Í2(È-E-MAM-SG01OÎ
1
9
0
8
-
E
-
M
A
M
-
S
G
0
1
MATHEMATICS STUDY GUIDE Grade 8
A member of the FUTURELEARN group
Mathematics Study guide
1908-E-MAM-SG01
Í3(È-E-MAM-SG01PÎ
Grade 8
CAPS aligned
DM Oost
Study Guide G08 ~ Mathematics
Contents UNIT 1: NUMBER SYSTEMS ......................................................................................... 2 Exercise 1: Where do number systems come from? ................................................... 2 Exercise 2: Relationships and Venn diagrams............................................................. 3 Exercise 3: Order of operations ................................................................................... 7 Exercise 4: Derivation from English to mathematics .................................................... 8 Exercise 5: Derivation from mathematics to English .................................................... 9 Exercise 6: Journal entry: Correction of errors........................................................... 10 Exercise 7: Representing number systems on number lines ..................................... 13 Exercise 8: Calculations with natural numbers .......................................................... 13 Exercise 9: Calculations with whole numbers (especially zero) ................................. 14 Exercise 10: Calculations with integers ..................................................................... 14 Exercise 11: Determining LCM, LCD and HCF .......................................................... 16 Exercise 12: Properties of division ............................................................................. 18 Exercise 13: Properties of calculations ...................................................................... 19 Exercise 14: Prime factors and composite numbers .................................................. 21 Exercise 15: Calculations with rational numbers........................................................ 23 Exercise 16: Equivalent fractions ............................................................................... 25 Exercise 17: Types of fractions .................................................................................. 27 Exercise 18: Percentages .......................................................................................... 29 Exercise 19: Irrational numbers ................................................................................. 31 Exercise 20: Mixed exercises .................................................................................... 33 Bibliography ............................................................................................................... 34 UNIT 2: EXPONENTS ................................................................................................... 36 Exercise 1: Expanded and exponential notation ........................................................ 36 Exercise 2: Laws of exponents .................................................................................. 38 Exercise 3: Negative exponents ................................................................................ 41 Exercise 4: Using prime factors to determine surds................................................... 43 Exercise 5: Difficult surds........................................................................................... 44 Exercise 6: Calculator work ....................................................................................... 47 Exercise 7: Scientific notation .................................................................................... 48 Exercise 8: Mixed exercises ...................................................................................... 50 Bibliography ............................................................................................................... 50 UNIT 3: ALGEBRA ....................................................................................................... 52 Exercise 1: Variables ................................................................................................. 52 Exercise 2: Addition, subtraction, multiplication and division of variables .................. 54 Exercise 3: Laws of algebra ....................................................................................... 57 Exercise 4: Words to mathematics ............................................................................ 60 Exercise 5: Substitution ............................................................................................. 62 Exercise 6: Mixed exercises ...................................................................................... 63 Exercise 7: Linear equations...................................................................................... 66 Exercise 8: Cross multiplication ................................................................................. 70
Š Impaq
i
Study Guide G08 ~ Mathematics
Exercise 9: Modelling (word problems) ...................................................................... 71 Exercise 10: Simple quadratic equations ................................................................... 80 Exercise 11: Mixed exercises .................................................................................... 81 Bibliography ............................................................................................................... 82 UNIT 4: NUMBER PATTERNS ..................................................................................... 84 Exercise 1: Visual representation of patterns............................................................. 84 Exercise 2: Interesting number patterns .................................................................... 86 Exercise 3: Formulae for number patterns ................................................................. 87 Exercise 4: Relationships: input and output diagrams ............................................... 89 Exercise 5: Interpreting graphs .................................................................................. 92 Exercise 6: Ordered pairs .......................................................................................... 94 Exercise 7: Mixed exercises ...................................................................................... 98 Bibliography ............................................................................................................. 100 UNIT 5.1: GEOMETRY: MEASUREMENT, SPACE AND SHAPE ............................. 102 Exercise 1: Measurement and the construction of angles ....................................... 102 Exercise 2: Types of angles ..................................................................................... 