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Reg. No.: 2011/011959/07
1. General
1.1
Make sure that you have received the following:
y A study guide with the relevant year’s study material.
SAMPLE
y A facilitator’s guide containing:
This letter of information.
The year plan for Grade 8.
The study guide memorandum that explains all the answers to the questions found in the study guide.
y A portfolio book containing:
The mark sheet with the due dates to capture marks.
All assessments for the year.
y A DVD containing the videos referred to in the study guide. Note: The videos must be viewed on a computer, as the format is not compatible with a DVD player.
1.2 Entrance (readiness) examination
In the event that a learner did not pass Grade 8 through Impaq or if a learner is enrolled in Grade 9 at Impaq for the first time, they should write the entrance examination found in this guide. You should grade the paper and the learner must make up any backlog as soon as possible. Note that the purpose of the entrance examination paper is simply so that you can get an indication of the learner’s readiness for Grade 9.
Learners can only achieve success in Grade 9 if they fully understand Grade 8 work and can handle it with ease. Every year the mathematics content relies on the learner’s prior knowledge, insight and ability to identify and complete work accurately.
2. Study guide
You will find explanations and activities in the study guide that help develop concepts, comprehension, skills and knowledge of mathematics.
The recommended calculator for learners is a CASIO fx-82ES (Plus), but any good scientific calculator will be sufficient.
How to use the study guide:
y Study the theory and examples with all the written explanations.
y Do some of the questions and mark them according to the memorandum.
y Refer to the videos indicated in the activities if any work is unclear.
Any other good sources could be useful as long as it meets the requirements of the CAPS curriculum.
3. Portfolio book
Refer to the portfolio book for all assessment tasks for the year.
4. Difficulty levels
Stars are used to indicate each question's level of difficulty. These levels are indicated in the study guide, and apply to all tests and exams.
SAMPLE
Level 1 *
Level 2 **
Knowledge of basic theory and procedures is necessary before any calculations can be done. This can be obtained by studying. This section counts 25% of a paper.
Routine work. Simple statements that are regarded as easy that can be learned through practice. Questions are asked directly. This section counts 30% of a paper.
Level 3 ***
Level 4 ****
Complex calculations based on an accumulation of acquired knowledge. Questions are asked indirectly with the emphasis on comprehension. These sums are not immediately recognised and calculations should first be considered before they become clear. This section counts 30% of a paper.
This tests learners’ insight, ability to reason and also to apply knowledge. It is the most difficult part of a paper. This is the part that is not found in books or study guides. This section counts 15% of a paper.
Level 5 ***** Enrichment. Not included in tests and examinations.
IMPORTANT: The year plan has been adapted to reflect changes to CAPS. The order in which topics are presented in the facilitator’s guide is not necessarily correct. ALWAYS refer to the year plan for the correct order in which topics must be covered.
For example: Although the order of topics to be covered in Term 1 is presented as Unit 1, Unit 2 and Unit 3 in the facilitator’s guide, the correct order according to the year plan is Unit 1, Unit 7 and
8.
Learners should be ready to write the June examination. Note that the exam covers work done in Term 1 and 2.
Unit 1: Number systems
Activity 1: Summary of previous work
N = {1; 2; 3; 4; 5; ...} = natural numbers
N0 = {0; 1; 2; 3; 4; ...} = whole 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 = {rational + irrational numbers} = real numbers
R' = {numbers that do not exist} = non-real numbers
1.1** Identify the numbers in the following table.
Answer:
Different notations
Example: All integers between –5 and 2.
Normal
4. Interval notation = only for real numbers
Graphical notation
Remember: –5 and 2 are not included.
Please note:
A round bracket = excluded
A square bracket = included In the example the same integers are notated.
1.3** Complete the following table for the different notations of groups of numbers.
Activity 2: One-dimensional graphs
Represent the following graphically. Give the answer in interval notation (use the example as guideline).
