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MATHEMATICS STUDY GUIDE Graad 9
A member of the FUTURELEARN group
Mathematics Study guide
1909-E-MAM-SG01
Í3)È-E-MAM-SG01RÎ
Grade 9
CAPS aligned
DM Oost
Study Guide G09 ~ Mathematics
Contents UNIT 1: NUMBER SYSTEMS ............................................................................................. 3 Exercise 1: Summary of previous work ............................................................................ 3 Exercise 2: One-dimensional graphs................................................................................ 5 Exercise 3: Rational and irrational numbers ..................................................................... 6 Exercise 4: Real and non-real numbers ........................................................................... 9 Exercise 5: Different types of fractions ........................................................................... 10 Exercise 6: Positioning of numbers ................................................................................ 12 Exercise 7: Mixed exercises ........................................................................................... 13 Bibliography ................................................................................................................... 13 UNIT 2: EXPONENTS ....................................................................................................... 15 Exercise 1: Simplifying with prime factors as bases ....................................................... 15 Exercise 2: Simplifying with compound factors as bases ............................................... 17 Exercise 3: Scientific notation ........................................................................................ 19 Exercise 4: Simple equations ......................................................................................... 20 Exercise 5: Mixed exercises ........................................................................................... 21 Bibliography ................................................................................................................... 22 UNIT 3: ALGEBRA............................................................................................................ 24 Exercise 1: Revision of basic algebra – Grade 8 ............................................................ 24 Exercise 2: Products of polynomials .............................................................................. 25 Exercise 3: Functional notation and substitution ............................................................ 28 Exercise 4: Invalidity in fractions .................................................................................... 30 Exercise 5: Invalidity in roots (non-real roots) ................................................................ 31 Exercise 6: Factorising: Common factors ....................................................................... 32 Exercise 7: Factorising: Difference between squares .................................................... 35 Exercise 8: Factorising: Grouping .................................................................................. 36 Exercise 9: Factorising: The trinomial pattern ................................................................ 37 Exercise 10: Factorising: Mixed factorising .................................................................... 38 Exercise 11: Simplification of fractions by factorising ..................................................... 41 Exercise 12: Equations................................................................................................... 43 Exercise 13: Formulas.................................................................................................... 44 Exercise 14: Mixed exercises ......................................................................................... 45 Bibliography ................................................................................................................... 46
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Study Guide G09 ~ Mathematics
UNIT 4: NUMBER PATTERNS AND RELATIONSHIPS................................................... 48 Exercise 1: Number patterns and formulas .................................................................... 48 Exercise 2: Relationship between x and y ...................................................................... 51 Exercise 3: Linear relationships ..................................................................................... 52 Exercise 4: The gradient or slope of straight lines.......................................................... 58 Exercise 5: Non-linear graphs ........................................................................................ 61 Exercise 6: Graphs with limitations (restrictions) ............................................................ 62 Exercise 7: Mixed exercises ........................................................................................... 64 Bibliography ................................................................................................................... 65 UNIT 5.1: GEOMETRY: MEASURING, SPACE AND SHAPE ......................................... 67 Exercise 1: Measuring instruments ................................................................................ 67 Exercise 2: Ratio between side lengths, areas and volumes, and fault analysis ............ 73 Exercise 3: Polyhedrons................................................................................................. 75 Exercise 4: Three-dimensional views ............................................................................. 78 Exercise 5: Constructions and compass direction .......................................................... 86 Exercise 6: Angles of elevation and angles of depression ............................................. 89 Exercise 7: Mixed exercises ........................................................................................... 91 Bibliography ................................................................................................................... 91 UNIT 5.2: GEOMETRY: EUCLIDIAN GEOMETRY ........................................................... 93 Exercise 1: Similarity ...................................................................................................... 93 Exercise 2: Congruent triangles ................................................................................... 106 Exercise 3: Congruent triangles and quadrilaterals ...................................................... 111 Exercise 4: Parallelograms........................................................................................... 113 Exercise 5: Mixed exercises ......................................................................................... 