraic language expressions
expressions -- recognize and interpret rules • Manipulating algebraic expressions in this term:
or relationships represented in
Algebraic expressions are done again in Term 2, where the focus is more
symbolic form
fully on manipulating algebraic expressions. In this term the focus is on 4,5 hours
-- identify variables and constants in interpreting algebraic expressions and introducing conventions of algebraic
given formulae and equations language through adding and subtracting like terms.
• Recognize and identify conventions Learners have opportunities to write and interpret algebraic expressions when they
for writing algebraic expressions write general rules to describe relationships between numbers in number patterns,
and when they find input or output values for given rules in flow diagrams, tables,
• Identify and classify like and unlike
formulae and equations.
terms in algebraic expressions
MATHEMATICS GRADES 7-9
Examples of interpreting algebraic expressions
• Recognize and identify coefficients
and exponents in algebraic a) What does the rule 2n mean for the following number sequence:
expressions 2; 4; 8; 16; 32...
Expand and simplify algebraic Here learners should recognize that 2n represents the general term in this
expressions sequence, where n represents the position of the term in the sequence. Thus, it
is the rule that can be used to find any term in the given sequence.
Use commutative, associative and
distributive laws for rational numbers b) The relationship between a boy’s age (x yrs old) and his mother’s age is given
and laws of exponents to: as 25 + x. How can this relationship be used to find the mother’s age when the
boy is 11 years old? Here learners should recognize that to find the mother’s
• Add and subtract like terms in
age, they must substitute the boy’s given age into the rule 25 + x. They should
algebraic expressions
also recognize that the given rule means the mother is 25 years older than the
• Determine the squares, cubes, boy.
square roots and cube roots of single
See further examples given for functions and relationships, as well as notes in
algebraic terms or like algebraic terms
Term 2.
• Determine the numerical value of
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
algebraic expressions by substitution
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS 2.4 Equations What is different to Grade 7? Time for
Algebraic
Algebraic • Revise the following done in Grade 7: Up to and including Grade 7 learners use the term 'numbers sentences'. From equations in
equations Grade 8 the term 'equations' is used. this term:
-- set up equations to describe
problem situations Solving equations using additive and multiplicative inverses as well as laws of
exponents
-- analyse and interpret equations that 3 hours
describe a given situation Algebraic equations are done again in Term 2 and Term 4. In this term the
focus is on solving equations that involve multiplication and division of
-- solve equations by inspection integers and numbers in exponential form, by inspection only.
-- determine the numerical value of an Learners have opportunities to write and solve equations when they write general
expression by substitution. rules to describe relationships between numbers in number patterns, and when
they find input or output values for given rules in flow diagrams, tables and
-- identify variables and constants in formulae.
given formulae or equations
See further notes in Term 2.
• Extend solving equations to include:
-- Using additive and multiplicative
inverses
-- Using laws of exponents
• Use substitution in equations to
generate tables of ordered pairs
REVISION/ASSESSMENT: Total time
for revision/
At this stage learners should have been assessed on: assessment
• calculating and solving problems with whole numbers and integers for the term
• representing, and calculating with, numbers in exponential form
• numeric and geometric patterns 6 hours
• functions and relationships
• algebraic expressions
• algebraic equations
91
MATHEMATICS GRADES 7-9
GRADE 8 – TERM 2
DURATION
92 CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
Patterns, 2.3 • Revise the following done in Term 1 Time for
functions and algebraic
algebra Algebraic Algebraic language Algebraic expressions were also done in Term 1. In this term the focus is expressions in
expressions on expanding and simplifying algebraic expressions. this term:
• Revise the following done in Grade 7:
Learners have opportunities to write and interpret algebraic expressions when
-- recognize and interpret rules
they write general rules to describe relationships between numbers in number
or relationships represented in 9 hours
patterns, and when they find input or output values for given rules in flow
symbolic form
diagrams, tables, formulae and equations.
-- identify variables and constants in
Examples of interpreting algebraic expressions
given formulae and equations
n
a) What does the rule 2 mean for the following number sequence:
• Recognize and identify conventions
2; 4; 8; 16; 32....
for writing algebraic expressions
MATHEMATICS GRADES 7-9
n
Here learners should recognize that 2 represents the general term in this
• Identify and classify like and unlike sequence, where n represents the position of the term in the sequence.
terms in algebraic expressions Thus, it is the rule that can be used to find any term in the given sequence.
• Recognize and identify coefficients b) The relationship between a boy’s age (x yrs old) and his mother’s age is
and exponents in algebraic given as 25 + x. How can this relationship be used to find the mother’s age
expressions when the boy is 11 years old?
Here learners should recognize that to find the mother’s age, they have to
substitute the boy’s given age into the rule 25 + x. They should also recognize
that the given rule means the mother is 25 years older than the boy.
See further examples given for functions and relationships.
Expand and simplify algebraic • Manipulating algebraic expressions
expressions
-- Make sure learners understand that the rule for operating with integers and
Use commutative, associative and rational numbers, including laws of exponents, applies equally when numbers
distributive laws for rational numbers are replaced with variables. The variables are numbers of a given type (e.g.
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
and laws of exponents to: whole numbers, integers or rational numbers) in generalized form.
• Add and subtract like terms in -- When multiplying or dividing expressions, make sure learners understand how
algebraic expressions the distributive rule works.
• Multiply integers and monomials by: -- The associative rule allows for grouping of like terms when adding.
-- monomials
-- binomials
-- trinomials
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS Patterns, 2.3 • Divide the following by integers or Look out for the following common misconceptions:
functions and monomials:
algebra Algebraic • x + x = 2x and NOT x2. Note the convention is to write 2x rather than x2
expressions -- monomials
• x2 + x2 = 2x2 and NOT 2x4
-- binomials
• a + b = a + b and NOT ab
-- trinomials
• (–2x2)3 = –8x6 and NOT – 6x5
• Simplify algebraic expressions
• –x(3x + 1) = –3x2 – x and NOT – 3x2 + 1
involving the above operations
2
6x + 1 1
2 2
• Determine the squares, cubes, • x = 6 + x and NOT 6 + 1
square roots and cube roots of single • If x = 2 then –3x2 = –3(2)2 = –3 x 4 = –12 and NOT (–6)2
algebraic terms or like algebraic
terms • If x = –2 then –x2 – x = –(–2)2 – 2 = –4 + 2 = –2 and NOT 4 + 2 = 6
• Determine the numerical value of
algebraic expressions by substitution Examples
• √25x2 – 9x2 = √16x2 = 4x and NOT 5x – 3x = 2x
a) Simplify: 2(5 + x – x2) – x(3x + 1) [multiply integer or monomial by polynomial]
b) If x = –2 determine the numerical value of 3x2 – 4x + 5 [using substitution]
3 2
2
c) Simplify: 6x +2x2x + 4 , x ≠ 0 [divide trinomial by monomial; reminder that
denominator cannot be 0]
3 2
(–x )(2x)
2
d) Simplify: 8x – –x , x ≠ 0 [calculations involving multiple operations; remind
learners that denominator cannot be 0]
It might help to remind learners that these variables (or x in this case) represent
e) Determine: √36x4 [square root of monomial]
numbers of a particular type – these may be rational, or integers, or perhaps
whole numbers; such a reminder also then implies that all the associated rules or
properties of these numbers apply here. In the above example, if x is an integer,
then x = a or x = –a because a4 = (– a)4
93
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
94 2.4 Equations What is different to Grade 7? Time for
algebraic
Algebraic • Revise the following done in Grade 7: • Solving equations using additive and multiplicative inverses as well as laws of equations in
equations exponents this term:
-- Set up equations to describe
problem situations Algebraic equations were also done in Term 1. In this term the focus is on
solving equations using additive and multiplicative inverses as well as the
-- Analyse and interpret equations 3 hours
laws of exponents. Algebraic equations are done again in Term 4.
that describe a given situation
Learners have opportunities to write and solve equations when they write general
-- Solve equations by inspection rules to describe relationships between numbers in number patterns, and when
-- Determine the numerical value of they find input or output values for given rules in flow diagrams, tables and
an expression by substitution. formulae.
-- Identify variables and constants in Examples of equations
given formulae and equations a) Solve x if x + 6 = –9
MATHEMATICS GRADES 7-9
• Extend solving equations to include: To solve the equation: add –6 to both sides of the equation
-- Using additive and multiplicative x + 6 – 6 = – 9 – 6, therefore x = –15
inverses
b) Solve x if –2x = 8
-- Using laws of exponents
To solve the equation; divide both sides of the equation by –2:
-- Use substitution in equations to
–2x 8
generate tables of ordered pairs –2 = –2 x = –4
c) Solve x if –x = –5
To solve the equation; divide both sides of the equation by –1
x
–1 = –5
–1 , therefore x = 5
d) Solve x if 3x + 1 = 7
To solve the equation requires two steps:
Add –1 to both sides of the equation:
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
3x + 1 – 1 = 7 – 1, therefore 3x = 6
Then divide both sides of the equation by 3
3x
3 = 63 , therefore x = 2
e) Provide an equation to find the area of a rectangle with length 2x cm and width
2x + 1 cm.
f) If the area of a rectangle is (4x2 – 6x) cm2, and its width is 2x cm, what will be
its length in terms of x?
g) If y = x3 + 1, calculate y when x = 4
h) Thandi is 6 years older than Sophie. In 3 years time Thandi will be twice as old
as Sophie. How old is Thandi now?
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS Space and shape 3.5 Constructions What is different to Grade 7? Total Time for
(geometry) constructions
Construction • Accurately construct geometric • All the constructions are new. In Grade 7 learners only constructed angles, of geometric
of figures appropriately using a perpendicular and parallel lines figures:
geometric compass, ruler and protractor,
• Using constructions to explore properties of triangles and quadrilaterals
figures including:
Constructions 8 hours
-- Bisecting lines and angles
• Constructions provide a useful context to explore or consolidate knowledge of
-- Perpendicular lines at a given point
angles and shapes
or from a given point
• Make sure learners are competent and comfortable with the use of a compass
-- Triangles
and know how to measure and read angle sizes on a protractor
-- Quadrilaterals
• Revise the constructions of angles if necessary before proceeding with the new
• Construct angles of 30°, 45° and 60° constructions
and their multiples without using a
• Start with the constructions of lines, so that learners can first explore angle
protractor
relationships on straight lines.
Investigating properties of geometric
• When constructing triangles learners should draw on known properties and
figures
construction of circles.
• By construction, investigate the
• Construction of special angles without protractors are done by:
angles in a triangle, focusing on:
-- bisecting a right angle to get 45°
-- the sum of the interior angles of
triangles -- drawing an equilateral triangle to get 60°
-- the size of angles in an equilateral -- bisecting the angles of an equilateral triangle to get 30°
triangle
-- the sides and base angles of an
isoceles triangle
• By construction, investigate sides
and angles in quadrilaterals, focusing
on:
-- the sum of the interior angles of
quadrilaterals
-- the sides and opposite angles of
parallelograms
95
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
96 Space and shape 3.1 Classifying 2D shapes What is different to Grade 7? Total time for
(geometry) geometry of
Geometry of • Identify and write clear definitions of • New properties in terms of angles of triangles 2D shapes
2D shapes triangles in terms of their sides and
• Write clear definitions of the properties of triangles and quadrilaterals
angles, distingushing between:
• Use definitions to solve geometric problems 8 hours.
-- equilateral triangles
• Investigate conditions for 2D shapes to be congruent or similar
-- isosceles triangles
Triangles
-- right-angled triangles
• Constructions serve as a useful context for exploring properties of triangles.
• Identify and write clear definitons of
See notes on Constructions above.
quadrilaterals in terms of their sides
and angles, distinguishing between: • Properties of triangles learners should know:
-- parallelogram -- the sum of the interior angles of triangles = 180°
MATHEMATICS GRADES 7-9
-- rectangle -- an equilateral triangle has all sides equal and all interior angles = 60°
-- square -- an isosceles triangle has at least two equal sides and its base angles are
equal
-- rhombus
-- a right-angled triangle has one angle that is a right-angle
-- trapezium
-- the side opposite the right-angle in a right-angled triangle, is called the
-- kite
hypotenuse
-- in a right-angled triangle, the square of the hypotenuse is equal to the sum of
the squares of the other two sides (Theorem of Pythagoras).
Quadrilaterals
• Constructions serve as a useful context for exploring properties of triangles.
See notes on Constructions above.
• The classification of quadrilaterals should include the recognition that:
-- rectangles and rhombi are special kinds of parallelograms
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
-- a square is a special kind of rectangle and rhombus.
• Properties of quadrilaterals learners should know:
-- the sum of the interior angles of quadrilaterals = 360°
-- the opposite sides of parallelograms are parallel and equal
-- the opposite angles of parallelograms are equal
-- the opposite angles of a rhombus are equal
-- the opposite sides of a rhombus are parallel and equal
-- the size of each angle of rectangles and squares is 90°
-- a trapezium has one pair of opposite sides parallel
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS Space and shape 3.1 • a kite has two pairs of adjacent sides equal
(geometry)
Geometry of Similarity and congruency
2D shapes
• This can be explored with any 2-D shape
• Learners should recognise that two or more figures are congruent if they are
Similar and congruent 2-D shapes equal in all respects i.e corresponding angles and sides are equal
• Identify and describe the properties • Learners should recognise that two or more figures are similar if they have.
of congruent shapes corresponding angles equal and their sides are proportionally longer or shorter.
See examples below.
• Identify and describe the properties
of similar shapes • Note that in Grade 9 learners will focus on the special cases of similarity and
congruence in triangles.
• Simiilar figures are further explored when doing enlargements and reductions.
Refer to “Clarification Notes” under 3.3 Transformation Geometry.
Examples:
• Comparing squares to other rectangles, learners can ascertain that having
corresponding angles equal does not necessarily imply that the sides will be
of proportional length. Hence having equal angles alone, is not a sufficient
condition for figures to be similar.
• Comparing rhombii with sides proportional, learners can ascertain that having
sides proportional does not necessarily imply that the corresponding angles will
Solving problems be equal. So only having sides of proportional length is not a sufficient condition
for similarity
• Solve geometric problems involving
unknown sides and angles in Solving problems
triangles and quadrilaterals, using • Learners can solve geometric problems to find unknown sides and angles
known properties and definitions. in triangles and quadrilaterals, using known definitions as well as angle
relationships on straight lines (see notes on Geometry of Straight lines.)
• For right-angled triangles, learners can also use the Theorem of Pythagoras to
find unknown lengths.
• Learners should give reasons and justify their solutions for every written
statement.
• Note that solving geometric problems is an opportunity to practise solving
equations.
97
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
98 Space and shape 3.1 Example:
(geometry) Geometry of
2D shapes
Â1, Â2, Â3 and are three angles on a straight line. Â2 = 75°, Â3 = 55°.