English Normal notation Tabulate notation Set notation Interval notation
1.3.1 Real numbers from –4 to 10
1.3.2 Rational numbers between 0 and 4
1.3.3 Integers from –6 up to 1
1.3.4 Real numbers greater than 4 and less than and equal to 8
1.3.5 Natural numbers less than 8
2.4** �� ≤ 5, �� ∈ N0
Answer: Number line R 0 ≤ �� ≤ 5, �� ∈ N0
Example 1
Represent the following graphically. Give the answer in interval notation. –2 < �� ≤ 1 or 0 ≤ �� ≤ 4 �� ∈ R –2 < �� ≤ 1 0 ≤ �� ≤ 4 Number line R
Answer: �� ∈ (–2 ; 4] ‘or’ means in both.
SAMPLE
Example 2
Represent the following graphically. Give the answer in interval notation. –2 < �� ≤ 1 and 0 ≤ �� ≤ 4 �� ∈ R –2 < �� ≤ 1 Number line R 0 ≤ �� ≤ 4
Answer: �� ∈ [0; 1] ‘and’ means intersecting.
Note the difference between ‘or’ and ‘and’ in the two examples.
Represent the following graphically. Give the answer in interval notation. 2.5**
Answer:
Answer:
2.9**
Answer:
2.10***
Answer:
Answer:
SAMPLE
2.12*** Can the collection of irrational numbers be represented graphically?
Remember: non-ending, non-recurring fractions are irrational numbers.
Answer:
The collection of irrational numbers looks just like the collection of rational numbers. There are many of these numbers, so we can also draw a solid line for this example.
Activity 3: Rational and irrational numbers
Addition, subtraction, multiplication and division of rational numbers can be done with or without a calculator. Learners must read the questions carefully to determine whether or not a calculator is allowed. If a calculator is not allowed, they must show all the steps followed to arrive at the answer.
With a calculator: All marks for the answer.
Fractions
Example 1
Calculate without a calculator.
Without a calculator: All marks for the steps.
Example 2
with a calculator.
Remember:
The bullet operator (∙) in calculations stands for ‘multiplication’. According to the order of operations, multiplication (×) and division (÷) is performed before addition (+) and subtraction (-).
SAMPLE
When dividing fractions, invert the second fraction and multiply instead of dividing.
Calculate without a calculator.
Invert the second fraction.
The 3 is multiplied using the distributive law.
Calculate the following with a calculator. (Do not change the questions to decimals. Use the �� y button on your scientific calculator.)
3.9***
Parenthesis (brackets) take precedence. The order of operations apply inside the brackets.
Irrational numbers are like variables, e.g.
SAMPLE
3
Simplify the following irrational numbers.
3.11* √ 3
Answer:
Remember:
Answer:
Answer:
Simplify the following using the technique in the given example:
Remember:
Answer:
Choose square numbers so that the root can be found.
Arrange the following rational numbers from smallest to largest. (Show the steps followed to arrive at the answer, an answer alone will not suffice.)
3.26* { 1 3 ; 1 2 ; 1 4 ; 1 5 }
Answer:
Find the lowest common denominator.
LCD = 60
∴ 1 3 = 20 60 and 1 2 = 30 60 and 1 4 = 15 60 and 1 5 = 12 60
= 12 60 ; 15 60 ; 20 60 ; 30 60 (from small to large)
= 1 5 ; 1 4 ; 1 3 ; 1 2
3.27** {–1 3 ; 1 2 ; –1 4 ; 1 5 }
Answer:
Find the lowest common denominator.
LCD = 60
∴ –1 3 = –20 60 and 1 2 = 30 60 and –1 4 = –15 60 and 1 5 = 12 60
= –20 60 ; –15 60 ; 12 60 ; 30 60 (from small to large)
= –1 3 ; –1 4 ; 1 5 ; 1 2
3.28** { 1 3 ;–1 2 ; –1 4 ; 1 5 }
Answer:
Find the lowest common denominator.
LCD = 60
∴ 1 3 = 20 60 and –1 2 = –30 60 and –1 4 = –15 60 and 1 5 = 12 60
= –30 60 ; –15 60 ; 12 60 ; 20 60 (from small to large)
= –1
2 ; –1
4 ; 1 5 ; 1 3
3.29** {4; –16 5 ; –17 5 ; 18 4 ; 4}
Answer:
Find the lowest common denominator.