119 Bibliography ................................................................................................................. 120 UNIT 5.3: GEOMETRY: AREA, CIRCUMFERENCE AND VOLUME ............................. 123 Exercise 1: Revision of area and circumference (Grade 8) .......................................... 123 Exercise 2: Volume ...................................................................................................... 128 Exercise 3: Outer areas of fixed bodies........................................................................ 133 Exercise 4: Mixed exercises ......................................................................................... 134 Bibliography ................................................................................................................. 136 UNIT 6: TRANSFORMATION GEOMETRY .................................................................... 139 Exercise 1: Symmetry .................................................................................................. 139 Exercise 2: Formulas for reflection ............................................................................... 140 Exercise 3: Formulas for translation ............................................................................. 141 Exercise 4: Formulas for rotation ................................................................................. 142 Exercise 5: Enlargement and reduction........................................................................ 146 Exercise 6: Mixed exercises ......................................................................................... 149 Bibliography ................................................................................................................. 151
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Study Guide G09 ~ Mathematics
UNIT 7: RATIO, PROPORTION AND RATE ................................................................... 154 Exercise 1: Equivalent fractions ................................................................................... 154 Exercise 2: Distribution of items (values) ..................................................................... 154 Exercise 3: Increasing and decreasing ratio................................................................. 156 Exercise 4: Direct and indirect (inverse) ratio ............................................................... 157 Exercise 5: Interpretation of graphs ............................................................................. 162 Exercise 6: Mixed exercises ......................................................................................... 168 Bibliography ................................................................................................................. 169 UNIT 8: FINANCES ......................................................................................................... 171 Exercise 1: Revision of Grade 8 ................................................................................... 171 Exercise 2: Interest....................................................................................................... 174 Exercise 3: Hire purchase ............................................................................................ 177 Exercise 4: Timeline problems ..................................................................................... 179 Exercise 5: Mixed exercises ......................................................................................... 181 Bibliography ................................................................................................................. 181 UNIT 9: STATISTICS ..................................................................................................... 184 Exercise 1: Identify the problem .................................................................................. 184 Exercise 2: Collection of data ...................................................................................... 185 Exercise 3: Deciding on a source ................................................................................ 188 Exercise 4: Recording and organising data ................................................................. 190 Exercise 5: The frequency table .................................................................................. 192 Exercise 6: Process data statistically .......................................................................... 195 Exercise 7: Presentation of data ................................................................................. 201 Exercise 8: Interpreting outcomes ............................................................................... 203 Exercise 9: Conclusions and decision making ............................................................ 209 Exercise 10: Apply solutions and make predictions .................................................... 210 Exercise 11: General statistics .................................................................................... 212 Exercise 12: Reliability of statistics .............................................................................. 213 Exercise 13: Correlation between two variable data ................................................... 215 Exercise 14: Mixed exercises ...................................................................................... 217 Bibliography ................................................................................................................. 218 UNIT 10: PROBABILITY ................................................................................................. 220 Exercise 1: Revision of Grade 8 ................................................................................... 220 Exercise 2: Probability of simple and compound events .............................................. 224 Exercise 3: Mutually exclusive and complementary events.......................................... 229 Exercise 4: The product of probabilities ....................................................................... 239 Exercise 5: Mixed exercises ......................................................................................... 243 Bibliography ................................................................................................................. 246
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Study Guide G09 ~ Mathematics
Topic
Exam paper
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5
Number systems Exponents Algebra Number patterns and relationships Geometry 5.1 Measurement, space and form 5.2 Euclidian geometry 5.3 Area, volume and circumference Unit 6 Transformation geometry Unit 7 Ratio, proportion and rate Unit 8 Finance Unit 9 Statistics Unit 10 Probability
1 1 1 1 2 2 2 2 1 1 2 2
The Greek numbers that we use in our Maths.
