Gr 10-Mathematics-Facilitator's Guide by Impaq

Í2*È-E-MAM-FG01+Î

MATHEMATICS FACILITATOR’S GUIDE Grade 10

Grade 10

Facilitator’s guide

Mathematics

'0 2RVW

CAPS aligned

Í2*È-E-MAM-FG01+Î 1810-E-MAM-FG01

CONTENTS

Memorandum of study guide

Entrance examination Paper 1 + memorandum Entrance examination Paper 2 + memorandum

Letter of information

Facilitator’s Guide G10 ~ Mathematics

1 – 700

38 – 55

15 – 37

3 – 14

Page

Facilitator’s Guide G10 ~ Mathematics

Levels of difficulty

4.1

4.2

o Further study in Mathematics (Specific Aims of Mathematics CAPS 2011)

o The study of related subject matter (other subjects)

o The application of Mathematics to physical, social and mathematical problems

Acquisition of specific knowledge and skills necessary for

Deep conceptual understandings in order to make sense of Mathematics

Recognition that Mathematics is a creative part of human activity

A spirit of curiosity and love for Mathematics

An appreciation for the beauty and elegance of Mathematics

Confidence and competence to deal with any mathematical situation without being hindered by a fear of Mathematics

A critical awareness of how mathematical relationships are used in social, environmental, cultural and economic relations

The teaching and learning of Mathematics aims to develop:

Year plan

Portfolio book

Study guide with activities and exercises

Learning outcomes

General

Guidelines for the facilitator

CONTENTS

Facilitator’s Guide G10 ~ Mathematics

x x x

Study the theory and examples with all the written explanations. Do some of the sums and mark it according to the memorandum. Use the video material if there is something you do not understand. It is available at the beginning of each exercise, explaining some of the problems.

The study guide contains references and exercises to develop concepts, comprehension, skills and knowledge of mathematics.

2. Study guide with activities and exercises

This paper is only meant to assure that learners understand Grade 9 work before they start with Grade 10 work. If the learners have a huge backlog, it is recommended the learner first master the Grade 9 work before continuing with Grade 10.

Success in Grade 10 cannot be achieved if Grade 9 work is not properly understood and handled with ease. Every Impaq Mathematics study guide relies on learners’ prior knowledge, insight and ability to identify and complete work accurately. A backlog in Mathematics increases with time and can become impossible to bridge if not dealt with quickly.

1.2 Entrance (readiness) Examination Included in this facilitator’s guide are two examination papers (Paper 1 and 2) that all learners who did not pass Grade 9 at Impaq, must write. If the learners are studying Grade 10 Mathematics through Impaq for the first time, you have to ensure that they also complete the entrance (readiness) examination. You must grade this paper and ensure that the learners catch up on any ‘weak’ points in the work that they may have.

1.1 Make sure that the following are in your possession: x A study guide covering the work that should be done in Grade 10. x A facilitator’s guide, which includes: ¾ this letter of information, ¾ the entrance (readiness) exam to identify possible problem areas, ¾ Year plan for Grade 10 ¾ The memorandums of the activities. x A DVD containing videos referred to in the study guide. Note: This only works on a laptop, not a DVD player.

1. General

Facilitator’s guide G10 ~ Mathematics

Patterns, Functions and Algebra

Measuring in Several Contexts Geometry and Trigonometry Data handling

Learning Outcome 2 LU2

Learning Outcome 3 LU3 25%

25%

For any queries, contact the education specialist at Impaq.

The number of stars, next to the number of each sum, indicates the level of difficulty. This indication is maintained in the guide, memorandum, tests and examination papers. Level * Knowledge of basic theory and procedures are 1 necessary before any calculations can be done. (Basic knowledge) This section of the question paper has a value of 20% and can be obtained by studying. Level ** Routine work Applying routine multi-step procedures in various contexts. Continuous 2 practise will enable you to master this section of (Routine work) the question paper. It has a value of 35%. Level *** Applying knowledge and procedures in a variety of contexts. Complex calculations based on a 3 combination of acquired knowledge. Questions (Complex work) are asked indirectly with the emphasis on comprehension. This part of the question paper has a value of 30%. Level **** Applying relevant knowledge to unfamiliar, nonroutine questions. The most difficult part of the 4 question paper – testing the learners’ ability to (Problem solving) apply all the knowledge they have gained. This part of the question paper has a value of 15%. Level ***** Advanced 5

4.2 Levels of difficulty

Learning Outcome 4 LU4

Numbers, Calculations and Relations

Learning Outcome 1 LU1

Explanation of the learning outcomes that are required in Grade 10,11 and 12

4.1 Learning outcomes

Refer to the portfolio book for all assessment.

