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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
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.
Each theme has the following structure:
Introduction
SampleTip: Each theme has the same structure in order to make it easier to navigate through them.
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 sub-themes.
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
End of theme exercise
Sample
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).
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.
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.
Sample
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.
Tip: Focus on the CAPS requirements and plan the year’s assessment accordingly. Learners must complete seven formal assessment tasks for school-based assessment.
• 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) 45 ± 3 Statistics (Theme 12)
Number patterns (Theme 3)
Finance, growth and decay (Theme 11)
± 3 Analytical geometry (Theme 4)
± 3 Trigonometry (Theme 6 and 9)
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.
± 3 Functions and graphs (Theme 5)
± 3 Euclidian geometry, and Measurement (Theme 7 and 8)
Probability (Theme 10) 20 ± 3
± 3
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.
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
• 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.
Sample
THEME 7
MEASUREMENT
CAPS learning requirements
Learners should:
1. Revise the volume and surface areas of right-prisms and cylinders.
2. Study the effect on volume and surface area when multiplying any dimension by a constant factor k.
3. Calculate the volume and surface areas of spheres, right pyramids and right cones.
Term 3
Duration 1 week
Paper 2
Weight
Measurement forms part of Euclidean geometry, of which the weight is 50 ± 3%.
Facilitator tips
Learners should revise basic results established in earlier grades and:
• know the formulae for perimeter and area of two-dimensional (2D) shapes
• know the formulae for calculating the volume and surface area of threedimensional (3D) objects, as well as understanding the effect on the surface area and volume when the dimensions of objects are multiplied by a factor k (the formulae are given in tables throughout the theme, with appropriate diagrams of the different 2D shapes and 3D objects)
Introduction
This theme specifically deals with:
1. the perimeter and area of 2D shapes
Sample
• be able to identify the different shapes or objects, as well as which formula is applicable to which shape or object
• be able to analyse shapes and objects before they routinely apply any formula to determine whether that formula is valid for that specific shape or object, which may not be a standard shape.
2. the volume and surface area of right prisms, cylinders, spheres, right pyramids, and right cones (3D objects)
3. the effect on volume and surface area when any dimension is multiplied by a constant factor k.
Prior knowledge
To master this theme, learners should already know:
• how to calculate the perimeter/circumference and area of:
◦ polygons
◦ circles
• that applying a factor k to any or all of the dimensions of a 2D shape affects perimeter and area
• how to calculate the surface area and volume of:
◦ cubes
◦ rectangular prisms
◦ triangular prisms
◦ cylinders
◦ pyramids
◦ cones
◦ spheres
• that applying a factor k to any or all of the dimensions of 3D objects affects volume and surface area.
Why is it important to be able to do measurement?
Measurement allows us to describe the physical world in numbers. We use measurement for many practical applications, for example:
• finding the widths and lengths of objects
• making sure that articles fit into certain spaces
• building structures according to correct sizes.
Polyhedrons
7.1 PERIMETER AND AREA OF TWO-DIMENSIONAL SHAPES
What is perimeter?
In mathematics, we define the perimeter of a 2D shape as the distance around the shape. Circumference specifically refers to the perimeter of curved shapes (circles, ovals or ellipses) or arcs.
What is area?
We can define the area of a 2D shape as the 2D space occupied by that shape. Area is measured in square units, such as cm 2 , m 2 , km 2, etc.
Angles of a regular polygon
The size of each interior angle of a regular polygon is 180° × ( n 2 ) n , where n represents the number of sides.
Prefixes that are used to describe polygons
Important formulae
Worked example 1
PQRS is a parallelogram with QR = 20 cm, h = 8 cm and RS = 12 cm.
Calculate:
a) The area of PQRS. b) The perimeter of PQRS. Solutions
Worked example 2
A prism on a square base has a height of 16 cm. The diagonal of the base is √ 288 cm. 16 cm
cm
Calculate:
a) The area of the base.
b) The length of a side of the base.