109 Exercise 3: Parallel lines .......................................................................................... 112 Exercise 4: Triangles ............................................................................................... 114 Exercise 5: Quadrilaterals ........................................................................................ 119 Exercise 6: Polygons ............................................................................................... 121 Exercise 7: Mixed exercises .................................................................................... 127 Bibliography ............................................................................................................. 128 UNIT 5.2: EUCLIDIAN GEOMETRY ........................................................................... 130 Exercise 1: Logic ..................................................................................................... 130 Exercise 2: Straight lines and angles ....................................................................... 132 Exercise 3: Parallel lines .......................................................................................... 137 Exercise 4: Triangles ............................................................................................... 142 Exercise 5: Similar and congruent triangles ............................................................. 149 Exercise 6: Mixed exercises .................................................................................... 159 Bibliography ............................................................................................................. 160 UNIT 5.3: GEOMETRY: AREA, CIRCUMFERENCE, PERIMETER AND VOLUME .. 163 Exercise 1: Convert units ......................................................................................... 163 Exercise 2: Basic rectangles, triangles and circles .................................................. 164 Exercise 3: Theorem of Pythagoras......................................................................... 169 Exercise 4: Combined figures .................................................................................. 175 Exercise 5: Polygons ............................................................................................... 177 Exercise 6: Basic volumes ....................................................................................... 180 Exercise 7: Mixed exercises .................................................................................... 182 Bibliography ............................................................................................................. 183 UNIT 6: TRANSFORMATION GEOMETRY ............................................................... 185 Exercise 1: Symmetry .............................................................................................. 185 Exercise 2: Reflection .............................................................................................. 187 Exercise 3: Translation ............................................................................................ 192 Exercise 4: Rotation ................................................................................................. 197
© Impaq
ii
Study Guide G08 ~ Mathematics
Exercise 5: Mixed exercises .................................................................................... 201 Bibliography ............................................................................................................. 202 UNIT 7: RATE, RATIO AND PROPORTION .............................................................. 204 Exercise 1: Ratio and equivalent fractions ............................................................... 204 Exercise 2: Division of values in specific ratios ........................................................ 205 Exercise 3: Cross multiplication ............................................................................... 207 Exercise 4: Direct proportion.................................................................................... 207 Exercise 5: Graphs of direct proportion ................................................................... 209 Exercise 6: Indirect proportion ................................................................................. 211 Exercise 7: Graphs of direct and indirect proportion ................................................ 212 Exercise 8: Rate of change ...................................................................................... 214 Exercise 9: Increasing and decreasing values ......................................................... 216 Exercise 10: Drawings to scale ................................................................................ 217 Exercise 11: Mixed exercises .................................................................................. 219 Bibliography ............................................................................................................. 220 UNIT 8: FINANCES .................................................................................................... 222 Exercise 1: Percentages .......................................................................................... 222 Exercise 2: Profit and loss ....................................................................................... 223 Exercise 3: Discount and marked price ................................................................... 225 Exercise 4: Simple interest ...................................................................................... 226 Exercise 5: Hire-purchase loans .............................................................................. 229 Exercise 6: Exchange rates ..................................................................................... 232 Exercise 7: Mixed exercises .................................................................................... 233 Bibliography ............................................................................................................. 234 UNIT 9: STATISTICS .................................................................................................. 237 Exercise 1: Theory: Terminology ............................................................................. 237 Exercise 2: Theory: Examples of diagrams/graphs.................................................. 238 Exercise 3: Criteria for central tendency and dispersion .......................................... 245 Exercise 4: Advantages and disadvantages of mean, mode and median................ 255 Exercise 5: Relative frequency ................................................................................ 256 Exercise 6: Reliability of statistics ............................................................................ 