3. Portfolio book

Recommended calculator for learners: CASIO fx82ES (Plus), however, any scientific calculator will be sufficient.

Facilitator’s guide G10 ~ Mathematics

Year plan: Mathematics Grade 10: Shortened version

Number patterns Trigonometry Functions Euclidean Geometry (revision grade 9)

1 2 2 2

Analytic Geometry Finance Statistics Space and Form Probability Revision

3 3 3 4 4 4

November Examinations: Two papers 100 each Paper 1: Unit 1 ; 2 ; 3 ; 4 ; 8 ; 10 ;11 (Application of Unit 6 in Unit 11) Paper 2: Unit 5 ; 9 ; 10 ; 12 ; 13 Note: Unit 7, Proportion and Ratio, is not formally dealt with in Grade 10. However it is applicable up to Grade 12.

Euclidean Geometry

June examination: Two papers 100 marks each Paper 1: Unit 1,2,3,4 and 11 (Application of Unit 6 in Unit 11) Paper 2: Unit 5, and 12

Algebra

Exponents

1 1

Number systems

Subject

Term

5.1

5.2

11 5.2

Unit

The order in which the different Units should be done is prescribed by the new curriculum. All schools have to comply with this and this will make it if necessary easier to move between different schools during the year.

Facilitator’s guide G10 ~ Mathematics

Exercises

Unit 2: Exponents LO1

Compound bases

Exponent equations without a calculator

Exponent equations with a calculator

Problem-based questions and exponents

Graphs of exponential graphs

Mixed exercise

Basic rules and definitions

Mixed exercise

Quadrates, roots and irrational numbers

Rounding off numbers

Types of fractions

Rules of divisibility

Terms, factors and multiples

Notations

Different types of numbers

Subject of exercises These exercises can be seen as lesson units

Unit 1: Number systems LO1 From January to 1 March will be enough time to 2 complete these 4 3 Units for Term 1 (42 days or lessons 4 are acquired) This is only a 5 guideline 6

Term 1

Year plan Grade 10 Mathematics

Facilitator’s guide G10 ~ Mathematics

The linear pattern Mixed exercise

3 4

(2) (2)

(1) [10]

(2)

(1)

[6]

(1)

(2)

(1)

(1) [16]

(1)

(2)

(3)

(2)

(1)

(2)

(1)

Days or lessons

Learners should be able to complete the first 4 units during January, February and March (making room for more or less 42 lessons). The formal test must be written after completion thereof. The test entails all the previous work. Note that the number of days allocated only serves as a guideline. However, do not waste time as the second term is also heavily loaded. The following assignments must be completed for Term 1 Assignment 1: Investigation 1 Assignment 2: Formal Test Assignment 6: Informal Test up to Unit 4 (remember that this assignment should be entered in the portfolio at the end of the year when all the units are completed

Formulae for the nth patterns 2

(1)

Visual patterns

Mixed exercise

Formulae

Linear inequalities

Solving equations

Fractions

Factorisation

Multinomial expressions with exponents

Unit 4: Number patterns LO1 and LO2

Multinomial expressions with fractions

Shortcut with multinomial expressions

2 3

Revision of Grade 9 work

(1) [10] Days or lessons

(2)

(1)

(2)

(1)

Days or lessons (1)

Time in terms of days

Unit 3: Algebra LO2

Facilitator’s guide G10 ~ Mathematics

Exercises

Subject of exercises These exercises can be seen as lesson units

Time in terms of days

This is a new unit in Maths and will be more or less 40% of the second paper from now on.

Unit 12: Trigonometry LO 3

Solution of right-angled triangles

Direction, angles of elevation and angles of depression

Trig-graphs

Mixed exercises

Standard angles

Determining of function values with a calculator

Determining the sizes of angles if the function values are known

Without a calculator with the help of Pythagoras

Definitions of Sin T , Cos T and Tan T

[22]

(1)

(3)

(2)

Days/ lessons (2)

This work acquires 52 days or lessons. The time-span from 12 April up to 22 June will be sufficient if the time allocation is kept in mind. Note that all the work done in the first two terms will be examined

Term 2

Facilitator’s guide G10 ~ Mathematics

The hyperbola The exponential function Mixed graphs

4 5 6

[2]

(1)

(1) [17]

(3)

(2)

(3)

(2)

Days/ lessons (3)