What is the size of Â1?
Learners can find Â1 by solving the following equation:
Â1 + 75° + 55° = 180° (because the sum of angles on a straight line = 180°)
MATHEMATICS GRADES 7-9
Â1 = 180° – 130°(add –55° and –75° to both sides of the equation)
Â1 = 50°
3.3 What is different to Grade 7? Total time for
Geometry of • In Grade 7 learners only defined line segment, ray, straight line, parallel lines geometry of
straight lines Angle relationships and perpendicular lines straight lines:
• Recognise and describe pairs of Angle relationships learners should know:
9 hours
angles formed by: • the sum of the angles on a straight line is 180°
-- perpendicular lines • If lines are perpendicular, then adjacent supplementary angles are each equal to
-- intersecting lines 90°.
-- parallel lines cut by a transversal • If lines intersect, then vertically opposite angles are equal.
• if parallel lines are cut by a transversal, then corresponding angles are equal
Solving problems • if parallel lines cut by a transversal, then alternate angles are equal
• Solve geometric problems using the • if parallel lines cut by a transversal, then co-interior angles are supplementary
relationships between pairs of angles The above angles have to be identified and named by learner
described above Solving problems
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
• Learners can solve geometric problems to find unknown angles using the
angle relationships above, as well as other known properties of triangles and
quadrilaterals.
• Learners should give reasons and justify their solutions for every written
statement.
• Note that solving geometric problems is an opportunity to practise solving
equations. Example:
Â, B and Ĉ are three angles on a straight line. Â = 55°, B = 75°. What is the size
of Ĉ ?
Learners can find Ĉ by solving the following equation:
55° + 75° + Ĉ = 180° (because the sum of angles on a straight line = 180°)
Ĉ = 180° – 130° (add –55° and –75° to both sides of the equation)
Ĉ = 50°
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS REVISION/ASSESSMENT: Total time
for revision/
At this stage learners should have been assessed on: assessment
• algebraic expressions for the term
• algebraic equations
• constructing geometric objects 8 hours
• geometry of 2D shapes
• geometry of straight lines
99
MATHEMATICS GRADES 7-9
GRADE 8 – TERM 3
DURATION
100 CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
Numbers, 1.4 What is different to Grade 7?
operations and
relationships Common • Divide by common fractions
fractions
• Squares, cubes, square roots and cube roots of common fractions Total time
for common
In Grade 8 learners consolidate number knowledge and calculation techniques for fractions:
common fractions, developed in Grade 7.
Calculations using fractions Calculations using fractions
7 hours
• Revise: • Learners should continue to do context free calculations and solve problems in
contexts.
-- addition and subtraction of
common fractions, including mixed -- By Grade 8 learners should be comfortable converting mixed numbers to
MATHEMATICS GRADES 7-9
numbers common fractions for calculations.
-- finding fractions of whole numbers Example
-- multiplication of common fractions, 5 12 = 11
2; 6 13 = 193
including mixed numbers
-- To simplify fractions, learners use knowledge of common factors i.e. what can
• Divide whole numbers and common divide equally into the numerator and denominator of a fraction. Emphasize that
fractions by common fractions when simplifying, the fractions must remain equivalent.
• Calculate the squares, cubes, square Addition and subtraction
roots and cube roots of common
• LCMs have to be found when adding and subtracting fractions with different
fractions
denominators. Here learners use knowledge of common multiples to find the
Calculation techniques LCM i.e. what number can both denominators be divided into.
• Revise: Multiplication
-- Convert mixed numbers to • For multiplication of fractions, learners should be encouraged to simplify
common fractions in order to fractions by dividing numerators and denominators by common factors.
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
perform calculations with them
• Learners should note the difference between adding or subtracting fractions,
-- Use knowledge of multiples and and multiplying fractions
factors to write fractions in the
Examples
simplest form before or after
calculations 3 2 15 8 23 3
4 + 5 = 20 + 20 = 20 = 120 (using LCM and equivalent fractions)
-- Use knowledge of equivalent 3 2 3
4 x 5 = 10 (divide 2 and 4 by common factor 2)
fractions to add and subtract
common fractions • Learners should recognize that finding a ‘fraction of a whole number’ or ‘finding
a fraction of a fraction’ means multiplying the fraction and the whole number or
• Use knowledge of reciprocal
the fraction with the fraction.
relationships to divide common
fractions
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS Numbers, 1.4 Solving problems • When learners find fractions of whole numbers, the examples can be chosen to
operations and result either in whole numbers or fractions or both.
relationships Common • Solve problems in contexts involving
fractions common fractions and mixed • Learners should also use the convention of writing the whole number as a
numbers, including grouping, sharing fraction over when multiplying.
and finding fractions of whole
Examples
numbers
4 4 20 80
Find 5 of 20 = 5 x 1 = 5 = 16
2 5 2 5 10 5
Find 3 of 6 = 3 x 6 = 18 = 9
Division
• The technique of ‘invert and multiply’ applies to division in general and not just
to division by fractions. Hence, a useful way of making learners comfortable with
division by fractions is to start with examples of division by whole numbers.
• Learners have to understand that dividing by a number is the same as
multiplying by the reciprocal of the number i.e. the reciprocal of n is 1n
Examples:
1
a) 10 ÷ 5 is the same as 10 x 5 = 2 (multiply by the reciprocal of 5)
1 1
b) 10 ÷ 5 = 10 x 5 = 50 (multiply by the reciprocal of 5 )
This can also be explained by using diagram models for fractions and asking,
1
how many times does 5 fit into 10? We know that 5 fifths fit into 1 whole, so (5
1
x 10) fifths will fit into 10 wholes. Hence, 10 ÷ 5 = 50
1
c) 20 ÷ 4 is the same as 20 x 4 = 5 (multiply by the reciprocal of 4)
1 1
d) 20 ÷ 4 = 20 x 4 = 80 (multiply by the reciprocal of 4 )
This can also be explained by using diagram models for fractions and asking,
1
how many times does 4 fit into 20? We know that 4 quarters fit into 1 whole,
1
so (4 x 20) quarters will fit into 20 wholes. Hence, 20 ÷ 4 = 80
e) Once learners have done a few of the above examples, they can use the
technique of multiplying by the reciprocal to divide fractions by fractions:
3 1 3 2 6 3 1 1
4 ÷ 2 = 4 x 1 = 4 = 2 = 1 2 (multiply by the reciprocal of 2 )
Squares, cubes, square roots and cube roots
• Knowing the rules of operations for calculating squares, cubes, square roots and
cube roots of common fractions is important
101
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
102 Numbers, 1.4 Percentages Examples:
2 2
operations and 9
Common • Revise: 2
a) ( 43 ) = 34 = 16
relationships fractions
-- finding percentages of whole 16 4
numbers
√ 16
25 = √ 25 = 5
-- calculating the percentage of part • Once learners are comfortable doing all the operations with fractions,
b) √
of a whole calculations do not have to be restricted to positive fractions.
-- calculating percentage increase or Calculation using percentages
decrease • Learners should continue to do context free calculations and solve problems in
• Calculate amounts if given contexts.
percentage increase or decrease • When doing calculations using percentages, learners have to use the equivalent
• Solve problems in contexts involving common fraction form, which is a fraction with denominator 100.
percentages • Learners should become familiar with the equivalent fraction and decimal forms
MATHEMATICS GRADES 7-9
1
of common percentages eg. 25% is equivalent to 4 or 0,25; 50% is equivalent to
1 3
2 or 0,5; 60% is equivalent to 5 or 0,6.
Equivalent forms
• To calculate percentage of part of a whole, or percentage increase or decrease,
1
• Revise equivalent forms between: learners have to learn the strategy of multiplying by 100. It is useful for learners
to learn to use calculators for some of these calculations where the fractions are
-- common fractions (fractions where
not easily simplified.
one denominator is a multiple of
the other) • When using calculators, learners can use the equivalent decimal fraction form
for percentages to do the calculations.
-- common fraction and decimal
fraction forms of the same number Examples:
a) Calculate of 60% of R105
-- common fraction, decimal fraction
60
and percentage forms of the same Amount = 100 x R105 = R63
number b) What percentage is 40c of R3,20?
40 100 100
Percentage = 320 x 1 = 8 = 12,5%
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
c) Calculate the percentage increase if the price of a bus ticket of R60 is
increased to R84.
24 100
Amount increased = R24. Therefore percentage increase = 60 x 1 = 40%
d) Calculate the percentage decrease if the price of petrol goes down from 20
cents a litre to 18 cents a litre.
2 100
Amount decreased = 2 cents. Therefore percentage decrease = 20 x 1 = 10%
e) Calculate how much a car will cost if its original price of R150 000 is reduced
by 15%
Calculation involves finding 15% of R150 000 and then subtracting that amount
15 R150 000
from the original price. i.e. 100 x 1 = R22 500
Hence new price of car = R150 000 – R22 500 = R127 500
85 R150 000
Or 100 x 1 = R127 500
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
1.5 What is different to Grade 7? Total time
for decimal
CAPS
Decimal • Multiplication by decimal fractions not limited to one decimal place fractions:
fractions
• Division of decimal fractions by decimal fractions
• Squares, cubes, square roots and cube roots of decimal fractions 6 hours
In Grade 8 learners consolidate number knowledge and calculation techniques for
decimal fractions, developed in Grade 7.
Ordering and comparing decimal Ordering, counting and comparing decimal fractions
fractions
• Learners should continue to practise counting, ordering and comparing decimal
• Revise: fractions. Counting should not only be thought of as verbal counting. Learners
can count in decimal intervals using:
-- Ordering, comparing and place
value of decimal fractions to at -- structured, semi-structured or empty number lines
least 3 decimal places
-- chain diagrams for counting
-- Rounding off decimal fractions to at
• Learners should be given a range of exercises
least 2 decimal place
• Arrange given numbers from the smallest to the biggest: or biggest to smallest.
• Fill in missing numbers in
-- a sequence
-- on a number grid
-- on a number line
-- fill in <, = or > e.g. 0,4 * 0,04 * 0,004
• Counting exercises in chain diagrams can be checked using calculators and
learners can explain any differences between their answers and those shown by
the calculator.
Calculations with decimal fractions Calculating using decimal fractions
• Revise: • Learners should continue to do context free calculations and solve problems in
contexts.
-- addition, subtraction and
multiplication of decimal fractions • Learners should estimate their answers before calculating, especially with
to at least 3 decimal places multiplication and division by decimal fractions. They should be able to judge
the reasonableness of answers in respect of how many decimal places and also
-- division of decimal fractions by check their own answers.
whole numbers
• Multiplication by decimal fractions should start with familiar numbers that
• Extend multiplication to 'multiplication learners can calculate by inspection, so that learners get a sense of how
103 by decimal fractions' not limited to decimal places are affected by multiplication.
one decimal place
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
104 1.5 • Extend division to 'division of decimal Examples:
fractions by decimal fractions'
Decimal a) 3 x 2 = 6
fractions • Calculate the squares, cubes, square
0,3 x 2 = 0,6
roots and cube roots of decimal
fractions 0,3 x 0,2 = 0,06
Calculation techniques 0,3 x 0,02 = 0,006
• Use knowledge of place value to 0,03 x 0,02 = 0,0006 etc
estimate the number of decimal
b) 15 x 3 = 45
places in the result before performing
calculations 1,5 x 3 = 4,5
• Use rounding off and a calculator to 0,15 x 3 = 0,45
check results where appropriate
MATHEMATICS GRADES 7-9
0,15 x 0,3 = 0,045
Solving problems
0,015 x 0,3 = 0,0045 etc
• Solve problems in context involving
decimal fractions • For division by decimal fractions without calculators, learners have to use
their knowledge of multiplication by 10 or multiples of 10 to make the divisor a
Equivalent forms whole number. Hence start with familiar numbers that learners can calculate by
inspection, so that learners get a sense of how decimal places are affected by
• Revise equivalent forms between:
division.
-- common fraction and decimal
Examples:
fraction forms of the same number
a) 54 ÷ 6 = 9
-- common fraction, decimal fraction
and percentage forms of the same 54 ÷ 0,6 = 540 ÷ 6 = 90
number
(multiply both numbers by 10 to make the decimal fraction a whole number)
54 ÷ 0,06 = 5 400 ÷ 6 = 900
(multiply both numbers by 100 to make the decimal fraction a whole number)
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
0,54 ÷ 0,06 = 54 ÷ 6 = 9
(multiply both numbers by 100 to make the decimal fraction a whole number)
b) 125 ÷ 5 = 25
125 ÷ 0,5 = 1 250 ÷ 5 = 250
(multiply both numbers by 10 to make the decimal fraction a whole number)
125 ÷ 0,05 = 12 500 ÷ 5 = 2 500
(multiply both numbers by 100 to make the decimal fraction a whole number)
1,25 ÷ 0,05 = 125 ÷ 5 = 25
(multiply both numbers by 100 to make the decimal fraction a whole number
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS 1.5 • For bigger and unfamiliar decimal fractions, learners should use calculators for
multiplication and division, but still judge the reasonableness of their solutions.
Decimal
fractions • Similarly, finding squares, cubes, square roots and cube roots for decimal
fractions should start with familiar numbers that learners can calculate by
inspection.
Examples:
a) 42 = 16
(0,4)2 = 0,4 x 0,4 = 0,16
(0,04)2 = 0,04 x 0,04 = 0,0016
b) (0,1)3 = 0,1 x 0,1 x 0,1 = 0,001
• Once learners are comfortable with all the operations using decimal fractions,
c) √0,04 = 0,2
calculations should not be restricted to positive decimal fractions.
Measurement 4.3 Develop and use the Theorem of • The theorem of Pythagoras is new in Grade 8. Total time for
Pythagoras the Theorem
The • It is important that learners understand that the Theorem of Pythagoras applies of Pythagoras:
Theorem of • Investigate the relationship between only to right-angled triangles.
Pythagoras the lengths of the sides of a right-
angled triangle to develop the • The Theorem of Pythagoras is basically a formula to calculate unknown length
of sides in right-angled triangles. 5 hours
Theorem of Pythagoras
• Determine whether a triangle is a • In the FET phase, the Theorem of Pythagoras is crucial to the further study of
right-angled triangle or not if the Geometry and Trigonometry
length of the three sides of the Examples of solving problems using the Theorem of Pythagoras:
triangle are known
• Use the Theorem of Pythagoras to using a calculator. Leave the answer in the simplest surd form.
• In ∆ABC, ∠B = 90o, AC = 4 cm, BC = 2 cm. Calculate the length of AB without
calculate a missing length in a right-
angled triangle, leaving irrational
answers in surd form
105
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
106 4.1 What is different to Grade 7? Total time
for area and
Area and • Areas of polygons by decomposition perimeter:
perimeter of
• Circumference and area of a circle
2D shapes
Area and perimeter • Formulae learners should know and use: 5 hours.
