Grade 10 • Study Guide 2/2 Mathematics
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Reg. No.: 2011/011959/07
Mathematics
Study Guide 2/2 – Grade 10
2310-E-MAM-SG02
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 you might choose mathematics as a subject, for example to prepare you 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 Grade 10 to 12 involves more abstract concepts and more complex procedures than in Grade 1 to 9. Mastering mathematics in Grade 10 to 12 requires more time, commitment, critical thought and reflection than in Grade 1 to 9.
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 you achieve success in your 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 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.
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; you and your facilitator 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.
The learning activities available in the OLP’s lesson structures involve different formats and levels of interaction. The resources not only support the learning process, but also offer you the opportunity to practise new knowledge.
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.
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.
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 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 guide.
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 your work and the ability to recall it. When you are able to recognise your work, you will often say “Oh, of course!” but you struggle to remember this when you are writing an examination. 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
SampleIn 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. 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.
5. Whoo-hoo, it’s 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 for the assessment plan and comprehensive information about the compilation and mark allocation of tests, assignments and examinations.
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.
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)
Note:
Paper 2
Euclidean geometry and measurement (Theme 8, 13 and 14)
Analytical geometry (Theme 9)
Trigonometry (Theme 5 and 7)
Statistics (Theme 11)
• 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 exams in Grade 10.
Supplementary books
Any other books can be used along with this textbook for extra exercises and explanations, including:
• Maths 4 Africa, available at www.maths4africa.co.za
THEME 9
ANALYTICAL GEOMETRY
Introduction
In this theme you will learn to:
1. find the distance between two points in the Cartesian plane
2. find the mid-point of a line segment
3. find the gradient of a line
4. find the equation of a line
5. find the equations of parallel and perpendicular lines and determine whether lines are parallel or perpendicular
6. work with collinear points (points that lie on the same line)
7. apply all this knowledge in mixed exercises.
Prior knowledge
In order to master this theme, you should already:
• know about the Cartesian plane and that it consists of four quadrants
• know about coordinate points, e.g. (3; 5), where the first coordinate (3) is always the ��-coordinate, and the second coordinate (5) is always the ��-coordinate
• know that the gradient or inclination of a line refers to the steepness of the line, and that it can be defined as: the change in the y-values between 2 points on the line the change in the ��-values between the same 2 points on the line
What is analytical geometry about?
Analytical geometry is also called coordinate geometry and was previously known as Cartesian geometry.
Sample
• know the properties of triangles and quadrilaterals.
It is the study of geometry using the principles of algebra and the Cartesian coordinate system. It involves the definition of geometric figures in a numerical way (using numbers) and finding numerical information when diagrams are given in the Cartesian plane.
The development of analytical geometry is sometimes considered the origin of modern mathematics.
Background
If we have the coordinates of the vertices of a figure, we can draw the figure in the Cartesian plane.
In the diagram below, for example, L(−5; − 2), M(−1; − 6) and K(5; 4) are the vertices of ∆ KLM in the Cartesian plane.
4)
5; − 2)
1; − 6)
In the following figure, the point (0; 0) has been indicated, which is the origin. The point P(3; 5) is also given.
Let us calculate the gradient of the line in the following figure: y (0; 3) (8; −1) ����
Gradient = the change in the ��-values between 2 points on a line the change in the ��-values between the same 2 points on the line
The gradient (inclination) of the line from (0; 0) to (3; 5)
= the change in the ��-values between 2 points on the line the change in the ��-values between the same 2 points on the line
Revision exercise
Consider the figure and answer the questions that follow.
1. Write down the coordinates of A, B, C and D.
Calculate the gradients of AD and BC. What do you notice?
Calculate the gradients of AB and DC. What do you notice?
4. What type of figure is ABCD? Give a reason for your answer.
REMEMBER
The order of the letters by which we name a figure is important.
Solution
1. A(−1; 3), B(2; −1), C(8; 1), D(5; 5)
2. Gradient of AD = 5 − 3 5 − (−1) = 2 6 = 1 3
Sample
The name ABCD tells us that we move from point A to point B, from B to point C, from C to point D and then back to point A in order to form figure ABCD.
Gradient of BC = 1 − (−1) 8 − 2 = 2 6 = 1 3
The gradients (inclinations) are equal.
3. Gradient of AB = −1 − 3 2 − (−1) = −4 3
Gradient of DC = 1 − 5 8 − 5 = −4 3
The gradients (inclinations) are equal.
4. It is a parallelogram, because both pairs of opposite sides are parallel.
9.1 THE DISTANCE FORMULA
In the diagram below, two points are given: (−3; −1) and (2; 3). We want to find the distance between them.
In order to find the distance between the points (−3; −1) and (2; 3), we will add a point vertically below (2; 3) and horizontally to the right of (−3; −1). This new point is therefore (2; −1). If we connect the three points, they form a rightangled triangle with the 90° angle at the new point.
We use the theorem of Pythagoras to calculate the distance between (−3; −1) and (2; 3).
–1)
–1)
The length of the vertical side is (3 − (−1)) = 4 units and the length of the horizontal side is (2 − (−3)) = 5 units. (distance)2 = (3 − (−1))2 + (2 − (−3))2 Pythagoras = 42 + 52 = 16 + 25 = 41 ∴ distance = √ 41
We can generalise this calculation and get a formula − a formula is simply the generalisation of a specific case.