Revise quickly

Days/ lessons

Revise the basics from grade 9. Include the following. Lines, triangles, polygons, parallel lines congruency and similarities. For Term 2 the following assignments are require: Assignment 3: Research project should be completed. Assignment 6: Fill in the marks of mixed exercises of completed Units. Assignments 4.1 and 4.2 June examination June examination: Two papers 100 marks each Paper 1: Unit 1 ; 2 ; 3 ; 4 and 11 Paper 2: Unit 5 and 13 Unit 6 necessary for Unit 11

Mixed exercise

The parabola

Straight lines and transformations

The straight line 2

Combined transformations

Rotations

Translations

Enlargements and reductions

Reflections

Symmetry

Unit 5.2 Euclidean Geometry

Unit 11: Functions LO2

This Unit is not formally asked any more Application in other parts of the Maths are still necessary

Unit 6: Transformation geometry LO3

Facilitator’s guide G10 ~ Mathematics

Exercises

Subject of exercises These exercises can be seen as lesson units

Time in terms of days

Study the formulas for analytical geometry by hart. No formula sheet will be provided

Midpoints of line segments

Gradients of line segments

Co- linear points

Mixed exercise

Vertical and horizontal lines

The distance formula

Remember that 1 Triangles prove of Theorems are included in the 2 Rectangles new curriculum and 3 Midpoint theorem is more or less 10% Answer questions 4 Mixed exercise formally using “Statements” and Reason” In grade 12 this part of paper 2 will be 34% Unit 13: Analytical geometry LO3

Hire purchase

(2)

Mixed exercise

Interpretation of data

Measures of dispersion

Measurements of central tendency

Term 4

Exercises

Subject of exercises These exercises can be seen as lesson units

Assignments for the third term are as follows: For the third term the following assignments should be completed Assignment 5: Formal Test 2 after completing all the units for term 3 Assignment 6: Informal Assessment for the completed units.

Representing data

Mixed exercises

The financial calculator

Rounding off cents

Exchange rates

Arranging data

(2)

(1) [12]

Types of volumes

The formal test includes all work previously done with emphasis on unit 5, 8, 9, 11 and 12

Unit 9: Statistics LO4

Compounded growth

Inflation

Percentages, VAT, income tax and simple interest

Formulae in Grade 10

Unit 8: Financial matters LO2

Facilitator’s guide G10 ~ Mathematics

(3)

(2)

Days/ lessons (2)

(4) [14]

(4)

(3)

Unit 7: Rate and ratio LO1 This unit is fully dealt with in Grade 8 and 9 and must be revised, because it can at any time be asked again in Grade 10, especially in Finance. The third term is from 15th July till the end of September and requires more or less 50 days or lessons. Unit 5.2: Euclidian Geometry LO3

Term 3

Facilitator’s guide G10 ~ Mathematics

Time in terms of days

(1) [14]

(2)

(3)

Days/ lessons (2)

[14]

(1)

(2)

(1)

(2)

(1)

(2)

Days/ Lessons (1)

The product of probability

Complimentary events

Mutually exclusive, not mutually exclusive and exhaustive events

Probabilities of simple and combined events

Mixed Exercises

Sphere

Terminology and theory

Pyramids

Cones

(2)

(3)

(2)

Days/ lessons (2)

[6]

(1)

(2)

(1)

Days or lessons

Mixed exercise

The weight of different content in grade 10 is more or less as follows. (It is only used as a guideline)

November examination: Two papers 100 marks each Paper 1: Units 1 ; 2 ; 3 ; 4 ; 8 ; 10 and 11 Paper 2: Units 5 ; 9 ; 12 and 13 Unit 6 and 7 as necessary in applications of questions in other units

buy the Examination Aid (Tel nr: 012 365 1824) Assignment 7: Test 4 Revision is two exam papers from Impaq’s previous years and should be included in the portfolio

You can buy exam papers for study purposes in educational bookshops or obtain it from the internet. Just make sure that the work covered in the papers is that of the new curriculum (started in 2012). One of the web pages is (www.thutong.gov.za) or

(1) [10] Do revision of all the work done through the year by working through all the mixed exercises, tests and exams again.

Unit 10: Probability LO4

Perimeter, Area and Volume

5.1 Measuring Space and Form LO 3

Unit 5: Geometry

Facilitator’s guide G10 ~ Mathematics

Total

% in final papers

15% 15% 40% 30% 100

10% 15% 30% 15% 100

30%

Also remember that if the question states: “without a calculator” then the different steps should be included so that proof is given that no calculator was used

Note that in the new “CAPS” curriculum no formula sheet is given. Pupils should know the different formulae by hart.

Before starting a test or exam paper, make sure you have the right calculator. It should be a non-programmable, non-graphic scientific calculator. You should know how to adjust the calculator to scientific -mode and statisticmode and use it when applicable.