259 Exercise 7: Correlation between two variables ........................................................ 261 Exercise 8: Mixed exercises .................................................................................... 262 Bibliography ............................................................................................................. 263 UNIT 10: PROBABILITY............................................................................................. 265 Exercise 1: Terminology .......................................................................................... 265 Exercise 2: Relative frequency of real outcomes ..................................................... 266 Exercise 3: The probability scale ............................................................................. 267 Exercise 4: Probability P(X) of outcomes X ............................................................. 268 Exercise 5: Mutually exclusive and complementary events ..................................... 271 Exercise 6: Mixed exercises .................................................................................... 273 Bibliography ............................................................................................................. 274 UNIT 11: GOLDEN RULES OF MATHEMATICS ....................................................... 275
© Impaq
iii
Study Guide G08 ~ Mathematics
Unit
1
Contents Unit 1: Number systems .................................................................................................... 2 Exercise 1: Where do number systems come from? ........................................................ 2 Exercise 2: Relationships and Venn diagrams ................................................................. 3 Exercise 3: Order of operations........................................................................................ 7 Exercise 4: Derivation from English to mathematics ........................................................ 8 Exercise 5: Derivation from mathematics to English ........................................................ 9 Exercise 6: Journal entry: Correction of errors ............................................................... 10 Exercise 7: Representing number systems on number lines.......................................... 13 Exercise 8: Calculations with natural numbers ............................................................... 13 Exercise 9: Calculations with whole numbers (especially zero) ..................................... 14 Exercise 10: Calculations with integers .......................................................................... 14 Exercise 11: Determining LCM, LCD and HCF .............................................................. 16 Exercise 12: Properties of division ................................................................................. 18 Exercise 13: Properties of calculations........................................................................... 19 Exercise 14: Prime factors and composite numbers ...................................................... 21 Exercise 15: Calculations with rational numbers ............................................................ 23 Exercise 16: Equivalent fractions ................................................................................... 25 Exercise 17: Types of fractions ...................................................................................... 27 Exercise 18: Percentages .............................................................................................. 29 Exercise 19: Irrational numbers...................................................................................... 31 Exercise 20: Mixed exercises ......................................................................................... 33 Bibliography ................................................................................................................... 34
Learning Outcome 1 LO1 Learning Outcome 2 LO2 Learning Outcome 3 LO3 Learning Outcome 4 LO4 Learning Outcome 5 LO5
Š Impaq
Numbers, operations and relationships Patterns, functions and algebra Space and shape (geometry) Measurement Data handling
1
40% 15% 15% 15% 15%
Study Guide G08 ~ Mathematics
Unit
1
Unit 1: Number systems Numbers and Relationships Topics • • • •
Where do number systems come from? Relationships and Venn diagrams Order of operations Properties of number systems. Associative, commutative and distributive laws
Exercise 1: Where do number systems come from? There is proof that the Ishango people of the Democratic Republic of the Congo (DRC) made marks on bones to count their cattle or family members. The oldest bone found to confirm these assumptions is approximately 20 000 years old. The number system that we use today is the Hindu-Arabic system and it was developed more than 1 000 years ago by the Hindu-Arabic mathematicians. The Egyptian number system consists of symbols. Examples of the two number systems are: Egyptian staff heel
coil of rope lotus finger frog man with both hands raised
đ?¤–đ?¤–
𓎆𓎆 �� 𓆟𓆟
đ“‚đ“‚ đ“†?đ“†? đ“ ?đ“ ?
Hindu-Arabic 1 10
100 1 000 10 000 100 000 1 000 000
1.1*
Write down the Egyptian equivalent for the number 3 516.
1.2*
Write down the Egyptian equivalent for the number 2 182.
Š Impaq
2
Study Guide G08 ~ Mathematics
Unit
đ“ ?đ“ ? 𓆟𓆟 𓆟𓆟
𓎆𓎆 𓎆𓎆
1
đ?¤–đ?¤– đ?¤–đ?¤– đ?¤–đ?¤–
1.3***
Write down the Hindu-Arabic equivalent for
1.4*
Study the number 234 654 365 123 987 341 236 687. Write down the number that is 10 000 more than the given number.
1.5***
Investigate: For fun Draw the following table in your answer book and translate the terms addition, subtraction, multiplication and division in any two other languages.
English Afrikaans Sepedi
+ Addition Optelling Go hlakantĹĄha
– Subtraction Aftrekking Go tloᚧa
Ă— Multiplication Vermenigvuldiging Go atiĹĄa
á Division Deling Go arola
Exercise 2: Relationships and Venn diagrams Study the following example to learn the theory. Example: If U = {1; 2; 3; 4; 5; 6; 7; 8; 9; 10} A = {2; 4; 6; 8; 10} B = {1; 2; 3; 4} Sketch a Venn diagram to illustrate the above:
Video exercise 2
U 5 6 4 A 8 2 10
Š Impaq
7
1 3 B
3
9
Study Guide G08 ~ Mathematics
Unit
1
Determine: This means the union of the two. 1. Which numbers are in A ∪ B? Answer: It means A united with B. We can also say A with B. Then the answer is {1; 2; 3; 4; 6; 8; 10}. The answer is given in curly brackets and each element is separated by a semicolon. This means the intersection of the two. 2. Which numbers are in A ∩ B? Answer: It means A intersecting B, and all the numbers that are in A as well as in B must be given in the answer. Then the answer is {2; 4}. 3.
