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Reg. No.: 2011/011959/07
Mathematics
Study Guide 1/2 − Grade 11 2411-E-MAM-SG01
Prof. C Vermeulen, Lead author P de Swardt H Otto M Sherman E van Heerden
Exercise
Exercise
5.1.8
Exercise
5.1.9
Exercise
5.2
5.2.1 The effect of the parameters p and
5.2.2 Characteristics of hyperbolic function f(x) =
Exercise 5.2.1: Properties of the hyperbolic
5.2.3 Drawing sketch graphs of hyperbolas ........................................................119
Exercise 5.2.2: Drawing sketch graphs of hyperbolas ...............................119
5.2.4 Determining the equation of a hyperbola ..................................................120
Exercise 5.2.3: Determining the equation of a hyperbola..........................120
5.2.5 Average gradient of a curve
Self-evaluation
5.3 Exponential function ...............................................................................122
5.3.1 The effect of the parameters p and q ........................................................122
5.3.2 Characteristics of the exponential function y = a . b x + p + q, b > 0 and b ≠ 1 and its graph ..............................................................................125
Exercise 5.3.2: Properties of the exponential function ..............................126
5.3.3 Drawing sketch graphs of exponential functions .......................................127
Exercise 5.3.3: Drawing sketch graphs of the exponential function
5.3.4 Determining the equation of an exponential function ...............................128
5.3.5 Average gradient of a curve .......................................................................129
5.4 Making deductions from sketch graphs.................................................130
Exercise 5.4: Deductions from sketch graphs ............................................131 Sample
6.6
6.6.1 Writing the trigonometric ratio of any angle as a trigonometric ratio of an acute angle ........................................................................................171
Exercise 6.6.1: Use reduction formulae to simplify expressions ...............177
6.6.2 Apply reduction formulae to work with special angles without using a calculator ..........................................................................177
Exercise 6.6.2: Using reduction formulae to determine the values of trigonometric ratios of special angles without the use of a calculator ......179
6.7.1 Equations of the type: Trigonometric ratio of an angle = a number .........181
Exercise 6.7.1: Solving trigonometric equations of the type: Trigonometric ratio of an angle = a number ..............................................184
6.7.2 Equations of the type where sin of an angle and cos of an angle are changed to tan of an angle ...................................................................184
Exercise 6.7.2: Solving trigonometric equations, where sin of an angle and cos of an angle are changed to tan of an angle .....................186
6.7.3 Equations of the type where factorisation is needed .................................186
Exercise 6.7.3: Solving trigonometric equations where factorisation is needed .... 187
6.7.4 Equations of the type where the trigonometric ratio of an angle = the same trigonometric ratio of another angle .........................187
6.7.5 The use of the co-ratio when the trigonometric ratios are co-ratios ..........188
Exercise 6.7.4: Solving trigonometric equations where the trigonometric ratio of an angle = the same trigonometric ratio of another angle, or a co-ratio .... 188
PREFACE
Sample
You will find the latest and most comprehensive information on assessment in the portfolio book and assessment plan.
Optimi's Grade 11 mathematics offering consists of two study guides and two facilitator’s guides, which are based on the concepts of Optimi’s GuidEd Learning™ model to help you achieve success in your study of mathematics. These books cover all work required for Grade 11 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 your 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 you can use these to achieve success in mathematics. The study guides and facilitator’s guides are divided into twelve themes. Study guide 1/2 and facilitator’s guide 1/2 cover themes 1 to 6 (terms 1 and 2) and study guide 2/2 and facilitator’s guide 2/2 cover themes 7 to 12 (terms 3 and 4). The themes correspond with the CAPS guidelines with regard to content and time allocation and represent the year 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; you and facilitators are at liberty to complete as much content per session and per week as your progress allows.
If you work at a slower pace, the necessary adaptations should be done so that you will still be able to master all the work in time.
Note that the teaching time referred to above does not include the time during which you should apply and practise the knowledge and concepts you 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. You should try to do all of these exercises. Complete solutions are provided in the facilitator’s guide.
Tip: The more exercises you do, the greater the chance that you will achieve success in mathematics.
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.
When one learns mathematics, there is a lot of new information your brain needs to process, which can easily exhaust your working memory. The study guides have been written and compiled in such a way that it does not overtax the working memory and therefore simplifies the process of learning mathematics.
Each theme has the same structure to make it easier for you to navigate through the study guide. The structure is as follows:
Introduction
This briefly tells you what the theme is about without providing details or using ‘difficult’ or unknown concepts. A comprehensive list of the learning outcomes you need to master in a specific theme is given as a summary at the end of the theme.
Prior knowledge
Revision
This may involve one of the following:
Sample
This section tells you what existing knowledge you need to master the theme involved.