Solutions
a) diagonal 2 = side 2 + side 2 Pythagoras (√ 288 ) 2 = 2 × side 2 Area = side 2 = 144 cm
b) side 2 = 144 side = 12 cm
Exercise 7.1:
Area and perimeter of two-dimensional shapes
1. ABCD is a rhombus with sides of length 4 3 x cm . The diagonals intersect at O and length DO = x cm. Express the area of ABCD in terms of x.
Diagonals bisect each other perpendicularly AO 2 = (4 3 x) 2 x 2 Pythagoras – diagonals intersect perpendicularly AO = √ 7 9 x 2 AO = √ 7 3 x
Area of rhombus = 1 2 × product of diagonals Rhombus area formula
2. You want to make new rectangular dishcloths for your kitchen. The dimension of each dishcloth is 30 cm × 45 cm. The dimensions of the piece of fabric you have available is 1 m × 1,8 m.
2.1 Calculate the area of one dishcloth in square metre.
Area of dishcloth = 0,3 × 0,45 Convert cm to m (÷ 100) = 0,135 m 2
2.2 You want to add a decorative border to each dishcloth. Calculate the perimeter of the dishcloth.
Perimeter of dishcloth = 2(0,3 + 0,45) = 2(0,75) = 1,5 m
2.3 Determine the number of dishcloths to be made from the available fabric.
Arrangement 1:
Number of dishcloths = 4 × 3 = 12
Arrangement 2:
Number of dishcloths = 6 × 2 = 12
3. Lauren’s mathematics textbook is 30 cm long and 20 cm wide. She notices that the dimensions of the top of her desk are in the same proportion as the dimensions of her textbook.
Both arrangements provide 12 dishcloths. Sample
3.1 If the top of her desk is 1,2 m long, calculate the area of the top of her desk. Proportion of desk is the same as textbook: lengthdesk : lengthbook = 120 : 30 = 4 : 1
∴ widthdesk : widthbook = x : 20 = 4 : 1
∴ x 20 = 4 1
∴ width of desk = x = 4 × 20 = 80 cm OR lengthbook widthbook = 30 20 = 3 2 lengthdesk widthdesk = 120 x = 3 2
∴ 120 × 2 = 3 × x
∴ widthdesk = x = 80 cm
Area of her desk = l × w = 1,2 × 0,8 = 0,96 m 2 (or 9 600 cm 2)
3.2 Lauren uses some cardboard to cover each corner of her desk with an isosceles triangle, as shown in the diagram.
Calculate, in metre or square metre, as applicable:
3.2.1 the new perimeter of the visible part of the top of her desk.
Hypotenuse 2 = 0,15 2 + 0,15 2 Calculate the length of the covered corner
Hypotenuse = √ 0,045
Hypotenuse = 0,21 m
Perimeter of visible part of desk = 2[(1,2 0,3) + (0,8 0,3)] + 4(0,21) = 2(0,9 + 0,5) + 4(0,21) = 3,64 m
3.2.2 the new area of the visible part of the top of her desk.
Area of covered corners = 4 × [1 2 × 0,15 × 0,15] each corner is a right-angled △ = 0, 045 m 2
Area of visible part of desk = 0,96 0,045 = 0,915 m 2
7.2 VOLUME AND SURFACE AREA OF THREEDIMENSIONAL OBJECTS
What is volume?
Volume can be defined as the 3D space enclosed by or occupied by a 3D object.
What is surface area?
This is the total area of the surfaces of a 3D object. The surface area is everything that will be covered in paint if an object is dipped in a can of paint.
Important formulae for volume
Sample
The base of a 3D object is one of the two parallel, congruent sides of the object.
What is a pyramid?
A pyramid is a geometric object that has a polygon as its base and sides that converge at a point called the apex. In other words, the sides are not perpendicular to the base.
What is a right pyramid?
A pyramid is a right pyramid if the line from the apex to the centre of the base is perpendicular to the base.
What is a cone?
Cones are similar to pyramids, but their bases are circles.
for
What is a sphere?
Spheres are objects that are perfectly round and look the same from any direction.
Sample
• Consider the net of the object. Open the object like a cardboard box and flatten it out to find all the 2D geometric shapes included.
• Calculate the area of each 2D geometric shape.
• Add these areas to find the total surface area.
• 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.