Which numbers are in A’? Answer: It means the numbers that are not in A. Then the answer is {1; 3; 5; 7; 9}.
4.
Which numbers are in B’? Answer: It means the numbers that are not in B. Then the answer is {5; 6; 7; 8; 9; 10}
5.
Which numbers are in A’ ∩ B? Answer: This means all the numbers that are not in A, but in B. The answer is {1; 3}.
2.1***
Sketch a Venn diagram of the following: U = {1; 2; 3; 4; 5; 6; 7; 8; 9} A = {1; 6; 7; 8} B = {6; 8; 4; 5} Determine: 2.1.1 A ∪ B 2.1.2 A ∩ B 2.1.3 A’ 2.1.4 B’ 2.1.5 A’ ∪ B’
© Impaq
4
Video exercise 2.1
Study Guide G08 ~ Mathematics
2.2**
Unit
Sketch a Venn diagram of the following: U = {11; 12; 13; 14; 15; 16; 17; 18; 19; 20} A = {11; 16; 17; 18} B = {11; 16; 20}
1
Video exercise 2.2
Determine: 2.2.1 A ∪ B 2.2.2 A ∩ B 2.2.3 A’ 2.2.4 B’ 2.2.5 A ∩ B’ 2.3**
Sketch a Venn diagram of the natural numbers, whole numbers and where the universal collection is the collection of integers. Use the following notation: natural numbers = N whole numbers = N0 integers = Z Determine: 2.3.1 N ∩ N0 2.3.2 Z ∪ N 2.3.3 Z ∩ N 2.3.4 N’
2.4*
Sketch a Venn diagram of the real number system, the rational number system and the irrational number system. Video exercise 2.4 Use the following notation: real numbers = R rational number system = Q irrational number system = Q’
2.5*
The sketched Venn diagram presents the following collections of numbers. Which letters denote each of the following? Whole numbers, natural numbers, integers, rational numbers, irrational numbers and real numbers. U D C B A E
© Impaq
5
Study Guide G08 ~ Mathematics
2.6 *****
© Impaq
Unit
1
A total of 150 guests attended a wedding. They were friends, family and colleagues of the bride and groom. • Fifteen of the guests were children and 10 of the children were family. • All 17 colleagues were adults and there were three more women than men. • Six of the 60 men present were family and 23 family members attended the wedding. Sketch a Venn diagram and determine how many female friends attended the wedding.
6
Study Guide G08 ~ Mathematics
Unit
Exercise 3: Order of operations
Video exercise 3
The order of operations is as follows: 1. Brackets (…) 2. Exponents 3. “of” which means multiplication 4. Multiplication and division from left to right 5. Addition and subtraction from left to right Example 1 Simplify the following by only doing one calculation per step: 2 + 3 × 10 – 6 + (3 – 2) +
Example 2 1 of (21 + 3) 3 1 = 2 + 23 × 2 – of 24 3 1 = 2 + 8 × 2 – of 24 3 =2+8 ×2–8 = 2 + 16 - 8 = 18 – 8 = 10 2 + 23 × 2 –
1 of 12 2
1 of 12 2 = 2 + 3 × 10 – 6 + 1 + 6 = 2 + 30 – 6 + 1 + 6 = 32 – 6 + 1 + 6 = 26 + 1 + 6 = 27 + 6 = 33 = 2 + 3 × 10 – 6 + 1 +
Remember: When the bracket is calculated, the bracket disappears. Simplify the following expressions by only doing one calculation per step: 3.1**
4+2 ×3–1
3.2**
1+3 ×2 ÷6–2
3.3**
9 – (12 – 8)
3.4**
3+3 ×3–3
3.5**
(3 + 2) × (6 – 2)
3.6** 3.7** 3.8** 3.9* 3.10** 3.11*** 3.12*** 3.13** 3.14**
© Impaq
Video exercise 3.2 2 + 3 × 4 = 14 and is never 20. First multiplication, then addition.
