Gr 9-Mathematics-Facilitator's Guide

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MATHEMATICS FACILITATOR’S GUIDE Grade 9

A member of the FUTURELEARN group

Mathematics Facilitator’s guide

1909-E-MAM-FG01

Í3)È-E-MAM-FG01*Î

Grade 9

CAPS aligned

DM Oost

Facilitator’s Guide G09 ~ Mathematics

Contents Guidelines for facilitators 1.

2.

3.

4.1 4.2 5.

General........................................................................................................................ 2 Study guide ................................................................................................................. 2 Portfolio book .............................................................................................................. 3 Learning outcomes ...................................................................................................... 3 Difficulty levels............................................................................................................. 3 Year plan ..................................................................................................................... 4

Study Guide Memorandum Unit 1: Number systems....................................................................................................... 2 Unit 2: Exponents .............................................................................................................. 28 Unit 3: Algebra ................................................................................................................... 54 Unit 4: Number patterns and relationships ....................................................................... 114 Unit 5.1: Geometry: Measuring, space and shape ........................................................... 159 Unit 5.2: Euclidean geometry ........................................................................................... 201 Unit 5.3: Geometry: Area, circumference and volume ..................................................... 248 Unit 6: Transformation geometry ..................................................................................... 274 Unit 7: Ratio, proportion and rate ..................................................................................... 292 Unit 8: Finances ............................................................................................................... 326 Unit 9: Statistics .............................................................................................................. 356 Unit 10: Probability ........................................................................................................... 418

Note that the page numbers of the memorandum start at page 1 again. The table of contents will guide you to easily access the units.

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Facilitator’s Guide G09 ~ Mathematics

5. Year plan Term

Topic

Unit in study guide

T1

Number systems

1

T1

Exponents

2

T1

Algebra

3

T2

Number patterns

4

T2

Measuring and constructions

5.1

T2

Euclidian geometry

5.2

T2

Area, volume, perimeter and circumference

5.3

June examination: Paper 1: Units 1, 2, 3 and 4 Paper 2: Units 5.1, 5.2 and 5.3 T3

Transformations

6

T3

Ratio, proportion and rate

7

T3

Finance

8

T3

Statistics

9

T4

Probability

10

Revision November examination: Paper 1: Units 1, 2, 3, 4, 7 and 8 Paper 2: Units 5, 6, 9 and 10

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Facilitator’s Guide G09 ~ Mathematics

Term 1

Unit 1 Number systems LO1

50 lessons are needed for this term.

Unit 2 Exponents LO1

Exercises

Topic These exercises can be regarded as lessons

Time in lessons or days

1 2

Summary of previous work One-dimensional graphs

1 2

3

Rational and irrational numbers

2

4

Real and non-real numbers

1

5

Different types of fractions

3

6

Positioning of numbers

2

7

Mixed exercises

1

1

Simplifying with prime factors as bases

2

2

Simplifying with compound factors as bases

4

3

Scientific notation

1

4

Simple equations

3

5

Mixed exercises

1

5

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Facilitator’s Guide G09 ~ Mathematics

Unit 3 Algebra LO2

1

Revision of basic algebra done in Grade 8

2

2

Products of polynomials

2

3

Functional notation and substitution

2

4

Invalidity in fractions

2

5

Invalidity in roots

2

6

Factorising: Common factors

2

7

Factorising: Difference between squares

2

8

Factorising: Grouping

2

9

Factorising: The trinomial pattern

3

10

Factoring: Mixed factorising

2

11

Simplification of fractions by factorising

3

12

Equations

3

13

Formulas

2

14

Mixed exercises

1

Note that the number of days allocated only serves as a guideline. However, do not waste time, since term 2 also comprises a lot of work.

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Facilitator’s Guide G09 ~ Mathematics

Term 2

Unit 4 Number patterns LO1 and LO2 38 lessons are needed for this term.

Term 2

Unit 5 Geometry 5.1 Measuring LO4

5.2 Euclidian geometry LO4

Exercises

Subject of exercises These exercises can be seen as lesson units

Time in lessons or days

1

Number patterns and formulas

2

2

Relationship between x and y

2

3

Linear relationships

2

4

The gradient or slope of straight lines

1

5

Non-linear graphs

2

6

Graphs with limitations

2

7

Mixed exercises

1

Exercises

Subject of exercises These exercises can be seen as lesson units

Time in lessons or days

1

Measuring instruments

1

2

2

3

Ratio between side lengths, areas and volumes, and fault analysis Polyhedrons

4

Three-dimensional views

2

5

Constructions and compass directions

1

6

Angles of elevation and angles of depression

1

7

Mixed exercises

1

1

Similarity

3

2

Congruent triangles

3

3

Congruent triangles and quadrilaterals

3

4

Parallelograms

3

5

Mixed exercises

1

7

2

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Facilitator’s Guide G09 ~ Mathematics

a. Area, perimeter and volume LO3

1

Revision of area and circumference (Grade 8)

2

2

Volume

2

3

Outer areas of fixed bodies

3

4

Mixed exercises

1

Learners should be ready to write the June examination. Note that the exam covers the work of term 1 and 2.