LCD = 5
∴ –20 5 ; –16 5 ; –17 5 ; 18 5 ; 20 5
= –20
5 ; –17 5 ; –16
5 ; 18 5 ; 20 5 (small to large)
= –4; –17
5 ; –16 5 ; 18 5 ; 4
Activity 4: Real and non-real numbers
y Real numbers include both rational and irrational numbers.
y Non-real numbers are unreal or ‘imaginary’.
y An example of this is √–number e.g. √ 2; √ 6
4.1** What number set is made up of a combination of all the whole numbers and all the integers?
i.e. {whole numbers} ⋃ {integers}
Answer:
The set of integers = Z
4.2**
Which set of numbers is larger: whole numbers or rational numbers?
Answer:
There is no right or wrong answer to this question. The goal here is to test the learner’s knowledge of numbers and their ability to reason and debate.
SAMPLE
4.3**
Whole numbers consist of all the positive integers from 0 to infinity. Rational numbers include all the integers (positive and negative), fractions, repeating decimals and terminal decimals. One could therefore argue that there are more rational numbers than whole numbers.
Which number set is made up of a combination of all the rational and irrational numbers?
i.e. {rational numbers} ⋃ {irrational numbers}
Answer:
The set of real numbers. The real number set contains all the numbers that exist.
4.4**
Which of the following are non-real numbers?
4 ; √ –8 ; 3√ 8 ; 3√ –8 ; 4√ 1 ; 4√ –1
Answer:
Those that do not have answers are non-real. Watch out for irrational numbers that could have an estimated value.
Calculate each one with your calculator – if you get an ‘error’, the number is non-real. √ 4 ; √ –8 ; 3√ 8 ; 3√ –8 ; 4√ 1 ; 4√ –1 = 2; √ –8 ; 2; –2; 1; 4√ –1
∴ Non-real numbers: {√ –8 ; 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.
Answer: ‘as well as’ means union.
Number line R
Take note of the ‘or’.
Activity 5: Different types of fractions
Change the following decimal fractions to normal fractions.
Change the following mixed fractions to improper fractions in their simplest form.
Remember:
Proper fractions e.g. 14 15 numerator < denominator
Improper fractions e.g. 15 14 denominator < numerator
Mixed fractions e.g. 4 2 3 (consist of integers and fractions)
Decimal fractions e.g. 12,456
Recurring decimals e.g. 234= 2,343434 ...
Convert the following recurring decimals to normal fractions.
5.10*** 0,6
Answer:
= 0,66666666...
10�� = 6,666666666...
�� = 6,666666... – 0,666666...
9�� = 6 Recurring decimal.
= 6 9 = 2 3 5.11*** 2,3 Answer:
�� = 2,3333333...
10�� = 23,333333333...
��
�� = 23,333333... – 0,33333...
= 21 9 Recurring decimal.
2 1 3 5.12*** 0,6 3 Answer:
�� = 0,636363636363...
100�� = 63,6363636363...
�� – �� = 63,636363... – 0,6363636363
99�� = 63 �� = 63 99 Recurring decimal. = 7 11
5.13**** 0,123 Answer: Let �� = 0,12333333333...
10�� = 1,23333333333... Recurring decimal. 10��
�� = 1,2333333... – 0,123333333
Provide two equivalent fractions for each normal fraction (There are many other possibilities.) 5.14* 1 3 Answer:
Activity 6: Positioning of numbers
Example
Place one fraction between 1 3 and 1 2 so that all three are evenly spaced.
3 ;
1 2 Use equivalent fractions until it's possible to place the correct fraction(s) between the given fractions.
6.1** Place seven fractions between
and
8 so that all nine fractions are evenly spaced.
Answer:
6.3** Place three fractions between 1 3 and 3 5 so that all five fractions are evenly spaced.
Answer:
6.4*** Place three rational numbers between 0,22 and 0,24, so that all five are evenly spaced.
Answer:
6.5** Which rational number is exactly between –0,15 and –0,13?
Answer:
6.2** Place three fractions between
Answer:
6.6** Place two rational numbers between 0,1 and 0,121 so that all four numbers are evenly spaced.
