1

Contents

Welcome!

1 Order of operations

2 Exponents

3 Products

4 Factorising

5 Algebraic fractions

6 Solving of linear equations

7 Linear inequalities

8 Straight lines

9 The theorem of Pythagoras

2

Welcome!

1. Write down the answer to each question in the square underneath the question:

3 + 4 × 2 √ √

Supplement of

110° 1 +

x + 3= –5

x = ?

11(-1)20

(3xy)0 2(a

3b

2)

4

6x2y – 3x

= 3x(□–1)

□ = ?

xy + xy + xy = Factorise:

a = ?

2y = 8

y = ?

Mean of

3 ; 4 ; 10 ; 11 ;

12

Value of

b2 – 4ac

if

a = b = –3; c = 0

×

(

)

2. Now use the key given below and write down the correct letter in the third row at no.1.

A B C D E F G H I J K L M N

1 14 20° - 4 x3y

3 1,3 a

2+b

2 11 2xy x 16a

12b

8

9

O P Q R S T U V W X Y Z ? !

70° 55° –9 8 3xy 2a12

b8 4 0,5 –8 –1

(a + b)2

50mm 1,03

3. Answer to the question? _____________________

? 4 cm

3 cm

a 55°

3

1 Order of operations

The following are the rules for the order of operations:

(i) When there are brackets in the mathematics problem, you always have to first

simplify that which is placed inside brackets.

(ii) Powers have the highest priority.

(iii) When the problem only involves division and/or multiplication and there are no

brackets, then you have to proceed from left to right.

(iv) When you have to do a problem that only involves addition and/or subtraction - with

no brackets - you may proceed from left to right.

(v) In case of a calculation problem that involves “something of everything”, and

there are no brackets, always first tackle the parts that require powers, division and

multiplication.

1. Do the following computations without a calculator.

Calculation task Answer In which order did you do your calculations?

a 18 6 3 =

18 (6 3) =

b 14 – 3 + 5 =

14 – (3 + 5) =

c 8 + 2 3 =

(8 + 2) 3 =

d 25 – 5 4 =

(25 – 5) 4 =

e 22 – 4 2 =

(22 – 4) 2 =

f 8 + 12 4 =

(8 + 12) 4 =

2. Do the following computations without a calculator.

Calculation task Answer In which order did you do your calculations?

a 18 + 6 – 9 + 3 =

(18 + 6) – (9 +3) =

b 3 6 2 3 =

(3 6) (2 3) =

c 4 – 3 × 3 + 4

(4 – 3) × (3 + 4)

d 3 + 9 3 + 3 =

(3 + 9) (3 + 3) =

e 5 × 2 + 3 × 2 =

5 × (2 + 3 × 2) =

f 10 – 8 + 4 2 =

(10 – 8 + 4) 2 =

4

3. Do the following computations without a calculator.

Calculation task Answer In which order did you do your calculations?

a

b

c 6 + 10 5 4 =

d 11 + 6 3 + 3 =

e 16 – 12 2 + 1 =

f (3 + 5)2 =

g 32 + 5

2 =

h 3 + 52 =

i 5 – 2 + 42 =

j (–4)2

k – (4)2

l – (–4)2

m (–3)2 – 4(–1)(3)

n (3)2 – 4(1)(3)

o √

p √

q √ √

4. What is wrong?

Calculation task Correct

answer What is wrong?

a 10 5 2 = 1

b 8 + 2 3 = 30

c 30 – 5 4 = 100

d 20 – 6 2 = 7

e

f

g (2 + 3)2 = 13

h (5)2 = 10

i –62 = 36

j (–10)2 = –100

k – (–4)2 = 16

l √

5

3 Exponents

1. Law 1

(i) 2 × 2 × 2 × 2 = 16 or 24

(ii) a3× a

2 = a × a × a × a × a = a

5

We also say a3× a

2 = a

3 + 2 = a

5

Use am

× an = a

m+n to determine

a302

× a245

We can add the exponents if we multiply and the bases are the same.

