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MATHEMATICS FACILITATOR’S GUIDE
Grade 7
A member of the FUTURELEARN group
Grade 7
Facilitator’s guide
Mathematics
DM Oost
CAPS aligned
Í2’ÈEMAMFG01%Î 1807EMAMFG01
Study Guide Memorandum
2.
CONTENTS
p. 1 – 369
p. 1 – 10
Page
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Note that the page numbers of the Study Guide Memorandum starts at p. 1 again. The table of contents will guide you to access the Units easily.
Information letter
Subject
1.
Facilitator’s Guide G07 ~ Mathematics
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Facilitator’s Guide G07 ~ Mathematics
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T2 T3 T3 T3 T3 T4 T4
Term T1 T1 T1 T2 T2
1
Content Number systems Exponents Geometry Number systems: Fractions Number patterns and relationships Perimeter, area and volume Algebra Transformations Ratio and rate Finance Statistics Probability Revision
Year plan: Abridged version
Subject advisor information
5.2 3 6 7 8 9 10
Unit in study guide 1 2 5.1 1 4
4.2
4.
6.
Difficulty levels
4.1
3.
Year plan
Year mark tasks
2.
5.
Study Guide with activities and exercises
Learning outcomes
General
Guidelines for facilitators
CONTENTS
1.
Facilitator’s Guide G07 ~ Mathematics
This letter serves as a guideline for facilitators concerning the following: The work that has to be covered. The year plan. The learning outcomes as prescribed by the National Curriculum. The five difficulty levels and the explanation of each. How to use study guide should be used.
1.2 o o o o o
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2
Refer to the portfolio book for detailed information regarding assignments, tests and tasks.
3. Year mark tasks
Recommended calculator: CASIO fx82ES (Plus)
Any other source can be used for revision and study aid purposes. Just keep in mind that the New Curriculum has new topics that must be mastered. Some material from the old syllabus is no longer applicable and you should take care not to waste time on topics that are no longer relevant. There is no change in standard; on the contrary, in more ways than one, the New Curriculum presents greater challenges than the previous one.
How to use the study guide: i Study the theory, the examples and the explanations. i Complete a few sums and mark them by referring to the answers in the memorandum.
The study guide contains references and exercises to develop concepts, comprehension, skills and knowledge of Mathematics. All references to page numbers and section divisions are done according to this prescribed guide.
2. Study guide with activities and exercises
Portfolio book that includes the following: Assessment planning All the tests and tasks with memoranda for the year Submission dates Examination papers will be sent to you at a later stage.
Make sure that you have the following: The study guide The facilitator’s guide, including: Letter of information Grade 7 year plan Contact information Study guide memorandum with answers and explanations for all the questions in the study guide.
General
i H H H H
1.1 i i H H H H
1.
Facilitator’s Guide G07 ~ Mathematics
Difficulty levels
30% 25% 25% 10% 10%
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*
**
***
****
*****
Level 1
Level 2
Level 3
Level 4
Level 5
3
Knowledge of basic theory and procedures is necessary before any calculations can be done. Studying theory is required. 25% of the question paper. Applying routine multistep procedures in various contexts. Continuous practice will enable learners to master this section of the question paper. 30% of the question paper. Applying knowledge and procedures in a variety of contexts. Complex calculations based on a combination of acquired knowledge. Questions are asked indirectly with the emphasis on comprehension. 30% of the question paper. Applying relevant knowledge to unfamiliar, nonroutine questions. The most difficult part of the question paper – testing the learners’ ability to apply all the knowledge they have gained. 15% of the question paper. Advanced
The number of stars next to the number of each sum indicates the level of difficulty. This indication is given in the guide, memorandum, tests and examination papers.
4.2
Explanation of the learning outcomes required in Grade 7, 8 and 9: Learning Outcome 1 LO1 Numbers, Calculations and Relationships Learning Outcome 2 LO2 Patterns, Functions and Algebra Learning Outcome 3 LO3 Space and Shape (Geometry) Learning Outcome 4 LO4 General Geometry Learning Outcome 5 LO5 Data Handling
4.1 Learning outcomes
Facilitator’s Guide G07 ~ Mathematics
i i i i i i i i i i i i i i i i
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Unit 1 Number systems LO1
Term 1
Term 4
Term 3
Term 2
Term 1
1 2 3 4 5 6 7 8 9 10
4
Content: Lessons/ These exercises can be used as lessons days Where do number systems come from? 1 Natural numbers 2 Counting numbers (Whole numbers) 2 Integers 3 Fractions (rational numbers) 4 Roots (irrational numbers) 2 Properties of calculations 2 Percentages 2 Calculator work 2 Mixed exercises 1 [20]
Year plan Grade 7 Mathematics Exercises
Number systems Exponents Construction of geometric figures Geometry of 2D figures Geometry of straight lines Fractions Number patterns and relationships Perimeter, area and volume Algebra Graphs Transformations Ratio and rate Finances Statistics Probability Revision
To comply with the National Curriculum and work schedule, the content set out below has priority at certain stages of the year. The following schedule will be important when setting papers for exams and tests. The year plan is based on this schedule, but some of the topics repeat and appear a few times during the year. It is very important to teach the curriculum in the same order as the rest of South Africa so that learners can move between schools with ease.
5. Year plan
Facilitator’s Guide G07 ~ Mathematics
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5
Revise the two Terms’ work for the June examination.
Expanded exponential notation Calculations with exponents Prime factors Calculator work Mixed exercises
1 2 3 4 5 6 7 8
Unit 8 Finance LO1
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1 2 3 4 5 6 7
6
Calculations with money Dividing money Percentages Profit, loss and discount VAT Interest Exchange rate Mixed exercises
Ratios and equivalent fractions Divide in specific ratio Rate Increase and decrease Distance, speed and time Scale drawings Mixed exercises
Reflection Translation Rotations Enlargements and reductions Mixed exercises
2 3 4 5 6
Unit 7 Ratio and rate LO1
Symmetry
1
Unit 6 Transformation geometry LO3
Variables Basic calculations Algebraic expressions English to Maths Substitution Linear equations Inequalities Basic word problems Mixed exercises
Content This exercises can be used as lessons
1 2 3 4 5 6 7 8 9
Exercises
Unit 3 Algebra LO2
Term 3
1 2 3 4 5
1 3 3 2 1 [10] Unit 5.1 1 Measurement and construction of lines 2 Geometry 2 Measuring and construction of angles 2 LO3 3 Different angles 2 4 Parallel lines 3 5 Triangle 3 6 Quadrilaterals 3 7 Polygons 3 8 Circle 1 9 Mixed exercise 1 [20] The lessons and time allocated are a guideline only. Do not waste time though, as the second Term is also very full. Term 2 Exercises Content Lessons This exercises can be used as lessons or days Unit 4 1 Visual presentation of patterns 1 Number patterns 2 Number patterns in a sequence 2 LO1 3 Types of number patterns (Sequences) 2 4 Determine formulas for general terms 2 5 Relations: 2 Input and output diagrams 6 Number coordinates 3 7 Mixed exercise 1 [13] Unit 5.2 1 Convert units 1 Geometry area, 2 Rectangle 3 perimeter and 3 Triangle 3 volume 4 Circles 3 LO3 5 Polygons 1 6 Volumes 2 7 Outside area 2 8 Mixed exercise 2 9 Summary of formulas 1 [18]
Unit 2 Exponents L02
Facilitator’s Guide G07 ~ Mathematics
Facilitator’s Guide G07 ~ Mathematics
1 1 1 2 2 2 [9] 2 2 2 2 2 2 1 [13] 1 2 2 2 2 2 2 1 [14]
1 3 2 2 3 2 2 2 2 [19]
Lessons or days
Subject advisor’s information
7
Contact the subject matter expert for more information or support.