Statistics Analytical Geometry Trigonometry Geometry: Euclidean and Form

Paper 2

Algebra, equations, factorising, fractions, inequalities and restrictions Patterns sequences and series Finance and growth Functions and graphs Probabilities

Paper 1

Facilitator’s guide G10 ~ Mathematics

3.2 3.3

Question 3 ** LO1 3.1

Question 2** LO2 2.1 2.2 2.3

1.4

1.2 1.3

Question 1 * LO1 1.1

Entrance Examination: Paper 1 of 2 LO1 and LO2.

Grade 10

Mathematics

Marks: 120

Explain in words what

0,13 means.

1 to a decimal fraction. 6 Change 4, 38 to a common fraction.

Change 4

2x - 6x (x - 4) + x – 5x2 – 2(x - 1)2 - (x2 - 1) (3x - 4y)2

Simplify.

1 1 1 u 2 2 2 Subtract – 10 from – 20. 2 4 3 8 What is 8 % of 600?

Simplify, without using a calculator:

[3]

(1) (1)

(1)

[9]

(2) (4) (3)

[4]

(1)

(1) (1)

(1)

General Instructions: Show all the steps. Use the = sign correctly. Calculators may not be used unless specified otherwise. A graph sheet is added to this question paper. Write your name and number on the graph sheet and submit it with your answer sheets. Number the questions correctly. Formulae sheets are prohibit

Time: 2 Hours

Facilitator’s Guide G10 ~ Mathematics

(2)

7.3***

7.2**

7.1*

LO2

3 x2

6 6x

2 1 2 2 x x 1 x 5x

x 1 2

4x 2 8x 2x 4

Simplify the following fractions. Assume that the denominators are not equal to zero.

Determine the invalidities of the expression.

6.3

Question 7

Determine the LCD of the numerators.

Determine the LCD of the denominators.

x2 2x 2

6.2

6.1

Study the following expression. Do not simplify, just answer the questions on the expression.

Question 6 *** LO2

(4)

m - n - n2 + m2

5.5

(7)

(5)

(3)

(6) [10]

(3)

(1)

[13]

(3)

1 - y4

(3)

(2)

(1)

[10]

(3)

(2)

5.4

2x2 - 16x - 2xy + 16y

27 xy

3x y

5.3

Determine

5.2

x2 + y2

Factorise in full.

Question 5*** LO2 5.1

f(x) ÷ g(x)

f(x) . g(x)

4.4

4.3

f(x) + g(x) f(x) - g(x)

4.2

If f(x) = -x2 + 6x and g(x) = x - 6

4.1

Question 4*** LO2

Facilitator’s Guide G10 ~ Mathematics

Sketch a graph of distance opposed to time.

Determine the distance that the car will travel in 4,5 hours.

Determine the hours it will take for the car to travel 260 kilometres.

Question 9 LO1

9.1**

9.2**

9.3**

11.1 17

Determine the 4th term of Pattern A.

- 4 ; - 7 ; -10 ; -13 ; ...

Study the following 2 number patterns.

Question 11 **LU1

15 ; 22 ; 29 ; ...

How much less would she have paid if she could have paid R999 cash for the radio?

10.3

Pattern B

Calculate the monthly instalment.

10.2

Pattern A

What is this type of transaction called?

10.1

Megan wants to buy a radio, but she cannot afford to pay the amount of R999 in full. She pays a deposit of R200 and on the balance she is charged 25 % simple interest per year over a period of three years. This means that she will have to pay 36 equal monthly payments.

A car travels at a constant speed of 80 kilometres per hour.

8.3***

Question 10** LO1

x 2 x 1 1 3x 4x 24 Test if y = 5 is an answer to the following equation. Do not solve the equation. 5 y 6y 2 1 y 1 100 50 y2

8.2**

12 8

x 2

8.1*

LO2

(1)

[8]

(1)

(6)

(1)

[8]

(2)

(4)

[14]

(6)

(2) 29

LO2

g: xy = 4

Total: 120

f: 2y + 4x -8 = 0

Sketch the graphs of f and g on the same set of axes.

Question 12**

Describe the tendency of the graph.

11.5

11.4

Solve for x in the following equations. Accept that the denominators are not zero.

Sketch Pattern A on a Cartesian plane with the aid of the following table.

11.3

Question 8

[21]

Determine the 6th term of Pattern B. Determine the general formula of Pattern A.