• Use appropriate formulae to calculate -- perimeter of a square = 4s
perimeter and area of:
-- perimeter of a rectangle = 2(l + b) or 2l + 2b
-- squares
-- area of a square = l2
-- rectangles
-- area of a rectangle = l x b
-- triangles 1
-- area of a triangle = 2(b x h)
-- circles
MATHEMATICS GRADES 7-9
-- diameter of a circle: d = 2r
• Calculate the areas of polygons,
to at least 2 decimal places, by -- circumference of circle: c = πd or 2πr
decomposing them into rectangles -- area of a circle: A = πr2
and/or triangles
Solving equations using formulae
• Use and describe the relationship
between the radius, diameter • The use of formulae provides a context to practise solving equations by
and circumference of a circle in inspection or using additive or multiplicative inverses.
calculations Examples:
• Use and describe the relationship 1. If the perimeter of a square is 32 cm, what is the length of each side?
between the radius and area of a Learners should write this as:
circle in calculations 32
4s = 32 and solve by asking: 4 times what will be 32 OR saying s = 4 ?
2. If the area of a rectangle is 200 cm2, and its length is 50 cm, what is its width?
Calculations and solving problems Learners should write this as:
• Solve problems, with or without a
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
50 x b = 200 and solve by inspection by asking: 50 times what will be 200 OR
calculator, involving perimeter and 200
saying b = 50 ?
area of polygons and circles
For areas of triangles:
• Calculate to at least 2 decimal places
• Make sure learners know that the height of a triangle is a line segment drawn
• Use and describe the meaning from any vertex perpendicular to the opposite side.
of the irrational number Pi (π) in
calculations involving circles
• Use and convert between appropriate
SI units, including:
-- mm2 ↔ cm2 ↔ m2 ↔ km2
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS 4.1 Example: AD is the height onto base BC of ∆ABC.
Area and A A
Perimeter of
2D shapes
B D C D B C
• Point out that every triangle has bases, each with a related height or altitude.
• For conversions, note:
-- If 1cm = 10 mm then 1 cm2 = 100 mm2
-- If 1m = 100 cm then 1 m2 = 10 000 cm2
Circles
• Make sure learners can identify the centre, radius, diameter and circumference
of the circle.
• Spend time investigating the relationship between radius, circumference and
diameter, so that learners develop a sense of where the irrational number Pi (π)
is derived from.
• Develop an understanding of π, making sure learners understand that:
-- π represents the value of the circumference divided by the diameter, for any
circle
-- π is an irrational number and is given as 3,141 592 654 correct to 9 decimal
places on the calculator
22
-- 7 or 3,14 are approximate rational values of π in everyday use.
107
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
108 4.2 Surface area and volume What is different to Grade 7? Total time for
surface area
Surface area • Use appropriate formulae to calculate • Surface area and volume of triangular prisms and volume:
and volume the surface area, volume and
• Formulae learners should know and use:
of 3D objects capacity of:
-- the volume of a prism = the area of the base x the height 5 hours
-- cubes
-- the surface area of a prism = the sum of the area of all its faces
-- rectangular prisms
-- the volume of a cube = l3
-- triangular prisms
-- the volume of a rectangular prism = l x b x h
• Describe the interrelationship
between surface area and volume of -- the volume of a triangular prism = (12b x h) x height of prism
the objects mentioned above
• For conversions, note:
Calculations and solving problems
MATHEMATICS GRADES 7-9
-- if 1 cm = 10 mm then 1 cm3 = 1 000 mm3 and
• Solve problems, with or without a
calculator, involving surface area, -- if 1 m = 100 cm then 1 m3 = 1 000 000 cm3 or 106 cm3
volume and capacity -- an object with a volume of 1 cm3 will displace exactly 1 ml of water
• Use and convert between appropriate -- an object with a volume of 1 m3 will displace exactly 1 kl of water.
SI units, including:
• Emphasize that the amount of space inside a prism is called its capacity; and
-- mm2 ↔ cm2 ↔ m2 ↔ km2 the amount of space occupied by a prism is called its volume.
-- mm3 ↔ cm3 ↔ m3 • Investigate the nets of cubes and rectangular prisms in order to deduce
-- ml (cm3) ↔ l ↔ kl formulae for calculating their surface areas.
Example of solving problems involving surface area and volume:
• Calculate the volume and surface area of the prism if AB = 8 cm, BC = 6 cm and
CF = 16 cm.
A C
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
B
D F
E
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
Data handling 5.1 What is different to Grade 7? Total time for
collecting and
CAPS
Collect, The following are new in Grade 8
organize organizing
• extremes data:
and
summarize • broken line graphs
data • dispersion of data 4 hours.
• error and bias in data
Collect data Data sets and contexts
• Pose questions relating to social, Learners should be exposed to a variety of contexts that deal with social and
economic, and environmental issues environmental issues, and should work with given data sets, represented in
• Select appropriate sources for a variety of ways, that include big number ranges, percentages and decimal
the collection of data (including fractions. Learners should then practise organizing and summarizing this data,
peers, family, newspapers, books, analysing and interpreting the data, and writing a report about the data.
magazines) Complete a data cycle
• Distinguish between samples and Learners should complete at least one data cycle for the year, starting with posing
populations, and suggest appropriate their own questions, selecting the sources and method for collecting, recording,
samples for investigation organizing, representing, analysing, summarizing, interpreting and reporting the
• Design and use simple data. Challenge learners to think about what kinds of questions and data need to
questionnaires to answer questions be collected to be represented on a histogram, a pie chart, a bar graph, or a line
with multiple choice responses graph.
Organize and summarize data
• Organize (including grouping where
appropriate) and record data using
-- tally marks
-- tables
-- stem-and-leaf displays
• Group data into intervals
• Summarize data using measures of
central tendency, including:
-- mean
-- median
-- mode
• Summarize data using measures of
dispersion, including:
109 -- range
-- extremes
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
110 5.5 Represent data Representing data Total time for
representing
Represent • Draw a variety of graphs by hand/ • Drawing pie charts to represent data do not have to be accurately drawn with a data:
data technology to display and interpret compass and protractor, etc. Learners can use any round object to draw a circle,
data including: then divide the circle into halves and quarters and eighths if needed, as a guide
to estimate the proportions of the circle that need to be shown to represent the 3 hours
-- bar graphs and double bar graphs
data. What is important is that the values or percentages associated with the
-- histograms with given and own data are shown proportionally on the pie chart.
intervals
• Drawing, reading and interpreting pie charts is a useful context to re-visit
-- pie charts equivalence between fractions and percentages, e.g. 25% of the data is
1
represented by a 4 sector of the circle.
-- broken-line graphs
• It is a context in which learners can find percentages of whole numbers e.g. if
25% of 300 learners like rugby, how many (actual number) learners like rugby?
MATHEMATICS GRADES 7-9
• Histograms are used to represent grouped data shown in intervals on the
horizontal axis of the graph. Point out the differences between histograms and
bar graphs, in particular bar graphs that represent discrete data e.g. favourite
sports, compared to histograms that show categories in consecutive, non-
overlapping intervals, e.g. test scores out of 100 shown in intervals of 10. The
bars on bar graphs do not have to touch each other, while in a histogram they
have to touch since they show consecutive intervals.
• Broken-line graphs refer to data graphs that represent data points joined by
a line and are not the same as straight line graphs that are drawn using the
equation of the line.
• Broken-line graphs are used to represent data that changes continuously over
time, e.g. average daily temperature for a month. Each day’s temperature is
represented with a point on the graph, and once the whole month has been
plotted, the points are joined to show a broken-line graph.
• Broken-line graphs are useful to read ‘trends’ and patterns in the data, for
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
predictive purposes e.g. will the temperatures go up or down in the next month.
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS 5.6 Interpret data Developing critical analysis skills Total time for
analysing,
Interpret, • Critically read and interpret data • Learners should compare the same data represented in different ways e.g. in a interpreting
analyse and represented in: pie chart or a bar graph or a table, and discuss what information is shown and and
report data what is hidden; they should evaluate what form of representation works best for summarising
-- words
the given data. data:
-- bar graphs
• Learners should compare graphs on the same topic but where data has been
-- double bar graphs collected from different groups of people, at different times, in different places 3,5 hours
or in different ways. Here learners should discuss differences between the data
-- pie charts
with an awareness of bias related to the impact of data sources and methods of
-- histograms data collection on the interpretation of the data.
-- broken-line graphs • Learners should compare different ways of summarizing the same data sets,
developing an awareness of how data reporting can be manipulated; they
Analyse data should evaluate which summary statistics best represent the data.
• Critically analyse data by answering • Learners should compare graphs of the same data, where the scales of
questions related to: the graphs are different. Here learners should discuss differences with an
-- data categories, including data awareness of how representation of data can be manipulated; they should
intervals evaluate which form of representation works best for the given data.
-- data sources and contexts • Learners should compare data on the same topic, where one set of data has
extremes, and discuss differences with an awareness of the effect of the
-- central tendencies – (mean, mode, extremes on the interpretation of the data, in particular, extremes affect the
median) range.
-- scales used on graphs • Learners should write reports on the data in short paragraphs.
-- samples and populations
-- dispersion of data
-- error and bias in the data
Report data
• Summarize data in short paragraphs
that include
-- drawing conclusions about the data
-- making predictions based on the
data
-- identifying sources of error and
bias in the data
111 -- choosing appropriate summary
statistics for the data (mean,
median, mode, range)
MATHEMATICS GRADES 7-9
-- the role of extremes in the data
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
112 REVISION/ASSESSMENT: Total time
for revision/
At this stage learners should have been assessed on: assessment
• calculating and solving problems with common fractions and decimal fractions for the term
• the Theorem of Pythagoras
• area and perimeter of 2D shapes 6,5 hours.
• surface area and volume of 3D objects
MATHEMATICS GRADES 7-9
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
GRADE 8 – TERM 4
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
CAPS (in hours)
Patterns, 2.2 Input and output values Functions and relationships were also done in Term 1. In this term the focus Time for
functions and Functions and • Determine input values, output is on using formulae to find output values from given input values, as well as Functions and
algebra equivalent forms of descriptions of the same relationship. Relationships
relationships values or rules for patterns and
in this term:
relationships using: See further notes and examples in Term 1.
-- flow diagrams Example
6 hours
-- tables Use the formula for the area of a rectangle: A = l x b to calculate the following:
-- formulae a) The area, if the length is 4,5 cm and the width is 2,5 cm
-- equations b) The length, if the area is 240 cm2 and the width is 4 cm
Equivalent forms c) The width, if the area is 14 cm2 and the length is 3,5 cm
• Determine, interpret and justify Learners can write these as number sentences, and solve by inspection.
equivalence of different descriptions
of the same relationship or rule
presented:
-- verbally
-- in flow diagrams
-- in tables
-- by formulae
-- by equations
2.4 Equations Algebraic equations were also done in Terms 1 and 2. In this term the focus is Time for
Algebraic • Revise the following done in Grade 7: on using substitution in equations to generate tables of ordered pairs. Algebraic
equations in
equations -- set up equations to describe See further notes and examples in Terms 1 and 2.
this term:
problem situations Examples of generating ordered pairs
-- analyse and interpret equations a) Complete the table below for x and y values for the equation: y = –3x + 2 3 hours
that describe a given situation
x –3 –1 0
-- solve equations by inspection
-- determine the numerical value of y –4 –10
an expression by substitution 2
b) Complete the table below for x and y values for the equation: y = x – 2
-- Identify variables and constants in
given formulae or equations x –3 –2 0
• Extend solving equations to include: y –2 2
-- using additive and multiplicative
inverses
113 -- using laws of exponents
• Use substitution in equations to
MATHEMATICS GRADES 7-9
generate tables of ordered pairs
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
114 2.5 What is different to Grade 7? Total time for
graphs:
Graphs • New features of global graphs: maximum and minimum; discrete and
continuous
• Plotting points to draw graphs 9 hours
Interpreting graphs
• Revise the following done in Grade 7: Examples of contexts for global graphs include:
-- Analyse and interpret global graphs • the relationship between time and distance travelled
of problem situations, with a special • the relationship between temperature and time over which it is measured
focus on the following trends and
features: • the relationship between rainfall and time over which it is measured, etc.
♦♦ linear or non-linear Examples of drawing graphs by plotting points
a) Complete the table of ordered pairs below for the equation: y = x + 3
MATHEMATICS GRADES 7-9
♦♦ constant, increasing or
decreasing
x –4 –3 –2 –1 0 1 2 3 4
• Extend the focus on features of
graphs to include: y
-- maximum or minimum Now, plot the above co-ordinate points on the Cartesian plane. Join points to
-- discrete or continuous form a graph.
Drawing graphs b) Complete the table of ordered pairs below for the equation: y = x2 + 3
• Draw global graphs from given x –4 –3 –2 –1 0 1 2 3 4
descriptions of a problem situation,
identifying features listed above y
• Use tables of ordered pairs to plot Now, plot the above co-ordinate points on the Cartesian plane. Join points to
points and draw graphs on the form a graph.
Cartesian plane
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
Space and Shape 3.4 What is different to Grade 7? Total time for
(Geometry) Transforma-
CAPS
Transformation • Transformations are done on a co-ordinate plane tions:
Geometry
• Co-ordinates of points and vertices
Co-ordinate plane 6 hours
Transformations • Doing transformations on the co-ordinate plane is an opportunity to practise
plotting points with ordered pairs, and links up with drawing algebraic graphs.
• Recognize, describe and perform
transformations with points on a • Learners have to learn how to plot points on the co-ordinate plane and read
coordinate plane, focusing on: the co-ordinates of points off the x-axis and y-axis. This is also done with
algebraic graphs.
-- reflecting a point in the y-axis or
x-axis • Learners have to know the convention for writing ordered pairs (x;y)
-- translating a point within and • Point out the differences between the axes in the four quadrants.
across quadrants
Focus of transformations
• Recognize, describe and perform
• Doing transformations on a co-ordinate plane focuses attention on the co-
transformations with triangles on a
ordinates of points and vertices of shapes.
co-ordinate plane, focusing on the
co-ordinates of the vertices when: • Learners should recognize that translations, reflections and rotations only
change the position of the figure, and not its shape or size.
-- reflecting a triangle in the x-axis or
y-axis • Learners should recognize that the above transformations produce congruent
figures.
-- translating a triangle within and
across quadrants • Learners do not have to learn general rules for the transformations at this
stage, but should explore the way the co-ordinates of points change when
-- rotating a triangle around the origin
performing different transformations with lines or shapes.
Enlargements and reductions
• Learners should recognize that enlargements and reductions change the size
• Use proportion to describe the effect of figures by increasing or decreasing the length of sides, but keeping the
of enlargement or reduction on area angles the same, produces similar rather than congruent figures.
and perimeter of geometric figures
• Learners should also be able to work out the factor of enlargement or
reduction of a figure.