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6.
Terminology Relative frequency of an outcome The probability scale Probability P(x) of outcomes x Mixed exercises
Content This exercises can be used as lessons Collecting data Organising and summarising data Presenting data Analysing and calculating data Interpretation and evaluating data Mixed exercises
Lessons or days 1 2 3 3 1 1 [11] 1 2 2 3 1 [9]
Use the mixed exercises at the end of each unit. Remember that the November examination covers the entire year’s work.
1 2 3 4 5
Unit 10 Probability LO5
Revision
1 2 3 4 5 6
Exercises
Unit 9 Statistics LO5
Term 4
Facilitator’s Guide G07 ~ Mathematics
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Facilitator’s Guide G07 ~ Mathematics
8
Algebra
Number patterns and relationships
Geometry
Area, perimeter and volume
Transformation geometry
Ratio and rate
Finance
Statistics
Probability
Unit 3
Unit 4
Unit 5.1
Unit 5.2
Unit 6
Unit 7
Unit 8
Unit 9
Unit 10
9
Exponents
Unit 2
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Number systems
Content
CONTENTS: MEMORANDUM
Unit 1
Facilitator’s Guide G07 ~ Mathematics
p. 355
p. 319
p. 301
p. 282
p. 256
p. 220
p. 170
p. 137
p. 94
p. 77
p. 1
Page
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Facilitator’s Guide G07 ~ Mathematics
10
Integers
Fractions (rational numbers)
Roots (irrational numbers)
Order of operations
Percentages
Calculator work
Mixed exercises
4
5
6
7
8
9
10
Simplify means..........
1
Counting numbers
3
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Natural numbers
2
Bibliography
Where do number systems come from?
Unit 1: Number systems Memorandum
CONTENTS
1
Exercise
Facilitator’s Guide G07 ~ Mathematics
7
7
7
7
7
7
7
7
7
7
Grade
Bibliography
p. 72 Exercise 10
p. 68 Exercise 9
p. 63 Exercise 8
p. 58 Exercise 7
p. 52 Exercise 6
p. 33 Exercise 5
p. 29 Exercise 4
p. 26 Exercise 3
p. 4 Exercise 2
p. 3 Exercise 1
Page
Unit
1
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Level of difficulty 5
Level of difficulty 4 Problemsolving 10%
Level of difficulty 3 Complex procedures 20%
Level of difficulty 2 Routine procedures 45%
Level of difficulty 1 Basic knowledge 25%
Facilitator’s Guide G07 ~ Mathematics
**** *
****
***
**
*
2
Assessment analysis Use mathematical facts and vocabulary Use the correct formulas Predict answers and round off values Theoretical knowledge Complete known procedures Apply knowledge and facts with more than one step i Come to conclusions from given information i Basic calculations as learned from examples and exercises i Complex calculations and high order arguments i Euclidean Geometry i No stipulated path to follow i Similarities and differences between presentations i Need conceptual and holistic approaches to problems i Unseen and notroutine problems. i Problems in different sections i Still in curriculum and study guide i Mostly aimed at practical and everyday situations. i High order of thinking Advanced Often regarded as acceleration of the curriculum. Not included in tests and exam papers. i i i i i i
Unit
1
100 000
1000 000
burbot
astonished man
3
Write down the Egyptian equivalent for the number 2182. Answer:
10 000
pointing finger
1.2*
1000
lotus
Write down the Egyptian equivalent for the number 3516. Answer:
100
HindoArabic
rope
1
10
Egyptians
heel
staff
1.1*
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Unit
1
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 20000 years old. The number system that we use today is the HinduArabic system and it was developed more than 1000 years ago by HinduArabic mathematicians. The Egyptian number system consists of symbols. Examples of the two number systems are:
Exercise 1: Where do number systems come from?
Content in this unit: i Where do number systems come from? i Natural numbers i Whole numbers (Counting numbers) i Integers i Fractions (Rational numbers). i Roots and surds (Irrational numbers) i Properties of number systems (associative, commutative and distributive laws)
Unit 1: Number systems
Facilitator’s Guide G07 ~ Mathematics
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YY
Unit
English Afrikaans Sepedi
Addition Optelling Go hlakantšha
+ Subtraction Aftrekking Go tlo a
–
Division Deling Go arola
h
The symbol for natural numbers is N
E
4
1
A represents the natural numbers. As the Unit progresses, the other number systems will be explained.
Natural numbers are the numbers from 1 to infinity. The symbol used is N. If tabulated: N = {1; 2; 3; 4; 5 ;..............} If we look at all the number systems together, then A in this diagram will represent the natural numbers:
Multiplication Vermenigvuldiging Go atiša
x
Answer: 1 million + 2 thousands + 2 tens + 3 units 100223 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 Investigate: For fun Draw the following table in your answer book and translate the terms addition, subtraction, multiplication and division in any other two languages.
Write the HinduArabic number for
Exercise 2: Natural numbers
1.5***
1.4*
1.3***
Facilitator’s Guide G07 ~ Mathematics
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4
Ten millions
3
Millions
1
Hundred thousands
7
Ten thousands
5
Thousands 9
0
6
Tens
Milliards
Unit
8
1 million= 1000 000 = 106 Six zeros 1 billion = 1000 000 000 = 109 Nine zeros 1 trillion =1000 000 000 000 =1012 Twelve zeros 1 quintillion = 1018 Eighteen zeros
Decimal system
5
(100 0000)3
(100 0000)2
A thousand million
(1000)2
Milliard and billion denote the same number. Milliard is almost never used in America and Britain, but is often used in Continental Europe.
Hundred millions
2
Hundreds
Each digit in natural numbers has its own meaning. Example:
Facilitator’s Guide G07 ~ Mathematics
Units
1
© Impaq
2.2**
2.1**
3452 7 825455 7 82 347 84556723
2.1.7** 2.1.8* 2.1.9****
2 7 82 527 8254 7 254164 9527 54123 1233452 7 3452 7 825455 7 82 347 84556723
2.1.2* 2.1.3** 2.1.4** 2.1.5*** 2.1.6* 2.1.7** 2.1.8* 2.1.9****
Example 2.2.1* 2.2.2* 2.2.3** 2.2.4* 2.2.5** 2.2.6*** 2.2.7****
6
Description 4 hundred thousands 6 tens 12 thousands 876 millions 9 units 47 hundreds 639 ten thousands 32456 hundreds
Number 400 000 60 12 000 876 000 000 9 4 700 6 390 000 3 245 600
hundred millions
hundreds
millions
units
hundred thousands
millions
ten thousands
hundreds
thousands
Description tens
Description tens
Write the number from the description given.
3452 7 823
2.1.1*
Example
Number 3452 7 8
1233452 7
2.1.6*
Answer:
7 254164 9527 54123
527 8254
2.1.5***
2 7 82
2.1.3** 2.1.4**
3452 7 823
2.1.2*
Number 3452 7 8
2.1.1*
Example
Complete the table by identifying the place value of the 7 .
Facilitator’s Guide G07 ~ Mathematics Unit
1
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2.4**
2.3***
2.2 Number 400 000 60 12 000 876 000 000 9 4 700 6 390 000 3 245 600
Number 65 382 1 234 87 592 96 795 9 328 999
Answer 65 392 1 244 87 602 96 805 9 329 009
Answer
Example 2.4.1** 2.4.2* 2.4.3** 2.4.4** 2.4.5**** 2.4.6**
Number 3 761 676 767 78 493 78 493 12 121 212 875462 867 444 444
7
Calculation Add 400 Add 1 million Subtract 50 Add 9000 Subtract 1 million Add 4 hundred thousand Add 5000
Complete the table by calculating what is asked.