June examination: Paper 1: Unit 1, 2, 3 and 4 (LO1 and LO2) Paper 2: Unit 5 (LO3 and LO4)

Term 3 Exercises

Unit 6: Transformation geometry LO3 55 lessons are needed for this term.

Unit 7 Ratio, proportion and rate LO1

Unit 8 Finance LO1

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Subject of exercises These exercises can be seen as lesson units

Time in lessons or days

1

Symmetry

1

2

Formulas for reflection

2

3

Formulas for translation

2

4

Formulas for simple rotation

2

5 6 1

Enlargement and reduction Mixed exercises Equivalent fractions

2 1 1

2 3

Distribution of items Increasing and decreasing

1 2

4 5 6 1

Direct and inverse ratio Interpretation of graphs Mixed exercises Revision of financial matters (Grade 8)

4 2 1 3

2 3 4 5

Interest Hire purchase Timeline problems Mixed exercises

2 3 2 1

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Facilitator’s Guide G09 ~ Mathematics

Unit 9 Statistics LO5

1

Identify the problem

1

2

Collection of data

1

3

Deciding on a source

1

4

Recording and organising data

1

5

The frequency table

2

6

Process data statistically

3

7

Presentation of data

3

8

Interpreting outcomes

2

9

Conclusions and decision-making

2

10

Apply solutions and make predictions

2

11

General statistics

3

12

Reliability of statistics

1

13

Correlation between two variable data

2

14

Mixed exercises

1

Term 4 Subject of exercises These exercises can be seen as lesson units

Exercises

Unit 10 Probability LO5

Time in lessons or days

1

Revision of probability (Grade 8)

2

2

Probability of simple and compound events

3

3

Mutually exclusive and complementary events

3

4

The product of probabilities

3

5

Mixed exercises

1

Any work that is done through the year should be revised.

November examination Paper 1: Units 1, 2, 3, 4, 7 and 8 Paper 2: Units 5, 6, 9 and 10

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Facilitator’s Guide G09 ~ Mathematics

Question 1

Unit 1: Number systems: Types of numbers Identify the following numbers by indicating if the numbers are an element of one (or more than one) of the following types of numbers: • real numbers • rational numbers • irrational numbers • integers • whole numbers (counting numbers) • natural numbers • non-real numbers

1.1

2

(1)

1.2

2

(1)

1.3

−2

(1)

1.4

2 −2

(1)

1.5

0,2

(1)

1.6

0 2

(1)

1.7

2 0

(1)

1.8

00

(1)

1.9

02

(1)

1.10

20

(1) [10]

Question 2

Unit 1: Number systems: Calculations, x ∈ Z

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Simplify the following without using a calculator.

12 – 5 - 8 - 1 + 6 - 3 (-4)(-2)(-1)(10) -3 (-2 - 2 - 2 - 2 - 2) 6(-10 - 4) - 3 (-1 - 7) 3 + 2 (- 4) (- 5) (-5 + 4) - 1 Subtract -10 from -5 −8 − 42

11

(1) (1) (1) (1) (1) (1) (1) (1)

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Facilitator’s Guide G09 ~ Mathematics

2.9 2.10

−8 ( −4) 2 (

(1)

−8 2 ) −4

(1) [10]

Question 3

Unit 1: Number systems: Calculations, x ∈ Q Simplify the following without using a calculator.

3.1 3.2 3.3 3.4 3.5

3

1 2 − 1 6 3

(2)

3

2 1 ÷ 1 3 6

(2)

2 + 2.2 - 2.2 ÷ 2 5 +

2 +

3

(2) (2)

9 1 1 − 1 .

14 9 6

(2)

3 of 48 8

[10] Question 4

Unit 1: Number systems: Calculations, x ∈ Q' Simplify the following without using a calculator.

Leave the answers in the simplest surd form.

4.1

48

4.2

2 2 +

4.3

−5 3

4.4 4.5

(2) 2. 2 −

16 − 2 − 2

(2) (2)

2 12 Write 1000 as the product of prime factors.