Examples of transformation problems
• Plot point A(4;3) and A', its image, after reflection in:
a) the x-axis
b) the y-axis.
• Write down the co-ordinates of T' if T(–2;3) is translated 4 units downwards.
• The perimeter of square abcd = 48cm
115 a) Write down the perimeter of the square if the length of each side is
doubled.
b) Will the area of the enlarged square be twice or four times that of the
MATHEMATICS GRADES 7-9
original square?
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
116 3.2 What is different to Grade 7? Total time for
geometry of
Geometry of 3D • Naming and comparing Platonic solids 3D objects:
objects
• Nets of pyramids
Platonic solids 7 hours
• Platonic solids are a special group of polyhedra that have faces that are
congruent regular polygons.
Classifying 3D objects
• There are only 5 Platonic solids:
• Describe, name and compare the 5
Platonic solids in terms of the shape -- Tetrahedron
and number of faces, the number of
-- Hexahedron (cube)
vertices and the number of edges
-- Octahedron
MATHEMATICS GRADES 7-9
-- Dodecahedron
Building 3D models
-- Icosahedrons
• Revise using nets to make models of
geometric solids, including: • The name of each Platonic solid is derived from its number of faces.
-- cubes • Platonic solids provide an interesting context in which to investigate the
relationship between the number of faces, vertices and edges. By listing these
-- prisms
properties for all the Platonic solids, learners can investigate the pattern that
-- pyramids emerges, to come up with the general rule: V – e + f = 2, where V = number
of vertices; e = number of edges; f = number of faces
Using and constructing nets
• Using and constructing nets are useful contexts for exploring or consolidating
properties of polyhedra.
• Learners should recognize the nets of different solids.
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
• Learners should make sketches of the nets using their knowledge of the
shape and number of faces of the solids, before drawing and cutting out the
nets to build models.
• Since learners have more knowledge about the size of angles in equilateral
triangles, and can measure angles, their constructions of nets should be more
accurate.
• Learners have to work out the relative position of faces of the nets in order to
build the 3D model.
DURATION
CONTENT AREA TOPICS CONCEPTS AND SKILLS SOME CLARIFICATION NOTES OR TEACHING GUIDELINES
(in hours)
CAPS Data handling 5.4 Probability Probability Probability experiments Total time for
probability:
• Consider a simple situation (with In the Intermediate Phase and Grade 7 learners did probability experiments with
equally likely outcomes) that can be coins, dice and spinners. In Grade 8 doing actual trials of experiments become
described using probability and: less important, and learners should consider probability for hypothetical events
e.g. the probability of white as a successful outcome on a roulette table, or the 4,5 hours
-- list all the possible outcomes probability of getting a Coca Cola at the shop if you know what the total number
of drinks is that they stock and how many cans of Coca Cola they have.
-- determine the probability of each
possible outcome using the Comparing relative frequency and probability
definition of probability • The relative frequency is the observed number of successful outcomes for a
-- predict, with reasons, the relative finite sample of trials.
frequency of the possible outcomes • For example, if you toss a coin 50 times, the results are 27 heads and 23 tails.
for a series of trials based on Define a head as a successful outcome. The relative frequency of heads is:
probability 27
50 = 54%
-- compare relative frequency with
• The probability of a head is 50% (one of two likely outcomes). The difference
probability and explain possible
between the relative frequency of 54% and the probability of 50% is due to
differences
small sample size.
• The more trials you do, the closer the relative frequency gets to the probability.
This can be compared in class by combining results from trials done in groups
or pairs.
REVISION/ASSESSMENT: Total time
for revision/
At this stage learners should have been assessed on: assessment
• functions and relationships for the term
• algebraic equations
• graphs 9, 5 hours
• transformation geometry
• geometry of 3D objects
• probability
117
MATHEMATICS GRADES 7-9
MATHEMATICS GRADES 7-9
Time allocation per Term: Grade 9
TERM 1 TERM 2 TERM 3 TERM 4
Topic Time Topic Time Topic Time Topic Time
4,5 Construction of 9 Functions and 5 Transformation 9
Whole numbers
hours geometric figures hours relationships hours geometry hours
4,5 Geometry of 2D 9 Algebraic 9 Geometry of 3D 9
Integers
hours shapes hours expressions hours objects hours
Collect, organize
4,5 Geometry of 9 Algebraic 9 4
Common fractions and summarize
hours straight-lines hours equations hours hours
data
4,5 Theorem of 5 12 3
Decimal fractions Graphs Represent data
hours Pythagoras hours hours hours
Area and Surface Area and
5 5 5 Interpret, analyse 3,5
Exponents Perimeter of 2D volume of 3D
hours hours hours and report data hours
shapes objects
Numeric and
4,5 4,5
geometric Probability
hours hours
patterns
Functions and 4
relationships hours
Algebraic 4,5
expressions hours
Algebraic 4
equations hours
Revision/ 5 Revision/ 8 Revision/ 5 Revision/ 12
Assessment hours Assessment hours Assessment hours Assessment hours
TOTAL: 45 hours TOTAL: 45 hours TOTAL: 45 hours TOTAL: 45 hours
118 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
3.3.3 Clarification of content for Grade 9
CAPS GRADE 9 – TERM 1
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
Numbers, 1.1 Properties of numbers What is different to Grade 8? Total time
operations and for whole
relationships Whole • Describe the real number system In Grade 9 learners consolidate number knowledge and calculation techniques for numbers
numbers by recognising, defining and whole numbers, developed in Grade 8.
distinguishing properties of:
• The focus in Grade 9 should be on developing an understanding of different
-- natural numbers number systems and the properties of operations that apply for different number 4, 5 hours.
systems.
-- whole numbers
• The contexts for solving problems should be more complex and varied, involving
-- integers
whole numbers, integers and rational numbers. Financial contexts are especially
-- rational numbers rich in this regard.
-- irrational numbers • Learners should be given a clear indication of when the use of calculators is
permissible or not. Calculators should be used routinely for calculations with big
Calculations using whole numbers
numbers and where knowledge of number facts or concepts are not explicitly
Revise: assessed. However, guard against learners becoming dependent on calculators
for all calculations. Calculators remain a useful tool for checking solutions.
• Calculations using all four operations
on whole numbers, estimating and • Competency in finding multiples and factors, and prime factorisation of whole
using calculators where appropriate numbers, remains important for developing competency in factorising algebraic
expressions and solving algebraic equations.
Calculation techniques
Properties of numbers
Use a range of strategies to perform
and check written and mental • By distinguishing the properties of different number systems, learners should
calculations of whole numbers recognize that natural numbers is a subset of whole numbers, which in turn is
including: a subset of integers, which in turn is a subset of rational numbers. All of these
numbers form part of the real number system.
• estimation
• Note that 0 may sometimes be included in the set of natural numbers.
• adding, subtracting and multiplying in
columns • Learners should recognize the following distinguishing features of the number
systems:
• long division
-- integers extend the natural and whole number systems by including the
• rounding off and compensating operation a – b, where a < b.
• using a calculator a
-- rational numbers extend the set of integers by including the operation b
Multiples and factors where a < b
a
Use prime factorisation of numbers to -- rational numbers are defined as numbers that can be written in the form b
find LCM and HCF where a and b are integers and b ≠ 0
119
-- since integers are a subset of rational numbers, every integer, can be
a
expressed as a rational number b
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
120 Numbers, 1.1 -- irrational numbers are numbers that cannot be expressed as rational
a
operations and numbers in the form b
relationships Whole
22
numbers -- Pi (π) is an irrational number, even though we use 7 or 3,14 as rational
number approximations for π in calculations
Ratio and rate problems
• Include problems involving speed, distance and time. Learners should be
familiar with the following formulae for these calculations:
distance
a) speed = time
b) distance = speed x time
distance
c) time = speed
MATHEMATICS GRADES 7-9
• Speed is usually given as constant speed or average speed.
• Make sure learners recognize and are able to convert correctly between units
for time and distance.
Solving problems Examples
• Solve problems in contexts involving a) A car travelling at a constant speed travels 60 km in 18 minutes. How far,
travelling at the same constant speed, will the car travel in 1 hour 12 minutes?
-- ratio and rate
b) A car travelling at an average speed of 100 km/h covers a certain distance in
-- direct and indirect proportion 3 hours 20 minutes. At what constant speed must the car travel to cover the
same distance in 2 hours 40 minutes?
Direct and Indirect proportion
Learners should be familiar with the following relationships:
x
• x is directly proportional to y if y = constant
• x and y are directly propotional if, as the value of x increases the value of y
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
increases in the same proportion, and as the value of x decreases the value of y
decreases in the same proportion
• The direct proportional relationship is represented by a straight line graph
• x is indirectly or inversely proportional to y if x x y = a constant. In other words
y = cx
• x and y are indirectly propotional if, as the value of x increases the value of y
decreases and as the value of x decreases the value of y increases
• an indirect proportional relationship is represented by a non-linear curve
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS Numbers, 1.1 • Solve problems that involve whole Financial contexts
operations and numbers, percentages and decimal
relationships Whole • Once learners have done sufficient calculations for simple and compound
fractions in financial contexts such
numbers interest through repeated calculations, they could use given formulae for these
as:
calculations.
-- profit, loss, discount and VAT
Examples
-- budgets
a) Calculate the simple interest on R600 at 7% p.a for 3 years using the
P.n.r r
-- accounts and loans formula SI = 100 or si = p.n.i for i = 100 .
-- simple interest and hire purchase b) R800 invested at r% per annum simple interest for a period of 3 years
yields R168. Calculate the value of r
-- exchange rates and commission
c) How long will it take for R3 000 invested at 6% per annum simple interest to
-- rentals
grow to R4 260?
-- compound interest
d) Temoso borrowed R500 from the bank for 3 years at 8% p.a. compound
interest. Without using a formula, calculate how much Temoso owes the
bank at the end of three years.
r n
e) Use the formula A = P(1 + 100 ) to calculate the compound interest on a
loan of R3 450 at 6,5% per annum for 5 years.
1.3 Calculations with integers What is different to Grade 8? Total time for
Integers:
Integers • Revise: In Grade 9 learners consolidate number knowledge and calculation techniques for
integers, developed in Grade 8.
-- perform calculations involving all
four operations with integers In Grade 9, learners work with integers mostly as coefficients in algebraic 4,5 hours
expressions and equations. They are expected to be competent in performing all
-- perform calculations involving all
four operations with integers and using the properties of integers appropriately
four operations with numbers that
where necessary.
involve the squares, cubes, square
roots and cube roots of integers
Properties of integers
• Revise:
-- commutative, associative and
distributive properties of addition
and multiplication for integers
-- additive and multiplicative inverses
for integers
Solving problems
121 • Solve problems in contexts involving
multiple operations with integers
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
122 1.4 Calculations using fractions What is different to Grade 8? Total time
for common
Common • All four operations with common In Grade 9 learners consolidate number knowledge and calculation techniques for fractions:
fractions fractions and mixed numbers common fractions, developed in Grade 8.
• All four operations, with numbers that In Grade 9, learners work with common fractions mostly as coefficients in algebra-
involve the squares, cubes, square ic expressions and equations. They are expected to be competent in performing 4,5 hours
roots and cube roots of common multiple operations using common fractions and mixed numbers, applying proper-
fractions ties of rational numbers appropriately. They are also expected to recognize and
use equivalent forms for common fractions appropriately in calculations and when
Calculation techniques
simplifying algebraic fractions.
• Revise:
-- Convert mixed numbers to
common fractions in order to
MATHEMATICS GRADES 7-9
perform calculations with them
-- Use knowledge of multiples and
factors to write fractions in the
simplest form before or after
calculations
-- Use knowledge of equivalent
fractions to add and subtract
common fractions
-- Use knowledge of reciprocal
relationships to divide common
fractions
Solving problems
Solve problems in contexts involving
common fractions, mixed numbers and
percentages
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
Equivalent forms
• Revise equivalent forms between:
-- common fractions where one
denominator is a multiple of
another
-- common fraction and decimal
fraction forms of the same number
-- common fraction, decimal fraction
and percentage forms of the same
number
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS 1.5 Calculations with decimal fractions What is different to Grade 8? Total time
for decimal
Decimal • Multiple operations with decimal In Grade 9 learners consolidate number knowledge and calculation techniques for fractions:
fractions fractions, using a calculator where decimal fractions, developed in Grade 8.
appropriate
In Grade 9, learners work with decimal fractions mostly as coefficients in algebraic
• Multiple operations, with or without expressions and equations. They are expected to be competent in performing 4,5 hours
brackets, with numbers that involve multiple operations using decimal fractions and mixed numbers, applying proper-
the squares, cubes, square roots and ties of rational numbers appropriately. They are also expected to recognize and
cube roots of decimal fractions use equivalent forms for decimal fractions appropriately in calculations.
Calculation techniques
• Use knowledge of place values to
estimate the number of decimal
places in the result before performing
calculations
• Use rounding off and a calculator to
check results where appropriate
Solving problems
Solve problems in context involving
decimal fractions
Equivalent forms
• Revise equivalent forms between:
-- common fraction and decimal
fraction forms of the same number
-- common fraction, decimal fraction
and percentage forms of the same
number
123
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
124 1.2 Comparing and representing What is different to Grade 8? Total time for
Exponents numbers in exponential form exponents:
• Additional laws of exponents involving integer exponents
• Revise:
• Scientific notation for numbers, including negative exponents
-- compare and represent integers in 5 hours
In Grade 9 learners consolidate number knowledge and calculation techniques for
exponential form
exponents, developed in Grade 8.
-- compare and represent numbers in
scientific notation
• Extend scientific notation to include
negative exponents
Calculations using numbers in
exponential form
MATHEMATICS GRADES 7-9
• Revise the following general laws of Laws of exponents
exponents:
• The laws of exponents should be introduced through a range of numeric
-- am x an = am + n examples first, then variables can be used.
-- am ÷ an = am - n, if m>n • The following laws of exponents should be known, where m and n are integers
-- (am)n = am x n and a and t are not equal to 0:
-- (a x t)n = an x t n am x an = am+n am ÷ an = am - n
-- a0 = 1
Examples Examples
• Extend the general laws of exponents
a) 23 x 24 = 23+4 = 27 = 128 a) 35 ÷ 32 = 33 = 27
to include:
b) x3 x x4 = x3+4 = x7 b) x5 ÷ x3 = x2
-- integer exponents
(am)n = am x n (a x t)n = an x t n
-- a–m = a1m
Examples Examples
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
• Perform calculations involving all
four operations using numbers in a) (23)2 = 26 = 64 a) (3x2)3 = 33x6 = 27x6
exponential form
a0 = 1 a–m = a1m
Examples Examples
–3 1 1
3
Solving problems a) (37)0 = 1 a) 5 = 5 = 75
3 5 –2 1 1
• Solve problems in contexts involving 2
b) (4x2)0 = 1 b) 7 ÷ 7 = 7 = 7 = 49
numbers in exponential form,
including scientific notation • Make sure learners understand these laws reading from both sides of the equal
sign i.e. if the LHS = RHS, then the RHS = LHS
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
• The law a0 = 1 can be derived by using the law of exponents for division in a few
axaxaxa
CAPS
examples e.g. a4 ÷ a4 = a x a x a x a = 1, therefore a4–4 = a0 = 1
• Learners should be able to use the laws of exponents in calculations and for
solving simple exponential equations as well as expanding and simplifying
algebraic expressions.