11.2

§1 x 2 24 x 25 24 ·¸ ¨ y ¨ x x 2 25 ¸ x 2 5x ¹ ©

7.4*** (6)

Facilitator’s Guide G10 ~ Mathematics

(1)

[7]

[13]

(2)

(6)

(3)

Mathematics Entrance Examination Paper 1 of 2 Grade 10 Graph sheet

Facilitator’s Guide G10 ~ Mathematics

0866714565 Memo Grade 10 Entrance Exam amFax: Paper 1 Grade 10 2011

Mrs. Susan Naudé E-mail: susanfn@vodamail.co.za Cell: 0768733829

Paper 1 of 2 Grade 10 LO1 and LO2

Mathematics Entrance Examination

Memorandum

Mrs. DonnaMarié Oost Cell: 0793827555 e-mail: oost@absamail.co.za

Facilitator’s Guide G10 ~ Mathematics

1.4

1.3

1.2

1.1

LO1

Question 1

Memorandum

1 2

(3x - 4y)2

4 8 16 12 24 4 1 24 6

8 of 600 100 6 . 100 8 x 100 1

Answer:

What is 8 % of 600?

2 3

Answer:

2 4 3 8

1 for the answer

[4]

(1)

4 ,38

38 100 19 4 50 4

1 for the answer

Change 4, 38 to a common fracture.

3.2

Answer:

1 to a decimal fraction. 6 Without using a calculator Answer : 1 for the answer 1 4 4 ,16 6 Change 4

1 mark for each term in the answer

1 mark for each term in the

1 mark for the middle term (median)

1 mark for each term in the answer

3.1

Question 3 LO1

9x2 - 24xy + 16y2

2.3

= - 3x2 + 4x - 1

= - 2x2 + 4x - 2 – x2 + 1

= - 2(x2 - 2x + 1) - x2 + 1

– 2(x - 1)2 - (x2 - 1)

Answer :

– 2(x - 1)2 - (x2 - 1)

= - 11x2 + 27x

2x - 6x2 + 24x + x - 5x2

Answer:

= - 10 (1)

(1)

2.2

Simplify.

2x - 6x (x - 4) + x – 5x2

LO2

Answer :

1 for the answer

(1)

2.1

Question 2

Facilitator’s Guide G10 ~ Mathematics

= -20 + 10

(- 20) – ( - 10)

Answer:

Subtract – 10 from – 20.

1 1 u 2 2 Answer: 1 1 u 2 2 1 1 2 4 1 4

Simplify, without using a calculator.

Facilitator’s Guide G10 ~ Mathematics

(1)

[9]

(3)

(4)

(2)

4.4

4.3

4.2

4.1

LU2

Question 4

3.3

0,13 means.

6 x ) ( x 6)

6x x 6

5x 6

x 2 6x x 6 x ( x 6) ( x 6) x

Answer:

f(x) ÷ g(x)

x 3 12x 2 36 x

No marks if terms are cancelled It must be factors that are cancelled out by division

2 marks for factorising 1 mark for the answer

1 mark for each term in the answer

1 mark for the correct interpretation of between f and g 1 mark for the answer

1 mark for the correct interpretation of + between f and g 1 mark for the answer

x 3 6 x 2 6 x 2 36 x

( x 2 6x ) . ( x 6 )

Answer:

f(x) . g(x)

( x

f(x) - g(x) Answer:

x 2 7x 6

( x 2 6x ) ( x 6 )

f(x) + g(x) Answer:

en g(x) = x - 6, determine

If f(x) = - x + 6x

Answer: It is a repetitive fraction where both the 1 and the 3 are repeated. Thus 0,1 3 0,1313131313 ...........

Explain in words what

Facilitator’s Guide G10 ~ Mathematics

[10]

(3)

(2)

[3]

(1)

5.5

5.4

5.3

5.2

5.1

LO2

Question 5

= (m – n ) (1 + m + n)

= (m – n )[1 + (m + n) ]

= (m - n ) + (m + n) (m – n)

= (m - n ) + (m2 - n2)

(m - n) + (-n2 + m2)

Answer:

m - n - n2 + m2

= ( 1 – y ) ( 1 + y ) ( 1 + y2 )

(1 – y2 ) ( 1 + y2)

Answer:

1 - y4

= 2(x – 8) (x - y)

= (x - 8) 2 (x - y)

= (x - 8) (2x - 2y )

= 2x (x – 8) - 2y (x – 8)

1 mark for grouping 1 mark for difference of squares 2 marks for the answer Note: Simplified factors in answer

1 mark for each of the 3 factors

1 mark for common factors 1 mark for difference of squares

It is a prime factor and it doesn’t have any further factors.

