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Reg. No.: 2011/011959/07
Mathematics
Facilitator's Guide 1/2 − Grade 10
2310-E-MAM-FG01
CAPS-aligned
Prof. C Vermeulen, Lead author P de Swardt H Otto
M Sherman E van Heerden L Young
PREFACE
In Grade 10, mathematics is an optional subject (as an alternative to mathematical literacy) for the first time. There may be various reasons why learners might choose mathematics as a subject, for example to prepare them for a field of study where Grade 12 mathematics is a prerequisite, or a career in which a background in mathematics would be advantageous.
In general, mathematics in the Further Education and Training (FET) phase from Grade 10 to 12 involves more abstract concepts and more complex procedures than in the General Education and Training (GET) phase from Grade 1 to 9. Mastering mathematics in the FET phase requires more time, commitment, critical thought and reflection than in the GET phase.
This product consists of two study guides and two facilitator’s guides, which are based on the concepts of Optimi’s GuidEd Learning™ model to help learners and facilitators achieve success in their study of mathematics. These books cover all work required for Grade 10 mathematics and have been compiled in accordance with the CAPS guidelines as required by the Department of Basic Education.
The study guides are supported by supplementary lesson structures on the Optimi Learning Platform (OLP), which is an online platform. These lesson structures offer continuous guidance to support and enrich learners’ learning process. This guidance is based on the latest insights in education, cognitive psychology and neuroscience. Note that the study guides can also be used independently of the OLP.
In the following section, we explain how the study guides and facilitator’s guides have been compiled and how learners and facilitators can use these to achieve success in mathematics. The study guides and facilitator’s guides are divided into 15 themes. Study guide 1/2 and facilitator’s guide 1/2 cover themes 1 to 8 (terms 1 and 2) and study guide 2/2 and facilitator’s guide 2/2 cover themes 9 to 15 (terms 3 and 4). The themes correspond with the CAPS guidelines with regard to content and time allocation and represent the year plan.
You will find the latest and most comprehensive information on assessment in the portfolio book and assessment plan.
Time allocation
According to the CAPS requirements, at least 4,5 hours should be spent on teaching mathematics per week. For example, 13,5 hours (three weeks × 4,5 hours per week) will be spent on teaching Theme 1 (algebraic expressions). Themes have not been sub-divided into lessons; learners and facilitators are at liberty to complete as much content per session and per week as learners’ progress allows. If learners work at a slower pace, the necessary adaptations should be done so that they will still be able to master all the work in time.
Tip: Use the suggested time allocation along with your learners’ progress to plan your lessons.
Note that the teaching time referred to above does not include the time during which learners should apply and practise the knowledge and concepts they have learned. For this purpose, various exercises are provided throughout each theme. These exercises involve different ways of applying and practising new knowledge and cover various degrees of difficulty. Learners should try to do all of these exercises. Complete solutions are provided in the facilitator’s guide.
Tip: Ensure that learners do as many of these exercises as possible. Follow up and offer support when learners struggle.
Structure of themes
Learning is a complex process. Millions of brain cells and neural pathways in our brains work together to store new information in the long-term memory so that we will be able to remember it later on. Long-term memory is not our only type of memory and when we learn, our working memory is just as important. Working memory is different from long-term memory and has a limited capacity. This means that one’s working memory can only handle a small amount of new information at a time.
compiled in such a way that it does not overtax the working memory and therefore simplifies the process of learning mathematics. Learners’ cognitive capacity is taken into account at all times.
SampleWhen one learns mathematics, there is a lot of new information your brain needs to process, which can easily exhaust your working memory. This is related to the cognitive load theory. The study guides have been written and
This means that various strategies are used to ensure that learners have the best possible chance of mastering every section of the work. Ultimately one can say that learning has taken place when learners have stored new information in their long-term memory and have the ability to recall and use this information. The structure of the study guides support this process and help learners master mathematics.
Tip: Each theme has the same structure in order to make it easier for learners to navigate through them.
Each theme has the following structure:
Introduction
What this theme is about
This briefly tells learners what the theme is about without providing details or using “difficult” or unknown concepts. A comprehensive list of the learning outcomes learners need to master in a specific theme is given as a summary at the end of the theme.
Prior knowledge
This section tells learners what existing knowledge they need to master the theme involved.
