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MATHEMATICS STUDY GUIDE
Grade 7
A member of the FUTURELEARN group
Mathematics Study guide
1807-E-MAM-SG01
Í2’È-E-MAM-SG01MÎ
Grade 7
CAPS aligned
DM Oost
Study Guide G07 ~ Mathematics
CONTENTS Content
Exam paper
Page
Unit 1
Number systems
1
p. 3
Unit 2
Exponents
1
p. 50
Unit 3
Algebra
1
p. 61
Unit 4
Number patterns and relationships
1
p. 93
Unit 5.1
Geometry
1
p. 115
Unit 5.2
Area, perimeter and volume
1
p. 157
Unit 6
Transformation geometry
1
p. 182
Unit 7
Ratio and rate
1
p. 203
Unit 8
Finance
1
p. 213
Unit 9
Statistics
1
p. 224
Unit 10
Probability
1
p. 254
Year plan: Abridged version Term T1 T1 T1 T2 T2
Content Number systems Exponents Geometry Number systems fractions Number patterns and relationships T2 Perimeter, area and volume T3 Algebra T3 Transformations T3 Ratio and rate T3 Finance T4 Statistics T4 Probability Revision November examination Paper covers the entire year’s work Š Impaq
Unit in study guide 1 2 5.1 1 4 5.2 3 6 7 8 9 10
Study Guide G07 ~ Mathematics
Unit
CONTENTS Exercise
Unit 1: Number systems
Grade
Page
1
Where do number systems come from?
7
p. 3 Exercise 1
2
Natural numbers
7
p. 4 Exercise 2
3
Counting numbers
7
p. 18 Exercise 3
4
Integers
7
p. 19 Exercise 4
5
Fractions (rational numbers)
7
p. 23 Exercise 5
6
Roots (irrational numbers)
7
p. 35 Exercise 6
7
Order of operations
7
p. 39 Exercise 7
8
Percentages
7
p. 42 Exercise 8
9
Calculator work
7
p. 45 Exercise 9
10
Mixed exercises
7
p. 48 Exercise 10
Bibliography
Bibliography
Simplify means..........
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Study Guide G07 ~ Mathematics
Unit
Level of difficulty 1 Basic knowledge 25%
*
Level of difficulty 2 Routine procedures 45%
**
Level of difficulty 3 Complex procedures 20%
***
Level of difficulty 4 Problem-solving 10%
****
Level of difficulty 5
**** *
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Assessment 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 · Come to conclusions from given information · Basic calculations as learned from examples and exercises · Complex calculations and high order arguments · Euclidean Geometry · No stipulated path to follow · Similarities and differences between presentations · Need conceptual and holistic approaches to problems · Unseen and not-routine problems. · Problems in different sections · Still in curriculum and study guide · Mostly aimed at practical and everyday situations. · High order of thinking Advanced Often regarded as acceleration of the curriculum. Not included in tests and exam papers.
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1
Study Guide G07 ~ Mathematics
Unit
1
Unit 1: Number systems Content in this unit: · Where do number systems come from? · Natural numbers · Whole numbers (Counting numbers) · Integers · Fractions (Rational numbers). · Roots and surds (Irrational numbers) · 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 20000 years old. The number system that we use today is the Hindu-Arabic system and it was developed more than 1000 years ago by Hindu-Arabic mathematicians. The Egyptian number system consists of symbols. Examples of the two number systems are: Egyptians
Hindo-Arabic
staff
1
heel
10
rope
100
lotus
1000
pointing finger
10 000
burbot
100 000
astonished man
1000 000
1.1*
Write down the Egyptian equivalent for the number 3516.
1.2*
Write down the Egyptian equivalent for the number 2182.
1.3***
Write the Hindu-Arabic number 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.
© Impaq
ÇÇ
3
Study Guide G07 ~ Mathematics
1.5***
Unit
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. + English Afrikaans Sepedi
–
Addition Optelling Go hlakantšha
¸
x
Subtraction Aftrekking Go tloṧa
Multiplication Vermenigvuldiging Go atiša
Division Deling Go arola
Exercise 2: Natural numbers 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: The symbol for natural numbers is E N A represents the natural numbers. As the Unit progresses, the other number systems will be explained.
Units
7
Tens
1
Hundreds
Millions
3
Thousands
Ten millions
4
Ten thousands
Hundred millions
2
Hundred thousands
Milliards
Each digit in natural numbers has its own meaning. Example:
5
9
0
6
8
Milliard and billion denote the same number. Milliard is almost never used in America and Britain, but is often used in Continental Europe.