Number 2.3.1* 65 382 2.3.2* 1 234 2.3.3** 87 592 2.3.4*** 96 795 2.3.5**** 9 328 999
Answer:
2.3.1* 2.3.2* 2.3.3** 2.3.4*** 2.3.5****
Answer 4 161
Add 10 to all the following values. Write your answers in the table.
Example 2.2.1* 2.2.2* 2.2.3** 2.2.4* 2.2.5** 2.2.6*** 2.2.7****
Description 4 hundred thousand 6 tens 12 thousand 876 million 9 units 47 hundreds 639 ten thousands 32456 hundreds
Write the number from the description given.
Facilitator’s Guide G07 ~ Mathematics Unit
1
© Impaq
2.7**
2.6**
2.5**
Number 3 761 676 767 78 493 78 493 12 121 212 875 462 867 444 444
Calculation Add 400 Add 1 million Subtract 50 Add 9000 Subtract 1 million Add 4 hundred thousand Add 5000
Answer 4 161 1 676 767 78 443 87 493 11 121 212 875 862 867 449 444
Unit
1
Number 1 234 567 648 33 333 54 545 454
8
Answer {2542; 2523; 2509; 2441; 154} = {154; 2441; 2509; 2523; 2542} Arrange the following natural numbers from big to small. {592; 523; 2600; 2699} Answer {592; 523; 2600; 2699} = {2699; 2600; 592; 523}
Example 2.5.1* 2.5.2* 2.5.3****
Answer = 1000000 + 200000 + 30000 + 4000 + 500 + 60 + 7 = 600 + 40 + 8 = 30000 + 3000 + 300 + 30 + 3 = 50000000 + 4000000 + 500000 + 40000 + 5000 + 400 + 50 + 4 2.5.4** 5 678 = 5000 + 600 + 70 + 8 2.5.5* 6 =6 2.5.6** 765 432 = 700000 + 60000 + 5000 + 400 + 30 + 2 Arrange the following natural numbers from small to big. {2542; 154; 2441; 2523; 2509}
Answer
Break down the natural numbers as shown in the example. Number Answer Example 1 234 567 = 1000000 + 200000 + 30000 + 4000 + 500 + 60 + 7 2.5.1* 648 2.5.2* 33 333 2.5.3**** 54 545 454 2.5.4** 5 678 2.5.5* 6 2.5.6** 765 432
Example 2.4.1** 2.4.2* 2.4.3** 2.4.4** 2.4.5**** 2.4.6**
Answer
Facilitator’s Guide G07 ~ Mathematics
<
Place a < or > symbol between the values in the table. Number 1 < or > Number 2 546 124 12 65 3232 6756 437 436 112233 112234 4356 11111 Answer Number 1 < of > Number 2 546 > 124 12 < 65 3232 < 6756 437 > 436 112233 < 112234 4356 < 11111
Example: 234 > 233 345 < 346
less than
Unit
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ã
2379 õ 6666 9045
2379 õ 6666
9
Make sure you know where the = sign is placed. Know how and where this sign is placed. 9379 ó 6666 ã 9379 ó 6666 2713
When doing addition and subtraction of natural numbers, your answer must have enough steps to show that you did not use a calculator. Example
2.8**
>
greater than
In Mathematics one uses symbols to show greater than or less than. If one looks at the following symbols from left to right, one can say:
Facilitator’s Guide G07 ~ Mathematics 1 Unit
The line is the same as the = sign
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1
Pay attention to all the zeros.
Answer
1 7 00 000 õ 25 0 000 ã 17 00 000 õ 25 0 000 1950 000
Plus
õ
Minus
ó
10
Multiplication
l
h Division
Pay attention to all the zeros.
Determine the sum of 17 hundred thousands and 25 ten thousands.
17 000 000 õ 25 000 000 ã 17 000 000 õ 25 000 000 42 000 000
Answer
Now use your answer and write down the correct answer for 17 million plus 25 million.
17 00 õ 2500 ã 1700 õ 2500 4200
Answer
Use your answer and write down the correct answer for 17 hundred plus 25 hundred.
17 õ 25 ã 17 õ 25 42
Answer
17 + 25
Calculations with natural numbers can only be one of the following:
2.12****
2.11**
2.10**
2.9.*
Simplify the following. Show your calculations. You are not allowed to use a calculator.
Facilitator’s Guide G07 ~ Mathematics
6765 õ 25863 32628
ó 24 23 ó 16 remainder 7
ó 16 26
ó 24 18
Long division 25863 h 8 3232 ã 8 25863
Subtraction 25863 ó 4444 ã 25863 ó 4444 21419
Multiplication 456 x 18 ã 456 x 18 3648 õ 4560 8208
Unit
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2.14**
2.13**
11
No calculators. Write down all the calculations to show that you did not use a calculator.
1346
2345 ó 999 ã 2345 ó 999
Answer
2345 – 999
Answer 2345 õ 999 ã 2345 õ 999 3344
2345 + 999
Use the prescribed methods and determine the answers of the following. You are not allowed to use a calculator.
ã
Addition 6765 õ 25863
Short division 25863 h 8 3232 res 7 ã 8 25863
Examples of calculations with natural numbers:
Facilitator’s Guide G07 ~ Mathematics 1
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2.18****
2.17***
2.16***
2.15** Remember the zero
ó 18 54 ó 54 05 ó 0 remainder 5
260 9 2345
Two numbers under each other without a +, –, x or h means nothing. In this case it is a minus (–) and you must write it down.
ó 198 365 ó 297 remainder 68
23 99 2345
ã
ó 1998 remainder 347
2 999 2345
Answer 2345 h 999
12
2345 h 999 with the long division method.
ã
Answer 2345 h 99
2345 h 99 with the long division method.
ã
Answer 2345 h 9
2345 h 9 with the long division method.
Answer 2345 x 99 ã 2345 x 99 21105 õ 211050 232155
2345 x 99
Facilitator’s Guide G07 ~ Mathematics Unit
1
Unit
Meaning Multiply Divide Subtract Add
Other English words Product of Quotient of Difference between Sum of
© Impaq
2.21**
2.20**
2.19**
If you cannot give the answers quickly…write the numbers below one other and add them.
672
12 x 56 72 õ 600
$
$
$
Marks are allocated for: 1 for the Maths sentence 1 for the 72 + 600 1 for the answer
ã 12
ó 48 remainder 8
4 56
Answer 56 h 12
13
The remainder is asked. That means that you have to do this question with long division. In future we will use decimals, but for now the remainder and quotient are both integers.
What is the quotient if 56 is divided by 12? Give the remainder.
ã
Answer 12 x 56
Calculate the product of 12 and 56.
12 õ 56 68
or
Answer 12 + 56 = 68
Determine the sum of 12 and 56.
Write the following English sentences in mathematical (number) sentences. Simplify them without using a calculator.
Symbol l h – +
Rewriting English sentences as mathematical sentences (number sentences) is a essential to solving word problems. The following terms for calculations may be used. Study them and use them.
Facilitator’s Guide G07 ~ Mathematics 1
© Impaq
2.23****
2.22***
Day 3
14
First day = 275 km Second day 275 + 35 km = 310 km Third day 275 + 35 + 35 km = 345 km Total at end of third day = 275 + 310 + 345 = 930 km
Answer
Day 2
Day 1= 275 km
Harbour
A yacht sails in one direction. On the first day it sails 275 km and then 35 kilometres further each day. How far will the yacht be from the harbour at the end of the third day?
Answer There were 23 working days in July 2013. 1 436 books per day. Ä1 436 x 23 for the month ã 1 436 x 23 4308 õ 28720 33028
A printing company prints 1 436 books per day. How many books did they print in July 2013 if they did not work on Saturdays and Sundays?