1 1 Which number is the largest: if x = 100? or 3 x 3x Use a calculator if necessary.

(2) (2)

[10]

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Facilitator’s Guide G09 ~ Mathematics

Question 5

Unit 2: Exponents Simplify the following without using a calculator.

Change all bases to prime numbers with positive exponents.

Indicate the use of exponent rules.

5.1

(2k3m2) (-3k5m)

5.2

5

5.3 5.4

(1) (1)

a100 b 50 17a44. a22 172 2

4

(1) (1)

6

a b c a −1 b −3 c 7

5.5

Write the following with positive exponents: − 2 −200

(1)

5.6 5.7 5.8 5.9 5.10

2. 23 2x . 4 4x+1 - (22 33 55) -2

(1) (1) (1) (1) (1)

2 .3 .5 .3 .5 .2 .3 .2 .5 .3 .2 .3 .2 .5 .6 .9

[10] Question 6

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

Unit 3: Algebra: Simplify Simplify in full: 3g + g2 - 4g + 6g2 -1 p (- 4p2 - 3p + 2) 2h(h - 1) - (h2 - 2h + 6) y 3y + 2 4 1 11 m 1 m − 3 18 What is the LCM of 12y and 24y2?

What is the HCD of 12y and 24y2?

Is 91 a prime number or a compound number?

1 When will the fraction be invalid? y

(1) (1) (1) (1) (1) (1) (1) (1 (1) (1)

( 12345 pqrstuvw ) 2

[10]

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Facilitator’s Guide G09 ~ Mathematics

Question 7

Unit 3: Algebra: Solve for y

7.1 7.2 7.3

2y – 12 8(3 - y ) y+4 = y

7.4

1 y y + 1 = −6 2 3 1000 000 y2 = 1

7.5

= - 4y + 24 - 2(3y -2) = 0 3 5

(2) (2) (2) (2) (2) [10]

Question 8

Unit 4: Number patterns

8.1

Determine the sixth term in the following pattern: 1; 4; 9;…

Determine the nth term of: 10; 12 ;14.

Determine the formula of the following input and output model.

8.2 8.3

(1) (2) (3)

x 3 4 5 6 8.4

y 10 13 16 19

?

Complete the table of the following model and plot the points on a Cartesian plane. Draw your own set of axes. x

0

(4)

1 2

y = -2x + 4

0

-2 [10]

Question 9

Unit 5.1: Measuring, area and shape

9.1

Draw a perpendicular line from a point onto a straight line, then (2) draw an angle of 60° on the straight line measured from the perpendicular line. Use only a ruler and a compass.

Use a protractor to measure and draw an angle of 80°. Bisect the angl (2) using a ruler and compass.

9.2

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Facilitator’s Guide G09 ~ Mathematics

9.3

Draw sketches to illustrate the following concepts:

(6)

A pair of right opposite angles that are equal.

A pair of corresponding angles that are equal.

A pair of co-interior angles with a sum of 180°.

An equilateral triangle.

Indicate in the sketch all that is equal.

An isosceles, right-angled triangle.

Three adjacent angles that are supplementary.

[10]

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Facilitator’s Guide G09 ~ Mathematics

Question 10

Unit 5.2: Euclidian geometry Study the given sketch and answer the questions. Write the number of the question on your answer sheet and then write down the answer that must be filled in on the given line in your question paper.

L M K

0 80° 2 3 80

1 0 70 70°

aa0°

Q

c°

1 4

E

P

2

b°

H

Given: KL//PQ and PM = PF PF//HG

3

F

Mˆ 1 = 70° Mˆ 3 = 80°

b0

d°

G 10.1

(1)

M̂1 + M̂2 + M̂3 = 180° M̂1 = 70° and M̂3 = 80° and

10.2

Given

M̂2 = a ∴

(1) ∴ a = 30°

10.3

(1)

M̂3 + Ê 4 = 180° Given

But M̂3 = 80° and

Ê 4 = c

10.4

∴

10.5 10.6

(1) (1)

M̂2 = PF̂M But

Already proven

(1)

∴ PF̂M = b = 30° 10.7

But PF̂E = HĜF

(1)

∴ d = 30°

10.8

In your own words, explain why ∆ EGQ can never be right angled if (3) GQ is the longest side.

[10]

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Facilitator’s Guide G09 ~ Mathematics

Question 11

Unit 5.3: Area, perimeter and volume Study the following prisms.

3 cm

Area = 12 cm2

8 cm

B

6 cm r=2

A

7 cm

C

2 cm 11.1

Calculate the volume of the prism in figure B.

(1)

11.2

Calculate the area of the circle in figure A.