• Look out for the following common misconceptions where:
-- Learners multiply unlike bases and add the exponents
Example: xm x yn = (xy)m + n
5 7 9 9
2 x 2 = 4 instead of the correct answer 2
-- Learners forget the middle term of the binomial e.g. (x + y)m = xm + ym
-- Learners confuse adding the exponents and adding the terms, e.g.
xm + xn = xm + n
-- Learners confuse the exponent of the variable and the coefficient e.g.
1 1
3
2x–3 = 2x instead of the correct answer 2 x 2
Calculations and simple equations using numbers in exponential form
• The calculations and equations should provide opportunities to apply the laws of
exponents and should not be unduly complex.
Examples
–1 3 –2
a) Calculate: 2 x 6 x 3
–2
b) Simplify: (–2x2)(–2x)
x
c) Solve x: 3 = 9
x 1
d) Solve x: 2 = 4
x+1
e) 5 =1
Scientific notation
• When writing numbers in scientific notation, learners have to understand the
relationship between the number of decimal places and the index of 10.
Example: 25 = 2,5 x 101; and 250 = 2,5 x 102
• Scientific notation that extends to negative exponents includes writing very small
numbers in scientific notation.
125
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
126 1.2 Example: 25 millionth = 2,5 x 10–5
Exponents
• Learners practise writing small and large numbers in scientific notation, which
they might already have encountered in Natural Science. It is useful to refer to
these contexts when discussing scientific notation.
• Calculations can be done with or without a calculator.
Examples
a) Calculate: 2,6 x 105 x 9 x 107 without using a calculator and give answer in
scientific notation.
b) Write in scientific notation: 0,00053
c) Calculate: 5,8 x 10–4 + 2,3 x 10–5 without using a calculator.
MATHEMATICS GRADES 7-9
Patterns, 2.1 Investigate and extend patterns What is different to Grade 8? Total time for
functions and numeric and
algebra Numeric and • Investigate and extend numeric Learners consolidate work involving numeric and geometric patterns done in geometric
geometric and geometric patterns looking for Grade 8. patterns
patterns relationships between numbers
• Investigating number patterns is an opportunity to generalize – to give general 4, 5 hours
including patterns:
algebraic descriptions of the relationship between terms and thier position in a
-- represented in physical or diagram sequence and to justify solutions.
form
Kinds of numeric patterns
-- not limited to sequences involving a
• Given a sequence of numbers, learners have to identify a pattern or
constant difference or ratio
relationship between consecutive terms in order to extend the pattern.
-- of learner’s own creation
Examples
-- represented in tables
Provide a rule to describe the relationship between the numbers in the
-- represented algebraically sequences below. Use this rule to give the next three numbers in the sequence:
• Describe and justify the general rules a) –1; –1,5; –2; –2,5 ... ...
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
for observed relationships between
Here learners should identify the constant difference between consecutive
numbers in own words or in algebraic
terms in order to extend the pattern. This pattern can be described in
language
learners’ own words as ‘adding –0,5’ or ‘counting in –0,5s ’ or ‘add –0,5 to
the previous number in the pattern’.
b) 2; –1; 0,5; –0,25 ; 0,125 ... ...
Here learners should identify the constant ratio between consecutive
terms. This pattern can be described in learners’ own words as ‘multiply the
previous number by –0,5 ’.
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS Patterns, 2.1 c) 1; 0; –2; –5; –9; –14 ... .
functions and
algebra Numeric and This pattern has neither a constant difference nor constant ratio. This
geometric pattern can be described in learners’ own words as ‘subtract 1 more than
patterns was subtracted to get the previous term’. Using this rule, the next 3 terms
will be –20, –27, –35
• Given a sequence of numbers, learners have to identify a pattern or
relationship between the term and its position in the sequence. This
enables learners to predict a term in a sequence based on the position of that
term in the sequence. It is useful for learners to represent these sequences in
tables so that they can consider the position of the term.
Examples
a) Provide a rule to describe the relationship between the numbers in this
sequence: 2; 5; 10; 17 ... . Use your rule to find the 10th term in this
sequence.
Firstly, learners have to understand that the ‘10th term’ refers to position 10 in
the number sequence. They have to find a rule in order to determine the 10th
term, rather than continuing the sequence up to the tenth term.
This sequence can be represented in the following table:
Position in sequence 1 2 3 4 10
Term 2 5 10 17 ?
Learners have to recognize that each term in the bottom row is obtained
by squaring the position number in the top row and adding. Thus the 10th
term will be '10 squared +1’ or 102 + 1 which is 101. Using the same rule,
learners can also be asked what term number or position will be 626? If the
term is obtained by squaring the position number of the term and adding 1,
then the position number can be obtained by subtracting, then finding the
square root of the term. Hence, 626 will be the 25th term in the sequence
b) Provide a rule to describe the relationship between the numbers in this
since 626 – 1 = 625 and √625 = 25.
sequence: –2; –5;–8; –11 ... .Use this rule to find the 20th term in this
sequence.
If learners consider only the relationship between consecutive terms, then
they can continue the pattern (‘add -3 to previous number’) up to the 20th
term to find the answer. However, if they look for a relationship or rule
between the term and the position of the term, they can predict the answer
127
without continuing the pattern. Using number sentences can be useful to
find the rule:
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
128 Patterns, 2.1 1st term: –2 = –3(1) + 1
functions and
algebra Numeric and 2nd term: –5 = –3(2) + 1
geometric
patterns 3rd term: –8 = –3(3) + 1
4th term: –11 = –3(4) + 1
The number in the brackets corresponds to the position of the term. Hence,
the 20th term will be: –3(20) + 1 = –59
The rule in learners’ own words can be written as –3 x the position of the
term + 1’ or –3n + 1, where n is the position of the term.
• These types of numeric patterns develop an understanding of functional
relationships, in which there is a dependent variable (position of the term) and
an independent variable (the term itself), and where you have a unique output
MATHEMATICS GRADES 7-9
for any given input value.
Kinds of geometric patterns
• Geometric patterns are number patterns represented diagrammatically. The
diagrammatic representation reveals the structure of the number pattern.
• Hence, representing the number patterns in tables, makes it easier for learners
to describe the general rule for the pattern.
Example
Consider this pattern for building hexagons using matchsticks. How many
matchsticks will be used to build the 10th hexagon? Provide an expression to
describe the general term for this number sequence.
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
The rule for the pattern is contained in the structure (construction) of the
successive hexagonal shapes:
(1) add on 1 matchstick per side
(2) there are 6 sides, so
(3) add on 6 matchsticks per hexagon as you proceed from a given hexagon to
the next one.
So, for the 2nd hexagon, you have 2 x 6 matches; for the 3rd hexagon you have
3 x 6 matches. Using this pattern for building hexagons, the 10th hexagon will
have 10 x 6matches.
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS Patterns, 2.1 Learners can also use a table to record the number of matches used for each
functions and hexagon. This way the number pattern is related to the number of matches used
algebra Numeric and for each new hexagon.
geometric
patterns
Position of hexagon in pattern 1 2 3 4 5 6 10
Number of matches 6 12 18
The nth term for this sequence can be written as 6n or 6 + (n – 1)6
2.2 Input and output values What is different to Grade 8?
Functions • Determine input values, output Learners consolidate work with input and output values done in Grade 8. They
and values or rules for patterns and should continue to find input or output values in flow diagrams, tables, formulae
relationships relationships using: and equations.
-- flow diagrams Functions and relationships are done again in Term 3.
-- tables In this phase, it is useful to start specifying whether the input values are natural
numbers, or integers or rational numbers. This builds learners’ awareness of the
-- formulae
domain of input values. Hence, to find output values, learners should be given the
-- equations rule/formula as well as the domain of the input values.
Equivalent forms Learners should begin to recognize equivalent representations of the same rela-
tionships shown as an equation, a set of ordered pairs in a table or on a graph.
• Determine, interpret and justify
equivalence of different descriptions Examples
of the same relationship or rule
a) If the rule for finding in the table below is: y = 12 x + 1, determine the values of y
presented:
for the given x values:
-- verbally
x 0 1 2 4 10 50 100
-- in flow diagrams
-- in tables y
-- by formulae b) Describe the relationship between the numbers in the top row and those in the
bottom row in the table. Then write down the value of m and n
-- by equations
-- by graphs on a Cartesian plane x –2 –1 0 1 2 12 n
y –7 –5 –3 –1 1 m 27
In tables such as these, more than one rule might be possible to describe the
relationship between x and y values. The rules are acceptable if they match the
given input values to the corresponding output values. For example, the rule
y = 2x – 3 describes the relationship between the given values for x and y. To find
129 m and n, learners have to substitute the corresponding values for x or y into the
rule and solve the equation by inspection.
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
130 2.3 Algebraic language What is different to Grade 8? Time for
algebraic
Algebraic • Revise the following done in Grade 8: • Algebraic manipulations which include: expressions in
expressions this term:
-- Recognize and identify conventions -- multiply integers and monomials by polynomials
for writing algebraic expressions
-- divide polynomials by integers or monomials
-- Identify and classify like and unlike 4,5 hours
-- the product of two binomials
terms in algebraic expressions
-- the square of a binomial
-- Recognize and identify coefficients
and exponents in algebraic Algebraic expressions are done again in Term 3. In this term the focus is on
expressions expanding and simplifying algebraic expressions. In Term 3 the focus is on
factorizing expressions.
• Recognize and differentiate between
monomials, binomials and trinomials Manipulating algebraic expressions
MATHEMATICS GRADES 7-9
Expand and simplify algebraic • Make sure learners understand that the rules for operating with integers and
expressions rational numbers, including laws of exponents, apply equally when numbers are
replaced with variables. The variables are numbers of a given type (e.g. integers
• Revise the following done in Grade
or rational numbers) in generalized form.
8, using the commutative, associative
and distributive laws for rational • When multiplying or dividing expressions, make sure learners understand how
numbers and laws of exponents to: the distributive rule works.
-- add and subtract like terms in • The associative rule allows for grouping of like terms when adding.
algebraic expressions
Look out for the following common misconceptions:
-- multiply integers and monomials
• x + x = 2x and NOT x2. Note the convention is to write 2x rather than x2
by:
• x2 + x2 = 2x2 and NOT 2x4
♦♦ monomials
• a +b = a + b and NOT ab
♦♦ binomials
• (–2x2)3 = –8x6 and NOT –6x5
♦♦ trinomials
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
• -x(3x + 1) = –3x2 – 1 and NOT –3x2 + 1
-- divide the following by integers or
monomials: 6x2 + 1 = 6 + 1 and NOT 6 + 1
• x2 x2
♦♦ monomials
• If x = 2 then –3x2 = –3(2)2 = 3 x 4 = 12 and NOT (–6)2
♦♦ binomials
• If x = –2 then –x2 – x = –(–2)2 – (–2)= –4 + 2 = –2 and NOT 4 + 2 = 6
♦♦ trinomials
-- Simplify algebraic expressions
• (x + 2)2 = x2 + 4x + 4 and NOT x2 + 4
• √25x2 –9x2 = √16x2 = 4x and NOT 5x – 3x = 2x
involving the above operations
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS 2.3 -- Determine the squares, cubes, Examples of expanding and simplifying expressions
square roots and cube roots of
Algebraic a) Simplify: –3(x3 + 2x2 – x) – x2(3x + 1) [multiply integer and monomial by
single algebraic terms or like
expressions polynomial]
algebraic terms
b) Determine/expand: (x + 2)(x – 3) [multiply binomial by binomial]
-- Determine the numerical value
of algebraic expressions by c) Determine/expand: (x + 2)(x – 2) [multiply binomial by binomial]
substitution
d) Determine/expand: (x + 3)2 [multiply out a perfect square]
• Extend the above algebraic
e) Simplify: 2(x – 3)2 –3(x + 1)(2x – 5) [multiple calculations involving product of
manipulations to include:
binomials]
-- multiply integers and monomials by
f) If x = –2 determine the numerical value of 3x2 – 4x + 5 [using substitution]
polynomials
6x4 – 8x3 – 2x2 + 4
-- divide polynomials by integers or g) Simplify: 2x2 for x ≠ 0 for [divide polynomial by monomial;
monomials remind learners that denominator cannot be 0]
-- the product of two binomials 8x3 – (–x3)(2x)
h) Simplify: -x2 for x ≠ 0 for [calculations involving multiple
-- the square of a binomial operations; remind learners that denominator cannot be ]
i)
It might help to remind learners that these variables (or x in this case) represent
Determine: √36x4 [square root of monomial]
numbers of a particular type – these may be rational, or integers, or perhaps
whole numbers; such a reminder also then implies that all the associated rules
or properties of these numbers apply here. So, in the above example, if x is an
integer, then x = a or x = –a because a4 = (–a)4
Factorize algebraic expressions
• Factorize algebraic expressions that
involve:
-- common factors
-- difference of two squares
-- trinomials of the form:
♦♦ x2 + bx + c
♦♦ ax2 + bx + c, where a is a common
factor
• Simplify algebraic expressions that
involve the above factorisation
processes
131
• Simplify algebraic fractions using
factorisation
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
132 2.4 Equations What is different to Grade 8? Time for
algebraic
Algebraic • Revise the following done in Grade 8: • Solving equations using factorization equations in
equations this term:
-- Set up equations to describe • Solving equations of the form: a product of factors = 0
problem situations
Algebraic equations are done again in Term 3. In this term the focus is on
-- Analyse and interpret equations consolidating solving equations using additive and multiplicative inverses 4 hours
that describe a given situation and the laws of exponents. In Term 3, the focus is on solving equations after
factorizing as well as generating tables of ordered pairs for linear equations.
-- Solve equations by:
Learners have opportunities to write and solve equations when they write general
♦♦ inspection rules to describe relationships between numbers in number patterns, and when
♦♦ using additive and multiplicative they find input or output values for given rules in flow diagrams, tables and
inverses formulae.
In Grade 9, learners can be given equations where they have to expand, simplify
MATHEMATICS GRADES 7-9
♦♦ using laws of exponents
or factorize expressions first, before solving the equation.