Facilitator’s Guide G07 ~ Mathematics Unit
1
15
To make the skirts, a single piece of 3 meters of material is required. The material can’t be joined. Explain how many rolls of material must be bought and how many meters will remain on each roll after 34 pieces of 3 meters have been cut.
2.25.2***
© Impaq
Show all your calculations and determine how much material she still needs to buy. Answer 34 girls each 3 metres gives in total 34 x 3 = 102 metres She bought 88 metres of material The difference: 102 ó 88 14 metres
Calculators are not allowed.
Sandra has to make 34 skirts for girls participating in a school play. She buys 2 rolls of material of 44 meters each. Each skirt requires 3 meters of material.
ó 1035 1250 metre
Answer Juan 1035 John 2285 Difference = 2285 – 1035 ã 2285
Juan drives 1 035 meter to a shop. John drives 2 285 meter to the same shop. How much further do John drive?
Unit
2.25.1***
2.25
2.24***
Facilitator’s Guide G07 ~ Mathematics 1
Number of pieces 3 metres each 14 pieces 14 pieces 6 pieces Total 34 pieces
Round off to the nearest 10 100 1000 10 100 1000
2 metres 2 metres 26 metres
Leftovers
4 is less than 5 34 is less than 50 234 is less than 500 5 is halfway to 10 95 is more than 50 795 is more than 500
1230 1200 1000 8800 8800 9000
© Impaq
Rounding off to 1 000 (It means that the number must be divided by 1 000 without leaving a remainder.)
Rounding off to 100 (It means that the number must be divided by 100 without leaving a remainder.)
16
Look at the units i When less than 5 round off downwards. i When 5 or more round off upwards. Look at the last two digits i When less than 50 round off downwards. i When 50 or more round off upwards Look at the last three digits i When less than 500 round off downwards. i When 500 or more round off upwards.
Reason
Answer
Rounding off to 10 (It means that the number must be divided by 10 without leaving a remainder.)
Question: number 1234 1234 1234 8795 8795 8795
Unit
To have 34 pieces you have to buy 3 rolls
ó 28 remainder 6
Each roll gives 14 pieces of 3 metres each 2 14 34
Rounding off natural numbers must be done in tens, hundreds and thousands. Examples
Each roll has 44 metres of material Roll 1 Roll 2 Roll 3
ó 3 14 ó 12 remainder 2
Answer Each roll has 44 metres 14 3 44
Facilitator’s Guide G07 ~ Mathematics 1
Answer
7 770 7 800 8 000 7 370 900 1000 000
10 100 1000 10 100 1000
Answer
To the nearest
To the nearest 10 100 1000 10 100 1000
6 is more than 5 66 is more than 50 719 is more than 500 1 is less than 5 90 is more than 50 Look at the last 3 digits. It is not necessary to round off
Reason
Reason
1
17
Round off 74 to the nearest 7. Answer
2.28****
© Impaq
Round off 123456 to the nearest 5. Answer 123456 e 123455
2.27***
e
Round off the following numbers to the nearest 5. Example: (It means that 5 has to be divided into the value without a remainder. Remember, the units that are a 1 or a 2 must be rounded off downswards and units that are 3 and 4 upwards.) i 62 rounded off to the nearest 5 is 60 i 63 rounded off to the nearest 5 is 65 i 84 rounded off to the nearest 5 is 85 When you round off in Mathematics, the following sign is used: e For clarity, the number you round off to is written at the end of the answer. For example: 26 e 30 to the nearest 10 3 456 e 3 500 to the nearest 100
Question: Number 7 766 7 766 7 719 7 371 890 1 million
Unit
Complete the table by rounding off the numbers. Read the headings of the table.
Question: Number 7 766 7 766 7 719 7 371 890 1 million
2.26**
Facilitator’s Guide G07 ~ Mathematics
© Impaq
2.29.3***
2.29.2**
2.29*** 2.29.1*
Round off upwards
First round off and then add.
Price R679
1
18
What is the difference between the answers in the previous two questions? Do you think one will always get to the same answers? Answer: The answer is the same in both cases. It seems that there is no difference between the two methods.
R922 e R920
What will the amount be if Shaun first adds the amounts before rounding off? Answer: R243 + R679 First added and ã 243 then rounded off. õ 679
Answer: R 243 e R 240 to the nearest R10 R 679 e R 680 to the nearest R10 In total: ã 240 õ 680 R 920
Price R243
Unit
74 rounded off to the nearest 7 will be 77.
Shaun buys a pair of trousers for R243 and a jacket for R679. Round off each to the nearest ten rand en give the total amount he paid for the trousers and the jacket.
Round off downwards
Multiples of 7 is..........63; 70; 77;......... all dividable by 7. Between 70 and 77 is where the middle value is between 73 and 74. 71; 72; 73; 74; 75; 76
Facilitator’s Guide G07 ~ Mathematics
Unit
The order of operations is as follows: 1. Brackets (…) 2. Exponents. 3. Multiplication and division from left to right. 4. Addition and subtraction from left to right.
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19
Remember: Once the calculation in the bracket has been completed, the bracket disappears.
ã 33
ã 27 õ 6
ã 26 õ 1 õ 6
ã 2 õ 30 ó 6 õ 1 õ 6 ã 32 ó 6 õ 1 õ 6
ã
ã
1 2 õ 3 l 10 ó 6 õ 1 õ of 12 2 2 õ 3 l 10 ó 6 õ 1 õ 6
1 of 12 2
Example 2: 23 – (12 + 8) + 2 = 23 – 20 + 2 =3+2 =5
2 õ 3 l 10 ó 6 õ ( 3 ó 2 ) õ
Example 4:
Example 1: 2+3x4 = 2 + 12 = 14
1 of ( 21 õ 3 ) 3 1 = 2 + 23 x 2 – of 24 3 1 =2+8x2– of 24 3 =2+8x2–8 = 2 + 16 – 8 = 18 – 8 = 10
2 + 23 x 2 –
Example 5:
Example 3: 30 h 3 x 2 + 15 h 5 = 10 x 2 + 15 h 5 = 20 + 15 h 5 = 20 + 3 = 23
Simplify the following by doing only one calculation for each step:
This sequence applies up to Grade 12. Study this now and you will never have to study it again.
Remember there are priorities for +, –, l and h
When questions are asked that include a combination of adding, subtracting, multiplying or division, it is important to remember the rules regarding the order of operations.
Order of operations
Facilitator’s Guide G07 ~ Mathematics 1
© Impaq
2.34**
2.33**
2.32**
2.31*
ã 20
ã 5 x 4
Answer: ( 3õ 2)l 6ó 2) ã 5 x ( 6 ó 2)
( 3 õ 2 ) l (6 ó 2 )
ã 9
ã 12 ó 3
ã 3 õ9 ó 3
Answer: 3 õ 3 l3 ó 3
3 õ 3 l3 ó 3
ã5
ã 9 ó 4
9 ó ( 12 ó 8 )
Answer:
9 ó (12 ó 8 )
ã0
ã 2ó2
ã1õ 6 h 6 ó 2 ã 1 õ1 ó 2
1õ 3 l 2 h 6 ó 2 Answer: 1õ 3 l 2 h 6 ó 2
20
Unit
The assignment is to do one calculation for each step. Do not do more than one!
Brackets first. A bracket disappears when the answer is written down.
x and h has the same priority. The first sign must be executed first.