(2 x2 - 16x) + (-2xy + 16y)

Answer:

2x2 - 16x - 2xy + 16y

= 3xy ( x – 3y ) ( x + 3y )

3xy ( x2 - 9 y2)

Answer:

3x3y - 27 xy3

1(x2 + y2)

Answer:

x2 + y2

Factorise in full.

Facilitator’s Guide G10 ~ Mathematics

[13]

( (4)

(3)

(2)

(1)

6.3

6.2

6.1

LO2

Question 6 1

x2 2x 2

2x 2 ( x 1) ( x 1)

1 mark for each factor in the answer. Take 2x2 as one.

? x

x2 = 0

Denominator 3 = 0

2 ( x 1) 0 0 ? x 1 ? x 1

Denominator 2 = 0

? Deno min ator 1 0 ? ( x 1)( x 1) 0 ? x 1 0 of x 1 ? x 1 or x 1 0

(6)

(3)

(1)

[10]

1 mark for denominator 1 = 0 1 mark for the “or” between the possibilities 1 mark for denominator 2 = 0 1 mark for the answer. The 2 is cancelled by division. 1 mark for Denominator 3 = 0 1 mark for the final answer

Solution = {0; 1; -1}

Final invalidity:

The fractions are invalid if the denominators are equal to zero.

Determine the invalidities of the expression. Answer:

x. x

LCD

( x 1) ( x 1) 2 ( x 1)

x2 1 2x 2

Determine the LCM of the numerators. Answer: The numerators are: 9x , x2 and 3 The LCD = 9x2 Determine the LCD of the denominators. Answer:

Study the following expression. Do not simplify, just answer the questions on the expression.

Facilitator’s Guide G10 ~ Mathematics

7.2

7.1

Question 7 LO2

6 6x

1mark for the answer

4 marks for 4 sets of factors at numerators and denominators

No marks if terms are cancelled instead of factors

1 mark for factors above 1 mark for factors below 1 for the answer

x2 x 1 ( x 1) ( x 1) x ( x 1) x ( x 1) ( x 1) 6 (1 x ) x 6

x 1 2

Answer:

x 1 2

4x 2 8x 2x 4 4 x ( x 2) 2 ( x 2) 2 . 2 x ( x 2) 2 ( x 2) 2x

Answer:

4 x 2 8x 2x 4

Assume the denominators are not equal to zero.

Simplify the following fractions.

Facilitator’s Guide G10 ~ Mathematics

(5)

(3)

7.4

7.3

Can also be factorise it (x - 25)(x+ 1)

2 for factorising in the first step 1 mark for changing division to multiplication 2 marks for the answer of the subtraction part 1 mark for the answer

x 2 25 24 x x (x 5 ) . x ( x 5) ( x 5) x 2 24 x 25 1 x 5

Answer:

x2 5x 8 x ( x 1 ) ( x 5)

2x 2 12x 10 x 2 5 x 2x 2 x ( x 1 ) ( x 5)

2( x 6 x 5 ) x 5 x 2x 2 x( x 1) ( x 5)

2 1 2 x x 1 x( x 5) 2( x 1 ) ( x 5) x ( x 5) 2 ( x 1) x ( x 1)( x 5)

Answer: 2 1 2 2 x x 1 x 5x

2 1 2 2 x x 1 x 5x

Facilitator’s Guide G10 ~ Mathematics

[21]

(6)

(7)

8.3

8.2

8.1

Question 8 LO2

1 24

x x

3 100

5 32 4 25 125 128 100

1 5 100 50 5 2 100 3 100

RHS :

1 mark for substitution LHS 2 marks for the answer of each term 1 mark for the answer 1 mark for the answer RHS 1 mark for the union LHS = RHS

5 5 1

6(5) 2

Answer: Test if y = 5 is an answer LHS: y 6y 2 y 1 y2

(6)

(2)

is an answer

LHS = RHS y =5

[14]

(6)

1 mark for the LCD 3 marks (1 mark for the answer of each term after multiplying with the LCD) 1 mark for simplifying 1 mark for the answer

1 mark for removing the fractions 1 mark for the answer

Test if y = 5 is an answer of the following equation. Do not solve the equation. y 6y 2 1 5 100 50 y 1 y2

x 2 x 1 1 3x 4x 24 kgv 24 x ? 8 ( x 2 ) 6( x 1 ) ? 8 x 16 6 x 6 ? 2x 10 x ? x 10

x 2 x 1 3x 4x Answer:

x 12 2 8 Answer: 12 x 2 8 24 ? 8x ? x 3

Solve for x in the following equations. Assume that the denominators are not equal to zero.