REMEMBER
The formula used for calculating the distance between two points, A (�� 1; ��1) and B (��2; ��2), is: AB = √ (�� 2
Worked example 1
B(3; 4) A(−5; −2)
Find the distance between A and B. Solution Let A(−5; −2) = (��1 ; ��1) and B(3; 4) = (��2; ��2) AB = √ (��2 − ��1 ) 2 + (��2 − ��1 ) 2 = √ (3 (−5))2 + (4 (−2))2 = √ (8)2 + (6)2 = √ 100 = 10 units
Worked example 2
The distance between A(��; 4) and B(2; 1) is 5 units. Find the value of ��.
Solution
5 = √ (2 − ��)2 + (1 − 4)2 5
0 = ��2 − 4�� − 12
0 = (�� − 6)(�� + 2)
�� = 6 or �� = −2
The coordinates of A can therefore be (6; 4) or (−2; 4).
There are two answers, because there are two possible positions for A. This is shown in the sketch below.
Important
Before you start solving a problem in analytical geometry, first draw a sketch showing the information given. Such a sketch helps you to use the information correctly and also to ensure that your answer is correct.
Exercise 9.1: The distance formula
1. Draw the points A(−1; 4) and B(7; −2) on a system of axes and calculate the distance from A to B.
2. Draw the points C(−4; 1) and D(3; −4) on a system of axes and calculate the distance from C to D correct to one decimal figure.
3. Calculate the lengths of the line segments in the following sketch. Leave your answers in the simplest surd form.
B(–1; 7)
F(–1; 1)
C(–1; –1)
A(−2; 4)
A(6; 4)
B(2; 1)
E(3; 9)
A(4; –2)
D(–2; –3)
4. The point B(b; −3) is 5 units from the origin. Find the value of b.
Note the difference between B and b.
The capital letter (B) indicates the point, and the lower case letter (b) indicates a coordinate (in this case the ��-coordinate).
5. Given: A(−7; m), B(−3; 4) and AB = 4√ 5 .
Calculate the value of m. Show your answer on a sketch.
6. A(−1;−1) is equally far from M(0; 2) and P(p; −2).
Find the value of p and also show your answer on a sketch.
7. A circle with the origin as its centre is given. A(3; 4) and B(−4; q) are points on the circumference of the circle.
Calculate the value of q.
y
q)
4)
8. Given: P(2; −3), M(−2; 1) and N(��; −7). N is equally far from P and M.
Draw a sketch to represent the information given. Then find the value of �� and show your answer on the sketch.
9. A(0; 0), B(p; q) and C(−q; p) are the vertices of a triangle.
Prove by calculation that ∆ABC is isosceles and right-angled.
10. A(0; 0), B(√ ; 1) and C(√ 3 ; −1) are the vertices of a triangle.
Prove by calculation that ΔABC is equilateral.
11. A parallelogram with vertices A(−2; 2), B(3; 2), C(1; −2) and D(−4; p) is given.
12. A circle with M(−2; 1) as its centre and radius √ 13 units are given.
12.1 Prove that the circle passes through the points A(−5; 3) and B(1; 3).
12.2 Calculate the circumference and area of this circle if the units are in centimetres. Round off your answers to the nearest integer.
Calculate the value of p. Sample
9.2 THE MID-POINT OF A LINE SEGMENT
In the diagram below, C(��; ��) is the mid-point of line segment AB, with A(��1; ��1) and B(��2; ��2).
; y₁) C(��������y) B(����₂; y₂) y y₁ y₂
The ��-coordinate of C is e��actly halfway between ��1 and ��2 on the ��-axis.
The value of the ��-coordinate of C is therefore ��1 + ��2 2 . This is the average value of ��1 and ��2.
In the same way, the value of �� is ��1 +
It follows that:
The formula for calculating the mid-point, C, of a line segment between two points, A(��1; ��1) and B(��2; ��2), is: CAB =
The word “mid-point” indicates that a line segment is bisected.
Worked example 3
Find the mid-point of the line segment between A(2; 5) and B(−8; −1) and show your answer on a sketch.
Solution
Worked example 4
2)
5)
8)
–1) Sample
If M(−2; 1) is the mid-point of the line segment between A(−6; 8) and B(a; b), find the coordinates of B.
1)
Exercise 9.2: The mid-point of a line segment
1. Consider the sketch below. Find the coordinates of the mid-point of each of the following line segments: AB, CD, EF and EH.
A(–3; 9)
C(–4; 5)
D(d; –4)
E(g; k) F(5; m)
B(11; –3)
H(a; b)
2. Given: Rectangle ABCD with A(−6; 2), B(−5; −2), C(5; 2) and D(4; 6). In addition, AE = ED.
6)
C(5; 2) B(–5; –2) A(–6; 2)
2.1 Prove that the mid-point of BC is Q(0; 0).
2.2 Now find the coordinates of E, the mid-point of AD.
2.3 Prove that AB = EQ = DC.
2.4 Is the following statement true?
“If the mid-points of a pair of opposite sides of a rectangle are connected, two rectangles are formed which are congruent.”
Give reasons for your answer.
Tip
In order to prove that two rectangles are congruent, you need to prove that the corresponding sides are equal and that the corresponding angles are equal.
3. If M(−3; −2) is the mid-point of A(−8; −5) and B(p; q), calculate the values of p and q.
4. Given: M is the mid-point of AB and N is the mid-point of BC. Find the coordinates of N.
A(–13; 5)
M(–6; 3)
y C(5; 8) N(r; s)
B(p; q)
• 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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