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Study Guide G07 ~ Mathematics
Unit
Decimal System 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
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(1000)2
A thousand million
(100 0000)2
(100 0000)3
5
1
Study Guide G07 ~ Mathematics
2.1**
Complete the table by identifying the place value of the 7 .
Example
2.2**
Number 3452 7 8
2.1.1*
3452 7 823
2.1.2*
2 7 82
2.1.3**
52 7 8254
2.1.4**
7 254164
2.1.5***
952 7 54123
2.1.6*
1233452 7
2.1.7**
3452 7 825455
2.1.8*
7 82
2.1.9****
34 7 84556723
Description tens
Write the number from the description given
Example 2.2.1* 2.2.2* 2.2.3** 2.2.4* 2.2.5** 2.2.6*** 2.2.7****
2.3***
Unit
Description 4 hundred thousands 6 tens 12 thousands 876 millions 9 units 47 hundreds 639 ten thousands 32456 hundreds
Add 10 to all the following values. Write your answers in the table.
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
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Number 400 000
Answer
6
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Study Guide G07 ~ Mathematics
2.4**
Unit
1
Complete the table by calculating what is asked.
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
Calculation Add 400 Add 1 million Subtract 50 Add 9000 Subtract 1 million Add 4 hundred thousand Add 5000
Answer 4 161
2.5**
Brake 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
2.6**
Arrange the following natural numbers from small to big: {2542; 154; 2441; 2523; 2509}.
2.7**
Arrange the following natural numbers from big to small: {592; 523; 2600; 2699}. In Mathematics we use symbols to show greater than or less than. If you look at the following symbols from left to right, you can say:
>
greater than
<
Example: 234 > 233 345 < 346
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7
less than
Study Guide G07 ~ Mathematics
2.8**
Unit
1
Place a < or > symbol between the next two values in the table. Number 1 < or > Number 2 546 124 12 65 3232 6756 437 436 112233 112234 4356 11111
When doing addition and subtraction of natural numbers, your answer must have enough steps to show that you did not use a calculator. Example 2379 + 6666 = 2379 + 6666 9045
Make sure you know where the = sign is placed. Know how and where this sign is placed.
9379 - 6666 = 9379 - 6666 2713
Simplify the following. Show your calculations. You are not allowed to use a calculator. 2.9.*
17 + 25
2.10**
Use your answer and write down the correct answer for 17 hundred plus 25 hundred.
2.11**
Use now your answer and write down the correct answer for 17 million plus 25 million.
2.12****
Determine the sum of 17 hundred thousands and 25 ten thousands.
-
Calculations with natural numbers can only be one of the following:
+
Plus
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´
Minus
Multiplication
8
¸ Division
Study Guide G07 ~ Mathematics
Unit
Examples of calculations with natural numbers: Short division 25863 ¸ 8 =
3232 res 7 8 25863
Long division 25863 ¸ 8 3232 = 8 25863 - 24 18 - 16 26 - 24 23 - 16 remainder 7
Adding 6765 + 25863 = 6765 + 25863 32628
Multiplication 456 x 18 = 456 x 18 3648 + 4560 8208
Subtraction 25863 - 4444 = 25863 - 4444 21419
Use the prescribed methods and determine the answers of the following. You are not allowed to use a calculator. No calculators. Write down all the actual calculations to show that you did not use a calculator.
2.13**
2345 + 999
2.14**
2345 – 999
2.15**
2345 x 99
2.16***
2345 ¸ 9 with the long division method.
2.17***
2345 ¸ 99 with the long division method.
2.18****
2345 ¸ 999 with the long division method.
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. Symbols ´ ¸ – +
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Meaning Multiply Divide Subtract Adding
Other English words Product of Quotient of Difference between Sum of
9
1
Study Guide G07 ~ Mathematics
Unit
Write the following English sentences in mathematical (number) sentences. Simplify them without using a calculator. The remainder is asked for question 2.21. 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.
2.19**
Determine the sum of 12 and 56.
2.20**
Calculate the product of 12 and 56.
2.21**
What is the quotient if 56 is divided by 12? Give the remainder.
2.22***
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?
2.23****
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? Harbour
Day 1= 275 km Day 2 Day 3
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Study Guide G07 ~ Mathematics
Unit
2.24***
Juan drives 1 035 meter to a specific shop. John drives 2 285 meter to the same shop. How much further do John drive?
2.25
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.
Calculators are not allowed.
2.25.1*** 2.25.2***
Show all your calculations and determine how much material she still needs to buy. 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.