Revision
This may involve one of the following:
1. revision of the concepts, definitions and procedures required as previous knowledge,
2. an exercise or activity with solutions so that you can test your previous knowledge yourself, or
3. a combination of the above.
Do not neglect this revision. It is important to work through this section thoroughly. Mathematical concepts often follow on one another and if basic knowledge is lacking or has not been mastered sufficiently, this will handicap the formation of new knowledge.
Following the introductory part of the theme, new knowledge is dealt with in sub-themes. Each sub-theme has the following structure:
SUB-THEME
Introduction
New concepts and procedures are explained. Relevant previous knowledge is also dealt with here if necessary.
Worked examples
Worked examples show learners how the new concepts and procedures are applied and help them understand and apply the newly taught concepts and procedures.
Exercises
The exercises give learners the opportunity to practise the concepts and procedures taught. It is important for them to try and complete all exercises. Complete solutions are provided in the facilitator’s guides.
Sample
Questions usually progress from easy (in order to master and practise basic concepts and procedures) to difficult (more complex operations).
Mixed exercises are also provided, where learners get the opportunity to practise different concepts and procedures and integrate these with previous themes.
Summary of theme
Here learners will find a summary of what they should have mastered in the theme. This is expressed in more formal mathematical language in order to be in keeping with the CAPS (the curriculum statement).
End of theme exercise
This is a mixed exercise involving all concepts and procedures dealt with in the theme, where this work can also be integrated with previous work. The degree of difficulty of this exercise varies. It is important that learners try and complete all the exercises. Complete solutions can be found in the facilitator’s guides.
Mixed exercises such as these in this textbook form a very important component of mastering mathematics. There is a big difference between the ability to recognise one’s work and the ability to recall it. When learners are able to recognise their work, they will often say “Oh, of course!” but they struggle to remember this when they are writing an examination. When they are able to recall their work, this means that they have captured that knowledge in their
long-term memory and are able to remember and use it. Mixed exercises enable learners to not only recognise the work, but also recall it from their long-term memory.
When learners practise the same type of sum or problem over and over, they often get lazy and do not reflect upon the exercise anymore. They are convinced that they know exactly what type of sum or problem they need to solve. But in a test or exam, all these problems are mixed up and then it might be difficult to know what to do. When mixed exercises form part of learners’ learning process, they learn to identify and complete a sum or problem correctly. This means that they are truly prepared for tests or exams, because they can recall their work instead of merely recognising it.
Self-evaluation
In each theme, and usually following each sub-theme, there is an activity where learners need to reflect critically about the extent to which they have mastered certain concepts and procedures. This activity has the following format:
Use the following scale to determine how comfortable you are with each topic in the table below:
1. Help! I don’t feel comfortable with the topic at all. I need help.
2. Alarm! I don’t feel comfortable, but I just need more time to work through the topic again.
3. OK! I feel moderately comfortable with the topic, but I still struggle sometimes.
4. Sharp! I feel comfortable with the topic.
Complete the table:
Sample
5. Whoo-hoo, it’s party time! I feel totally comfortable with the topic and can even answer more complicated questions about it.
Facilitators should use this evaluation to ascertain whether learners need more help in the theme or sub-theme involved. If so, it is recommended to do revision or more exercises immediately in order to ensure that learners master the essential concepts and procedures. The self-evaluation can also be used to plan enrichment. If learners have mastered the work in that theme or subtheme, enrichment exercises can be done.
It is important not to move on to the next theme or sub-theme before the topic involved has been completely taught and mastered, even if this means that you spend more time on a specific theme than recommended by the CAPS. Be flexible in adapting the time allocation according to the learners’ needs. However, it is also important to complete the themes involved before a test or exam is written.
Tip: Use learners’ self-evaluation to decide whether they need assistance with the section involved, what the nature of such assistance should be, and whether you could move on to the next section.
Assessment
criteria
Visit Impaq’s online platform for the assessment plan and comprehensive information about the compilation and mark allocation of tests, assignments and examinations. The number of assignments, mark allocation and relative weighting are subject to change.
Tip: Be aware of the CAPS requirements and plan the year’s assessment accordingly.
Learners complete seven formal assessment tasks for school-based assessment.