-- Determine the numerical value of For equations of the form: a product of two factors = 0, learners have to
an expression by substitution. understand that if the product of two factors equals 0, then at least one of the
-- Use substitution in equations to factors must be equal to 0. Hence to solve the equation, each factor must be
generate tables of ordered pairs written as an equation equal to 0, and therefore more than one solution for x is
possible.
• Extend solving equations to include:
When working with algebraic fractions, learners must be reminded that the
-- using factorisation denominator cannot equal 0, so any value of x that makes the denominator 0
cannot be a solution to the equation.
-- equations of the form: a product of
factors = 0 Examples of equations
a) Solve: x if 3(x – 2) = x + 2
3x – 6 = x + 2 (expand LHS first)
3x – x – 6 = x – x + 2 (add –x to both sides of the equation)
therefore 2x – 6 = 2
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
2x – 6 + 6 = 2 + 6 (add 6 to both sides of the equation)
therefore 2x = 8
2x 8
2 = 2 (divide both sides of the equation by 2)
x=4
b) Solve x if (x – 1)(x + 3) = 0
x – 1= 0 or x + 3 = 0 (at least one factor must be equal to 0)
Thus x = 1 (add –1 to both sides of the equation) or x = –3 (add –3 to both
sides of the equation)
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
2.4 x 2x – 1
c) Solve x if 3 + 4 =1
CAPS
Algebraic 4x + 3(2x – 1) = 12 (multiply each term on both sides of the equation by the
equations LCM, 12)
4x + 6x –3 = 12 (expand expression on LHS)
10x = 15 (add 3 to both sides of the equation, and add like terms to the LHS)
2
x = 3 (divide both sides of the equation by 10)
d) If y = 2x2 + 4x + 3, calculate y when x = –2
e) Thandi is 6 years older than Sophie. In 3 years time Thandi will be twice as old
as Sophie. How old is Thandi now?
REVISION/ASSESSMENT: Total time
for revision/
At this stage learners should have been assessed on: assessment
• the properties of different number systems for the term
• calculating and solving problems with whole numbers, integers, common fractions and decimal fractions, numbers in exponential form
• numeric and geometric patterns 7 hours
• functions and relationships
• algebraic expressions
• algebraic equations
133
MATHEMATICS GRADES 7-9
GRADE 9 – TERM 2
DURATION
134 CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
Space and shape 3.5 Constructions What is different to Grade 8? Total time for
(geometry) constructions
Construction • Accurately construct geometric • Bisecting angles in a triangle of geometric
of geometric figures appropriately using a figures:
• Constructing 30° without a protractor
figures compass, ruler and protractor,
including bisecting angles of a • Investigation of new properties of triangles, quadrilaterals and polygons
triangle 9 hours
Constructions
• Construct angles of 45°, 30°, 60°and
• Constructions provide a useful context to explore or consolidate knowledge of
and their multiples without using a
angles and shapes.
protractor
• Make sure learners are competent and comfortable in the use of a compass and
Investigating properties of geometric
know how to measure and read angle sizes on a protractor
figures
MATHEMATICS GRADES 7-9
• Revise the constructions of angles if necessary, before proceeding with the new
• By construction, investigate the
constructions.
angles in a triangle, focusing on the
relationship between the exterior • Start with the constructions of lines, so that learners can first explore angle
angle of a triangle and its interior relationships on straight lines.
angles
• When constructing triangles learners should draw on known properties and
• By construction, explore the minimum construction of circles.
conditions for two triangles to be
congruent • Construction of special angles without protractors are done by:
• By construction, investigate -- bisecting a right-angle to get 45°
sides, angles and diagonals in -- drawing an equilateral triangle to get 60°
quadrilaterals, focusing on the
diagonals of rectangles, squares, -- bisecting the angles of an equilateral triangle to get 30°
parallelograms, rhombi and kites
• By construction explore the sum of
the interior angles of polygons
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS Space and shape 3.1 Classifying 2D shapes What is different to Grade 8? Total time for
(geometry) geometry of
Geometry of • Revise properties and definitions of • Properties of diagonals of quadrilaterals 2D shapes
2D shapes triangles in terms of their sides and
• Minimum conditions for congruent and similar triangles
angles, distinguishing between:
• Triangles 9 hours
-- equilateral triangles
• Constructions serve as a useful context for exploring properties of triangles.
-- isosceles triangles
See 3.5 Construction of Geometric figures.
-- right-angled triangles
• Properties of triangles learners should know:
-- the sum of the interior angles of triangles = 180°
-- an equilateral triangle has all sides equal and all interior angles = 60°
-- an isosceles triangle has at least two equal sides and its base angles are
equal
-- a right-angled triangle has one angle that is a right-angle
-- the side opposite the right-angle in a right-angled triangle, is called the
hypotenuse
-- in a right-angled triangle, the square of the hypotenuse is equal to the sum of
the squares of the other two sides (Theorem of Pythagoras).
-- the exterior angle of a triangle = the sum of the opposite two interior angles
• Revise and write clear definitions of Quadrilaterals
quadrilaterals in terms of their sides, • Constructions serve as a useful context for exploring properties of triangles.
angles and diagonals, distinguishing See notes on constructions above.
between:
• The classification of quadrilaterals should include the recognition that:
-- parallelogram
-- rectangles and rhombi are special kinds of parallelograms
-- rectangle
-- a square is a special kind of rectangle and rhombus.
-- square
Properties of quadrilaterals learners should know:
-- rhombus
• the sum of the interior angles of quadrilaterals = 360°
-- trapezium
• the opposite sides of parallelograms are parallel and equal
-- kite
• the opposite angles of parallelograms are equal
• the opposite angles of a rhombus are equal
• the opposite sides of a rhombus are parallel and equal
135
• the size of each angle of rectangles and squares is 90°
• a trapezium has one pair of opposite sides parallel
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
136 Space and shape 3.1 • a kite has two pairs of adjacent sides equal
(geometry)
Geometry of • the diagonals of a square, rectangle, parallelogram and rhombus bisect each
2D shapes other
• the diagonals of a square, rhombus and kite are perpendicular
Congruent triangles
• Constructions are a useful context for establishing the minimum conditions for
Similar and congruent triangles
two triangles to be congruent. See notes on Constructions above.
• Through investigation, establish the
• Conditions for two triangles to be congruent:
minimum conditions for congruent
triangles -- three corresponding sides are equal (S,S,S)
• Through investigation, establish -- two corresponding sides and the included angle are equal (S,A,S)
MATHEMATICS GRADES 7-9
the minimum conditions for similar
-- two corresponding angles and a corresponding side are equal (A,A,S)
triangles
-- right-angle, hypotenuse and one other corresponding side are equal (R,H,S)
Similar triangles
• Constructions are a useful context for establishing the minimum conditions for
two triangles to be similar. See notes on Constructions above.
• Condition for two triangles to be similar: corresponding angles are equal and
corresponding sides are proportional
Solving problems Solving problems
Solve geometric problems involving • Learners can solve geometric problems to find unknown sides and angles
unknown sides and angles in in triangles and quadrilaterals, using known definitions as well as angle
triangles and quadrilaterals, using relationships on straight lines.
known properties of triangles and
• For right-angled triangles, learners can also use the Theorem of Pythagoras to
quadrilaterals, as well as properties of
find unknown lengths.
congruent and similar triangles.
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
• Learners should give reasons and justify their solutions for every written
statement.
• Note that solving geometric problems is an opportunity to practise solving
equations.
Example:
In ∆ABC, Â = x, angle B = 50° and angle Ĉ = 80°. What is the size of Â?
Learners can solve x in the following equation:
x + 50° + 80° = 180° (because the sum of the angles in a triangle = 180°)
x = 180° – 130° (add –50° and –80° to both sides of the equation)
x = 50°, hence  = 50°
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS 3.3 What is different to Grade 8? Total time for
geometry of
Geometry of Learners revise and write clear descriptions of angle relationships on straight lines straight lines
straight lines
Angle relationships learners should know:
• the sum of the angles on a straight line is 180° 9 hours
• If lines are perpendicular lines, then adjacent supplementary angles are each
Angle relationships
equal to 90° .
• Revise and write clear descriptions • If lines intersect, then vertically opposite angles are equal.
of the relationship between angles
• if parallel lines are cut by a transversal, then corresponding angles are equal
formed by:
• if parallel lines cut by a transversal, then alternate angles are equal
-- perpendicular lines • if parallel lines cut by a transversal, then co-interior angles are supplementary
-- intersecting lines The above angles have to be identified and named by learners.
-- parallel lines cut by a transversal
Solving problems Solving problems
Solve geometric problems using the • Learners can solve geometric problems to find unknown angles using the
relationships between pairs of angles angle relationships above, as well as other known properties of triangles and
described above. quadrilaterals.
• Learners should give reasons and justify their solutions for every written
statement.
• Note that solving geometric problems is an opportunity to practise solving
equations.
Example:
B1 and B2 are two angles on a straight line. B1 = 35°. What is the
size of B2 ?
Learners can find B2 by solving the following equation:
35° + B2 = 180° (because the sum of angles on a straight line
180°)
B2 = 180° – 35° (add –35° to both sides of the equation)
B2 = 145°
137
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
138 Measurement 4.3 Solve problems using the Theorem • The Theorem of Pythagoras was introduced in Grade 8 Total time for
of Pythagoras the Theorem
The Theorem • It is important that learners understand that the Theorem of Pythagoras applies of Pythagoras:
of Use the Theorem of Pythagoras to only to right-angled triangles.
Pythagoras solve problems involving unknown
• The Theorem of Pythagoras is basically a formula to calculate unknown length
lengths in geometric figures that con- 5 hours
of sides in right-angled triangles.
tain right-angled triangles
• In particular, the Theorem of Pythagoras can be the first step in calculations of
perimeters or areas of composite figures, when one of the figures is a right-
angled triangle with an unknown length. See example below.
• In the FET phase, the Theorem of Pythagoras is crucial to the further study of
Geometry and Trigonometry.
Examples of solving problems using the Theorem of Pythagoras:
MATHEMATICS GRADES 7-9
ABCD is a rectangle where BD = 15 cm and BC = 9 cm.
Determine the:
a) perimeter of ABCD
b) area of ABCD
B
15 cm
9 cm
A
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS) D C
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
Measurement 4.1 Area and perimeter What is different to Grade 8? Total time
for Area and
CAPS
Area and • Use appropriate formulae and • The calculations are the same as in Grade 8, but learners can find perimeters Perimeter
perimeter of conversions between SI units, and areas of more composite and complex figures
2D shapes to solve problems and calculate
• Polygons can include trapeziums, parallelograms, rhombi and kites
perimeter and area of: 5 hours
• Formulae learners should know and use:
-- polygons
-- circles Perimeter of a square: P = 4s
• Investigate how doubling any or all of Perimeter of a rectangle: P = 2(l + b) or P = 2l + 2b
the dimensions of a 2D figure affects
its perimeter and its area Area of a square: A = l2
Area of a rectangle: A = length x breadth
Area of a rhombus: A = length x height
1
Area of a kite: A = 2 (diagonal1 x diagonal2
Area of a parallelogram: A = base x height
1
Area of a trapezium A = 2 (sum of parallel sides) x height
1
Area of a triangle: A = 2 (b x h)
Diameter of a circle: d = 2r
Circumference of circle: c = πd or c = 2πr
Area of a circle: A = πr2
Solving equations using formulae
The use of formulae provides a context to practise solving equations by inspection
or using additive or multiplicative inverses.
Example:
a) If the perimeter of a square is 32 cm, what is the length of each side? Learners
should write this as:
32
4s = 32 and solve by asking: 4 times what will be 32? or saying s= 4 .
b) If the area of a rectangle is 200 cm2, and its length is 50 cm, what is its width?
Learners should write this as:
50 x b = 200 and solve by inspection by asking: 50 times what will be 200? or
139 200
saying b = 50
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
140 Measurement 4.1 For areas of triangles:
Area and • Make sure learners know that the height of a triangle is a line segment drawn
perimeter of from any vertex perpendicular to the opposite side.
2D shapes
Example: AD is the height onto base BC of ∆ABC.
A A
B C D B C
D
MATHEMATICS GRADES 7-9
• Point out that every triangle has 3 bases, each with a related height or altitude.
• For conversions:
-- If 1 cm = 10 cm then 1 cm2 = 100 mm2
-- If 1 cm = 100 cm then 1 m2 = 10 000 cm2
REVISION/ASSESSMENT: Total time
for revision/
At this stage learners should have been assessed on: assessment
• constructing geometric objects for the term
• geometry of 2D shapes
• geometry of straight lines 8 hours
• the Theorem of Pythagoras
• area and perimeter of 2D shapes
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
GRADE 9 – TERM 3
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
CAPS (in hours)
Patterns, 2.2 Input and output values Functions and relationships were also done in Term 1. The focus in this term is on Time for
functions & finding output values for given equations, and recognising equivalent forms be- Functions and
algebra Functions • Determine input values, output relationships
and tween different descriptions of the same relationship.
values or rules for patterns and in this term:
relationships relationships using: See additional notes and examples in Term 1.
5 hours
-- flow diagrams
-- tables
-- formulae
-- equations
Equivalent forms
• Determine, interpret and justify
equivalence of different descriptions
of the same relationship or rule
presented:
-- verbally
-- in flow diagrams
-- in tables
-- by formulae
-- by equations
-- by graphs on a Cartesian plane
141
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
142 Patterns, 2.3 Algebraic language Total time
functions & for algebraic
algebra Algebraic • Revise the following done in Grade 8: Algebraic expressions were also done in Term 1. The focus in this term is on expressions:
expressions factorizing expressions.
-- recognize and identify conventions
for writing algebraic expressions See additional notes and examples in Term 1.
9 hours
-- identify and classify like and unlike Factorizing expressions
terms in algebraic expressions
• Make sure learners understand that factorizing is the reverse of expanding an
-- recognize and identify coefficients expression through multiplication e.g.
and exponents in algebraic
Expand: 2x(x + 3) = 2x2 + 6
expressions
Factorize: 2x2 + 6 = 2x(x + 3)
• Recognize and differentiate between
monomials, binomials and trinomials • Note that 1 and –1 are common factors of every expression e.g.