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Study Guide G09 ~ Mathematics
Unit
1
Contents Unit 1: Number systems .................................................................................................... 3 Exercise 1: Summary of previous work ............................................................................ 3 Exercise 2: One-dimensional graphs................................................................................ 5 Exercise 3: Rational and irrational numbers ..................................................................... 6 Exercise 4: Real and non-real numbers ........................................................................... 9 Exercise 5: Different types of fractions ........................................................................... 10 Exercise 6: Positioning of numbers ................................................................................ 12 Exercise 7: Mixed exercises ........................................................................................... 13 Bibliography ................................................................................................................... 13
Irrational
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Study Guide G09 ~ Mathematics
Unit
Learning outcome 1 LO1
Numbers, calculations and ratios (algebra)
40%
Learning outcome 2 LO2
Patterns, functions and algebra
15%
Learning outcome 3 LO3
Space and shape (geometry)
15%
Learning outcome 4 LO4
Measurement in a variety of contexts
15%
Learning outcome 5 LO5
Data handling
15%
Level of difficulty 1 Basic knowledge 25% Level of difficulty 2 Routine procedures 45%
Level of difficulty 3 Complex procedures 20%
Level of difficulty 4 Problem solving 10%
Level of difficulty 5
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Assessment systematic • Use Maths facts and vocabulary • Use the correct formulas • Predict answers and round off values • Theoretical knowledge ** • Completing known procedures • Apply knowledge and facts with more than one step • Come to conclusions from given information • Basic calculations as learnt from examples and exercises *** • Complex calculations and high order arguments • Euclidean geometry • No stipulated path to follow • Similarities and differences between presentations • Needs conceptual and holistic approaches to problems **** • Unseen and non-routine problems • Problems in different sections • Still in curriculum and study guide • Mostly projected to practical and everyday situations • High order of thinking ***** Advanced Mostly seen as knowledge of the next grade. These situations do occur to understand where the topic will end up, but should not be included in tests and exam papers. *
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Study Guide G09 ~ Mathematics
Unit
1
Unit 1: Number systems LO1 Exercise 1: Summary of previous work N = { 1;2;3;4;5.......} = natural numbers N0 = {0;1;2;3;4.....} = counting numbers Z = {.....-2;-1;0;1;2;3;.....} = integers Q = {all numbers that can be written as
integer } non - zero integer
= rational numbers Q’ = {all numbers that cannot be written as
integer } non - zero integer
= irrational numbers R = real numbers = {rational plus irrational numbers} R’ = non-real numbers = {numbers that do not exist} 1.1**
Recognise the following numbers in the table: N
N0
Z
Q
Q’
R
R’
9 3 −8 −8
π 0,3 −1 5 4,7 13
17 1.2*
Arrange the following from smallest to largest: −7 − 15 15 7 ; ; ; 15 7 7 15 Video exercise 1 Video exercise 1.1 Video exercise 1.2
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Study Guide G09 ~ Mathematics
Unit
Example x ∈ (-3;5] NOTE: A round bracket means excluded, and a square bracket means included.
Different notations Example: All integers between -5 and 2. 1. Normal notation -5 < x < 2 2. Tabulate notation = {-4;-3;-2;-1;0;1 } 3. Set notation = {x / − 5 < x < 2, x ∈ Z } 4. Interval notation This is only valid for x ∈R 5. Graphically -4
-3
-2
-1
0
1
R
-5 < x < 2
Remember: only integers. –5 and 2 are not included.
Video of notations Video exercise 1.3
1.3**Complete the following table for the different notations of groups of numbers.
English 1.3.1 1.3.2
1.3.3 1.3.4
1.3.5
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Normal notation
Tabulate notation
Real numbers from -4 to 10 Rational numbers between 0 and 4 Integers from -6 up to 1 Real numbers greater than 4 and less than and equal to 8 Natural numbers less than 8
4
Set notation
Interval notation
Study Guide G09 ~ Mathematics
Unit
1
Exercise 2: One-dimensional graphs LO1 Present the following collections graphically as the examples are given about the different notations. −3 ≤ x < 4 x∈ R 2.1 3 < x ≤ 8, x ∈ R 2.2 Video exercise 2.1 to 2.4 − 1 < x ≤ 5, x ∈ Z 2.3 2.4 x ≤ 5 , x ∈ N0 Example 1: Use graphs to get the answers of the following. Give the answer in interval notation. −2 < x ≤ 1
-2
or
-1
0 ≤ x ≤ 4 x∈ R
0
1
2
3
4
R −2 <x ≤ 1 0≤ x ≤ 4
“Or” means in both
Answer: x ∈ ( − 2 ; 4 ] Example 2: Use graphs to get the answer of the following. Give the answer in interval notation. −2 < x ≤ 1 and 0 ≤ x ≤ 4 x ∈ R -2
-1
0
1
2
3
4
R
−2< x ≤ 1 0≤ x ≤ 4
Answer:
“And“ means intersecting.
x ∈ [0 ; 1 ]
Note the difference between the “or” and the “and” in the two examples.