The two papers at the end of the year are compiled as follows:
Paper 1
Algebraic expressions, equations and inequalities, exponents (Theme 1, 2 and 4)
Number patterns (Theme 3)
Functions and graphs (Theme 6)
Finance and growth (Theme 10)
Probability (Theme 15)
Paper 2
Note:
• Only one project/investigation should be done per year.
• No graphing or programmable calculators are allowed (for example to factorise or find the roots of equations). Calculators should only be used to do standard numeric calculations and to verify calculations done by hand.
• Formula sheets are not provided during tests and final examinations in Grade 10.
Tip: This table only indicates the formal assessment (i.e. the assessment used for promotion). Informal continuous assessment should also take place to monitor each learner’s progress so that gaps in learners’ knowledge are seen and rectified timeously.
Sample
Euclidean geometry and measurement (Theme 8, 13 and 14)
Analytical geometry (Theme 9)
Trigonometry (Theme 5 and 7)
Statistics (Theme 11)
Tip: You need to know which themes are covered in which papers, as well as the relative weighting of each. Make sure that papers meet the requirements of this distribution.
Note: The themes covered in the examination papers are subject to change. Always refer to the portfolio book and assessment plan for updated information about the composition of the examination papers.
Supplementary books
Any other books can be used along with this textbook for extra exercises and explanations, including:
• Maths 4 A��rica, available at www.maths4africa.co.za
• The Si��avula textbook, available online for free at www.siyavula.com
• P��thagoras, available at www.fisichem.co.za.
Tip: Help learners obtain and use supplementary resources efficiently.
Calculator
We recommend the CASIO fx-82ES (Plus) or CASIO fx-82ZA. However, any scientific, non-programmable and non-graphing calculator is suitable.
Tip: Ensure that each learner has a suitable calculator.
THEME 1
NUMBERS AND ALGEBRAIC EXPRESSIONS
Learning requirements according to CAPS
Learners should be able to:
1. understand that real numbers can be rational or irrational
2. establish between which two integers a given simple surd lies
3. round real numbers to an appropriate degree of accuracy
4. multiply a binomial by a trinomial
5. factorise expressions using techniques taught in Grade 9 as well as:
• trinomial (more advanced trinomials)
• grouping in pairs
• sum and difference of two cubes
6. simplify algebraic fractions by factorising denominators, which may include cubes (limited to sum and difference of cubes).
Term 1
Duration 3 weeks
Paper 1
Weight
Algebraic expressions form part of algebra, of which the weighting is 30±3 of Paper 1.
Introduction
In this theme learners will learn more about:
1. different types of numbers
Sample2. how to estimate the values of certain numbers
3. how to round off numbers
4. how to multiply algebraic expressions
5. how to find factors of algebraic expressions
6. how to simplify algebraic fractions.
Prior knowledge
In order to master this theme, learners should already know:
• what types of numbers there are
• how we classify numbers
• how to multiply simple algebraic expressions
• how to simplify simple fractions.
1.1 THE NUMBER SYSTEM
Introduction
This sub-theme is a summary of work covered in Grades 8 and 9. If learners should find it difficult to complete this section, you should first revise the work covered in these grades.
Different types of numbers
ℕ = { 1; 2; 3; 4; 5; …} = natural numbers
ℕ 0 = { 0; 1; 2; 3; 4; 5; …} = whole numbers
ℤ = { … ; − 2; − 1; 0; 1; 2; 3; …} = integers
ℚ = { numbers that can be written as an integer a non-zero integer } = rational numbers
ℚ ′ = { numbers that cannot be written as an integer a non-zero integer }= irrational numbers (non-terminating and non-repetitive decimal numbers)
ℝ = {rational and irrational numbers} = real numbers
ℝ ′ = {numbers that do not exist in the real number system} = non-real numbers
Summary of the real number system
Rational numbers
Irrational numbers Real numbers
Integers Fractions
• Negative integers
• Zero
• Positive integers (natural numbers)
REMEMBER
Note
• A rational number is any number that can be written as a b , where a and b are integers, where b ≠ 0.
Sample
0 �� = 0 √ − 3 is non-real �� 0 is undefined
• The following are rational numbers:
◦ fractions of which both the numerator and denominator are integers, e.g. 3 7
◦ integers, e.g. − 5
◦ decimal numbers that end, e.g. 0,125
◦ decimal numbers that repeat, e.g. 0,151515…
• Irrational numbers are not rational. They cannot be written with an integer numerator and denominator, e.g. 0,8672345…
• If the nth root of a number cannot be written as a rational value, this nth root is called a surd, e.g. 3√ 5 .