MATHEMATICS GRADES 7-9
Expand and simplify algebraic Expand: a – 4b = 1(a – 4b)
expressions
Factorize: 4b – a = 1(a – 4b)
• Revise the following done in Grade
8, using the commutative, associative Examples of expressions with common factors that can be factorized
and dstributive laws for rational a) 6a4 – 4a2
numbers and laws of exponents to:
b) ax – bx + 2a – 2b
-- add and subtract like terms in
algebraic expressions c) 2x(a – b) – 3(a – b)
-- multiply integers and monomials d) 2x(a – b) – 3(b – a)
by: e) (a + b)2 – 5(a + b)
♦♦ monomials Examples of expressions with a difference of two squares that can be
factorized
♦♦ binomials
a) 25a2 = 1
♦♦ trinomials
b) a4 – b4
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
-- divide the following by integers or
monomials: c) 9(a + b)2 – 1
♦♦ monomials d) 3x 3 – 27
♦♦ binomials Examples of algebraic fractions that can be factorized
♦♦ trinomials 2x + 6y
a) x + 3y
-- simplify algebraic expressions
3x – 3y
involving the above operations b) 6x – 6y
-- determine the squares, cubes, 9a2 – 1
square roots and cube roots of c) 3a + 1
single algebraic terms or like
algebraic terms
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS Patterns, 2.3 -- determine the numerical value Examples of trinomials that can be factorized
functions & of algebraic expressions by
algebra Algebraic a) x2 + 5x + 6
substitution
expressions b) x2 – 5x + 6
• Extend the above algebraic
manipulations to include: c) x2 – x – 6
-- multiply integers and monomials by d) x2 – 6x + 9
polynomials
e) 2x2 + 10x + 12
-- divide polynomials by integers or
monomials
-- the product of two binomials
-- the square of a binomial
Factorize algebraic expressions
• Factorize algebraic expressions that
involve:
-- common factors
-- difference of two squares
-- trinomials of the form:
♦♦ x2 + bx + c
♦♦ ax2 + bx + c, where a is a common
factor
• Simplify algebraic expressions that
involve the above factorisation
processes
• Simplify algebraic fractions using
factorisation
143
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
144 2.4 Equations Time for
algebraic
Algebraic • Revise the following done in Grade 8: Algebraic equations were done in Term 1. The focus in this term is on solving equations in
equations using factorization, and equations with a product of factors. This term this term:
equations -- set up equations to describe also focuses on using equations to generate tables of ordered pairs.
problem situations
See additional notes and examples in Term 1.
-- analyse and interpret equations 9 hours
that describe a given situation Examples of equations
-- solve equations by: a) Solve x if x2 – 3x = 0
♦♦ inspection x(x – 3) = 0 (factorize LHS)
♦♦ using additive and multiplicative x = 0 or x – 3 = 0 (at least one factor = 0)
inverses Therefore, x = 0 or x = 3 (add 3 to both sides of the second equation)
MATHEMATICS GRADES 7-9
♦♦ using laws of exponents b) Solve x if x2 – 25 = 0
-- determine the numerical value of (x + 5)(x – 5) = 0 (factorize the difference of two squares on LHS)
an expression by substitution.
x + 5 = 0 or x – 5 = 0 (at least one factor = 0)
-- use substitution in equations to
generate tables of ordered pairs Therefore, x = – 5 (add – 5 to both sides of the equation) or x = 5 (add 5 to
both sides of the equation)
• Extend solving equations to include:
c) Write an equation to find the volume of a rectangular prism with length 2x cm;
-- using factorisation width (2x + 1) cm and height (2x + 3) cm
-- equations of the form: a product of d) If y = 2x2 + 4x + 3, calculate y when x = –2
factors = 0
e) Thandi is 6 years older than Sophie. In 3 years' time Thandi will be twice as old
as Sophie. How old is Thandi now?
Examples of generating ordered pairs
a) Complete the table below for x and y values for the equation: y = 2x2 – 3
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
x –2 –1 0 1 2
y
b) Complete the table below for x and y values for the equation: y = x2 – 2
x –3 –2 0
y –2 2
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS 2.5 What is different to Grade 8? Total time for
graphs:
Graphs • x-intercept, y-intercept and gradient of linear graphs
• Draw linear graphs from given equations
12 hours
• Determine equations of linear graphs
Learners should continue to analyse and interpret graphs of problem situations.
Interpreting graphs Investigating linear graphs
• Revise the following done in Grade 8: • To sketch linear graphs from given equations, learners should first draw up a
-- analyse and interpret global table of ordered pairs, that includes the intercept points (x; 0) and (0; y), and
graphs of problem situations, with then plotting the points.
a special focus on the following vertical change
trends and features: • Learners should investigate gradients by comparing horizontal change between
any two points on a straight line graph.
♦♦ linear or non-linear
• Learners should also investigate the relationship between the value of the
♦♦ constant, increasing or gradient and the coefficient of x in the equation of a straight line graph.
decreasing
• Learners should compare y-intercepts of linear graphs to the value of the
♦♦ maximum or minimum constant in the equation of the straight line graph.
♦♦ discrete or continuous Examples of linear graphs
• Extend the above with special focus a) Sketch and compare the graphs of: y = 4 and x = 4
on the following features of linear
graphs: b) Sketch and compare the graphs of: y = x and y = –x
-- x-intercept and y-intercept c) Sketch and compare the graphs of: y = 2x; y = 2x + 1; y = 2x – 1
-- gradient d) Sketch and compare the graphs of: y = 3x; y = 4x; y = 5x
Drawing graphs e) Sketch the graphs of: y = –3x + 2; using the table method
• Revise the following done in Grade 8: f) Determine the equation of the straight line passing through the following
points:
-- draw global graphs from given
descriptions of a problem situation, x –4 –3 –2 –1 0 1 2 3 4
identifying features listed above.
y –1 0 1 2 3 4 5 6 7
-- use tables of ordered pairs to plot
points and draw graphs on the
Cartesian plane
• Extend the above with a special
focus on:
145 -- drawing linear graphs from given
equations
MATHEMATICS GRADES 7-9
-- determine equations from given
linear graphs
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
146 Measurement 4.2 What is different to Grade 8? Total time for
surface area
Surface area • Surface area and volume of cylinders and volume:
and volume
Surface area and volume • Formulae learners should know and use:
of 3D objects
• Use appropriate formulae and -- the volume of a prism = the area of the base x the height 5 hours
conversions between SI units to
-- the surface area of a prism = the sum of the area of all its faces
solve problems and calculate the
surface area, volume and capacity of: -- the volume of a cube = l3
-- cubes -- the volume of a rectangular prism = l x b x h
-- rectangular prisms 1
-- the volume of a triangular prism = ( 2 b x h) x height of the prism
-- triangular prisms
-- the volume of a cylinder = (πr2) x the height of the cylinder
MATHEMATICS GRADES 7-9
-- cylinders
• For conversions, note:
• Investigate how doubling any or all
the dimensions of right prisms and -- if 1 cm = 10 mm then 1 cm3 = 1 000 mm3; and
cylinders affects the volume -- if 1 m = 100 cm then 1 m3 = 1 000 000 or 106 cm3
-- an object with a volume of 1 cm3 will displace exactly 1 ml of water; and
-- an object with a volume of 1 m3 will displace exactly 1 kl of water.
Examples of solving problems involving surface area and volume of
cylinders
• Calculate the volume of a cylinder, without using a calculator if its diameter is
28 cm, its height is 30 cm and π = 22
7
• Calculate the surface area of a cylinder, correct to 2 decimal places, if its height
is 65 cm and the circumference of its base is 47,6 cm.
REVISION/ASSESSMENT: Total time
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
for revision/
At this stage learners should have been assessed on: assessment
• functions and relationships for the term
• algebraic expressions
• Algebraic equations 5 hours
• graphs
• volume and surface area
GRADE 9 – TERM 4
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
CAPS (in hours)
Space and shape 3.4 What is different to Grade 8? Total time for
(geometry) transforma-
Transforma- • Reflection in the line y = x tions:
tion
• Identify transformations from co-ordinate points of the image
Geometry
• Co-ordinates of vertices 9 hours
Co-ordinate plane
Transformations • Doing transformations on the co-ordinate plane is an opportunity to practise
• Recognize, describe and perform reading and plotting points with ordered pairs, and links to drawing algebraic
transformations with points, line graphs.
segments and simple geometric • Make sure learners know how to plot points on the co-ordinate plane and can
figures on a co-ordinate plane, read the co-ordinates of points off the x-axis and y-axis.
focusing on:
• Make sure learners know the convention for writing ordered pairs (x; y)
-- reflection in the y-axis or x-axis
• Point out the differences between the axes in the four quadrants.
-- translation within and across
quadrants Focus of transformations
-- reflection in the line y = x • Doing transformations on a co-ordinate plane focuses attention on the co-
ordinates of points and vertices of shapes.
• Identify what the transformation of a
point is, if given the co-ordinates of • Learners should recognize that translations, reflections and rotations only
its image change the position of the figure, and not its shape or size.
Enlargements and reductions • Learners should recognize that the above transformations produce congruent
figures.
• Use proportion to describe the effect
of enlargement or reduction on area • Learners should begin to see patterns in terms of the co-ordinate points, for the
and perimeter of geometric figures different transformations, such as:
• Investigate the co-ordinates of the -- for translations to the right or left, the x–value changes and y–value stays the
vertices of figures that have been same
enlarged or reduced by a given scale -- for translations up or down, the y–value changes and the x–value stays the
factor same
-- for reflections in the y–axis, the x–value changes sign and the y–value stays
the same
-- for reflections in the x–axis, the y–value changes sign and the x–value stays
the same
-- for reflections in the line y = x, the x–value and y–value are interchanged.
• Learners should recognize that enlargements and reductions change the size of
147 figures by increasing or decreasing the length of sides, but keeping the angles
the same, produces similar rather than congruent figures.
• Learners should also be able to work out the factor of enlargement or reduction
MATHEMATICS GRADES 7-9
of a figure.
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
148 3.2 What is different to Grade 8? Total time for
geometry of
Geometry of • Properties of spheres and cylinders 3D objects:
3D objects
• Nets of cylinders
Platonic solids 9 hours
Classifying 3D objects • Properties of Platonic solids should be revised
• Revise properties and definitions • Platonic solids are a special group of polyhedra that have faces that are
of the 5 Platonic solids in terms of congruent regular polygons.
the shape and number of faces, the
• There are only 5 Platonic solids:
number of vertices and the number
of edges -- tetrahedron
• Recognize and describe the -- hexahedron (cube)
MATHEMATICS GRADES 7-9
properties of:
-- octahedron
-- spheres
-- dodecahedron
-- cylinders
-- icosahedrons
Building 3D models
• The name of each Platonic solid is derived from its number of faces.
• Use nets to create models of
• Platonic solids provide an interesting context in which to investigate the
geometric solids, including:
relationship between the number of faces, vertices and edges. By listing these
-- cubes properties for all the Platonic solids learners can investigate the pattern that
emerges, to come up with the general rule:
-- prisms
V – E + F = 2, where V = number of vertices; E = number of edges; F = number of
-- pyramids faces
-- cylinders Using and constructing nets
• Using and constructing nets are useful contexts for exploring or consolidating
properties of polyhedra.
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
• Learners should recognize the nets of different solids.
• Learners should make sketches of the nets using their knowledge of the the
shape and number of faces of the solids, before drawing and cutting out the nets
to build models.
• Since learners have more knowledge about the size of the internal angles of
polygons, and can measure angles, their constructions of nets should be more
accurate.
• Learners have to work out the relative position of faces of the nets in order to
build the 3D model.
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS Data handling 5.1 What is different to Grade 8? Total time for
collecting and
Collect, • Organizing data according to more than one criteria organizing
organize and data:
summarize • Outliers
data • Scatter plots
4 hours
Collect data Data sets and contexts
• Pose questions relating to social, Learners should be exposed to a variety of contexts that deal with social and
economic, and environmental issues environmental issues, and should work with given data sets, represented in
a variety of ways, that include big number ranges, percentages and decimal
• Select and justify appropriate sources
fractions. Learners should then practise organizing and summarizing this data,
for the collection of data
analysing and interpreting the data, and writing a report about the data.
• Distinguish between samples and
Complete a data cycle
populations, and suggest appropriate
samples for investigation Learners should complete at least one data cycle for the year, starting with posing
their own questions, selecting the sources and method for collecting, recording,
• Select and justify appropriate
organizing, representing, then analysing, summarizing, interpreting and reporting
methods for collecting data
on the data. Challenge learners to think about what kinds of questions and data
Organize and summarize data need to be collected to be represented on a histogram, a pie chart, a bar graph, a
line graph or a scatter plot.
• Organize numerical data in different
ways in order to summarize by
determining:
-- measures of central tendency
-- measures of dispersion including
extremes and outliers
• Organize data according to more
than one criteria
149
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
150 5.2 Represent data Representing data Total time for
representing
Represent • Draw a variety of graphs by hand/ • Pie charts to represent data do not have to be accurately drawn with a compass data:
data technology to display and interpret and protractor, etc. Learners can use any round object to draw a circle, then
data including: divide the circle into halves and quarters and eighths if needed, as a guide to
estimate the proportions of the circle that need to be shown to represent the 3 hours
-- bar graphs and double bar graphs
data. What is important is that the values or percentages associated with the
-- histograms with given and own data, are shown proportionally on the pie chart.
intervals
• Drawing, reading and interpreting pie charts is a useful context to re-visit
-- pie charts equivalence between fractions and percentages, e.g. 25% of the data is
1
represented by a 4 sector of the circle.
-- broken-line graphs
• It is also a context in which learners can find percentages of whole numbers e.g.
-- scatter plots
if 25% of 300 learners like rugby, how many (actual number) learners like rugby?
MATHEMATICS GRADES 7-9
• Histograms are used to represent grouped data shown in intervals on the
horizontal axis of the graph. Point out the differences between histograms and
bar graphs, in particular bar graphs that represent discrete data (e.g. favourite
sports) compared to histograms that show categories in consecutive, non-
overlapping interval, (e.g. test scores out of 100 shown in intervals of 10). The
bars on bar graphs do not have to touch each other, while in a histogram they
have to touch since they show consecutive intervals.
• Broken-line graphs refer to data graphs that represent data points joined by
a line and are not the same as straight line graphs that are drawn using the
equation of the line.
• Broken-line graphs are used to represent data that changes continuously over
time, e.g. average daily temperature for a month. Each day’s temperature is
represented by a point on the graph, and once the whole month has been
plotted, the points are joined to show a broke- line graph.
• Broken-line graphs are useful to read ‘trends’ and patterns in the data, for
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
predictive purposes e.g. Will the temperatures go up or down in the next month?
• A scatter plot is used to represent data that involves two different criteria and
the graph is used to look at the relationship between the two criteria. e.g. How
does the performance of learners in Mathematics compare to their performance
in English? Each point on the graph represents the results of one learner in
Mathematics and English. After all the results have been plotted, you can
compare the relationship between performance in English and Mathematics for
all the learners i.e. if they score high in Mathematics, do they also score high in
English? or, if they score high in Mathematics do they score low in English, or
is there no relationship between what they score on Mathematics to what they
score in English?
• The scatter plot allows one to see trends and make predictions, as well as
identify outliers in the data.