From am

× an = a

m+n :

(i) (a3)

2 = a

3× a

3 = a

6 or a

3× 2

(ii) (p3)

25 = p

3× 25 = p

75 by using law 1

(am

)n = a

mn

2. Law 2

(i)

by cancelling out

(ii) To simplify

will take too long and that is why we apply law 2

We can subtract the exponents if we divide and the bases are the same.

From

:

by cancelling

Therefore any number (except 0) to power zero is equal to 1

a0 = 1 , a ≠ 0

by cancelling

Therefore

3. Law 3

√

½ = or

√

4. Simplify: (give answers with positive exponents)

(a) 3-2

+ 30 + 3

3 (b) (–3x

-2)(2x

3) (c) x

-3

(d)

(e) 5x0 + (5x)

0 (f)

(g)

(h) x3 × x

-7 × x × x

2 (i) (2x

3) (3x

-8)(x)(-3x

2)

(j) (k)

(l)

6

5. Simplify and rewrite in exponential form. There must be no variable in the denominator. See example.

√

=

=

=

(a)

(b)

(c) √

(d) √ (e)

√ (f) √

(g)

√ (h)

√ (i)

(j)

(k) √

(l)

√

(m) √

(n)

(o)

√

(p)

7

3 PRODUCTS

1.

(a) What is the area of A ?

(b) What is the area of B ?

(c) What is the area of A+B ?

(d) Area of big rectangle = l × b

Thus the area = 5 (5 + 2) = 5(7) = 35 square units

We can also get 35 square units by doing this:

5(5 + 2) = (5×5) + (5×2)

a(a + 2) = (a × a) + (a × 2)

= a² + 2a

2.

(a) What is the area of A?

(b) What is the area of B?

(c) What is the area of C?

(d) What is the area of D?

(e) What is the area of A+ B + C + D ?

(f) Area of big rectangle = l × b

Thus the area = (5 + 3) (5 + 2) = (8)(7) = 56 square units

We can also get 56 square units by doing this:

(5 + 3)(5 + 2) = (5×5) + (5×2) + (3 × 5) + (3 × 2)

3. What is the area of the figure with length (a + 3) units

and width (a + 2) units?

(a + 3)(a + 2) = (a × a) + ……….. + (3 × a) + ……..

= a2 + 2a + 3a + 6

= a2 + 5a + 6

5

5

2

A B

a

a

2

5 2

5

A

B

3

A B

C D

a

a

2

A

B

3

8

Can you maybe do the following?

(x - 3)(x + 2)

(x - 5)(x - 4)

(x + 3)2

(x + 3)(x2 + 2x - 5)

4. Simplify:

(a) 3(2m – 5) (b) 5(x + 2)

(c) –3(2y2 – 3y + 5) (d) -5(x – 2)

(e) 2x(3x2 – 4x + 3) (f) 2a(a – b)

(g) (x – 4)(x + 7) (h) -3a(a + b)

(i) (2a + 7)(3a - 5)4 (j) 3a(a2 + 3a – 5)

(k) (5a – 3)(5a + 3)( –2) (l) (x + 1)(x + 3)

(m) (2y – 5)(2y + 5) (n) (x + 2)(4 – x)

(o) –4(3q – 5)(3q + 5) (p) (2x + 5)(2x – 3)

(q) (2x – 1)(3x + 2) (r) (2x + y)(2x – y)

(s) (2p – 3)2 (t) (x +

3

1)(x –

3

1)

(u) –2(2m – 5)2 (v) (3x – 2y)(5x + y)

(w) 2(2x – 3) – 4(3x + 4)

(x) (2

1x +

3

1)(

4

1x + 3)

(y) (p – 5)(p + 6) – (p + 5)(p + 2) (z) (x + 2)(x2 + 4x + 4)