Worked example 1
Rewrite 0, 1 ˙ 2 ˙ as a common fraction.
Solution
In order to rewrite a recurring fraction as a common fraction, you need to manipulate the recurring fraction to lose the recurring “tail”.
Let ��= 0,1212121212…
∴ 100��= 12,1212121212… × 100 to get integer + recurring “tail”
�� = 0,1212121212… Subtract
12
Worked example 2
Rewrite 2,51 ˙ 2 ˙ as a common fraction.
�� = 2,512121212…
1 000�� = 2512,121212… × 1 000 and × 10 to get integer + recurring “tail”
10�� = 25,121212… Remember that the 5 is not recurring
Worked example 3
Using your knowledge of the number system, complete the following table by making a in the appropriate block(s):
Sample
Solution
In order to determine where these numbers fit into the number system, you can use your calculator to find the decimal fraction where applicable:
This number is written in the form a b ; therefore it is a rational number (ℚ).
It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ).
This number cannot be written in the form a b therefore it is an irrational number (ℚ').
It exists (your calculator does not give you a math error message); therefore it is a real number (ℝ).
Note that real numbers (ℝ) are either rational (ℚ) or irrational (ℚ') numbers.
Exercise 1.1: The number system
1. Is the number zero a positive or negative number?
Zero is neither positive nor negative.
2. What type of number is √ 8 ?
√ = √ 4 × 2 = 2 √ 2
∴ a real, irrational number
3. What type of number is √ −8 ?
The square root of a negative number does not exist in the real number system.
∴ a non-real number
4. What type of number is 3√ 8 ? 3
= 2
∴ a real number, a rational number, an integer, a whole number and a natural number
5. What type of number is 3√ −8 ?
3√ −8 = 2
∴ a real number, a rational number and an integer
6. Without using a calculator, determine all the number types that 2 10 27 belongs to. 2 10 27 = 64 27
∴ a real, rational number (64 ÷ 27 gives a non-finite, recurring decimal number as an answer)
7. Rewrite the following as common fractions:
Always remember to simplify completely.
8. For which value(s) of �� will ��(��) be non-real if:
��(��)= √ 9 11
�� and�� ∈ {−5; 0; 11}?
For the expression to be non-real, the value in the root will be a negative number. Substitute the given elements to determine which value will give a negative answer:
If the nth root of a number cannot be simplified to a rational number, we call it a surd. For example, √ 2 and 6√ 3 are surds, but √ 4 is not a surd because it can be simplified to the rational number 2.
Consider surds of the form n√ a , where a is any positive number, for example √ or 3√ 5 . It is very common for n to be 2, so we usually do not write 2√ a . Instead, we write the surd as simply √ a . This is called the square root of a. It is sometimes useful to know the approximate value of a surd without having to use a calculator.
For example, let us estimate where √ 3 lies on the number line:
Using a calculator, we know that √ = 1,73205…
It is easy to see that √ 3 is greater than 1 and smaller than 2.
Sample
But to estimate the values of other surds, such as √ 18 , without using a calculator, learners should first understand the following:
• If a and b are positive integers and a < b, then n√ a < n√ b .
• A perfect square is the number obtained when an integer is squared. For example, 9 is a perfect square since 3 2 = 9.
• A perfect cube is a number which is the cube of an integer. For example, 27 is a perfect cube, because 3 3 = 27.
Worked example 4
Determine between which two integers the irrational number √ 62 lies.
Solution
Find the two perfect squares to the left of (just smaller than) and to the right of (just bigger than) 62 on the number line.
• The perfect square to the left of 62 is 49.
• The perfect square to the right of 62 is 64.
• Revision exercises to refresh prior knowledge.
• Detailed explanations of concepts and techniques.
• Worked examples help learners to better understand new concepts.
• Varied exercises to entrench theory and practise mathematical skills.
• Test papers and memorandums for exam preparation
• Formula sheets and accepted geometrical reasons for quick reference.
• Index of mathematical terms.
• The facilitator’s guide contains step-by-step calculations and answers.