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS 5.3 Interpret data Developing critical analysis skills Total time for
analysing,
Interpret, • Critically read and interpret data • Learners should compare the same data represented in different ways e.g. in a interpreting,
analyse and represented in a variety of ways. pie chart or a bar graph or a table, and discuss what information is shown and summarizing
report data what is hidden; they should evaluate which form of representation works best for and reporting
• Critically compare two sets of data
the given data. data:
related to the same issue
• Learners should compare graphs on the same topic but where data has been
Analyse data
collected from different groups of people, at different times, in different places 3,5 hours
• Critically analyse data by answering or in different ways. Here learners should discuss differences between the data
questions related to: with an awareness of bias related to the impact of data sources and methods of
data collection on the interpretation of the data.
-- Data collection methods
• Learners should compare different ways of summarizing the same data sets,
-- Summary statistics of data
developing an awareness of how data reporting can be manipulated; they
-- Sources of error and bias in the should evaluate which summary statistics best represent the data.
data
• Learners should compare graphs of the same data, where the scales of
Report data the graphs are different. Here learners should discuss differences with an
awareness of how representation of data can be manipulated; they should
• Summarize data in short paragraphs evaluate which form of representation works best for the given data.
that include
• Learners should compare data on the same topic, where one set of data has
-- drawing conclusions about the data extremes or outliers, and discuss differences with an awareness of the effect of
-- making predictions based on the the extremes or outliers on the interpretation of the data. In particular, extremes
data affect the range and outliers which are identified on scatter plots.
-- making comparisons between two • Learners should write reports on the data in short paragraphs.
sets of data
-- identifying sources of error and
bias in the data
-- choosing appropriate summary
statistics for the data (mean,
median, mode, range)
-- the role of extremes and outliers in
the data
151
MATHEMATICS GRADES 7-9
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
152 5.4 Probability Probability experiments Total time for
Probability probability:
• Consider situations with equally In Grades 8 and 9 probability experiments are less important, and learners should
probable outcomes, and: consider probability for hypothetical events. In Grade 9, learners have to consider
outcomes of compound events and use two-way tables and tree diagrams to work
-- determine probabilities of out the probability of an outcome. 4,5 hours
compound events using two-way
Probability of compound events
tables and tree diagrams
For example, what is the probability of a woman giving birth to two boys after each
-- determine the probabilities of other? A two-way table can be used:
outcomes of events and predict
their relative frequency in simple
2nd birth
experiments 1st birth
-- compare relative frequency with Boy Girl
probability and explain possible
MATHEMATICS GRADES 7-9
Boy BB BG
differences
Girl GB GG
Two boys after each other (BB) is 1 of 4 possible outcomes, so the probability of
two boys is 1 out of 4, or 25%.
How does this probability change if you ask what is the probability of giving birth to
three boys after each other? A tree diagram can then be used:
1st birth
2nd birth
3rd birth
Boy [BBB]
Boy
Girl [BGG]
Boy
Girl [BBG]
Girl
CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
Boy [BGB]
Girl [GBG]
Boy [GBB]
Boy
Girl
Boy [GGB]
Girl
Girl [GGG]
DURATION
CONTENT AREA TOPICS CONCEPTS and SKILLS SOME CLARIFICATION NOTES or TEACHING GUIDELINES
(in hours)
CAPS 5.4 Total outcomes 8
Probability Three boys after each other, [BBB], is 1 out 8 of possible outcomes, hence the
probability of three boys after each other is 18 or 12,5%.
REVISION/ASSESSMENT: Total time
for revision/
At this stage learners should have been assessed on: assessment
• transformation geometry for the term
• geometry of 3D objects
• collecting, organizing, representing, analysing, summarizing, interpreting and reporting data 12 hours
• probability
153
MATHEMATICS GRADES 7-9
MATHEMATICS GRADES 7-9
section 4: assessment
4.1 Introduction
Assessment is a continuous planned process of identifying, gathering and interpreting information regarding the per-
formance of learners, using various forms of assessment. It involves four steps: generating and collecting evidence
of achievement; evaluating this evidence; recording the findings and using this information to understand and thereby
assist the learner’s development in order to improve the process of learning and teaching. Assessment should be
both informal and formal. In both cases regular feedback should be provided to learners to enhance their learning
experience. This will assist the learner to achieve the minimum performance level of 40% to 49% required in Math-
ematics for promotion purposes.
4.2 Types of assessment
The following types of assessment are very useful in Mathematics; as a result teachers are encouraged to use them
to serve the purpose associated with each.
Baseline assessment: Mathematics teachers who might want to establish whether their learners meet the basic
skills and knowledge levels required to learn a specific Mathematics topic will use baseline assessment. Knowing
learners’ level of proficiency in a particular Mathematics topic enables the teacher to plan her/his Mathematics lesson
appropriately and to pitch it at the appropriate level. Baseline assessment, as the name suggests, should therefore
be administered prior to teaching a particular Mathematics topic. The results of the baseline assessment should not
be used for promotion purposes.
Diagnostic assessment: It is not intended for promotion purposes but to inform the teacher about the learner’s
Mathematics problem areas that have the potential to hinder performance. Two broad areas form the basis of diag-
nostic assessment: content-related challenges where learners find certain difficulties to comprehend, and psycho-
social factors such as negative attitudes, Mathematics anxiety, poor study habits, poor problem-solving behaviour,
etc. Appropriate interventions should be implemented to assist learners in overcoming these challenges early in their
school careers.
Formative assessment: Formative assessment is used to aid the teaching and learning processes, hence assess-
ment for learning. It is the most commonly used type of assessment because it can be used in different forms at any
time during a Mathematics lesson, e.g. short class works during or at the end of each lesson, verbal questioning dur-
ing the lesson. It is mainly informal and should not be used for promotion purposes. The fundamental distinguishing
characteristic of formative assessment is constant feedback to learners, particularly with regard to learners’ learning
processes. The information provided by formative assessment can also be used by teachers to inform their methods
of teaching.
Summative assessment: Contrary to the character of formative assessment, summative assessment is carried out
after the completion of a Mathematics topic or a cluster of related topics. It is therefore referred to as assessment of
learning since it is mainly focusing on the product of learning. The results of summative assessment are recorded
and used for promotion purposes. The forms of assessment presented in Table 4.1 are examples of summative as-
sessment.
154 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
MATHEMATICS GRADES 7-9
4.3 Informal or daily assessment
Assessment for learning has the purpose of continuously collecting information on learner performance that can be
used to improve their learning.
Informal assessment is a daily monitoring of learners’ progress. This is done through observations, discussions, prac-
tical demonstrations, learner-teacher conferences, informal classroom interactions, etc. Informal assessment may be
as simple as stopping during the lesson to observe learners or to discuss with learners how learning is progressing.
Informal assessment should be used to provide feedback to learners and to inform planning for teaching, but need
not be recorded. It should not be seen as separate from the learning activities taking place in the classroom.
Self-assessment and peer assessment actively allow learners to assess themselves. This is important as it allows
learners to learn from, and reflect on their own performance. The results of the informal daily assessment tasks are
not formally recorded unless the teacher wishes to do so. The results of daily assessment tasks are not taken into
account for promotion purposes.
4.4 Formal assessment
Formal assessment comprises School-Based Assessment (SBA) and End of the Year Examination. Formal
assessment tasks are marked and formally recorded by the teacher for promotion purposes. All Formal Assessment
tasks are subject to moderation for the purpose of quality assurance and to ensure that appropriate standards are
maintained. The SBA component may take various forms. However, tests, examinations, projects, assignments
and investigations are recommended for Mathematics. The Senior Phase Mathematics minimum formal programme
of assessment tasks are outlined in Table 4.1
Table 4.1: Minimum requirements for formal assessment: Senior Phase Mathematics
Minimum Requirements per term Number of
Forms of
Tasks per Weighting
Assessment
Term 1 Term 2 Term 3 Term 4 Year
Test 1 1 1 3
Examination 1 1
SBA Assignment 1 1 1 3
40%
Investigation 1 1 2
Project 1 1
Total 2 3 3 2 10*
End of the year
1 60%
Examination
*To be completed before the End of the year Examination
Tests and examinations are individualised assessment tasks and should be carefully designed to ensure that
learners demonstrate their full potential in Mathematics content. The questions should be carefully spread to cater
for different cognitive levels of learners. Tests and examinations are predominantly assessed using a memorandum.
The Assignment, as is the case with tests and examinations, is mainly an individualised task. It can be a collection
of past questions, but should focus on more demanding work as any resource material can be used, which is not the
case in a task that is done in class under supervision.
CAPS 155
MATHEMATICS GRADES 7-9
Projects are used to assess a range of skills and competencies. Through projects, learners are able to demonstrate
their understanding of different Mathematics concepts and apply them in real-life situations. Caution should, how-
ever, be exercised not to give projects that are above learners’ cognitive levels. The assessment criteria should be
clearly indicated on the project specification and should focus on the Mathematics involved and not on duplicated
pictures and facts copied from reference material. Good projects contain the collection and display of real data, fol-
lowed by deductions that can be substantiated.
Investigation promotes critical and creative thinking. It can be used to discover rules or concepts and may involve
inductive reasoning, identifying or testing patterns or relationships, drawing conclusions, and establishing general
trends. To avoid having to assess work which is copied without understanding, it is recommended that whilst initial
investigation could be done at home, the final write-up should be done in class, under supervision, without access to
any notes. Investigations are assessed with rubrics, which can be specific to the task, or generic, listing the number
of marks awarded for each skill. These skills include:
• organizing and recording ideas and discoveries using, for example, diagrams and tables.
• communicating ideas with appropriate explanations
• calculations showing clear understanding of mathematical concepts and procedures.
• generalizing and drawing conclusions,
The forms of assessment used should be appropriate to the age and cognitive level of learners. The design of these
tasks should cover the content of the subject and designed to achieve the broad aims of the subject. Appropriate
instruments, such as rubrics and memoranda, should be used for marking. Formal assessments should cater for a
range of cognitive levels and abilities of learners as shown in Table 4.2:
156 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
MATHEMATICS GRADES 7-9
Table 4.2: Cognitive levels
DESCRIPTION AND EXAMPLES OF COGNITIVE LEVELS
Cognitive levels Description of skills to be demonstrated Examples
• Estimation and appropriate rounding of 1. Estimate the answer and then calculate with a
62 816
numbers calculator: 325 + 279 [Grade 7]
• Straight recall 2. Use the formula A = πr2 to calculate the area
Knowledge
• Identification and direct use of correct formula of a circle if the diameter is equal to 10 cm.
(≈25%) [Grade 8]
• Use of mathematical facts
3. Write down the y-intercept of the function
• Appropriate use of mathematical vocabulary y = 2x + 1 [Grade 9]
• Perform well-known procedures 1. Determine the mean of 5 Grade 7 learners’
marks if they have respectively achieved 25;
• Simple applications and calculations which
40; 21; 85; 14 out of 50. [Grade 7]
might involve many steps
Routine 2. Solve x in x – 6 = 9 [Grade 8]
procedures • Derivation from given information may be
involved 3. R600 invested at r% per annum for a period
(≈45%) of 3 years yields R150 interest. Calculate the
• Identification and use (after changing the P.n.r
value of r if SI = 100 . [Grade 9]
subject) of correct formula
• Generally similar to those encountered in class
• Problems involving complex calculations and/or 1. Mr Mnisi pays R75 for a book which he marks
higher order reasoning up to provide 20% profit. He then sells it for
cash at 4% discount. Calculate the selling
• Investigate elementary axioms to generalize
price. [Grade 7]
them into proofs for straight line geometry,
congruence and similarity 2. A car travelling at a constant speed travels
Complex 60 km in 18 minutes. How far, travelling at the
procedures • No obvious route to the solution
same constant speed, will the car travel in 1
(≈20%) • Problems not necessarily based on real world hour 12 minutes? [Grade 8]
contexts
3. Use investigation skills to prove that the
• Making significant connections between angles on a straight line are supplementary.
different representations [Grade 9]
• Require conceptual understanding
• Unseen, non-routine problems (which are not 1. The sum of three consecutive numbers is 87.
necessarily difficult) Find the numbers. [Grade 7]
• Higher order understanding and processes are 2. Mary travels a distance of km in 6 hours if she
often involved travels at an average speed of 20 km/h on her
Problem solving bicycle. What should be her average speed
• Might require the ability to break the problem
if she wants to cover the same distance in 5
(≈10%) down into its constituent parts
hours? [Grade 8]
3. The combined age of a father and son is 84
years old. In 6 years time the father will be
twice as old as the son was 3 years ago. How
old are they now? [Grade 9]
4.5 Recording and reporting
Recording is a process in which the teacher documents the level of a learner’s performance in a specific assessment
task. It indicates the learner’s progress towards the achievement of the knowledge as prescribed in the National
Curriculum and Assessment Policy Statements. Records of learner performance should provide evidence of the
learner’s conceptual progression within a grade and her/his readiness to be promoted to the next grade. Records of
learner performance should also be used to verify the progress made by teachers and learners in the teaching and
learning process.
CAPS 157
MATHEMATICS GRADES 7-9
Reporting is a process of communicating learner performance to learners, parents, schools, and other stakeholders.
Primary schooling is a critical period for the acquisition of foundational Mathematics skills and conceptual knowledge.
Reporting of learner performance is therefore essential and should not be limited to the quarterly report card. Other
methods of reporting should be explored, e.g. parents’ meetings, school visitation days, parent-teacher conferences,
phone calls, letters. These extreme, but worthwhile modalities will ensure that any underperformance is communi-
cated promptly and appropriate measures of intervention are implemented collaboratively by teachers and parents.
Formal reporting is done on a 7-point rating scale (see Table 4.3)
Table 4.3: Scale of achievement for the National Curriculum Statement Grades 7 - 9
RATING CODE DESCRIPTION OF COMPETENCE PERCENTAGE
7 Outstanding achievement 80 – 100
6 Meritorious achievement 70 – 79
5 Substantial achievement 60 – 69
4 Adequate achievement 50 – 59
3 Moderate achievement 40 – 49
2 Elementary achievement 30 – 39
1 Not achieved 0 – 29
4.6 Moderation of assessment
Moderation refers to the process that ensures that the assessment tasks are fair, valid and reliable. Moderation should
be carried out internally at school and/or externally at district, provincial and national levels. Given that the promotion
of learners in the Senior Phase is largely dependent upon the SBA (which contributes 40%); the moderation process
should be intensified to ensure that:
• learners are not disadvantaged by the invalid and unreliable assessment tasks,
• quality assessment is given and high but achievable standards are maintained.
4.7 General
This document should be read in conjunction with:
4.7.1 National policy pertaining to the programme and promotion requirements of the National Curriculum
Statement Grades R-12; and
4.7.2 National Protocol for Assessment Grades R-12.
158 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
MATHEMATICS GRADES 7-9
CAPS 159
MATHEMATICS GRADES 7-9
160 CURRICULUM AND ASSESSMENT POLICY STATEMENT (CAPS)
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