8 – 2 × 3 + 6 – (5 – 1) 10 – 4 + (3 – 1) × 3 3 ×0+2 ×0 1 of 12 – (6 ÷ 3 + 1) 3 1 of 23 × (14 – 5) 2 24 – 4 – 3 – 2 – 1 100 – 50 ÷ 10 × 6
Video exercise 3.6 Video exercise 3.7 and 3.8
2 + 2 × 2 ÷ 2 – 2 + (2 + 2)
(6 – 2 + 7) + (3 – 2 + 12)
Video exercise 3.4 and 3.5
Video exercise 3.9 Video exercise 3.10 Example: 1 of 23 × (14 – 5) 2 1 = of 8 × (9) 2 =4 ×9 = 36
7
Video exercise 3.11 Video exercise 3.12 Video exercise 3.13 and 3.14
1
Study Guide G08 ~ Mathematics
Unit
3.15**
5 ×2–4×2+7
3.16**
2 ×2 ×2 ×2–2 ×2 ×3
3.17**
10 –
1 2
Video exercise 3.15 and 3.16
of 18 + (6 – 2)
3.18**
5+3–2–1×6
3.19**
18 + 24 ÷ 4 ÷ (3 – 2 × 0)
3.20***
1 1 of 16 + of 8 – 2 × 2 4 8
1
Video exercise 3.17 and 3.18 Video exercise 3.19 and 3.20
Exercise 4: Derivation from English to mathematics Symbol × ÷ – +
Meaning Multiplication Division Subtract Addition
Other English Words Product Quotient Difference Sum
Video exercise 4
Write the following in mathematical expressions. Follow the order of operations and simplify the expressions. Example: Subtract the sum of 32 and 70 from the product of 15 and 10 and divide the answer by a half. Answer: {(15 × 10) – (32 + 70)} ÷ = {150 – 102} × 2 = 48 × 2 = 96
1 2
4.1*
The sum of (20 – 3 × 6) and (5 – 2).
4.2*
The product of (2 + 4) and (10 – 8).
Video exercise 4.1
4.3**
The quotient if 20 is divided by (4 + 6).
Video exercise 4.2
4.4**
The difference between 34 and (10 – 2 + 3).
Video exercise 4.3
4.5*
Subtract 10 from (20 + 2).
4.6*
Subtract (10 + 2) from 24.
Video exercise 4.4 to 4.6
4.7*
Subtract 24 from (20 + 6).
4.8**
Subtract 14 from 26 + 3.
4.9**
Subtract 10 + 6 from 40 – 10.
4.10**
The product of (12 – 3) and (8 + 2).
4.11**
The difference between 18 and half of 10.
© Impaq
Video exercise 4.8 and 4.9 Video exercise 4.10 and 4.11
8
Study Guide G08 ~ Mathematics
Unit
1
Write the sentences in mathematical expressions. Follow the order of operations and simplify the expressions. • Each step may include only one calculation. • No calculators may be used. 4.12***
4.14***
Add 6 and 50 and then multiply it with the difference between 5 and 3. Subtract (5 + 6 – 2) from (10 + 8) and then add (3 + 9) to the answer. Half of 60’s product with (2 + 6 – 1).
4.15***
The quotient of (24 – 8) if it is divided by half of 16.
4.16**
Multiply the difference between 28 and 15 with the sum of 10 and 2. Subtract 5 from 8 and then divide it by half of 6.
4.13***
4.17*** 4.18***
Video exercise 4.12 and 4.13 Video exercise 4.14 Video exercise 4.15 Video exercise 4.16 to 4.18
Add the sum of (10 – 4) and (8 + 12) and then multiply the answer by the difference between 12 and 6.
Exercise 5: Derivation from mathematics to English • • • •
Write the following mathematical expressions in English. Then simplify the expression by using preference of calculations. Only one calculation per step. No calculators may be used. Remember the = signs.
5.1*
4 + 12 + 1 000
5.2*
57 – 15
5.3*
18 ÷ 6 × 3 1 24 ÷ of 12 2 55 – 4(2 + 9) 1 3(5 + 4) – of 20 × 4 4 55 – (7 × 5) + 100
5.4** 5.5** 5.6*** 5.7** 5.8* 5.9*** 5.10*
© Impaq
Video exercise 5.1 and 5.2
0 – (6 – 6) 2 3 �2. of 45 – 1� + (5 × ) 15 5 2+2–2×2
9
Study Guide G08 ~ Mathematics
Unit
1
Exercise 6: Journal entry: Correction of errors In the following exercise, the calculation was done incorrectly. Write an explanation in English of the mistake(s) found in each step. Then do the calculation correctly.
