Gr 10-Mathematics-Facilitator's Guide

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MATHEMATICS FACILITATOR’S GUIDE Grade 10

A member of the FUTURELEARN group

Facilitator’s Guide G10 ~ Mathematics

CONTENTS Page 1 2 3 4

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Letter of information

3 – 14

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

15 – 37 38 – 55

Memorandum of study guide

1 – 700

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Facilitator’s Guide G10 ~ Mathematics

CONTENTS Guidelines for the facilitator 1.

General

2.

Study guide with activities and exercises

3.

Portfolio book

4.

4.1

Learning outcomes

4.2

Levels of difficulty

5.

Year plan

The teaching and learning of Mathematics aims to develop: A critical awareness of how mathematical relationships are used in social, environmental, cultural and economic relations Confidence and competence to deal with any mathematical situation without being hindered by a fear of Mathematics An appreciation for the beauty and elegance of Mathematics A spirit of curiosity and love for Mathematics Recognition that Mathematics is a creative part of human activity Deep conceptual understandings in order to make sense of Mathematics Acquisition of specific knowledge and skills necessary for o The application of Mathematics to physical, social and mathematical problems o The study of related subject matter (other subjects) o Further study in Mathematics (Specific Aims of Mathematics CAPS 2011)

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Facilitator’s guide G10 ~ Mathematics

Year plan Grade 10 Mathematics Term 1

Exercises

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

Subject of exercises These exercises can be seen as lesson units

Different types of numbers

Days or lessons (1)

Notations

(1)

Terms, factors and multiples

(1)

Rules of divisibility

(1)

Types of fractions

(2)

Rounding off numbers

(1)

7

Quadrates, roots and irrational numbers

(2)

8

Mixed exercise

1

Basic rules and definitions

(1)

2

Compound bases

(2)

3

Exponent equations without a calculator

(2)

4

Exponent equations with a calculator

(1)

5

Problem-based questions and exponents

(1)

6

Graphs of exponential graphs

(2)

7

Mixed exercise

(1) [10]

Unit 2: Exponents LO1

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Time in terms of days

7

(1) [10] Days or lessons

Facilitator’s guide G10 ~ Mathematics

Days or lessons

Unit 3: Algebra LO2 1

Revision of Grade 9 work

(1)

2

Shortcut with multinomial expressions

(1)

3

Multinomial expressions with fractions

(2)

4

Multinomial expressions with exponents

(1)

5

Factorisation

(2)

6

Fractions

(2)

7

Solving equations

(3)

8

Linear inequalities

(2)

9

Formulae

(1)

10

Mixed exercise

(1) [16]

Unit 4: Number patterns LO1 and LO2 1

Visual patterns

(1)

2

Formulae for the nth patterns

(2)

3

The linear pattern

(2)

4

Mixed exercise

(1) [6]

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

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Facilitator’s guide G10 ~ Mathematics

Term 2

Exercises

Subject of exercises These exercises can be seen as lesson units

Time in terms of days

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 Unit 12: Trigonometry LO 3

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

Days/ lessons (2)

1

Definitions of Sin θ , Cos θ and Tan θ

2

Without a calculator with the help of Pythagoras

(2)

3

Determining of function values with a calculator

(2)

4

Determining the sizes of angles if the function values are known

(3)

5

Standard angles

(3)

6

Solution of right-angled triangles

(3)

7

Direction, angles of elevation and angles of depression

(3)

8

Trig-graphs

(3)

9

Mixed exercises

(1) [22]

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Facilitator’s guide G10 ~ Mathematics

Unit 6: Transformation geometry LO3

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

Days/ lessons

1

Symmetry

2

Reflections (1)

3

Translations

4

Rotations

5

Enlargements and reductions

6

Combined transformations

Revise quickly

(1) [2]

Unit 11: Functions LO2

Days/ lessons (3)

1

The straight line

2

Straight lines and transformations

(2)

3

The parabola

(3)

4

The hyperbola

(3)

5

The exponential function

(2)

6

Mixed graphs

(3)

7

Mixed exercise

(1) [17]

Unit 5.2 Euclidean Geometry 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

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Facilitator’s guide G10 ~ Mathematics

Term 3

Exercises

Subject of exercises These exercises can be seen as lesson units

Time in terms of days

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 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

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

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Days/ lessons (2)

1

The distance formula

2

Vertical and horizontal lines

(2)

3

Midpoints of line segments

(3)

4

Gradients of line segments

(2)

5

Co- linear points

(2)

6

Mixed exercise

(1) [12]

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Facilitator’s guide G10 ~ Mathematics

Unit 8: Financial matters LO2

Days/ Lessons (1)

1

Formulae in Grade 10

2

Percentages, VAT, income tax and simple interest

(2)

3

Compounded growth

(2)

4

Types of volumes

(1)

5

Hire purchase

(2)

6

Inflation

(1)

7

Exchange rates

(2)

8

Rounding off cents

(1)

9

The financial calculator

(1)

10

Mixed exercises

(1) [14]

Unit 9: Statistics LO4 The formal test includes all work previously done with emphasis on unit 5, 8, 9, 11 and 12

Days/ lessons (2)

1

Arranging data

2

Representing data

(3)

3

Measurements of central tendency

(3)

4

Measures of dispersion

(3)

5

Interpretation of data

(2)

6

Mixed exercise

(1) [14]

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.