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 subthemes. Each sub-theme has the following structure:
1.1 SUBTHEME
Introduction
New concepts and procedures are explained. Relevant previous knowledge is also dealt with here if necessary.
Worked examples
Worked examples show you how the new concepts and procedures are applied and help you understand and apply the newly taught concepts and procedures.
Exercises
The exercises give you the opportunity to practise the concepts and procedures taught. It is important for you to try and complete all exercises. Complete solutions are provided in the facilitator’s guides.
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 you get the opportunity to practise different concepts and procedures and integrate these with previous themes.
Summary of theme
Here you will find a summary of what you 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 you 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 you are able to recognise your work, you will often say ‘Oh, of course!’ but you may struggle to remember this when you are writing an examination.
Sample
When you are able to recall your work, this means that you have captured that knowledge in your long-term memory and are able to remember and use it. Mixed exercises enable you to not only recognise the work, but also recall it from your long-term memory.
When you practise the same type of sum or problem over and over, you often get lazy and do not reflect upon the exercise anymore. You are convinced that you know exactly what type of sum or problem you 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 your learning process, you learn to identify and complete a sum or problem correctly. This means that you are truly prepared for tests or exams, because you can recall your work instead of merely recognising it.
Self-evaluation
In each theme, and usually following each sub-theme, there is an activity where you need to reflect critically about the extent to which you 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. Alarm! I don’t feel comfortable, but I just need more time to work through the topic again.
2. Help! I don’t feel comfortable with the topic at all. I need help.
3. OK! I feel moderately comfortable with the topic, but I still struggle sometimes.
4. Sharp! I feel comfortable with the topic.
5. Party time! I feel totally comfortable with the topic and can even answer more complicated questions about it.
Complete the table:
Tip: Complete each self-evaluation as honestly as possible. If there are aspects which you have not mastered, revisit these and make sure that you do master them. Ask the facilitator for help. It is important not to move on to a next theme or sub-theme before you have mastered the topic involved, even if this means that you spend more time on a specific theme than recommended by the CAPS.
Assessment criteria
Visit Impaq’s online platform (OLP) 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.
The two papers at the end of the year are compiled as follows:
Exponents and surds, and Quadratic equations and inequalities (Theme 1 and 2)
(Theme 10)
Tip: Make sure that you know which themes are covered in which paper. 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.
Note:
Sample
• 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 11.
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.
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 you have a suitable calculator.
THEME 1
EXPONENTS AND SURDS
Introduction
In this theme you will learn more about:
1. simplifying exponential expressions with exponents containing fractions
2. solving equations with exponents containing fractions
3. addition, subtraction, multiplication and division of simple surds
4. rationalising the denominator of a fraction that contains a surd.
Prior knowledge
To master this theme, you should already:
• know the laws and definitions of exponents, and be able to apply them when combined and composite bases are given, where x, y > 0 and m, n ∈ Z:
◦ the laws of exponents:
x
x
◦ the definitions of exponents:
x
• be able to solve equations with:
◦ an unknown in the base (simple equations with fraction exponents and factorising)
◦ an unknown in the exponent (exponential equations).
Sample
• be able to simplify exponential expressions by using laws of exponents for exponents containing fractions
• be able to simplify exponential expressions by factorising
Revision
The definition of a power is as follows:
power
Worked
revision examples
Simplify the following expressions and write the answers with positive exponents. You may not use a calculator (show which law of exponents you used, to prove that you are not using a calculator).
Assume that the values of all variables are greater than zero.
REMEMBER
Revision exercise
1. Simplify the following expressions, then write the answers with positive exponents. You may not use a calculator. Assume that the values of all variables are greater than zero.
2. Solve the following equations:
1.4
2 p + 1 × 2 p 1 ÷ 2 −2p + 3
= 2 (p + 1) + (p 1) (−2p + 3)
= 2 p + 1 + p 1 + 2p 3 = 2 4p 3
1.5 (2p −1) −3
= (2 1 p −1) −3 = 2 −3p 3
Remember that the exponent of 2 must also be multiplied by −3 = p 3 2 3 = p 3 8 1.6 3 √ 2 6p −12 = 2 6 ÷ 3 p −12 ÷ 3 3√x can be written as x 1 3 according to the definition given above, then the laws of exponents are applied = 2 2 p −4 = 4 p 4 1.7 √ 3 10m + 4 + (3 2m) −4 − 3 12m 4 3 2m 1 = 3 5m + 2 + 3 −8m 3 12m 4 (2m 1) = 3 5m + 2 + 3 −8m 3 10m 3 1.8 3
.
