Í2(È-E-MAM-FG01’Î
1
9
0
8
-
E
-
M
A
M
-
F
G
0
1
MATHEMATICS FACILITATOR’S GUIDE Grade 8
A member of the FUTURELEARN group
Mathematics Facilitator’s guide
1908-E-MAM-FG01
Í3(È-E-MAM-FG01(Î
Grade 8
CAPS aligned
DM Oost
Facilitator’s Guide G08 ~ Mathematics
Contents Information letter 1. 2. 3. 4.1 4.2 5.
General ..................................................................................................................... 3 Study guide ............................................................................................................... 3 Portfolio book ............................................................................................................ 3 Learning outcomes .................................................................................................... 4 Levels of difficulty ...................................................................................................... 4 Year plan ................................................................................................................... 5
Study Guide Memorandum Unit 1: Number systems....................................................................................................... 3 Unit 2: Exponents .............................................................................................................. 92 Unit 3: Algebra ................................................................................................................. 122 Unit 4: Number patterns ................................................................................................... 175 Unit 5.1: Geometry: Measurement, space and shape ...................................................... 198 Unit 5.2: Euclidian geometry ............................................................................................ 240 Unit 5.3: Geometry: Area, circumference, perimeter and volume .................................... 284 Unit 6: Transformation geometry ..................................................................................... 320 Unit 7: Rate, ratio and proportion ..................................................................................... 345 Unit 8: Finances ............................................................................................................... 389 Unit 9: Statistics ............................................................................................................... 424 Unit 10: Probability ........................................................................................................... 464
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.
Š Impaq
1
Facilitator’s Guide G08 ~ Mathematics
1.
General Make sure that you received the following: • The study guide. • The facilitator’s guide including the following: o Letter of information. o The year plan for Grade 8. o The study guide memorandum in which all the answers to the questions in the study guide are explained. • The portfolio book that includes the following: o Assessment planning o All the tasks for the year • The portfolio book memorandum that includes the following: o Assessment planning o Memoranda of all the tasks and tests
2.
Study guide The study guide contains explanations and exercises to develop concepts, comprehension, skills and knowledge of Mathematics. The recommended calculator for learners is the CASIO fx-82ES (Plus), however, any good scientific calculator will be sufficient. • • •
Study the theory and examples with all the explanations. Do some of the questions and mark them according to the memorandum. Refer to the videos indicated in the exercises if any work is unclear.
Any other good sources could be useful. Note that any additional sources must meet the requirements of the CAPS curriculum.
3.
Portfolio book Refer to the portfolio book for all assessment tasks for the year.
© Impaq
3
Facilitator’s Guide G08 ~ Mathematics
4.1
Learning outcomes Explanation of the learning outcomes that are required in Grade 8 and 9. Learning Outcome 1 LO1 Learning Outcome 2 LO2 Learning Outcome 3 LO3 Learning Outcome 4 LO4 Learning Outcome 5 LO5
4.2
Numbers, operations and relationships Patterns, functions and algebra Space and shape (geometry) Measurement Data handling
40% 15% 15% 15% 15%
Levels of difficulty These levels are applied to all tests and exams, and are also used in the study guide. The stars next to each calculation indicates the level of difficulty. • Use mathematical facts and vocabulary Level • Use correct formulas * 1 • Estimating and rounding values • Theoretical knowledge • Follow familiar procedures Level • Apply concepts consisting of various steps ** 2 • Deduce from given information • Basic calculations learned from examples and exercise • Complex calculations and high-order reasoning • Euclidian geometry Level • No clear method to get the answer *** 3 • Indicate links, similarities and differences between different representations • Requires conceptual and holistic understanding • Unfamiliar non-routine problems that are not necessarily difficult Level **** • Problems must usually be solved in different parts 4 • Usually about practical problems in everyday life • High-order understanding and processes Enrichment Level ***** Often regarded as acceleration of the curriculum. 5 Not included in tests and exams.
© Impaq
4
Facilitator’s Guide G08 ~ Mathematics
Unit
1
These levels are applied to all tests and exams, and are also used in the study guide. The stars next to each calculation indicates the level of difficulty. • Use mathematical facts and vocabulary Level • Use correct formulas * 1 • Estimating and rounding values • Theoretical knowledge • Follow familiar procedures Level • Apply concepts consisting of various steps ** 2 • Deduce from given information • Basic calculations learned from examples and exercise • Complex calculations and high-order reasoning • Euclidian geometry Level • No clear method to get the answer *** 3 • Indicate links, similarities and differences between different representations • Requires conceptual and holistic understanding • Unfamiliar non-routine problems that are not necessarily difficult Level • Problems must usually be solved in different parts **** 4 • Usually about practical problems in everyday life • High-order understanding and processes Enrichment Level ***** Often regarded as acceleration of the curriculum. 5 Not included in tests and exams.
© Impaq
2
Facilitator’s Guide G08 ~ Mathematics
Unit
1
This unit comprises: Addition, subtraction, multiplication and division of real numbers. Real numbers include: N Natural numbers
All positive integers
1; 2; 3; 4; ...