(aa) –3(m + 4)(3m – 2) + (2m + 3)(2m – 1) (bb) ( x + 2)(x2 – 2x + 4)

(cc) (2h + 3 )(2h – 3) – 4(h – 5)(h – 3) (dd) (x – y)(x2 + xy + y2)

(ee) (x – 3)(x + 2)(x – 1) (ff) (x – 3)(x2 + 3x + 9)

(gg) (2x–- 3)(x – 2)(x + 1) (hh) (x +

3

1)2

5. Determine the area (in terms of x) of the following :

(a) Area of rectangle (b) Area of square

(c) Area of triangles

(i) (ii)

2x +7

2x + 10

2x +3

2x - 3

2x + 6

x - 1

2x + 3

9

(iii)

6. A photo is in a frame

with a border of 1 cm.

Determine the area of

the photo (in terms of x).

7. A garden (rectangular shape) has a lawn, bordered on two sides by flowerbeds.

Determine the area of lawn in terms of x.

8. Determine the volume (in terms of x) of the following :

Cube Rectangular prism

9. Karabo does not have a calculator and has to find the value of the following expression:

(a) 20082 + 2009 × 2007 – 2006 × 2010 – 2016 × 2000 (Hint: Let 2008 = x)

(b) 19998882 - 1999890 × 1999886 – 1999891 × 1999885 + 1999898 × 1999878

Frame (x + 10) cm

(x + 5) cm

flowerbeds

lawn

6 m

x

x

9 m

3 – x

x + 2

(x + 2) cm (x + 5) cm

(x + 3) cm

(x + 2) cm

10

4 Factorising

We get the result if

2(3a – 5)

= 2 × 3a – 2 × 5

= 6a – 10

Here the number in front of the brackets is

multiplied with every term inside the

brackets.

You can do the converse by starting on the left hand side: then you

write the common number with which everything is multiplied in

front of the brackets: we say “you take out a common factor”:

6a – 10 = 2(3a – 5)

If :

(x – 3)(x + 2)

= x × x + x × 2 – 3 × x – 3 × 2

= x2 – x – 6

We want know what did we multiply to get x2 – x – 6. We multiplied (x

– 3)(x + 2) to get x2 – x – 6.

(x – 3)(x + 2) = x2 – x – 6

Factorisation is therefore the converse operation of finding a product.

1. What factors belong to which expression?

Expression Factors

2a – 8 a(a + 2)

a2 + 2a (a + 4)(a + 1)

a2 – 4 (a + 2)(a + 1)

a2 + 5a + 4 2(a – 4)

a2 + 3a + 2 (a + 4)(a – 4)

a2 – 16 (a + 2) (a – 2)

2. What pairs of factors from Table B belong to which expressions in Table A. (If you work correctly, you will use

all the factors in Table B.)

Ta

ble

A

Expressions

x2 – x – 6 x

2 + 9x + 14

x2 + x – 6 x

2 – 9x + 14

x2 + 5x + 6 x

2 + 5x – 14

x2 – 5x + 6 x

2 – 5x – 14

x2 + 3x + 2 x

2 + 4x + 3

x2 – 3x + 2 x

2 – 2x – 3

x2 – x – 2 x

2 – 4x + 3

x2 + x – 2 x

2 + 2x – 3

Ta

ble

B

Factors

(x + 1) (x + 1) (x + 1) (x + 1)

(x – 1) (x – 1) (x – 1) (x – 1)

(x + 2) (x + 2) (x + 2) (x + 2)

(x + 2) (x + 2) (x – 2) (x – 2)

(x – 2) (x – 2) (x – 2) (x – 2)

(x + 3) (x + 3) (x + 3) (x + 3)

(x – 3) (x – 3) (x – 3) (x – 3)

(x + 7) (x + 7) (x – 7) (x – 7)

Products (left to right)

Factorisation (right to left)

11

3. Factorise completely:

3.1 4x – 12 3.2 x(p – q) + y(p – q) 3.3 x(a – b) + y(b – a)