Term 4

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Exercises

Subject of exercises These exercises can be seen as lesson units

12

Time in terms of days

Facilitator’s guide G10 ~ Mathematics

Days or lessons

Unit 5: Geometry 5.1 Measuring Space and Form LO 3 1

Perimeter, Area and Volume

(1)

2

Cones

(1)

3

Pyramids

(2)

4

Sphere

(1)

5

Mixed Exercises

(1) [6]

Unit 10: Probability LO4

Days/ lessons (2)

1

Terminology and theory

2

Probabilities of simple and combined events

(2)

3

Mutually exclusive, not mutually exclusive and exhaustive events

(3)

Complimentary events 4

The product of probability

(2)

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

5

Mixed exercise

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 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 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

The weight of different content in grade 10 is more or less as follows.

(It is only used as a guideline)

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Facilitator’s Guide G10 ~ Mathematics

Mathematics Grade 10 Entrance Examination: Paper 1 of 2 LO1 and LO2.

Time: 2 Hours      

Marks: 120

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

Question 1 * LO1 1.1 1.2 1.3 1.4

Simplify, without using a calculator: (1)

1 1 1 − × 2 2 2 Subtract – 10 from – 20.

2 4 − 3 8 What is 8 % of 600?

(1) (1) (1) [4]

Question 2** LO2 2.1 2.2 2.3

Simplify.

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

(2) (4) (3) [9]

Question 3 ** LO1 3.1 3.2 3.3

1 to a decimal fraction.

6 Change 4, 38 to a common fraction.

Change 4

••

Explain in words what

0,13 means.

(1) (1) (1) [3]

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Facilitator’s Guide G10 ~ Mathematics

Question 4*** LO2

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

4.1

f(x) + g(x)

(2)

4.2

f(x) - g(x)

(2)

4.3

f(x) . g(x)

(3)

4.4

f(x) ÷ g(x)

(3)

Determine

[10] Question 5*** LO2

Factorise in full.

5.1

x2 + y2

5.2

3x3y

5.3

2x2 - 16x - 2xy + 16y

(3)

5.4

1 - y4

(3)

5.5

m - n - n2 + m2

(4)

-

(1) 27 xy3

(2)

[13] Question 6 *** LO2

Study the following expression.

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

9x

− x2 − 1

−

x2 2x + 2

+

3 x2

6.1

Determine the LCD of the numerators.

(1)

6.2

Determine the LCD of the denominators.

(3)

6.3

Determine the invalidities of the expression.

(6) [10]

Question 7

Simplify the following fractions.

Assume that the denominators are not equal to zero.

LO2 7.1*

4x 2 − 8x 2x − 4

7.2**

(x − 1)2 x2 − 1

7.3***

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(3)

÷

(5)

6 − 6x x2 + x

(7)

2 1 2 − + 2 x x −1 x − 5x

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Facilitator’s Guide G10 ~ Mathematics

7.4***

 x 2 − 24 x − 25 24  1 − ÷  x x 2 − 25  x 2 − 5x  

(6)

[21] Question 8 LO2

Solve for x in the following equations.

Accept that the denominators are not zero.

8.1*

x − 12 = 2 8

(2)

8.2**

x−2 x −1 1 − = 3x 4x 24 Test if y = 5 is an answer to the following equation.

Do not solve the equation. y 6y + 2 −5 1 − = + y −1 100 50 y2

(6)

8.3***

(6)

[14] Question 9 LO1

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

9.1**

Sketch a graph of distance opposed to time.

(4)

9.2**

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

(2)

9.3**

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

(2) [8]

Question 10** 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.

10.1

What is this type of transaction called?

(1)

10.2

Calculate the monthly instalment.

(6)

10.3

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

(1) [8]

Question 11 **LU1

11.1 © Impaq

Study the following 2 number patterns.

Pattern A

15 ; 22 ; 29 ; ...

Pattern B

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

Determine the 4th term of Pattern A. 17

(1)

Facilitator’s Guide G10 ~ Mathematics

11.2

Determine the 6th term of Pattern B.

(1)

11.3

Determine the general formula of Pattern A.

(3)

11.4

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

(6)

11.5

n

1

2

3

Tn

15

22

29

4

5

Describe the tendency of the graph.