Rounding off natural numbers must be done in tens, hundreds and thousands. Examples Question: number 1234 1234 1234 8795 8795 8795
Round off to the nearest 10 100 1000 10 100 1000
Answer
Reason
1230 1200 1000 8800 8800 9000
4 is smaller than 5 34 is smaller than 50 234 is smaller than 500 5 is halfway to 10 95 is more than 50 795 is more than 500
Rounding off to 10 (It means that the number must be divided by 10 without leaving a remainder.)
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Look at the units · When less than 5 round off downwards. · When 5 or more round off upwards.
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Study Guide G07 ~ Mathematics
Rounding off to 100 (It means that the number must be divided by 100 without leaving a remainder.) Rounding off to 1000. (It means that the number must be divided by 1 000 without leaving a remainder.)
2.26**
Unit
1
Look at the last two digits · When less than 50 round off downwards. · When 50 or more round off upwards. Look at the last three digits · When less than 500 round off downwards. · When 500 or more round off upwards.
Complete the table by rounding off the numbers. Read the headings of the table.
Question: Number To the nearest Answer Reason 7766 10 7766 100 7719 1000 7371 10 890 100 1 million 1000 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 downwards and units that are 3 and 4 upwards.) · 62 rounded off to the nearest 5 is 60 · 63 rounded off to the nearest 5 is 65 · 84 rounded off to the nearest 5 is 85
»
When you round off in Mathematics, the following sign is used: For clarity, the number you round off to is written at the end of the answer. For examples: 26 » 30 to the nearest 10 3456 » 3 500 to the nearest 100 2.27***
Round off 123456 to the nearest 5.
2.28****
Round off 74 to the nearest 7.
2.29***
Shaun buys a pair of trousers for R243 and a jacket for R679. Round off each to the nearest ten Price R243 Price R679 rand en give the total amount paid for the trousers and the jacket. What will the amount be if Shaun first adds the amounts together before rounding off? What is the difference between the answers in the previous two questions? Do you think one will always get to the same answers?
2.29.1*
2.29.2** 2.29.3***
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12
Study Guide G07 ~ Mathematics
Unit
Order of operations 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.
The sequence of calculations is as follows: 1. Brackets (…) 2. Exponents. 3. Multiplication and division from the left to the right. 4. Adding and subtraction from left to the right.
This sequence applies up to Grade 12. Study this now and you will never have to study it again.
Simplify the following by doing only one calculation for each step: Example 1: 2+3x4 = 2 + 12 =14
Example 2: 23 – (12 + 8) +2 = 23 – 20 + 2 =3+2 =5
Example 4
2 + 3 ´ 10 - 6 + ( 3 - 2 ) + = =
1 of 12 2
Example: 5
1 of 12 2 2 + 3 ´ 10 - 6 + 1 + 6
1 of ( 21 + 3 ) 3 1 = 2 + 23 x 2 – of 24 3 1 =2+8x2– of 24 3 = 2 + 8 x 2 -8 = 2 + 16 – 8 = 18 – 8 = 10
2 + 23 x 2 –
2 + 3 ´ 10 - 6 + 1 +
= 2 + 30 - 6 + 1 + 6 = 32 - 6 + 1 + 6 = 26 + 1 + 6 = 27 + 6 = 33 Remember: Once the calculation in the bracket has been completed, the bracket disappears.
© Impaq
Example 3: 30 ¸ 3 x 2 + 15 ¸ 5 = 10 x2 + 15 ¸ 5 = 20 + 15 ¸ 5 = 20 + 3 = 23
13
1
Study Guide G07 ~ Mathematics
Unit
Simplify the following expression by doing only one calculation per step. 4 + 2 ´ 3 -1 2.30* Do one calculation for each step. It will teach you the preference and the use of the = sign. Remember that before and after the = sign everything must be identical. 2.31*
1+ 3 ´ 2 ¸ 6 - 2 Brackets first. A bracket disappears when the answer is written down.
x and ¸ has the same priority. The first sign must be executed first.
2.32**
9 - (12 - 8 )
2.33**
3 + 3 ´3 - 3
2.34**
( 3 + 2 ) ´ (6 - 2 )
It is easy to multiply by 0. The answer stays zero.
REMEMBER! (0)4 = 0 and not 4 6 x 0 = 0 and not 6 etc.
Be aware All the number 2’s can be confusing. Work systematically … one calculation for each step.
2.35**
2 + 2 ´ 2 ¸ 2 - 2 + ( 2 + 2)
2.36**
8- 2 ´ 3 + 6 - (5 - 1 )
2.37**
10 - 4 + ( 3 - 1) ´ 3
© Impaq
The assignment is to do one calculation for each step. Do not do more than one!