Simplify the following expression by doing only one calculation per step. 4 õ 2 l 3 ó1 2.30* Do one calculation for each step. Answer: It will teach you the preference and 4 õ 2 l 3 ó1 the use of the = sign. Remember ã 4 õ 6 ó1 that before and after the = sign ã 10 ó 1 everything must be identical. ã 9
Facilitator’s Guide G07 ~ Mathematics 1
© Impaq
2.39**
2.38**
2.37**
2.36**
2.35**
6 õ 6
(6 – 2 +7) +( 3 – 2 + 12) Answer: (6 – 2 +7) +( 3 – 2 + 12) = (4 + 7) + (3 – 2 + 12) = 11 + (1 + 12) = 11 + 13 = 24
ã 0 õ 0 ã 0
ã 0 õ 2x 0
Answer: 3l0 õ 2l0
3l0 õ 2l0
ã 12
ã
ã 10 ó 4 õ 2 x 3 ã 10 ó 4 õ 6
Answer: 10 ó 4 õ ( 3 ó 1) l 3
10 ó 4 õ ( 3 ó 1) l 3
ã 8 ó 4 ã 4
21
ã8ó6õ6ó4 ã 2õ 6 ó 4
All the number 2’s can be confusing. Work systematically … one calculation for each step.
Be aware
For some or other reason Grade 7 learners do not get this right. (0)4 = 0 and not 4 6 x 0 = 0 and not 6 etc.
It is easy to multiply by 0. The answer stays zero.
Answer: 8ó 2 l 3 õ 6 ó (5 ó 1 ) ã 8ó2 x 3 õ 6 ó4
8ó 2 l 3 õ 6 ó (5 ó 1 )
ã 6
ã 2õ4
ã 2 õ 2 ó2õ4 ã 4 ó2 õ4
Answer: 2 õ 2 l 2 h 2 ó 2 õ ( 2 õ 2) ã 2õ 2x2h2 ó2 õ4 ã 2 õ 4h 2ó2 õ 4
2 õ 2 l 2 h 2 ó 2 õ ( 2 õ 2)
Facilitator’s Guide G07 ~ Mathematics Unit
1
( 60 õ 4) h 8 (16 ó 8)x2
( 60 õ 16 h 4) h 8 (16 ó 8)x 2
=
8 1 = 8x2 2
64 h 8 (16 ó 8)x 2 8 = (16 ó 8)x2 =
=
=
Answer: (5 x12 õ 16 h 4) h 8 (16 ó 8)x 2
The division line must be treated as two large brackets. i The numerator must be simplified on its own and the denominator must be simplified on its own. i Then the numerator must be divided by the denominator.
Simplify by doing only one calculation for each step. (5 x12 õ 16 h 4) h 8 (16 ó 8)x 2
Unit
1
© Impaq
4 is written as 2 x 2 12 is written as 2 x 2 x 3 24 is written as 2 x 2 x 2 x 2 x 3
22
Prime numbers that are multiplied are called composite factors.
Numbers that can be divided by more than 2 factors are called composite numbers. For example, 24 is composite because it can be divided by 1, 2, 3, 4, 6, 8, 12 and 24. Composite numbers have three or more factors. Example: Factors of 24 = {1; 2; 3; 4; 6; 8; 12; 24} = {the number 1} + {prime numbers} + {composite numbers} = {1} + {2; 3} + {4; 6; 8; 12; 24}
Composite factors
1 is neither a prime number nor a composite number.
The number 1
A prime number is a number that has exactly two factors, namely the number 1 and the number itself. This means that only two numbers can be divided into a prime number without a remainder. There are an infinite number of prime numbers, but no formula exits to determine the number of prime numbers. The set of prime numbers is = {2; 3; 5; 7; 11; 13; 17; 19 ;………….}
Prime factors
This is a general term which means these are numbers that divide exactly into a bigger number, with an integer as an answer. Therefore 10 will be a factor of 20 because 10 can be divided into 20.
Factors
2.40***
Facilitator’s Guide G07 ~ Mathematics
35
Answer Factors of 36 = {1; 2; 3; 4; 6; 9; 10; 12; 18; 36} = {1} + {2; 3} + {4; 6; 9; 10; 12; 18; 36}
36
Answer Factors of 30 = {1; 2; 3; 5; 6; 10; 15; 30} = {1} + {2; 3; 5} + {6; 10; 15; 30}
30
Answer Factors of 12 = {1; 2; 3; 4; 6; 12} = {1} + {2; 3} + {4; 6; 12}
12
2 2 2 3 3 5
360 180 90 45 15 5 1
© Impaq
Write the following numbers as the product of their prime numbers. Use the ladder method. This means the same as writing the number as the product of prime factors.
Example: 360 =2 x 2 x 2 x 3 x 3 x 5
23
Unit
Divide the number by the first prime number that can be divided into that number. Keep on dividing by the same number until you end up with a remainder. Then use the next prime number. Prime numbers = {2; 3; 5; 7; 11; 13;....}
Answer Factors of 35 = {1; 5; 7; 35} = {1} + {5; 7} + {35} Use the ladder method to determine prime factors.
2.44**
2.43**
2.42*
2.41*
Use the example given.
Write each number given as sets of 1, prime numbers and composite factors.
Facilitator’s Guide G07 ~ Mathematics 1
2 2 17
68 34 17 1
2 2 2 3
24 12 6 3 1
Determine the largest prime number that can be divided into 39 600. Answer 2 39600 2 19800 39600 2 9900 = 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 x 11 2 4950 3 2475 The largest prime number is 11 3 825 5 275 5 55 11 11 1
68 Answer 68 = 2 x 2 x 17
24 Answer 24 = 2 x 2 x 2 x 3
Unit
© Impaq
24
Determine the multiple and factors of the following: 2.53* 6 Answer Factors of 6 = {1; 2; 3; 6} Multiples of 6 = {6; 12; 18; 24; …….}
First learn the following: i Multiples. Numbers into which the given number can be divided. i Example: multiples of 6 are {6; 12; 18; 24; 30; …} i Factors. These are numbers that can be divided in the given numbers: Example: factors of 12 is {1; 2; 3; 4; 6; 12} Example: Multiples are large numbers. Factors of 8 = {1; 2; 4; 8} Factors are small numbers. Multiples of 8 = {8; 16; 24; 32; …….}
2.52*
2.51**
2.50*
Facilitator’s Guide G07 ~ Mathematics 1
12 Answer: Factors of 12 = {1; 2; 3; 4; 6; 12} Multiples of 12 = {12; 24; 36…….} 18 Answer: Factors of 18 = {1; 2; 3; 6; 9; 18} Multiples of 18 = {18; 36; 54…….} 24 Answer: Factors of 24 = {1; 2; 3; 4; 6; 8; 12; 24} Multiples of 24 = {24; 48; 72…….}
Unit
1
© Impaq
2.59***
2.58**
25
Determine the HCF of 20 and 30 Answer: Factors of 20 = {1; 2; 4; 5; 10; 20} Factors of 30 = {1; 2; 3; 5; 6; 10; 15; 30} Ä HCF = 10 Determine the LCM of 9 and 12 Answer: Multiples of 9 = {9; 18; 27; 36; ….} Multiples of 12 = {12; 24; 36; 48;..} Ä LCM = 36
Example 1: Determine the HCF of 6 and 8 Factors of 6 = {1; 2; 3; 6} Factors of 8 = {1; 2; 4; 8} Common factors = {1; 2} Ä HCF = 2 (Highest common factor. This is the biggest factor that can divide into both numbers) Example 2: HCF means highest Multiples of 6 = {6; 12; 18; 24; …….} common factor. Multiples of 8 = {8; 16; 24; 32; 40 ……..} LCM means lowest Ä LCM = 24 common multiple. (Smallest common multiple. LCD means lowest The smallest number that 6 and 8 can divide into) common denominator. 2.57** Determine the HCF of 16 and 24 Answer: Factors of 16 = {1; 2; 4; 8; 16} Factors of 24 = {1; 2; 3; 4; 6 ; 8; 12; 24} Ä HCF = 8
Determine the LCM (lowest common multiple) and HCF (highest common factor) of two or more integers.