Facilitator’s Guide G10 ~ Mathematics

9.2

9.1

LO1

Question 9

X-axis

4,5

= direct ratio

The longer the time the further the distance

Hours

Distance in km

but must be calculated.

Note that the answer cannot be read off from the graph,

Answer:

3 240

y = 80x

Determine the distance that the car will travel in 4,5 hours.

160

240

2 160

2 marks for the 2 points calculated and plotted correctly 1 mark for straight line (shape) 1 mark for general information

Distance y

Y-axis

1 80

Hours x

Answer:

Sketch a graph of distance opposed to time.

A car travels at a constant speed of 80 kilometres per hour.

Facilitator’s Guide G10 ~ Mathematics

(2)

(4)

Reading off is no answer

There are other methods also

2 marks for calculating the answer

Hours

Hire-purchase

Answer:

What do we call this type of transaction?

10.1

LO1

Megan wants to buy a radio, but she cannot afford to pay the amount of R999 in full. She pays a deposit of R200 and on the balance she is charged 25% simple interest per year over a period of three years. This means that she will have to pay 36 equal monthly payments.

? direct ratio 80 1 ? 260 x ? 80 x 260 ? x 3,25 hours

The longer the distance the more the hours.

260

Distance in km

Determine the hours it will take for the car to travel 260 kilometres. Answer:

1 4,5 80 x 4,5 360 km

Question 10

9.3

80 ? x ? x ? x

Facilitator’s Guide G10 ~ Mathematics

(1)

[8]

(2)

Question 9 ***

Ê 5

and B̂ 3

180 0

B̂ 2 D̂ 4 Ê 5

180 0

Asked: Give a complete statement with reasons and prove that interior angles of a triangle = 1800 . In other words, prove that:

and ABC // DE

Given: 'BDE

Alternative int erior angles ABC // DE

Total: 80

ABC is a straight line

? Interior angles of a triangle 180 0

? D̂ 4 B̂ 2 Ê 5

D̂ 4

but B̂1

Answer: B̂1 B̂ 2 B̂ 3

Study the following sketch:

Facilitator’s Guide G10 ~ Mathematics

(5)

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5.1 Unit 5.2 Unit 6 Unit 7 Unit 8 Unit 9 Unit 10 Unit 11 Unit 12 Unit 13

Contents

Number systems Exponents Algebra Number patterns and relations Geometry: measurement, space and form Geometry: Euclidian geometry Transformation geometry Ratio and rate Finances Statistics Probability Functions Trigonometry Analytical geometry

Facilitator’s Guide G10 ~ Mathematics

1 1 1 1 2 2 2 1 1 2 1 1 2 2

Exam paper.

1 27 66 135 153 194 226 257 266 335 404 482 560 650

Page

1 2 3 4 5 6 7 8

Exercise

Unit 1: Memorandum Numbers and Number Relationships

CONTENTS

Real numbers Notation Terms, Factors and Multiples Rules of Divisibility Types of Fractions Approximating Decimals Roots, squares and Irrational Numbers Mixed Exercises Bibliography

Facilitator’s Guide G10 ~ Mathematics

4 5 7 10 12 17 18 24 26

Page

Unit

Rating Code 7 6 5 4 3 2 1

Level

Level 1

Facilitator’s Guide G10 ~ Mathematics

*****

****

***

Rating Outstanding Meritorious Substantial Moderate Adequate Elementary Not Achieved

Marks % 80 - 100 70 – 79 60 – 69 50 – 59 40 – 49 30 – 39 0 – 29

Assessment Knowledge of basic theory and procedures are necessary before any calculations can be done. This section of the question paper has a value of 25% and can be obtained by studying. Applying routine multi-step procedures in various contexts. Continuous practising will enable you to master this section of the question paper and it has a value of 30%. Applying knowledge and procedures in a variety of contexts. Complex calculations based on a combination of acquired knowledge. Questions are asked indirectly with the emphasis on comprehension. This part of the question paper has a value of 30%. Applying relevant knowledge to unfamiliar, nonroutine questions. The most difficult part of the question paper seeing that the candidates’ ability to apply all the knowledge they’ve gained, is tested. This part of the question paper comprises 15%. Advanced

Unit

Facilitator’s Guide G10 ~ Mathematics

Unit

1 Unit

an int eger } non zero int eger

x x x

Fractions

http://www.mathsisfun.com/whole-numbers.html

["Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject.]