3.4 px – py + y – x 3.5 x2 – 4 3.6 3d

2 – 75

3.7 –2x2 + 8 3.8 x

2 – 3x – 10 3.9

3.10

3.11 12x

2 + 11x + 2

3.12 x2 – 8x + 4

3.13 x2 + 6x + 9 3.14 y

2 + 9 3.15 x

2 – 7x + 6

3.16 a4 – 1 3.17 2x

2 + 10x + 12 3.18 4x

2 + 8x – 60

3.19 2m2 + 8 3.20 x

3 – x

2 – 2x

4. The area of the rectangle in the diagram is

(x2 + 3x – 18 )cm² .

Determine the length and width of the rectangle in terms of x.

5. The area of the square in the diagram is

(x2 + 8x + 16) cm².

Determine the length of the sides of square in terms of x.

6. The area of the triangle in the diagram is

cm².

Determine the basis and height of the triangle in terms of x.

7. The volume of the right angle prism in the diagram is

(x3 + 5x

2 + 4x) m

3

Determine the length, width and the height of the prism in

terms of x.

8. Factorise completely:

8.1 9a2 – 25b

2

8.2 9sin

2 x – 25cos

2 x [ Hint : sin

2 x = (sin x)

2 ]

8.3 cos2 x + 3cos x + 2

9. Simplify:

9.1 √ 9.2 √

x2 + 3x – 18

x2 + 8x + 16

𝟏

𝟐𝒙𝟐 𝟒𝒙 𝟔

12

5 Algebraic fractions

1. Below are a few methods that can be used to simplify

. Which one is wrong?

(a)

(b)

(c)

(d)

2. Learners made errors below. What did they do wrong? Rectify it.

(a)

(b)

=

(c)

=

=

=

3. Simplify

(a)

(b)

(c)

(d)

(e)

√

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

4. Simplify

(a)

(b)

(c)

13

6 Solving of linear equations

1. Examples:

a – 7 = – 3

Add 7 both sides:

[a – 7] + 7 = –3 + 7

a – 7 + 7 = –3 + 7

a = 4

Times each side by 15:

[

] [

]

30 – 10y = 12

Subtract 30 from each side:

[30 – 10y] – 30 = 12 – 30

– 10y = – 18

Divide both sides by -10:

y = 1,8

5m = 48

Divide both sides by 5:

[5m] ÷ 5 = 48÷ 5

m = 9,6

2.1 The solutions to the equations are in the answer cloud, but there is one extra answer. Find

the extra answer and write an equation to match it.

2a + 5 = 17 5d + 2 = - 18

3b – 5 = 10 -6e – 3 = - 15

4c – 1 = -13 -12f + 4 = -8

2.2 Select any TWO of your above solutions and check if they are correct.

3. Find the length of each side in the two shapes:

Perimeter = 62 cm

Perimeter = 260 mm

10 cm

5x

3x

3x

2x

4y

7y

6y

2y

1

3 -4 6

-3

5 2

14

4. Solve for x and discuss the three solutions:

A. 7(x – 2) = 4x – 14 + 3x

B. 7(x – 2) = x – 3

C. 7(x – 2) = (x + 1)2 + 5x

5. Determine the values of the unknowns in the following equations:

5.1 2x = 6 5.2

a = 6

5.3 0,5q = 6 5.4 5x – 15 = 3x – 7

5.5 m + 7 = 3m – 15 5.6

x + 5 = 14

5.7 7(2x + 1) = 5 – 4(2x – 3) 5.8 2(b – 2) – (b – 1) = 2b – 2

5.9 5(x – 2) –

(8x + 8) = (x - 4)3 5.10 (4p – 3) – (3p – 4) = (7p – 6) – (4p – 3)

6. What is wrong? Why?

3a – 5 = 7

3a = 12

a = - 4

(x + 1) = 12

3:

2 ( 3x + 3) = 36

5y = 8

y =

q – 1 =

q

6:

3q – 1 = 4q

7. Determine the values of the unknowns in the following equations:

7.1 2(x + 4)2 = (x + 1)

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

7.2

–

(x – 3) = 5

7.3

–

= x –

7.4

=

7.5

7.6

= 3

7.7

+ 1 = 0

8. The marked sides are equal. The longest

side is four times the length of one of the

equal sides. The perimeter of the shape

is 150 mm. Find the length of the longest

side and the area of the shape.

a

35 mm

15

7 Linear inequalities 1. Match the sign(A) to the correct description(B).

A = < >

B Is greater

than

Is less or equal

to Is equal to Is less than

Is greater than

or equal to

2. Write in maths language:

You are not allowed to

drive faster than 80

km per hour!

You must be taller

than 1,5m to enter.

For readers younger

than 10 years.

Price = R35

3. Match the description (C) to the correct number line (D):

C x = - 2 x < - 2 x > - 2 - 2 - 2

D

4.1 Complete the table:

2 < 5 -2 5 -5 -2 5 -2

+ 3 on both sides 2+3 ? 5+3

5 < 8

-4 on both sides -2 – 4 5 – 4

-6 1

5 on both sides 5 5 -2 5

25 -10

-6 on both sides !!! -5 -6 -2 -6

30

- 1 on both sides !!!

4.2 What do you notice?

-2 -2 -2 -2 -2

16

5. Solve the following inequalities:

5.1 2x < 8

5.2

a ≥ –6

5.3 –2x ≤ 4

5.4 –

x > –8

5.5 m + 4 > 10

5.6 3k + 5 ≤ –7

5.7 –7m – 9 ≥ 4m + 13

5.8 3(2h + 5) – 3(3h + 5) > 10 – 19

6.1 Represent each solution to Question 5 on a number line if x R.

6.2 What will change if x N (natural numbers) or x Z (integers)?

7. Write in maths language:

Socks for children

from 3 to 5 years old.

Free entry for adults

over 50 and children

under 7.

Only taxis between 6

en 9 a.m.

Mass is more than

3kg, but not more

than 5kg.

8. Match the descriptions (C) to the correct number lines (D):

C – 4 < x < 7

or : (– 4; 7 )

– 4 ≤ x ≤ 7

or: [– 4; 7 ]

–4 < x ≤ 7

or: (– 4; 7 ]

–4 ≤ x < 7

or: [– 4; 7 )

D

9. Solve the following inequalities:

9.1 10 < 5a < 25

9.2 –9 < 3a < 21

9.3 14 ≤ –7a ≤ 49

9.4 –7 2a + 1 < 10

9.5 –1 < –a – 3 < 1

9.6 –4 <

≤ 8

10. Represent each solution to Question 9 on a number line if a R.

7 -4 7 -4 7 -4 7 -4

17

8 Straight lines

1. Fill in the missing coordinates:

2. Write in your own words what you notice.

3. Match Column B to column A:

Column A Column B

3.1 Equation of the y-axis P.

x = k (k is a number)

3.2 Equation of a line parallel to the y-axis Q.

y = 0

3.3 Equation of the x-axis R.

y = c (c is a number)

3.4 Equation of a line parallel to the x-axis S.

x = 0

4. Make sketches of the following lines on the same system of axes. (Remember to write down the equation of

each line.)

x = 4 y = - 3 x + 1 = 0 y – 2 = 0 x + 3 = 4 – 1

5.1 Write down the equation of:

5.1.1 AB

5.1.2 BC

5.1.3 DC

5.1.4 AD

5.2 Write down the coordinates of A, B, C and D.

5.3 Write down the length of each line segment.

5.4 Calculate the area of the figure.

18

6. EXAMPLES OF STRAIGHT LINES:

A. 2y + 5x + 10 = 0

To find the y-intercept, let x = 0 (Why?!?)