2.56*
2.55*
2.54*
Facilitator’s Guide G07 ~ Mathematics
© Impaq
3.1**
26
0 ã 0 2 2 is undefined 0
The zero is added. Remember
(2 + 0 – 1) + (3 – 0) Answer: (2 + 0 – 1) + (3 – 0) =(2 – 1) +( 3 – 0) = 1 + (3 – 0) =1+3 =4
E
Unit
The letter used for counting numbers is N0.
Natural numbers A = {1; 2; 3; 4; 5; 6;......}
Counting numbers B include the natural numbers and zero. B = {0; 1; 2; 3; 4; 5; 6;....}
Determine the LCM of 7, 9 and 21 Answer: Multiples of 9 = {9 ;18; 27 ; 36; 45; 54; 63; 72; …….} Multiples of 7 = {7; 14; 21; 28; 35; 42 ; 49; 56; 63; 70; ……..} Multiples of 21 = {21; 42; 63;........} Ä LCM = 63
Ä LCM = 30
Determine the LCM of 5, 6, and 15 Answer: Multiples of 5 = {5; 10; 15; 20; 25; 30;...} Multiples of 6 = {6; 12; 18; 24; 30; 36...} Multiples of 15 = {15; 30; 45;.....}
Exercise 3: Counting numbers
2.61***
2.60***
Facilitator’s Guide G07 ~ Mathematics 1
2l7 õ 6 6ó2l3 ó 5l4 3 Answer:
3.6***
© Impaq
2l7 õ 6 6 ó 2l3 ó 5l4 3 14 õ 6 6ó6 ã ó 20 3 20 0 ã ó 20 3 ã 1ó 0 ã 1
2õ0 3ó3 Answer: 2õ0 3ó3 2 ã 0 ã undefined
3.4**
3.3*
22 x 0 + 4 – 0 + (10 – 0) Answer: 22 x 0 + 4 – 0 + (10 – 0) = 22 x 0 + 4 – 0 + 10 = 0 + 4 – 0 + 10 = 4 – 0 + 10 = 4 + 10 = 14 4l0 2 Answer: 4l0 2 0 ã 2 ã 0
3.2**
Facilitator’s Guide G07 ~ Mathematics
27
Answer
ã
ã
14 0 undefined
2 õ (3 l 4 ) 12 ó ( 3 l 4 ) 2 õ 12 ã 12 ó 12
3.7** 2ó 2 3ó3 l 5 7 Answer: 2ó 2 3ó3 l 5 7 0 0 ã x 5 7 0 ã 35 ã0
3.5*
2 õ (3 l 4 ) 12 ó ( 3 l 4 )
Do not be afraid of the zero.
Unit
1
© Impaq
3.10****
3.9****
3.8** Remember that you are not allowed to divide by zero. The numerator does not matter; it is the denominator that cannot be zero.
Unit
8
7
6
5
4
3
From 9 to 5 is 4 units to the right. The change is then positive 4°C.
9
Sunday 2
1
0
1
Right is + Left is –
10
8
7
6
5
4
Saturday 3
2
1
0
28
1
Right is + Left is –
From 5 to 9 is 4 units left. The change is negative 4°C.
9
Sunday
Answer Study the temperature on the number line.
If the temperature in Sutherland was 5°C on Saturday and 9 0C on Sunday, what was the change in temperature from Saturday to Sunday?
10
Saturday
Answer Study the temperature on the number line.
The temperature in Sutherland was 9°C on Saturday and 5°C on Sunday. What was the change in temperature from Saturday to Sunday?
2 õ 6 ó 4 õ 100 ( 2 ó 2) õ ( 8 ó 8 ) Answer: 2 õ 6 ó 4 õ 100 (2 ó 2) õ (8 ó 8) 8 ó 4 õ 100 ã 0 õ 0 4 õ 100 ã 0 104 ã 0 ã undefind
Facilitator’s Guide G07 ~ Mathematics 1
= {1; 2; 3; 4; 5 ;.......}
=
+
Positive numbers
Zero +
{0} + {1; 2; 3; 4; 5 ;.......}
+
Negative numbers
Natural numbers A = {1; 2; 3; 4; ......}
Counting numbers B include the natural numbers and zero. B = {0; 1; 2; 3; 4; 5; 6;....}
Integers C include counting numbers. C = {.....3; 2; 1; 0; 1; 2; 3....}
Unit
x
x
õ
© Impaq
X
X
õ
ó
õ
=
=
ó
õ
Addition of negative numbers Example: – 2 – 3 =–5 i Subtraction of negative numbers Example: (– 2) – (– 3) =–2+3 =1 i Multiplication of negative numbers Example: (– 2)(– 3) =+6 i Division of negative numbers ó4 ã 2 Example: ó2
i
29
x
x
ó ó õ
ó =
=
ó
õ
Signs also have to be divided.
Signs also have to be multiplied.
If you pay a debt, it means that you have to add.
Read it as: “Owe 2 and owe another 3”. That means that you owe 5.
The minus sign in front of a counting number means that it works in the same way as a “debit”.
Integers
E
The letter used for integers is a Z.
Exercise 4: Integers
Facilitator’s Guide G07 ~ Mathematics 1
More on negative numbers
Unit
3
2
1
0
1
2
3
3
2
1
0
1
2
3
7
6
5
4
3
2
7
6
5
4
3
2
1
1
0
0
2
1
2
Number line
1
5
Number line
5
Number line
4
4
Number line
© Impaq
Example 5: 50 + (– 10) = 50 – 10 = 40
Example 6: – 30 + (+ 2) = – 30 + 2 = – 28
30
Example 7: 40 –(– 3) = 40 + 3 = 43
Example 8: – 30 – – 8 = – 30 + 8 = – 22
Example 9: (– 2)(– 3) – – 5 =6+5 = 11
The number line method is only used to explain addition and subtraction and it is not always necessary to graph it. When you find it difficult to use the negative and positive numbers, then use the number line and make sure that you understand this section of the work. Sometimes, multiplication can also cause problems:
8
4 units to the left
Example 4: – 8 – (– 4) = – 8 + 4.......multiply the two signs – x – = + Start at – 8 and move 4 units to the right =–4
8
4 units to the left
Example 3: – 2 – 4 = – 6. Start at – 2 and move 4 units to the right.
4
6 units to the right
Example 2: – 2 + 6 = 4. Start at – 2 and move 6 units to the right.
4
5 units to the left
Study the number line. When you add, move to the right of the number line. When you subtract, move to the left on the number line. Example 1: 2 – 5 = – 3. Start at 2 and move 5 units to the left.