N0 Whole numbers
All positive integers, including 0
0; 1; 2; 3; 4; ...
Z Integers
Q Rational numbers
All positive and negative numbers, including 0
Numbers that can be expressed as a fraction where the denominator is not 0
...; -2; -1; 0; 1; 2; ...
1
- ; -2,3333; 0; 0,75; 6 4
Q’ Irrational numbers Cannot be expressed as a normal fraction; non-repeating decimals đ?œ‹đ?œ‹ = 3,14159 ...;
√2 ; -√11 (note: not all square roots are irrational)
The following basic concepts are covered in this unit: • Properties of division • LCM, LCD and HCF • Prime factors • Order of operations • Types of fractions
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.
Š Impaq
3
Facilitator’s Guide G08 ~ Mathematics
Unit
1
The Egyptian number system consists of symbols. Examples of the two number systems are: Egyptian staff heel
coil of rope
finger frog man with both hands raised
10
100 1 000
đ“‚đ“‚ đ“†?đ“†? đ“ ?đ“ ?
�� �� �� �� �� 𓎆𓎆
10 000 100 000 1 000 000
𓆟𓆟 𓆟𓆟 𓆟𓆟
đ?¤–đ?¤– đ?¤–đ?¤– đ?¤–đ?¤– đ?¤–đ?¤– đ?¤–đ?¤– đ?¤–đ?¤–
�� 𓎆𓎆 𓎆𓎆 𓎆𓎆 𓎆𓎆 𓎆𓎆 𓎆𓎆 𓎆𓎆 𓎆𓎆
Write down the Egyptian equivalent for the number 2 182.
Answer: 1.3***
𓎆𓎆 ��
Write down the Egyptian equivalent for the number 3 516.
Answer: 1.2*
đ?¤–đ?¤–
𓆟𓆟
lotus
1.1*
Hindu-Arabic 1
𓆟𓆟 𓆟𓆟
Write down the Hindu-Arabic equivalent for
đ?¤–đ?¤– đ?¤–đ?¤–
đ“ ?đ“ ? 𓆟𓆟 𓆟𓆟
Answer: 1 million + 2 thousands + 2 tens + 3 units 1 002 023 1.4*
Š Impaq
𓎆𓎆 𓎆𓎆
đ?¤–đ?¤– đ?¤–đ?¤– đ?¤–đ?¤–
Study the number 234 654 365 123 987 341 236 687. Write down the number that is 10 000 more than the given number. Answer: 234 654 356 123 987 341 246 687
4
Facilitator’s Guide G08 ~ Mathematics
1.5***
Unit
1
Investigate: For fun Learners must draw the following table in their answer book and translate the terms addition, subtraction, multiplication and division in any two other languages. + – × ÷ English Addition Subtraction Multiplication Division Afrikaans Optelling Aftrekking Vermenigvuldiging Deling Sepedi Go hlakantšha Go tloṧa Go atiša 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: U 5
6 4
1
8
A
7 3
2
9
10 B Determine: 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. 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}.
© Impaq
5
Facilitator’s Guide G08 ~ Mathematics
Unit
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} 2
U
3
9
1 6
4
A
B 7
5 8
Determine: 2.1.1 A ∪ B Answer: It is A with B. {1; 4; 5; 6; 7; 8} 2.1.2 A ∩ B Answer: The numbers that appear in both. {6; 8}
Remember this work is about compiling sets and the answers are given in curly brackets with semicolons between the terms.
2.1.3 A’ Answer: We read it as “not in A”. {2; 3; 4; 5; 9} 2.1.4 B’ Answer: We read it as “not in B”. {1; 2; 3; 7; 9} © Impaq
6
1
Facilitator’s Guide G08 ~ Mathematics
Unit
1
2.1.5 A’ ∪ B’ Answer: We read it as “not in A with not in B”. The union of all the numbers that are not in A together with those that are not in B. {1; 2; 3; 4; 5; 7; 9} 2.2**
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} 12 13 14 15
U
19 A 17
16
B
11 20
Determine: 2.2.1 A ∪ B Answer: It is A with B. {11; 16; 17; 18; 20}
18
Remember the answers in curly brackets.
