Apart from any fair dealing for the purpose of research, criticism or review as permitted in terms of the Copyright Act, no part of this publication may be reproduced, distributed, or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system without prior written permission from the publisher.
The publisher has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
There are instances where we have been unable to trace or contact the copyright holder. If notified, the publisher will be pleased to rectify any errors or omissions at the earliest opportunity.
Reg. No.: 2011/011959/07
Mathematics
Study Guide 2/2 − Grade 11
2411-E-MAM-SG02
CAPS-aligned
Prof. C Vermeulen, Lead author P de Swardt H Otto
M Sherman E van Heerden
PREFACE
Sample
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.
SampleWhen 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:
2 Exponents and surds, and Quadratic equations and inequalities (Theme 1 and 2)
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 7
MEASUREMENT
Introduction
This theme specifically deals with:
1. the perimeter and area of 2D shapes
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, you 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.
previously studied
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.
that are used to describe polygons:
Worked example 1
Calculate:
a) The area of PQRS.
Solutions
a) Area PQRS = b × ⊥h Area formula = 20 × 8
b) The perimeter of PQRS. Sample
Substitute values = 160 cm 2 Remember unit
b) Perimeter PQRS = 2(b + w) Area formula = 2(20 + 12)
Substitute values = 64 cm
Worked example 2
A prism on a square base has a height of 16 cm. The diagonal of the base is √ 288 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 2
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.
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.
3.1 If the top of her desk is 1,2 m long, calculate the area of the top of her desk.
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:
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.
2.2 You want to add a decorative border to each dishcloth. Calculate the perimeter of the dishcloth.
2.3 Determine the number of dishclothsto be made from the available fabric.
3.2.1 the new perimeter of the visible part of the top of her desk.
3.2.2 the new area of the visible part of the top of her desk.
7.2 VOLUME AND SURFACE AREA OF 3D 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
The base of a 3D object is 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.
More formulas for volume
What is a sphere?
Spheres are objects that are perfectly round and look the same from any direction.
Surface area
• 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.
Cylinder
(h = slant height)
Worked example 3
SampleArea of base + 1 2 × perimeter of base × ⊥ height of triangular face
Alternative method
area of base + area of each triangular face × number of triangular faces
A cylindrical aerosol can has a lid in the shape of a hemisphere that fits exactly on the top of the can. The height of the can without the lid is 32 cm and the radius of the base of the can is 5,8 cm.
Volume of sphere = 4 3 πr 3
Surface area of sphere = 4πr 2
1
2 Figure 1 Figure 2
a) Calculate to two decimal places the surface area of the can with the lid in place, as shown in Figure 1.
b) If the lid is 80% filled with a liquid, as shown in Figure 2, calculate to two decimal places the volume of the liquid in the lid.
Solutions
a) Surface area of cylinder
= π r 2 + 2πr × h
= π( 5,8) 2 + 2π(5,8) × 32
= 1 271,84 cm 2
Surface area of hemisphere = 1 2 × 4π(5,8) 2
= 211,37 cm 2
Surface area of can
= 1 271,84 + 211,37
= 1 483,21 cm 2
Figuur
Figuur
b) 80% volume of hemisphere
= 0,8 × 1 2 × 4 3 π (5,8) 2
= 56,36 cm 3
Worked example 4
A right pyramid with a square base with a side length (DC) of 12 cm is shown.
P lies on the square base directly below O. The volume of the pyramid is 360 cm 3 .
b) PT = 1 2 × 12 = 6 cm
OT 2 = OP 2 + PT 2
OT 2 = (6) 2 + (7,5) 2 = 369 4
OT = √ 369 4 cm
Volume of pyramid = 1 3 base area × ⊥height
a) Show that the perpendicular height of the pyramid is 7,5 cm.
b) Determine the total surface area of the pyramid to two decimal places.
Solutions
a) Volume = 1 3 × base area × ⊥height
360 = 1 3 × (12) 2 h
h = 360 × 3 144
h = 7,5 cm
Pythagoras
Sample
Total surface area = area of base + 4 × (1 2 × base of ∆ OBC × OT)
= (12) 2 + 4(1 2 × 12 × √ 369 4 )
= 374,51 cm 2
Exercise 7.2: Surface area and volume
1. The perpendicular height, AC, of the cone shown in the diagram is 2 m and the radius is r = 1,45 m. AB is the slant height.
B C r h s A
Calculate to two decimal places the total surface area of the cone.
2. A solid metallic hemisphere has a radius of 6 cm. It is made of metal A. To reduce its weight, a conical hole is drilled into the hemisphere (as shown in the diagram) and is filled with a lighter metal B. The conical hole has a radius of 3 cm and a by depth of 8 9 cm.
Calculate the ratio of the volume of metal A to the volume of metal B.
3. After 4 billion years, the dwarf planet Ceres still has a surprising amount of surface water – as much as 30%. The average radius of the dwarf planet is 469,73 km.
3.1 Calculate the surface area of the dwarf planet Ceres to two decimal places.
3.2 Calculate the area of the dwarf planet’s surface that is not covered by water to the nearest square kilometre.
7.3 EFFECT ON VOLUME AND SURFACE AREA WHEN DIMENSIONS ARE INCREASED BY A FACTOR k
When one or more of the dimensions of a 3D object is multiplied by a constant, the surface area and volume will change. It is important to see a relationship between the change in dimensions and the resulting change in surface area and volume. These relationships make it simpler to calculate the new volume or surface area of an object when its dimensions are scaled up or down. The new surface area and volume can be calculated by using the formulae from the previous section. For example, consider a rectangular prism of dimensions l, b and h. By multiplying one, two and three of its dimensions by a constant factor of 5 the new volume and surface area can be calculated as done in the below table.
Multiply two dimensions by five
Multiply one dimension by five
Multiply all three dimensions by five
Multiply all three dimensions by k
Worked example 5
The Khumalo family wants to build a television room onto their house. Dad draws up the plans for the new square room of length k metres. Mum looks at the plans and decides that the area of the room needs to be doubled.
To achieve this:
• Dad suggests doubling the length of the sides of the room
• Mum recommends adding 2 m to the length of the sides
• the son suggests multiplying the length of the sides by a factor of √ 2
• the daughter suggests doubling only the width of the room.
Whose suggestion will double the area of the square room? Show all calculations.
Solution
Draw sketches of each suggestion.
Therefore, both the son and daughter’s suggestion would give double the area of the original room, but with the daughter’s suggestion the room would no longer be square. Therefore, the son had the best suggestion.
Exercise 7.3:
Sample
Multiplying dimensions by a factor k
1. A sphere has a radius of 12 cm.
If the radius is increased by a factor of 3, calculate:
1.1 the surface area of the sphere to two decimal places
1.2 the volume of the sphere to two decimal places
2. A gift company wants to increase the volume of their bestselling gift boxes. Currently the gift box is in the shape of a cube which has a length of 5 cm. The length is increased by a factor of 5.
Calculate the volume of the enlarged gift box.
3. A rectangular box has a length of 3 m, a width of 4 m and a height of 5 m. Calculate the volume of the box if the length is multiplied by a factor of 2, the width is multiplied by a factor of 1,5 and the height is multiplied by a factor of 0,5.
Self-evaluation
Use the following scale to determine how comfortable you are with each topic in the table that follows:
1. Alarm! I don’t feel comfortable with the topic at all. I need help.
2. Help! 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.
5. Party time! I feel totally comfortable with the topic and can even answer more complicated questions about it.
• 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.
