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Summary Grade 9 Mathematics Study Guide
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MIND ACTI
ON SERI
ES
MATHEMATI
CS
9 Text
book
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AssessmentPolicyStat
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D.Phi
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ips,J.Basson&
J.Odendaal
, ALL RIGHTS RESERVED
©COPYRIGHT BY THE AUTHOR
The whole or any part of this publication may not be reproduced or transmitted in any
form or by any means without permission in writing from the publisher. This includes
electronic or mechanical, including photocopying, recording, or any information storage
and retrieval system.
Every effort has been made to obtain copyright of all printed aspects of this publication.
However, if material requiring copyright has unwittingly been used, the copyrighter is
requested to bring the matter to the attention of the publisher so that the due
acknowledgement can be made by the author.
Maths Textbook Grade 9 NCAPS
ISBN 13: 978-1-869-21973-4 ( Perpetual book )
978-1-869-21775-4 ( 1 Year License )
Product Code: MAT 145
Authors: M.D Phillips
J. Basson
J. Odendaal
First Edition: August 2014
Second Edition: November 2014
Third Edition: August 2015 (Minor revisions)
PUBLISHERS
ALLCOPY PUBLISHERS
P.O. Box 963
Sanlamhof, 7532
Tel: (021) 945 4111, Fax: (021) 945 4118
Email: info@allcopypublishers.co.za
Website: www.allcopypublishers.co.za
i
, MATHS TEXTBOOK
GRADE 9 NCAPS INFORMATION
The AUTHORS
MARK DAVID PHILLIPS
B.SC, H Dip Ed, B.Ed (cum laude) University of the Witwatersrand.
Mark Phillips has over twenty-five years of teaching experience in both state and private schools and has a
track record of excellent matric results. He has presented teacher training and learner seminars for various
educational institutions including Excelearn, Isabelo, Kutlwanong Centre for Maths, Science and
Technology, Learning Channel, UJ and Sci-Bono. Mark is currently a TV presenter on the national
educational show Geleza Nathi, which is broadcast on SABC 1. He has travelled extensively in the United
Kingdom, Europe and America, gaining invaluable international educational experience. Based on his
experiences abroad, Mark has successfully adapted and included sound international educational
approaches into this South African textbook. The emphasis throughout the textbook is on understanding
the processes of Mathematics.
JURGENS BASSON
B.A (Mathematics and Psychology) HED RAU (UJ)
Jurgens Basson is a Mathematics consultant with more than 25 years experience and expertise in primary,
secondary and tertiary education. His passion is the teaching of Mathematics to teachers, students and
learners. Jurgens founded and started the RAU / Oracle School of Maths in 2002 in partnership with
Oracle SA. Due to the huge success of this program in the community, there are now several similar
programs running in the previously disadvantaged communities that are being sponsored by corporate
companies. During 2012, one of his projects sponsored by Anglo Thermal Coal in Mpumalanga was voted
Best Community Project of the Year. Jurgens had the privilege to train and up skill more than 15 000
teachers countrywide since 2006. He was also part of the CAPS panel that implemented the new syllabus
for Grade 10 - 12 learners. He successfully co-authored the Mind Action Series Mathematics textbooks,
which received one of the highest ratings from the Department of Education.
JACO ODENDAAL
B.Sc (Mathematics and Applied Mathematics)
Jaco Odendaal has been an educator of Mathematics and Advanced Programme Mathematics for the past
ten years. He is currently the Head of Mathematics at Parktown Boys’ High School in Parktown,
Johannesburg. He has worked as a teacher trainer in many districts in Limpopo and Mpumalanga. Jaco
has a passion for Mathematics and the impact it can have in transforming our society. He has been
successful in teaching and tutoring Mathematics at all levels, from primary school to university level. He
has also worked with learners at both ends of the spectrum, from struggling to gifted learners. He is known
for his thorough, yet simple explanation of the foundational truths of Mathematics.
The CONCEPT
The focus in this textbook is on providing learners with crucial background knowledge and skills needed to
cope with Mathematics in the higher grades. The emphasis is on the understanding of concepts which are
reinforced through more than enough quality examples and exercises. Each exercise is based on given
concepts at a “standard grade” level in order to ensure that learners master the basics. At the end of each
chapter, revision exercises consolidate all of the concepts by providing learners with an opportunity to
tackle questions of a mixed nature. These are the typical examination-type questions. The challenge
questions are of a “higher grade” nature and serve to extend the learners and develop problem-solving
skills. The Grade 8 and 9 publications serve to bridge the gap between the Senior Phase and FET phase.
The educator’s guide contains approaches to the teaching of the particular topic, detailed solutions to the
exercises, and various assessment tasks. The publication is also available in e-book/e-pub format.
ENDORSEMENT
This textbook is truly amazing. It is such pleasure to be able to work through a book that has thorough
explanations and excellent examples. I am blown away with the different approaches and methodologies
used and the layout is user friendly. The exercises are lengthy and cater for students of varied abilities.