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Study Guide G09 ~ Mathematics
Unit
Get the following answers by making use of graphs. Video 2.5 Video 2.6 Give the answers in interval notation. −3 ≤ x < 4 or 4 ≤ x < 6 , x ∈ R 2.5** Video 2.7 Video 2.8 −1< x < 5 or 4 ≤ x ≤ 7 , x ∈ R 2.6** −1< x < 5 and 4 ≤ x < 7 , x ∈ R 2.7** Video 2.9 Video 2.10 x ≥ − 2 and 0 < x ≤ 3 , x ∈ R 2.8** x ≥ − 2 or 0 < x ≤ 3 , x ∈ R 2.9** Video 2.11 Video 2.12 −1≤ x < 5 or 4 ≤ x ≤ 6 , x ∈ Z 2.10*** 3 ≤ x < 5 and 4 ≤ x ≤ 8 , x ∈ N 2.11*** 2.12*** Can one present the collection of irrational numbers graphically? Remember that non-ending, non-repetitive fractions are irrational numbers.
Exercise 3: Rational and irrational numbers LO1 Addition, subtraction, multiplication and division of rational numbers can be done with or without a calculator. Read carefully and before you start with your work, first determine whether or not you may use a calculator.
F
With a calculator: All marks for the answer.
r a c t i o n s
Without a calculator: All marks for the steps.
Example 1 Determine the answer without using a calculator: 4 1 1 1 2 −1 . ÷ 3 4 2 3 9 3 1 3 = − . . 4 2 3 4 Example 2: 9 3 = − Determine with a calculator: 4 8 1 1 3 18 − 3 22 − 8 . = 3 5 4 8 971 calculator) 15 = ((with with acalculator) = 60 8 11 7 = 16 = 1 60 8
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Study Guide G09 ~ Mathematics
Unit
1
Determine without using a calculator: 3.1** 3.2** 3.3***
1 3 − 4 8 1 3 ÷ 4 8
Video 3.1 to 3.3 Video 3.4 and 3.5
8 3 − ) 3 8 3.4** 1 3 1 3 + − 3 8 2 3.5*** 1 7 3 . − + 3 12 9 8 Determine the following by using a calculator: 3(
(Do not change the question to decimals. Use the 3.6* 3.7** 3.8*** 3.9***
2 3 − 5 3 7 2 5 1 4 ÷ 1 x 2 13 9 13 1 1 7 (23 − 100 ) . 4 3 8 64 1 23 1 13 x ÷ +1 + 8 8 24 69 2 2
x button on your scientific calculator.) y Video 3.6 Video 3.7 Video 3.8 Video 3.9 Video 3.10
3.10***
3 3 3 3 3 + 4( . ÷ − ) 4 4 4 4 4
Irrational numbers are like variables. e.g. 3 2 + 2 = 4 2
3 2 −
2 = 2 2
3 2 x
2
= 3 .2 = 6
3 2 ÷
2
=
3 2 2
7
= 3
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Study Guide G09 ~ Mathematics
Unit
Simplify the following irrational numbers: 3.11* 3 + 6 3 − 8 3 3.12*
12 2 . 2
3.13**
2 2 . 2 2 2
3.14**
12 5 6 3
.
2 3
2 2
.
8 5
3.15**
4 2 ÷ 3 2 2
3.16**
7 49 +
3.17**
4 +3 8 8
Remember: x x = x
2 2 3
Video 3.11 and 3.12 Video 3.13 and 3.14
25 7
Video 3.15 to 3.17
Simplify the following by making use of the technique in the given example: 3.18***
18 + 4 2
3.19***
27 +
36.3
3.20***
48 −
75
3.21***
( 19 )2 2 − 50
3.22***
2 8 − 3 32
3.23***
108 −
3.24***
25 − 16 4 9
3.25***
121
3 100 − 64
Example: Remember the following: x.y = x y
∴ 12 =
3
Video 3.18
4 3
= 2 3 Choose square numbers so that the root can be drawn.