Example:
3 + 4×7 12 × 7 .........Step 1 19
.........Step 2
Step 1: The = sign is missing. The order of operations is incorrect. The multiplication should be done first. Step 2: The = sign is missing. Since the order of operations was wrong in step 1, step 2 is automatically incorrect. Correct way: 3+4×7 = 3 + 28 = 31 6.1**
49 ÷ (3 + 4) + 2 – 3 49 ÷ 6 + 2 ……….. step 1 = 49 ÷ 8 ….…….... step 2 16 ……………...… step 3
6.2**
4 ×3 +2 4 + 5 ……..…….. step 1 9 ………………... step 2
6.3**
15 – 2 + 4 ÷ 2 = 13 – 2 + 4 ÷ 2 …….. step 1 = 13 – 2 + 2 ……….... step 2 13 – 4 ………………... step 3 = 9 ……………………. step 4
6.4**
24 ÷ (2 + 4) + 2 24 ÷ 2 + 6 ……………. step 1 = 24 ÷ 8 ……….……... step 2 = 16 …………………... step 3
6.5**
(6 + 4) × 2 of 20 – 6 1 = 10 × of 14 ……… step 1 2 = 5 of 14 ….……….... step 2 = 70 ……..…………... step 3
© Impaq
Video exercise 6.1
Video exercise 6.2
1
Video exercise 6.5
10
Study Guide G08 ~ Mathematics
Unit
1
Different notations in presenting number systems 1
Natural numbers Video on N and N0
Symbol: N Tabulated: N = {1; 2; 3; ‌} Graphic representation on a number line: 0
1
2
3
4
5
Number line R N
2
Number lines are always R. R means real.
Whole numbers Symbol: N0 Tabulated: N0 = {0; 1; 2; 3; ‌} Graphic representation on a number line: 0
1
2
3
4
5
Number line R N0
3
Integers
0 is a natural number
Video on Z
Symbol: Z Tabulated: Z = {‌ -2; -1; 0; 1; 2; 3; ‌} Graphic representation on a number line: -2
-1
0
1
2
3
Number line R Z
4
Rational numbers
đ?‘Žđ?‘Ž
where đ?‘Žđ?‘Ž đ?‘?đ?‘? and đ?‘?đ?‘? are integers. Note that đ?‘?đ?‘? cannot be zero. (If đ?‘?đ?‘? = 0, the answer is undefined.)
Fractions and integers together. Any number that can be written as
Symbol: Q or Ra (We prefer Q) Video on Q Tabulated: Rational numbers cannot be tabulated because there are too many. Graphic representation on a number line: -2
-1
0
1
2
Number line R Q
Š Impaq
11
Study Guide G08 ~ Mathematics
5
Unit
1
Non-real numbers (also called imaginary numbers) These are numbers that do not exist, e.g., √-3 and are therefore not defined.
Symbol: Doesn’t exist. Tabulated: Cannot be tabulated. Graphic representation on a number line: Not possible, because the numbers don’t exist. 6
Irrational numbers
Video on Q'
These are numbers that do not end and are not repeated, thus they are not rational. Symbol: Q’ (called not Q) or IRa Tabulated: Irrational numbers cannot be tabulated because there are too many, and their exact value is not known. Examples: √2, √3, √5 ‌ as well as đ?œ‹đ?œ‹ (The number of times the centre of a circle is divided into the circumference of the same circle.) Graphic representation on a straight line: No possible presentation, otherwise it will look like a rational line. If you generate the value of an irrational number, it can go on forever without repeating. The square root of a prime number is irrational. Example: √2 = 1,414213562 ‌
7
Real numbers
Video on R
These are numbers that exist and are thus defined. Real numbers include all rational and irrational numbers. Symbol: R Tabulated: Cannot be tabulated, because there are too many. Graphic representation on a number line: -2
-1
0
1
2
Number line R R
Note that a number line is always a Real line.
Š Impaq
12