Facilitator’s Guide G07 ~ Mathematics 1
–2–3–4–5–6 Answer: =–5–4–5–6 =–9–5–6 = – 14 – 6 = – 20 6–7–1+3 Answer: =–1–1+3 =–2+3 =1
(– 3)(+ 6) Answer: (– 3)(+ 6) = – 18 (– 2)(– 7) Answer: = + 14 (+ 4)(– 2) + (2)(6) Answer: = – 8 + (2)(6) = – 8 + 12
4.7*
4.11**
© Impaq
4.15**
4.13*
4.9*
= 4
(10 – 12) + 0 Answer: (10 – 12) + 0 =–2+0 = –2
4.5**
4.3*
–4–3 Answer: –4–3 =–7 5–4 Answer: 5 ó 4 ã 1
4.1*
Facilitator’s Guide G07 ~ Mathematics
31
=–1
4.10** (10 – 15) + (6 – 5) Answer (10 – 15) + (6 – 5) = – 5 + (6 – 5) =–5+1 =–4 4.12* (+ 3)(– 2) Answer: (+3)(– 2) = –6 4.14* (– 33)(–1) Answer: = + 33 4.16 * 5 +(– 2)(3) + 0 Answer: 5+–6 +0 = 5–6+0
4.8* 10 – 9 – 8 Answer 10 – 9 – 8 =1–8 = –7
4.6** 2 + 3 – 4 – 5 Answer: 2+3–4–5 =5–4–5 = 1–5 =–4
4.4* 2 + 3 – 6 Answer: 2 õ3 ó6 ã 5 ó 6 ã ó1
4.2 * – 4 + 3 Answer: –4+3
Unit
1
© Impaq
Answer =–1 2 ó ó 3 ó 10 ó2 Answer 2 ó ó 3 ó 10 ó2 2 õ 3 ó 10 ã ó2 5 ó 10 ã ó2 ó5 ã ó2 1 ã2 2
8 ó8
4.19*
Answer = +2
ó4 ó2
4.20***
+ –2 +
0 +
+
+
+
9 + –2 + 6 =–5
32
Two signs next to each other mean they have to be multiplied. Example 2 – – 1 =2+1 =3
Remember: A minus sign multiplied by a minus gives a positive. A minus sign multiplied by a plus sign will give a minus.
Complete the table by filling in the blank spaces with integers. Answer –9 + 0 + 9 + + + –2 + 5 + –8 + + + 6 + –7 + –4 =–5 =–2 =–3
Study the word puzzle. + + –2 + + 6 + =–5
4.18*
4.17****
Facilitator’s Guide G07 ~ Mathematics
=–5
=–5
=0
=–5
=–5
=0
Unit
1
õ
Answer: 10 ó 20 õ ó 5 ó 4 10 20 18 ã õ õ 5 4 9 ã 2 õ 5 õ 2 ã 7 õ 2 ã 9 õ 18 ó 9
ó 20 ó4 ó
õ 18 ó 9
© Impaq
The letter for rational numbers is a Q.
E
D is all the fractions. It includes the integers.
33
2 3 Denominators
Numerators
Natural numbers A = {1; 2; 3; 4; 5; 6;......}
12 3
Counting numbers B include the natural numbers. B = {0; 1; 2; 3; 4; 5; 6;....} The zero is added.
i Integers C include the counting numbers. Counting numbers. C = {.....3; 2; 1; 0; 1; 2; 3....}
rational number. The word rational comes from ratio and all rational numbers can be written as ratios or fractions. Fractions have a numerator and a denominator. The denominator may not be zero.
The definition of fractions (rational numbers): 4 Integers like 4 = which is a numerator divided by a denominator, is called a 1
Exercise 5: Fractions (rational numbers)
Answer: 26 ó 2 l ( ó3 ) ó2 ã ó 13 ó 2 x( ó3 ) ã ó 13 õ 6 ã ó7
10 5
1 2
...........Numerator < denominator
Unit
© Impaq
5.1*
0,
3 100
2 10
Change the answer back to mixed fractions.
4 1000
4
Decimal fractions
Complete the calculations
6 6 100 000
5 5 10 000
34
Give the numerical value of the letter in the equation. Equation Value of letter 5.1.1* a 4 5 3,45 = 3+ õ a 100 5.1.2* b 7 8 9 12,789 = b + õ õ 10 100 1000 5.1.3* c c 328,012 = 328 + 1000
3
2
Change the mixed fractions to improper fractions if necessary.
1 i Mixed fractions: example 7 .............Integer plus proper fraction 2 11 i Improper fractions: example ....................Numerator > denominator 2 i Decimal fractions: example 23,5..............Use a comma and not a dot Remember the following: Answers are always given as mixed fractions. However, improper fractions are used to do the calculations.
i Proper (common) fractions: example
The different types of fractions are:
Units
4.22 * * *
Tenths
ó 2 l ( ó3 )
Hundreds
26 ó2
Facilitator’s Guide G07 ~ Mathematics
Thousands
4.21***
1
10 Thousands
Unit
100 Thousands
Facilitator’s Guide G07 ~ Mathematics
7 1000 000
7
Millions
1
c = 12
b = 12
Value of letter a = 10
5.2.1**
Question 9 5 4+ õ 10 100 5.2.2** 8 145 + 100 5.2.3** 9 5 5454 + õ 10 1000
Answer
2 13
© Impaq
ã
35
Prime factors will work every time. Composite factors are also correct if you can get them equal. Remember that calculators cannot be used and all steps must be shown. Show all factors that cancel out.
Factorise the values and cancel out all factors that are equal above and below. Example: 12 2 12 2 78 78 2 6 3 39 2 x 2 x3 3 3 13 13 ã 2x3 x13 1 1
5454,905
145,08
4,95
Answer
Write the following as a decimal. Do not use a calculator. Question Answer 5.2.1** 9 5 4+ õ 10 100 5.2.2** 8 145 + 100 5.2.3** 9 5 5454 + õ 10 1000
4 5 3,45 = 3 + õ a 100 7 8 9 12,789 = b + õ õ 10 100 1000 c 328,012 = 328 + 1000
Equation
Simplify the following fractions:
5.2**
5.1.3*
5.1.2*
5.1.1*
Answer:
Facilitator’s Guide G07 ~ Mathematics Unit
1
© Impaq
5.4**
Unit
It is difficult to explain how you got to 78 = 13 x 6. Prime factors have a method and can be used to show that you did not use a calculator. If the values are easier then it will be easy to explain.
225 150 Answer 225 150 3 x3 x3 x5 ã 2x3 x3 x5 3 ã 2 1 ã1 2
3 3 3 5
36
2 3 3 5
Ladder method……..