2y + 5(0) + 10 = 0

2y = –10

y = –5 (0; –5)

To find the x-intercept, let y = 0 (Why?!?)

2(0) + 5x + 10 = 0

5x = –10

x = –2 (–2; 0)

B. y – 4x = 0

To find the y-intercept, let ______

(0; ___)

To find the ________the x-intercept, let y = 0

(___; 0)

What next ???

Select any value for x and calculate the coordinates of a point that is on the line, e.g. if x = 1, then?

7. Two special lines:

7.1 Fill in the missing coordinates.

7.2 What is the size of the angle that is formed between the positive x-axis and the line?

7.3 What is the equation of each line?

8. Make sketches of the following straight lines: show all calculations and clearly indicate intercepts with axes.

8.1 3y – 2x = 12

8.2 y = x

8.3 3y + 2x = 0

8.4 2y – 6 = 0

8.5 5y + 4x – 15 = 0

8.6 3x = 9

8.7 2y + 2x = 0

8.8 5y + 4x +20 = 0

19

9 The theorem of Pythagoras

Warming up 1-2-3!

1. Completing:

(5) (5) =

(–5) (–5) =

Thus:

if x2 = 25

then

x = +5 OR x = – 5

(7) ( 7 ) =

( ___ ) (– 7) = 49

Thus:

if y2 = ____

then

y = +7 OR y = ____

(1) (1) =

(– 1 ) ( ____ ) = 1

Thus:

if a2 = 1

then

a = _____ OR a = – 1

(

) (

) =

(

) (

) =

Thus:

if p2 =

then

p =___ OR p = ___

2. Complete:

(√ ) (√ )

=( 3) ( 3)

= 9

√ √

= ____ ____

= 81

√ . √

= ____ . ____

= ____

Thus: (√ √ ) = 17 ; (√ √ ) = ____ ; √ √ = ____

3.

The sum of the three interior angles of a triangle is _____ In any triangle the longest side is always opposite the

_______ angle and the ________ side is opposite the smallest angle. In a right angled triangle the biggest angle is

equal to _____ . The side opposite this angle is named the __________ and it is always the _______________ side of

the triangle.

The theorem of Pythagoras

In Δ ABC with ̂ = 90 : c2 = a

2 + b

2

Examples:

Given: a = 24; b = 7

c2 = a

2 + b

2 (Pyth)

= (24)2 + (7)

2

= 576 + 49

= 625

c = √

= 25 (c > 0)

(Why is c > 0?)

Given : c = 14; b = 5

c2 = a

2 + b

2 (Pyth)

(14)2 = (a)

2 + (5)

2

196 = a2 + 25

196 – 25 = a2

171 = a2

a = √

= 13,08 (a > 0)

(Why is a > 0?)

A

B

C

a2

b2

c2

20

4. Answer the following two questions for each of the triangles:

(a) Which side is the hypotenuse?

(b) What is the missing length (in cm)?

4.1

4.2 4.3 4.4

4.5

4.6 4.7 4.8 ABCD is 'n rhombus.

DE = 20

EC = 21

4. The antenna is situated next to the Klipheuwelroad between Stellenbosch and Malmesbury. At heights of 11m, 22m,

33m and 44 m there are THREE steel cables attached to the antennae to keep it vertical. (Only TWO sets of cables

can be seen on the photo.)

Make use of the information in the Side View and calculate:

4.1 the total length of steel cable necessary to keep the antenna vertical.

4.2 the total cost if the price of steel cable is R350 per meter.

SIDE VIEW

A

B

3

D

E

F

E

15 8

B C D 60

Q

P

E

D

C

4

R

13

P

Q5

K

0,7

M

K

2,5

L

A

61

11

P

RS

R

3

46

C

D

F

E

9

C

40

9C

15

A

B

C E

55m

A

B

C

D

E

F

O

44 m

33m

22 m

11 m

20m