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 learners and facilitators achieve success in the 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 the 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 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 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.
When 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 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.
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 longterm memory and have the ability to recall and use this information. The structure of the study guides support this process and helps learners to master mathematics. Tip: Each theme has the same structure in order to make it easier to navigate through them.
Each theme has the following structure:
Introduction
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 learners can test their prior knowledge by themselves, 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 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 learners 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 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 may struggle to remember this when writing an examination.
When learners 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. Learners 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 the learning process, learners 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 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:
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 sub-theme, 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 learners spend more time on a specific theme than recommended by the CAPS. Be flexible in adapting the time allocation according to 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 they 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: Focus on the CAPS requirements and plan the year’s assessment accordingly. Learners must complete seven formal assessment tasks for school-based assessment.
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 11.
Tip: This table only indicates the formal assessment (i.e. the assessment used for promotion). Informal continuous assessment should also take place to monitor learners’ progress so that gaps in their knowledge are seen and rectified timeously.
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.
The two papers at the end of the year are compiled as follows: Paper 1 Paper 2
Exponents and surds, and Quadratic equations and inequalities (Theme 1 and 2)
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
Sample
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.
• 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 learners have a suitable calculator.
Sample
Typically, learners find it especially difficult to apply the laws of exponents to fractions. They will need considerable practice to embed this skillset.
Learners also typically find the trinomial-type equations with fraction exponents more difficult.
• Learners must be able to do operations with fractions.
• Learners must know the different methods of factorising.
• Learners must know how to break up composite numbers into prime factors.
Introduction
In this theme learners 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, learners 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
◦ the laws of exponents:
◦ the definitions of exponents:
• be able to simplify exponential expressions by using laws of exponents for exponents containing fractions
• be able to simplify exponential expressions by factorising
• 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).
Revision
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.
Remember that the exponent of 2 must also be multiplied by −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
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1.1 SIMPLIFYING EXPONENTIAL EXPRESSIONS WITH RATIONAL EXPONENTS
Introduction
In this section, we will focus on the simplification of exponential expressions with rational exponents. Specifically, powers with fraction exponents, for example, x 2 3 and 3 1 5. Note the following definition:
Worked example 1
the brackets, by multiplying the exponents within the brackets by those outside the brackets
Worked example 2
Simplify without using a calculator: (0,125)
Write the base as the product of prime factors
Remove the brackets, by multiplying the exponents
Worked example 3
Remove the brackets, by multiplying the exponents within the brackets by those
Worked example 4
Simplify without using a calculator: (
Sample
Write mixed numbers as improper fractions
By taking the reciprocal, the sign of the exponent changes
Write the bases as the product of prime factors
Remove the brackets, by multiplying the exponents within the brackets by those outside the brackets
Worked example 5
Simplify without using a
Determine the LCM of the denominators and write the expression as one fraction
Worked example 6
Simplify without using a calculator:
Exercise 1.1:
Simplifying exponential expressions with rational exponents
Simplify each of the following, without using a calculator. Give your answers with positive exponents.
1.2 SIMPLIFYING MORE DIFFICULT EXPONENTIAL EXPRESSIONS
Introduction
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
Note that it is important to always work with prime factors. If the base is not a prime number, you must break it up into the product of its prime factors. Then, write these prime factors in brackets, in the place of the composite base.
Worked
example 7
Worked example 8
Write each base as the product of its prime
• 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.
• Use in school or at home.
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