225 75 25 5 1
150 75 25 5 1
2 13 Factorise and simplify the following fractions. Calculators may not be used. 5.3* 64 80 Answer Alternative: 64 8 x8 2 64 2 80 ã 64 80 8 x10 2 32 2 40 80 8 2 16 2 20 ã 2 x 2 x 2 x 2 x 2 x 2 10 ã 2 8 2 10 2 x2 x2x 2x5 2x 4 2 4 5 5 ã 2x 2 2x 5 ã 2 2 1 5 4 ã 1 4 5 ã 5 ã
Alternative: 12 78 2x 6 ã 6 x13
Facilitator’s Guide G07 ~ Mathematics 1
30 x 49 14 x 21x 25 Answer 30 x 49 14x 21x 25 2x3 x5 x7 x7 ã 2 x 7 x 3 x 7 x5 x5 1 ã 5
5.6***
2 2 3 3
2 3 5
36 18 9 3 1
30 15 5 1
2 2 2 3
24 12 6 3 1
15 5 1
18 9 3 1
2 3 3
1 2
© Impaq
34 7
8
Mixed fraction
Improper fraction
5x13 .........f actorise 5x20 13 ã .......... ......canc el out 20
37
1 17 8 = .......numerator > denominator 2 2 (8 x 2 + 1= 17) 34 6 =4 7 7 (34 h 7 = 4 remainder 6)
ã
Unit
3 5
It is important to know how to convert fractions. Examples: Fraction Convert to Answer Decimal fraction 1 0,5 1 = 2 1, 0 2 2 With a calculator 1 divided by 2 0, 65 Proper fraction 65 0,65 = 100
36 x 24 15 x18 Answer 36 x 24 15 x18 2 x 2 x 3 x 3 x 2 x 2x 2x 3 ã 5x3 x 2x3 x3 2x 2x 2x 2 ã 5 16 ã 5 1 ã3 5
5.5***
Facilitator’s Guide G07 ~ Mathematics 1
© Impaq
5.7*
Unit
38
2 1226 ã 3 3
1 17 ã 8 8
5 68 ã 9 9
408
2
1 8
2 3
7
Improper fraction 4 103 11 ã 9 9
5 9
408
2
7
Answer Mixed fraction 4 11 9
48 15
720 33
65 24
720 27 ã 21 33 33 3 x9 ã 21 3 x11 9 ã 21 11 48 3 ã3 15 15 3 ã3 3 x5 1 ã3 5
65 17 ã2 24 24
Improper fraction Mixed fraction 16 16 7 ã1 9 9 9
Change the improper fractions to mixed fractions and mixed fractions back to improper fractions in the following tables. Improper fraction Mixed fraction Mixed fraction Improper fraction 16 4 11 9 9 65 24 5 7 9 720 33 1 2 8 48 15 2 408 3
Facilitator’s Guide G07 ~ Mathematics 1
© Impaq
5.8**
Unit
7
1 9
39
12 20
5
0,125
1,0101
0,02
5 8
3
2 3
3,375
Change the following by completing the table. Simplify the answers if necessary. The use of calculators is not allowed. Proper fraction Decimal fraction Decimal fraction Proper fraction 0,24 7 25
Facilitator’s Guide G07 ~ Mathematics 1
© Impaq
12 20
5 8
1 7 9
2 3
5
3
7 25
Answer Proper fraction
Facilitator’s Guide G07 ~ Mathematics
ó 100 120 ó 120
5,6 ã 20 112,0
12 112 ã 20 20
ó 24 50 ó 48 20 ó 16 40 ó 40
3,625 ã 8 29 ,000
5 29 ã 8 8
= 7,1ú
1 64 7 ã 9 9 7,11... ã 9 64
40
Recurring decimal
0,66666... 2 ã 3 2,0000 3 ã 0,6ú
5
3
Decimal fraction 0,28 7 ã 25 7, 00 25 ó 50 200 200
0,125
1,0101
0,02
2 2 2 5
1000 500 250 125 25
3,375 3 375 5 125 5 25
Decimal fraction 0,24
2 10
5 x5 x5 2x 2x2 x5 x5 x5 1 ã 2x 2x 2 1 ã 8 ã
125 1000
1,0101 101 =1 10000
2 2x5 1 ã 5 ã
0,02 =
Proper fraction 24 0,24 ã 100 4x 6 ã 4 x 25 6 ã 25 3,375 375 3 1000 3 x5 x25 = ã3 2x 2x2 x5 x25 3 ã3 8
Unit
1
.......... 8
Unit
© Impaq
41
The denominators are correct. To get the numerators, you have to divide the denominator into the LCM and multiply by the numerator. ( 8 h 8x17) õ ( 8 h 4x3 ) Alternative: 8 17 3 2 õ x 17 õ 6 8 4 2 ã 8 17 6 ã õ Make denominators equal 23 8 8 ã 8 17 õ 6 23 7 ã ã ã2 7 ã2 8 8 8 8
The LCM of 8 and 4 is 8. Write the LCM below a long line
Explanation: 17 3 õ 8 4
Proper fractions Example without a calculator: 1 3 2 õ Method 8 4 1. Change all 17 3 numbers to improper ã õ 8 4 fractions. 17 õ 6 2. Determine the ã 8 LCM of the Place all the commas below 23 numerators. Write as ã one another. Write “0” in the 8 one fraction. open spaces. 7 4. Simplify. ã2 8 1 3 2 ó 350,246 – 29,96 8 4 = 350,246 17 3 ã ó – 29,960 8 4 320,286 17 ó 6 ã 8 11 Place commas below one another. ã 8 Write “0” in the open spaces. 3 ã1 8
Adding and subtracting fractions
Decimal fractions Example without a calculator: 23,056 + 9,8 = 23,056 + 9,800 32,856
Facilitator’s Guide G07 ~ Mathematics 1
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5.12***
5.11***
5.10**
1 7 1 ó õ 4 8 2 Answer 1 7 1 2 ó õ 4 8 2 9 7 1 ã ó õ 4 8 2 18 ó 7 õ 4 ã 8 15 ã 8 7 ã1 8 2
i i
i i i i
42
Improper fractions LCM of 8 Long line with LCM below Numerators: Divide the LCM with the denominator and multiply b numerator Simplify Give answer in mixed fraction
2,0008 – 0,34 + 999,9 Answer Place zeros in the spaces where 2,0008 numbers are not given. Commas must – 0,3400 be below one another. 1,6608 +999,9000 1001,5608 1 1 5 ó2 16 3 Answer 1 1 5 ó2 16 3 81 7 81 81x3 243 ã ó ã ã 16 3 16 16 x3 48 243 ó 112 7 7 x16 112 ã ã ã 48 3 3 x16 48 131 ã 48 35 ã 2 48
Simplify the following fractions without using a calculator. 5.9** 0,123 + 4,56 + 0,789 Answer i Commas below one another 0,123 i Now add the first two +4,560 i Then the answer plus the last two 4,683 decimals +0,789 5,472
Facilitator’s Guide G07 ~ Mathematics Unit
1
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Answer 15 7 3 1 3ó õ õ ó 16 8 4 2 48 ó 15 õ 14 õ 12 ó 8 ã 16 51 ã 16 3 ã 3 16
No calculator
15 7 3 1 3ó õ õ ó 16 8 4 2
5.15*
5.17***
No calculator
6 3 õ4 7 7 Answer 6 3 1 õ4 7 7 13 31 ã õ 7 7 13 õ 31 ã 7 44 ã 7 2 ã 6 7 5 3 õ 6 4 Answer 5 3 õ 6 4 10 õ 9 ã 12 19 ã 12 7 ã1 12
1
5.13**
Facilitator’s Guide G07 ~ Mathematics
43
2
No calculator
Answer 3 3 2 ó1 5 20 13 23 ã ó 5 20 52 ó 23 ã 20 29 ã 20 9 ã1 20
5.16*
1 12 No calculator
No calculator
3 3 ó1 5 20
6 ó3
Answer 1 6 ó3 12 6 37 ã ó 1 12 72 ó 37 ã 12 35 ã 12 11 ã 2 12
5.14**
Unit
1
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h
Division
Unit
44
i Improper fractions. 345 i Flip the fraction and change division to multiplication. i Factorise and simplify.
1 1 5 h 10 8 4 41 41 ã h 8 4 41 4 ã x 4x 2 41 1 ã 2
1218
ó 203 316 ó 203 1136 ó 1015 1218
23,4668 h 2,03 23,4668 100 ã x 2,03 100 2346,68 ã 203 11 , 56 ã 203 2346,68
Examples without using a calculator. Proper fractions Decimal fractions 2,45 x 12,3 1 1 4 x3 = 2,45 8 11 x 12,3 33 34 ã x 735 8 11 + 4900 3 x11 2x17 ã x +24500 2x 2 x 2 11 30,135 3 x17 ã 2x 2 51 ã i Multiply the ones 4 (the 3) by 2,45. 3 ã 12 i Multiply the tenths 4 (20) by 2,45. i Create improper i Multiply the fractions. hundreds (100) by 2,45. i Factorise in prime factors. i Add the answers. i Divide the same factors into the top and bottom. i Change the answer back to a mixed fraction.
Multiplication and division of fractions
Division is the opposite of multiplication. 1 10 h ã 10 x 2 ã 20 2 3 4 8 2 2h ã 2x ã ã 2 4 3 3 3
Multiplication
Facilitator’s Guide G07 ~ Mathematics 1