(1)

11.3

Calculate the volume of the prism in figure C.

(1)

11.4

Calculate the size of each angle of the base of the prism in figure C. Accept that it is a regular hexagon.

(2)

11.5

Which one of the given prisms’ volumes is the largest?

(2)

11.6

If the volume of the prism in figure A stays the same, but its height doubles, calculate the new radius of the circle.

(3) [10]

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Facilitator’s Guide G09 ~ Mathematics

Question 12

Unit 6: Transformation Study figure A. Y-axis

A X-axis

12.1

On the same set of axes, sketch the reflection image of figure A in the x-axis. Name this figure B.

(1)

12.2

On the same set of axes, sketch the reflection image of A in the y-axis. Name this figure C.

(1)

12.3

Translate figure A according to the following instruction:

(2)

(x ; y) →(x - 8; y + 1). Name this figure D. (2)

12.4

Rotate figure A 180° about the origin. Name this figure E.

12.5 12.5.1 12.5.2 12.5.3 12.5.4

If the following transformations occur, describe in your own words exactly what will take place. (x ; y) →(x + 2 ; y - 3) (x ; y) →(-x ; y) (x ; y) →(x ; -y) (x ; y) →(-x ; -y)

Question 13

Unit 7: Proportion and ratio

13.1

Give four equivalent fractions for 0,25.

(2)

13.2

Divide R600 in the ratio 9:6.

(2)

13.3

Increase R120 with 40%.

(2)

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Facilitator’s Guide G09 ~ Mathematics

13.4

During a building project it takes 10 hours for six men to lay 7800 bricks.

Calculate: 13.4.1 How many bricks does each man lay per hour?

13.4.2 How long will it take six men to build a wall with 23 400 bricks?

13.4.3 How many men are needed to lay 7800 bricks in 7,5 hours?

13.4.4 How many bricks will 12 men lay in 10 hours?

(4)

[10] Question 14

Unit 8: Financial matters

14.1

Study the following exchange rate: $1 = R10,17 and £1 = R14,80 Calculate how many $ can be bought for £1000.

14.2

Study the following two formulas with which interest can be calculated: E = B (1 +

r n ) 100

E = B (1 +

(2)

r ) 100

n

14.2.1

Calculate the future amount if R100 is invested for five years at 10% simple interest per year.

14.2.2

Calculate the present amount, if the future amount is R180 after investing it at 10% simple interest over a period of five years.

(1)

14.2.3

Determine the interest rate for R150 to grow to R240 over a period of four years. Simple interest is added annually.

At what interest rate was R120 invested if it yielded R180 after a period of four years at compounded interest added annually?

(1)

14.2.4

(1)

(1)

14.3

Calculate the selling price of an article if the cost price is R100 and the profit is 12%.

(1)

14.4

Calculate the percentage loss if a product is sold at R120while the cost price is R150.

(1)

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Facilitator’s Guide G09 ~ Mathematics

14.5

Calculate the cost price of an article if the purchase price is R100 and the profit is 10%.

(1)

14.6

Explain in your own words how a monthly instalment is calculated if simple interest is charged on the outstanding amount.

(1)

[10] Question 15

Unit 9: Statistics The following frequency table represents the marks learners scored on a test. The true values are not given and the grouping is already done.

Class interval

Count

0 – 19 20 – 39 40 – 59 60 – 79 80 – 99

1111 1111 11 1111 1

15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10

Frequency x

Class median y

Total

1 1111

Complete the table.

Determine the mode interval.

Determine the median interval.

Which value does the mode represent?

Which value does the median represent?

What do you think is the range of the values?

Calculate the mean of the values as it is illustrated in the table.

Sketch a pie diagram of the test results.

Sketch a histogram of the given data.

Do you think that the test the learners wrote was a difficult one?

Give a reason for your answer.

(1) (1) (1) (1) (1) (1) (1) (1) (1) (1) [10]

Question 16 Unit 9: Probability

16.1 16.2 16.3

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Study the following set of numbers and answer the questions.

S = {1; 2; 3; 4; 5; 6; 7; 8; 9; 10} Calculate n(S).

Calculate P (even number in S).

If a number should randomly be drawn from S, what is the probability of drawing a prime number?

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Facilitator’s Guide G09 ~ Mathematics

Question 17 A non-transparent bag contains two red balls and three blue balls.

17.1 Sketch a tree diagram to illustrate the outcomes if one ball is drawn from the bag twice, and after each draw the ball is placed back into the bag.

17.2 What is the probability of drawing three blue balls consecutively?

(3)

(2) [10]

TOTAL: 160

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