2.2.2 A ∩ B Answer: The numbers that appear in both. {11; 16} 2.2.3 A’ Answer: Not in A. {12; 13; 14; 15; 19; 20} 2.2.4 B’ Answer: Not in B. {12; 13; 14; 15; 17; 18; 19}
© Impaq
7
Facilitator’s Guide G08 ~ Mathematics
Unit
1
2.2.5 A ∩ B’ Answer: Watch out for this one! We read it as “In A, but not in B”. {17; 18} 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 U = {integers} = Z Positive + negative + zero
negative integers
whole numbers (N0) natural numbers (N)
Determine 2.3.1 N ∩ N0 Answer: The intersection between the natural numbers and the whole numbers are the natural numbers. ∴ N = {1; 2; 3; 4; 5; …} 2.3.2 Z ∪ N Answer: All the integers united with the natural numbers are the integers. {… -2; -1; 0; 1; 2; 3; …}
2.3.3 Z ∩ N Answer: The integers’ intersection with the natural numbers are the natural numbers. ∴ N = {1; 2; 3; 4; 5; …} 2.3.4 N’ Answer: Non-natural numbers are the negative integers plus zero. {0; -1; -2; -3; -4; …} © Impaq
8
Facilitator’s Guide G08 ~ Mathematics
2.4*
Unit
1
Sketch a Venn diagram of the real number system, the rational number system and the irrational number system. Use the following notation: real numbers = R rational number system = Q irrational number system = Q’ R = {real numbers}
QQ Q' = {non-rational numbers} = {irrational numbers}
Q = {rational Numbers}
Note that there is no intersection in the two collections. 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 Answer: A = {natural numbers} B = {whole numbers} C = {integers} D = {rational numbers} E = {irrational numbers} U = {real numbers}
Š Impaq
9
Facilitator’s Guide G08 ~ Mathematics
2.6 *****
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. Colleagues = 17 Men = 7 Family = 23 Women = 10 Men = 6 Women = 23 – 6 – 10 = 7 Children = 10 Friends = 47 + 58 + 5 = 110 Men = 47 Women = 58 Children = 15 – 10 = 5 In this question, the Venn diagram is used to keep calculations apart. There is no intersection and numbers must be filled in systematically. The answer is 58 female friends.
Exercise 3: Order of operations 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 +
© Impaq
10
Facilitator’s Guide G08 ~ Mathematics
Unit
Remember: When the bracket is calculated, the bracket disappears. Simplify the following expression by only doing one calculation per step: 3.1**
4 +2 ×3–1 Answer: 4+2 ×3–1 =4+6–1 = 10 – 1 =9
3.2**
1+3 ×2 ÷6–2 Answer: 1+3 ×2 ÷6–2 =1+6 ÷6–2 =1+1–2 =2–2 =0
3.3**
× and ÷ are equal. The one that appears first must be done first.
9 – (12 – 8) Answer: 9 – (12 – 8) =9–4 =5
3.4**
Learners must do one calculation per step. They will learn the preference sequence and the use of the = sign. Remember that in front of and after the = sign the answers are identical.
Brackets first. Once the answer is written, the brackets disappear.
3+3 ×3–3 Answer: 3+3 ×3–3 =3+9–3 = 12 – 3 =9
3.5**
(3 + 2) × (6 – 2)
Answer: (3 + 2) × (6 – 2) = 5 × (6 – 2) =5 ×4 = 20
© Impaq
The task is to do one calculation per step. Do not do more than one!
11
1
Facilitator’s Guide G08 ~ Mathematics
3.6**
Unit
2 + 2 × 2 ÷ 2 – 2 + (2 + 2) Answer: 2 + 2 × 2 ÷ 2 – 2 + (2 + 2) =2+2 ×2 ÷2–2+4 =2+4 ÷2–2+4 =2+2–2+4 =4–2+4 =2+4 =6
3.7**
Watch out! The many 2s may confuse you! Work systematically.
8 – 2 × 3 + 6 – (5 – 1) Answer: 8 – 2 × 3 + 6 – (5 – 1) =8–2 ×3+6–4 =8–6+6–4 =2+6–4 =8–4 =4
3.8**
10 – 4 + (3 – 1) × 3 Answer: 10 – 4 + (3 – 1) × 3 = 10 – 4 + 2 × 3 = 10 – 4 + 6 =6+6 = 12
3.9*
3 ×0+2 ×0 Answer: 3 ×0+2 ×0 =0+2 ×0 =0+0 =0
© Impaq
It is very easy to multiply by 0. The answer remains 0. 0(4) = 0 and not 4 6 × 0 = 0 and not 6 ... and so forth!
12
1
Facilitator’s Guide G08 ~ Mathematics
3.10**
Unit
(6 – 2 + 7) + (3 – 2 + 12)
Answer: (6 – 2 + 7) + (3 – 2 + 12) = (4 + 7) + (3 – 2 + 12) = 11 + (3 – 2 + 12) = 11 + (1 + 12) = 11 + 13 = 24 3.11***
1 of 12 – (6 ÷ 3 + 1) 3 Answer: 1 of 12 – (6 ÷ 3 + 1) 3 1 = of 12 – (2 + 1) 3 1 = of 12 – 3 3 =4–3 =1
3.12***
“of” also means multiplication. It takes preference over ordinary multiplication and must be done prior to multiplication.
1 of 23 × (14 – 5) 2
()
Answer: 1 of 23 × (14 – 5) 2 1 = of 23 × 9 2 1 = of 8 × 9 2 =4 ×9 = 36 3.13**
"of" × and ÷ + and –
24 – 4 – 3 – 2 – 1 Answer: 24 – 4 – 3 – 2 – 1 = 16 – 4 – 3 – 2 – 1 = 12 – 3 – 2 – 1 =9– 2–1 =7–1 =6
© Impaq
Exponents
24 = 2.2.2.2
13
1