+ 144 + 25
Video 3.19 and 3.20
Video 3.21 and 3.22
Video 3.23 to 3.25
Arrange the following rational numbers from smallest to largest: (Show the steps to confirm your answer. An answer alone will not be enough.) 3.26* 1 1 1 1 } ; ; ; { 5 3 2 4 Video 3.26 3.27*** 1 1 1 1 } ; ; ; − { − 5 4 3 2 Video 3.27 and 3.28 3.28** 1 1 1 1 ; } { ;− ; − 4 5 3 2 Video 3.29 3.29**
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{ -4 ; −
16 17 18 ; − ; ; 4 } 5 5 5
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Study Guide G09 ~ Mathematics
Unit
1
Exercise 4: Real and non-real numbers LO1
• • •
Real numbers consist of all the rational and irrational numbers together. Non-real numbers are numbers that do not exist. An example is −number : −2 ,
−6
4.1**
Which collection of numbers is obtained if all the whole numbers and all the Integers are put together? i.e. {whole numbers} { integers}
4.2**
Which collection of numbers is the greatest: the integers or the rational numbers?
4.3**
Which collection is obtained when the rational numbers and irrational numbers are put together? {rational numbers} {irrational numbers}
4.4**
Which of the following are non-real numbers? 4 ; − 8 ; 3 8 ; 3 − 8 ; 4 1 ; 4 −1
4.5**
Present the following real numbers graphically. All the numbers between -3 and 0, as well as from 2 up to 4.
Video 4.1
Video 4.2
Video 4.3
9
Video 4.4
Video 4.5
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Study Guide G09 ~ Mathematics
Unit
Exercise 5: Different types of fractions LO1 In this section we will do the following: • Convert a normal fraction to a decimal fraction • Change decimal fractions to normal fractions • Change normal fractions to repetitive fractions • Change repetitive fractions to regular fractions • Mixed, real, improper and equivalent fractions
3 2 8
53 24
0,123
0,241 Do you know the different fractions, and can you do the following with them?
Examples: Normal Fraction 1
Decimal Fraction (with a calculator) 7 8 = 7÷ 8 = 0,875
7 8
2
o , 125 125 = 1000 25.5 = 25 . 8.5 1 = 8
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0 ,125
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Study Guide G09 ~ Mathematics
Unit
Normal Fraction
3
Recurring Fraction 1 9 =1÷ 9 = 0,1
1 9
Let x = 0,1111111..... 1,111111.... ∴ 10x =
4
10x = − x 1,11111.... − 0,1111..... 1 ∴ 9x = 1 11 0 ,1 = = x = ∴∴0,1 9 99
6
5.6**
0 , 1 2
1 ÷ 33 = 0 , 0 3
1 33
Change the following decimal fractions to normal fractions: 2,04 0,003 230, 35 12,0045 24,2 Change the following mixed fractions to improper fractions in their simplest form: 1 33 Example 1: 4 = 8 8 1 9 25 Example 2: 2 , 25 = 2 = 2 = 100 4 4
5.8**
1 4 3 12 8 10 , 375
5.9**
7,6
5.7**
0,1
Let x = 0, 12121212.... ∴ 100x = 12,12121212... ∴ 100x − x = 12 ∴ 99x = 12 12 ∴ x = 99 4 = 33
5
5.1* 5.2* 5.3* 5.4* 5.5*
1
12
Video 5.1 to 5.4 Video 5.6 to 5.8
11
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Study Guide G09 ~ Mathematics
Unit
5.10*** 5.11*** 5.12*** 5.13****
Change the following repetitive fractions to normal fractions: 0 , 6 These are all 2 , 3 Video 5.10 Video 5.11 recurring fractions. 0 , 63 Video 5.12 Video 5.13 0 , 123 Give two equivalent fractions for each normal fraction.
5.14* 5.15* 5.16*
Examples: 50 46 1 2 = = = 92 2 4 100 All these fractions are equivalent.