Each section is well presented and thoroughly researched. It is such a breath of fresh air to work with a
book that is completely CAPS aligned. The challenges at the end of the chapter really extend my learners
and I am able to alleviate the boredom of my bright learners while working with my weaker learners.
This textbook is truly in a league of its own and will help to raise the standard of Maths in this country.
Belinda Pretorius (Curro Inter-schools Head of Mathematics)
ii
, MATHEMATICS
TEXTBOOK
GRADE 9 NCAPS
CONTENTS
CHAPTER 1 WHOLE NUMBERS 1
CHAPTER 2 INTEGERS 34
CHAPTER 3 COMMON FRACTIONS 39
CHAPTER 4 DECIMAL FRACTIONS 47
CHAPTER 5 EXPONENTS 56
CHAPTER 6 NUMERIC AND GEOMETRIC PATTERNS 69
CHAPTER 7 FUNCTIONS AND RELATIONSHIPS 80
CHAPTER 8 ALGEBRAIC EXPRESSIONS 94
CHAPTER 9 ALGEBRAIC EQUATIONS 110
CHAPTER 10 CONSTRUCTIONS 127
CHAPTER 11 GEOMETRY OF 2D SHAPES 143
CHAPTER 12 GEOMETRY OF LINES 175
CHAPTER 13 THEOREM OF PYTHAGORAS 181
CHAPTER 14 AREA AND PERIMETER OF 2D SHAPES 191
CHAPTER 15 GRAPHS 212
CHAPTER 16 SURFACE AREA AND VOLUME OF 3D SHAPES 238
CHAPTER 17 TRANSFORMATIONS 254
CHAPTER 18 GEOMETRY OF 3D SHAPES 273
CHAPTER 19 DATA HANDLING 290
CHAPTER 20 PROBABILITY 308
iii
,
, CHAPTER 1: NUMBERS, OPERATIONS AND RELATIONSHIPS
TOPIC: WHOLE NUMBERS
What is a number? This is a very important question in mathematics, but not an easy one to answer.
If you could ask someone living in 500 BC, you would not get the same answer as when you ask a
21st century mathematician. Many numbers we work with today were totally unheard of or
considered very strange in earlier times. Just think of your own journey in Mathematics so far.
Before you started school, you may have thought of numbers merely as tools for counting. Your
entire number system would have been: 0; 1; 2; 3; 4; ... etc. During your primary school career, you
learnt about negative numbers, fractions and decimals. In Grade 8 you came across interesting
numbers like 2 and π . In fact, the way in which humanity discovered new kinds of numbers is not
all too different from the way a typical child finds out about these numbers today. It just happens
much quicker now, because it’s all been done before. We will now see how one type of number leads
to the next by asking the right questions.
THE NUMBER SYSTEM
Some of the most important building blocks of mathematics are numbers, operations and
relationships. Think of the simple sentence: 1 2 3 . Here we have all three of the building blocks.
The ‘1’, ‘2’ and ‘3’ are numbers, the ‘+’ is the operation and the ‘ ’ is the relationship. You will see
that, as we introduce more operations, we need new types of numbers to answer the questions and
this is how our number system grows.
Whole Numbers
Whole numbers are the numbers we use to count: 0; 1; 2; 3; 4; 5; ... You should remember, from
Grade 8:
The set of natural numbers N 1; 2;3; 4;5;...
The set of whole numbers N0 0;1; 2;3; 4;5;...
Some mathematicians don’t distinguish between natural numbers and whole numbers and include 0
in the natural numbers. We will keep the distinction. Any two whole numbers can be added or
multiplied and the result will still be a whole number. When we subtract or divide, we may run into
trouble. What is 4 – 7 or 2 5 , for example? To find the answers to these questions, we need new
types of numbers. Let’s start with the subtraction issue.
Integers
In order to get an answer for something like 4 – 7 (where we subtract a greater number from a
smaller number), we need negative numbers. You should know that 4 7 3 . The number –3 does
not belong to the set of whole numbers. If we take all the whole numbers, together with all the
negative numbers, we get a new set of numbers called integers. The symbol for integers is Z . This
comes from the German word ‘zahlen’ which means number. Z ...; 3; 2; 1;0;1; 2;3;... .
Note that the integers include the whole numbers. All whole numbers are also integers, but careful –
not all integers are whole numbers.
We can add, multiply or subtract any two integers and the result will always be another integer, but
what about division? We have no integer to give as an answer to 2 5 .
Rational Numbers
So what is the answer to 2 5 ? From your knowledge of fractions and decimals, you should know
2
that the answer is or 0,4. If we allow for fractions, we can perform any division of two integers:
5
a
a b where a and b are integers. The only exception is division by 0. We cannot divide by 0.
b
This is undefined.
1
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