(2) [13]

Question 12**

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

LO2 f: 2y + 4x -8 = 0 g: xy = 4

[7]

Total: 120

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Facilitator’s Guide G10 ~ Mathematics

Mathematics Entrance Examination Paper 1 of 2 Grade 10 Graph sheet

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Facilitator’s Guide G10 ~ Mathematics

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Facilitator’s Guide G10 ~ Mathematics

Memorandum Mathematics Entrance Examination Paper 1 of 2 Grade 10 LO1 and LO2 Mrs. DonnaMarié Oost Cell: 0793827555 e-mail: oost@absamail.co.za

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

0866714565 Memo Grade 10 Entrance ExamFax: Paper 1 Grade 10 2011

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Facilitator’s Guide G10 ~ Mathematics

Memorandum Question 1

Simplify, without using a calculator.

LO1 1.1

1.2

1 1 − × 2 2 Answer: 1 1 − × 2 2 1 1 = − 2 4 1 = 4

1 2

(1)

1 2

1 for the answer

Subtract – 10 from – 20.

Answer:

(1) 1 for the answer

(- 20) – ( - 10) = -20 + 10 = - 10 1.3

(1)

2 4 − 3 8 Answer: 2 3 = =

1.4

4 8 16 − 12 24 4 1 = 24 6 −

1 for the answer

(1)

What is 8 % of 600?

Answer: 8 of 600 100 8 6 . 100 = x 100 1

1 for the answer = 48 [4]

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Facilitator’s Guide G10 ~ Mathematics

Question 2 2.1

LO2

Simplify.

2x - 6x (x - 4) + x – 5x2

(2) 1 mark for each term in the answer

Answer: 2x - 6x2 + 24x + x - 5x2 = - 11x2 + 27x 2.2

– 2(x - 1)2 - (x2 - 1)

(4) 1 mark for the middle term (median)

Answer : – 2(x - 1)2 - (x2 - 1) = - 2(x2 - 2x + 1) - x2 + 1

1 mark for each term in the

= - 2x2 + 4x - 2 – x2 + 1 = - 3x2 + 4x - 1 2.3

(3x - 4y)2

(3) 1 mark for each term in the answer

Answer : 9x2 - 24xy + 16y2

[9] Question 3 LO1 3.1

3.2

1 to a decimal fraction.

6 Without using a calculator Answer : 1 for the answer 1 4 = 4 ,16 6

(1)

Change 4, 38 to a common fracture.

(1)

Change 4

Answer: 38 100 19 = 4 50

4 ,38 = 4

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1 for the answer

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Facilitator’s Guide G10 ~ Mathematics

3.3

••

Explain in words what

(1)

0,13 means.

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

Thus 0,1 3 = 0,1313131313 ...........

[3] Question 4

If f(x) = - x2 + 6x

LU2

en g(x) = x - 6, determine

4.1

f(x) + g(x) Answer:

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

(− x 2 + 6x ) + ( x − 6 )

(2)

= − x 2 + 7x − 6 4.2

f(x) - g(x) Answer:

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

( − x 2 + 6 x ) − ( x − 6) = − x 2 + 6x − x + 6 = − x 2 + 5x + 6 4.3

f(x) . g(x)

(2)

(3)

Answer: 1 mark for each term in the answer

(− x 2 + 6x ) . ( x − 6 ) = − x 3 + 6 x 2 + 6 x 2 − 36 x = − x 3 + 12x 2 − 36 x 4.4

f(x) ÷ g(x)

(3) 2 marks for factorising 1 mark for the answer

Answer: − x 2 + 6x x − 6 − x ( x − 6) = ( x − 6) = −x

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

[10]

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Facilitator’s Guide G10 ~ Mathematics

Question 5

Factorise in full.

LO2 5.1

x2 + y2 Answer:

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

(1)

1(x2 + y2) 5.2

3x3y - 27 xy3

(2) 1 mark for common factors 1 mark for difference of squares

Answer: 3xy ( x2 - 9 y2) = 3xy ( x – 3y ) ( x + 3y ) 5.3

2x2 - 16x - 2xy + 16y

(3)

Answer: (2 x2 - 16x) + (-2xy + 16y) = 2x (x – 8) - 2y (x – 8)

1 mark for each of the 3 factors

= (x - 8) (2x - 2y ) = (x - 8) 2 (x - y) = 2(x – 8) (x - y) 5.4

1 - y4

(3)

Answer: 1 mark for each of the 3 factors

(1 – y2 ) ( 1 + y2) = ( 1 – y ) ( 1 + y ) ( 1 + y2 ) 5.5

m - n - n2 + m2

(4)

Answer:

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

(m - n) + (-n2 + m2) = (m - n ) + (m2 - n2) = (m - n ) + (m + n) (m – n) = (m – n )[1 + (m + n) ] = (m – n ) (1 + m + n)

[13]

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