5 2 . 3
= 25 3 1.11 45 x 3 . 3 . 75 4 x 25 x . 15 x + 2 = (3 2 . 5) x 3 . 3 . (3 . 5 2) 4 x (5 2) x . (3 . 5) x + 2 = 3 2x 6 . 5 x 3 . 3 . 3 4 x . 5 8 2x 5 −2x . 3 x + 2 . 5 x + 2
1.1 SIMPLIFYING EXPONENTIAL EXPRESSIONS WITH RATIONAL EXPONENTS
In this section, we will focus on the simplification of exponential expressions with rational exponents. Specifically, powers with fraction exponents, for example, x 2
Note the following definition:
Worked example 1
Worked example 3
Worked example 2
Worked example 4
Remove the brackets, by multiplying the exponents within the brackets by those outside the brackets
Worked example 5
Simplify without using a calculator: (1 p + 1 q) −1
Determine the LCM of the denominators and write the expression as one fraction
Reciprocal
Simplify
Worked example 6
Simplify each of the following, without using a calculator. Give your answers with positive exponents. Sample
1.2 SIMPLIFYING MORE DIFFICULT EXPONENTIAL EXPRESSIONS
In this section you will learn how to simplify more difficult exponential expressions, for example:
1. exponential expressions in fraction form, with a number of powers that have unknowns or composite numbers as their base
2. exponential expressions in fraction form, containing more than one term in the numerator and/or in the denominator, which must be factorised.
1.2.1 Simplifyin g expressions with composite bases
Exercise 1.1:
Simplifying exponential expressions with rational exponents
REMEMBER
It is important that you must always work with prime factors. If the base is not a prime number, you therefore have to break it up into the product of its prime factors. Then, write these prime factors in brackets in place of the composite base.
Worked example 7 Simplify:
Give
Simplify:
Worked example 8
2
Simplify:
Worked example 9
Remove the brackets, by multiplying the exponents within the brackets by those outside the brackets
Exercise 1.2.1:
Simplifying expressions with composite bases
Simplify each of the following, without using a calculator. Give answers with positive exponents.
1.2.2 Simplifyin g expressions by factorising
• Factorising by means of a common factor, the difference of two squares, trinomials, and the difference of cubes, can also be applied to expressions with exponents.
• First, determine if there is a common factor, then take it out by factorising.
• The numerator and denominator of exponential expressions in fraction form must first be factorised. Then, divide equal factors.
Examples of factorising:
• Factorise by taking out a common factor.
2 x + 1 − 2 x = 2 x(2 − 1) = 2 x
• Factorise by using the difference of squares.
2 2x − 1 = (2 x − 1)(2 x + 1)
• Factorise by using the trinomial method.
2 2x − 2 . 2 x + 1
Let 2 x = k.
∴ k 2 − 2k + 1 = (k − 1)(k − 1) Substitute back in
∴ (2 x − 1)(2 x − 1)
• Factorise by grouping.
x 1 2 + x 1 2 y 1 2 + 1 + y 1 2 = x 1 2(1 + y 1 2) + (1 + y 1 2)
• Factorise by using the sum or the difference of cubes.
Simplify:
Worked example 10
Worked example 11
= (1 + y 1 2)(x 1 2 + 1) Sample
Worked example 12
3
Worked example 13 Simplify:
Simplify:
Worked example 14
Sample
Exercise 1.2.2: Simplifying expressions by factorising
1. Simplify each of the following expressions:
REMEMBER
Simplify by factorising.
• Keep on factorising until factors can divide out.
• Always attempt to take out the common factors first, before considering a difference of squares or factorising trinomials.
2. Prove that
3. Show that
4. Prove that
Self-evaluation
Determine how comfortable you are with each topic by using the scale that follows:
1. Alarm! I don’t feel comfortable with the topic at all. I need help.
2. Help! I don’t feel comfortable yet, 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 with some concepts.
4. Sharp! I feel comfortable with the topic.
5. Party time! I feel totally comfortable with the topic and can even answer more complicated questions about it.
Complete the table:
I can rewrite any surd in the exponent form with a rational exponent.
I know how to write any composite number as the product of prime factors.
I know the four laws of exponents and definitions and can apply them to simplify exponential expressions.
I can identify the different factorisation options and apply them when I have to simplify an exponential expression with more than one term in the numerator and denominator.
1.3 EXPONENTIAL EQUATIONS
1.3.1 Equations with the unknown in the base (with rational exponents) x 1 2 = 16 is an example of an equation where the unknown is in the base.
Method
• Raise both sides of the equation to a power that is the reciprocal of the exponent of the unknown.
• Remember that you cannot raise individual terms to a power – you must raise the entire left-hand side and the entire right-hand side of the equation to a power to keep it balanced.
• Write the equation so that the term with the unknown is on its own on one side of the equation, with a coefficient of 1.
Keep an eye out for trinomials. For example, 3x 2 3 − 7x 1 3 + 4 = 0, where the exponent of the first term is always twice the exponent of the
term
• 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.