Whole numbers

Irrational numbers

Negative integers Zero Positive integers (Counting numbers )

Integers

Rational numbers

Real Numbers

Summary of the relationships among the sets of numbers

Q’ = {numbers that cannot be written as

an int eger } non zero int eger = Irrational numbers (non-terminating, non-repeating decimals) R = Real numbers = { Rational and irrational numbers} R’ = Non-real numbers = {numbers that do not exist}

= Rational numbers

Q = {numbers that can be written as

N = { 1;2;3;4;5.......} = Natural numbers N0 = { 0;1;2;3;4.....} = Whole numbers Z = {.....-2;-1;0;1;2;3;.....} = Integers

Exercise 1: Number systems

L01 This Unit is a summary of the work done in Grade 8 and Grade 9. The idea is to support you with examples and not so much as to provide exercises that must be done. Should you experience any difficulty in completing this section, you should go back to the work done in Grade 8 and Grade 9. Use the supplied links.

Unit 1: Numbers and Number Relationships

Facilitator’s Guide G10 ~ Mathematics

Identify the number 3 8 ? Answer: 3 8 = -2 ? A Real, Rational number and an Integer

1.5**

2.1

-3 d x < 6

-3 -2 -1 0 1 2 3 4 5 6

Number line R

Answer : 1. English : Real numbers from -3 to 6. The -3 are included but the 6 are not “ 2. Standard form : -3 d x < 6 3. Roster method (table notation) : cannot be written in this way. 4. Set-builder notation: { x / -3 d x < 6 x R } 5. Interval notation: x [ 3 ; 6) 6. Graphical number line:

“Real numbers from -3 to 6. The -3 are included but the 6 are not “.

Give the other 5 forms of notation:

LO1 en LO2

Exercise 2: Different forms of notation

What type of number is 3 8 ? Answer: 38= 2 ? A Real, Rational, Whole number, Integer and a Natural number Natural number implicates – Real, Rational, Whole number and Integer.

Is zero a positive or negative number? Answer: Zero is neither positive nor negative. What type of number is 8 ? Answer: 8 4 x2 2 2 ? A Real, Irrational number What type of number is 8 ? Answer: The square root of a negative number doesn’t exist ? A Non-Real number.

Unit

1.4**

1.3**

1.2**

1.1**

Facilitator’s Guide G10 ~ Mathematics 1

2.5**

2.4**

2.3 **

2.2**

30 < x d 33

28 29 30 31 32 33

-1

{ x / -1 < x d 7, x R}

-2

-1

Answer:

x [0 ; 4)

7 R

Study the graph and give the answer in interval notation.

-2

Answer:

Study the graph and give the answer in set-builder notation.

Answer:

Draw the graph for 30 < x d 33 if x R

Answer: The bracket ”)” shows that the 10 is excluded from the set. If it was included then x [ 1 ; 10] should have been given.

If x [ 1 ; 10) , is 10 included or excluded from the set?

Facilitator’s Guide G10 ~ Mathematics Unit

Unit

Adding and subtracting

Order of Calculations

1 of 12 3

1 4 3 x 6 y 9 2 14 of 12 3 1 4 3 x 6 y 9 16 14 of 12 3 4 3 x 6 y 9 16 14 4 4 18 y 9 16 14 4 4 2 16 14 4 2 16 14 4 18 14 4 32 4 28

4 3 x 6 y 9 2 4 (10 24)

1 of 12 3

Multiplying and division

Exponents

Brackets

4 3 x 6 y 9 2 4 (10 8 x 3 )

Answer:

1 4 3 x 6 y 9 2 (10 8 x 3 ) of 12 3

LO1 and LO2 3.1 Do the next calculation, without a calculator and prove that you know the calculation order by doing only one calculation per step.

Exercise 3: Terms, Factors and multiples

Facilitator’s Guide G10 ~ Mathematics 1

3.4.2**

3.4** 3.4.1**

3.3**

3.2.2

Before it is simplified?

3.2.1

1; 2; 3; 4; 6 ; 12

9 ; 36 ; 18

F36

Common factors = {1;2;3;4;6;12}

Answer:

diagram.

Give the common factors of 24 and 36 from your

8 ; 24

F24

Answer: Prime numbers = {2;3;5;7;11;13;17;19;23;29} Take the numbers 24 and 36. Draw a Venn diagram to illustrate the factors of these two numbers. Answer: Factors of 24 = {1;2;3;4;6;8;12;24} Factors of 36 = {1;2;3;4;6;9;12;18;36}

Write down the first 10 prime numbers.

Has 2 terms.

(x – 1) (4x) = 4x2 - 4x

Answer:

After it was simplified?

Only 1

Answer:

How many terms does (x – 9) (4x) consist of

3.2**

Facilitator’s Guide G10 ~ Mathematics

Intersection F24 F36 Union of F24 and F36 F24 F36

Unit