1 3 3 5 7 8
5.17**
0,25
5.18**
0,04
5.19**
6,1
5.20**
2,3
Video 5.14 to 5.16 First change these decimal fractions to normal fractions.
Video 5.17 and 5.18 Video 5.19 and 5.20
Exercise 6: Positioning of numbers LO1 Example: 1 1 so that all three are equally and 3 2 apart from each other in value.
Place one fraction between
1 3
2 = ; 6 4 = ; 12 4 ∴ ; 12 6.1**
1 2 3 .....; 6 6 ..... ; 12 5 6 ; 12 12
; ........;
Place seven fractions between
Use equivalent fractions until it becomes possible to put the required fraction between the given fractions.
1 3 and 8 8
so that all nine fractions are equally
apart in value. 6.2**
Place three fractions between apart in value.
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1 1 so that all five fractions are equally and 3 2 Video 6.1 Video 6.2
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Study Guide G09 ~ Mathematics
6.3**
1
Unit
Place three fractions between
1 3 so that all five fractions are equally and 3 5
apart in value. 6.4***
Place three rational numbers between 0, 22 and 0, 24, so that all five are equally apart in value.
6.5**
Which rational number is exactly between -0,15 and – 0,13?
6.6**
Place two rational numbers between 0, 1 and 0,121, so that all four numbers are equally apart in value. Video 6.3 and 6.4 Video 6.5 and 6.6
Exercise 7: Mixed exercises LO1 7.1* 7.2* 7.3** 7.4** 7.5** 7.6**
7.7*
7.8**
Give an example of an irrational and a non-real number. Why is 3 − 8 a rational number? Present the following number line: {x / -3 < x < 5 and 0 < x ≤ 3 , x ∈ R } Present the following number line: {x / -3 < x < 5 and 0 < x ≤ 3 , x ∈ Z } Change 0, 4 to a normal fraction. Simplify without using a calculator: 1 3 1 2 − + 5 10 2 Simplify without using a calculator: 15 8 1 ÷ 17 17 Use a calculator and determine the following to two decimal places: 32,45 − 11 , 8 12,58
7.9**
Marks 25 (2) (2) (3) (3) (2)
(3)
(3)
(2)
Determine the LCM and HCF of 124 and 2232. Note: LCM and HCF were done in Grade 8. Go back to your study guide of Grade 8 and revise this section if you do not remember the methods.
(5)
Bibliography Burger, E. Tree Vooruit: Wiskunde Graad 9. Bateleur Books. Couvaras, E. (2006). Maths on the Move Grade 9. Nolwazi Educational Publishers. Education, D. (n.d.). GTA Graad 9. 2008. Eksamenhulp. (2008). Graad 9 Vraestelle met oplossings. en.wikipedia.org/wiki/Amoeba. (n.d.). Heinemann. (2006). Kollig op Wiskunde. Heinemann Uitgewers (Edms) Bpk. Ina Nel, M. S. (2006). My Clever Wiskunde Graad 9 Leerdersboek. CleverBooks (Edms) Bpk. Johnson, Y. (2000). Maths for All Learners Activity Book. Laridon. (1992). Wiskunde vir die Klaskamer 7. SA, N. E. (2008). Gemeenskaplike Taak vir Assessering (GTA). Sarah, O. (2008). Kollig op Wiskunde Graad 9 (proefeksemplaar). Seeliger. PracMaths Gr 9. (2008). 13
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Study Guide G09 ~ Mathematics
Unit
Contents Unit 2: Exponents ............................................................................................................ 15 Exercise 1: Simplifying with prime factors as bases ....................................................... 15 Exercise 2: Simplifying with compound factors as bases ............................................... 17 Exercise 3: Scientific notation ........................................................................................ 19 Exercise 4: Simple equations ......................................................................................... 20 Exercise 5: Mixed exercises ........................................................................................... 21 Bibliography ................................................................................................................... 22
From Grade 8... do you remember this?
Learning outcome 1 LO1 Learning outcome 2 LO2
Numbers, calculations and ratios (algebra) Patterns, functions and algebra
40% 15%
Learning outcome 3 LO3
Space and shape (geometry)
15%
Learning outcome 4 LO4
Measurement in a variety of contexts
15%
Learning outcome 5 LO5
Data handling
15%
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