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1
Study Guide G07 ~ Mathematics
2.38**
3´0 + 2´0
2.39**
(6-2+7) +( 3 – 2 + 12)
2.40***
Unit
Simplify by doing only one calculation for each step. (5 x12 + 16 ¸ 4) ¸ 8 (16 - 8)x2
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.
Prime factors 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 ;………….}
The number 1 1 is not a prime number and also not a composite number.
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} 4 is written as 2 x 2 6 is written as 2 x 3 12 is written as 2 x 2 x 3 24 is written as 2 x 2 x 2 x 2 x 3
Prime numbers that are multiplied are called composite factors.
Write each number given as sets of 1, prime numbers and composite factors. Use the example given. 2.41*
12
2.42*
30
2.43**
36
2.44**
35
© Impaq
First determine the prime factors.
15
1
Study Guide G07 ~ Mathematics
Unit
The use of the ladder method to determine prime factors. Example: 360 =2 x 2 x 2 x 3 x 3 x 5
2 2 2 3 3 5
360 180 90 45 15 5 1
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;....}
Use the ladder method 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. 2.50*
24
2.51**
68
2.52*
Determine the largest prime number that can be divided into 39 600.
First learn the following: · Multiples. Numbers into which the given number can be divided. · Example: multiples of 6 are {6; 12; 18; 24; 30; …} · 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; …….} Determine the multiple and factors of the following: 2.53* 6 2.54*
12
2.55*
18
2.56*
24
© Impaq
Use the ladder method
16
1
Study Guide G07 ~ Mathematics
Unit
Determine the LCM (lowest common multiple) and HCF (highest common factor) of two or more integers. 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} \ HCF = 2 (Highest common factor. This is the biggest factor that can divide into both numbers) Example 2: Multiples of 6 = {6; 12; 18; 24; …….} Multiples of 8 = {8; 16; 24; 32; 40 ……..} \ LCM= 24 (Smallest common multiple is the smallest number that 6 and 8 can divide into) 2.57** Determine the HCF of 16 and 24. 2.58**
Determine the HCF of 20 and 30.
2.59***
Determine the LCM of 9 and 12.
2.60***
Determine the LCM of 5, 6, and 15.
2.61***
Determine the LCM of 7, 9 and 21.
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17
HCF means highest common factor. LCM means lowest common multiple. LCD means lowest common denominator.
1
Study Guide G07 ~ Mathematics
Unit
Exercise 3: Counting numbers Counting numbers B include the natural numbers and zero. B = {0; 1; 2; 3; 4; 5; 6;....}
E
Natural numbers A = {1; 2; 3; 4; 5; 6;......} The letter used for counting numbers is N0.
The zero is added. Remember
0 = 0 2 2 is undefined 0 3.1**
( 2 + 0 – 1) +( 3 – 0)
3.2**
22 x 0 + 4 – 0 +(10 – 0)
3.3*
4´0 2
3.4**
2+0 3-3
© Impaq
Do not be afraid of the zero.
3.5*
18
2 + (3 ´ 4) 12 - ( 3 ´ 4 )
1
Study Guide G07 ~ Mathematics
Unit
2´7 + 6 6-2´3 5´4 3
3.6***
3.7** 2- 2 3-3 ´ 5 7
2 + 6 - 4 + 100 ( 2 - 2) + ( 8 - 8 )
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.
3.9****
The temperature in Sutherland was -9°C on Saturday and -5°C on Sunday. What was the change in temperature from Saturday to Sunday?
3.10****
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?
Exercise 4: Integers
Integers C include counting numbers. C = {.....-3; -2; -1; 0; 1; 2; 3....}
Counting numbers B include the natural numbers and zero. B = {0; 1; 2; 3; 4; 5; 6;....}
E
Natural numbers A = {1; 2; 3; 4; ......}
Integers
=
Positive Numbers
= {1; 2; 3; 4; 5 ;.......}
+
+
+
{0} + {-1;-2;-3;-4;-5;.......}
The letter used for integers is a Z.
© Impaq
Zero
19
Negative numbers
1
Study Guide G07 ~ Mathematics
Unit
The minus sign in front of an counting number means that it works in the same way as a “debit”. ·
Read it as: “Owe 2 and owe another 3”. That means that you owe 5.
Addition of negative numbers Example: – 2 – 3 =–5 Subtraction of negative numbers Example: (– 2) – (– 3) =–2+3 =1 Multiplication of negative numbers Example: (– 2)(– 3) =+6 Division of negative numbers -4 = 2 Example: -2
=
+ X
+
X
+
x
=
-
x
=
If you pay a debt, it means that you have to add. Signs also have to be multiplied.
Signs also have to be divided.
-
+
-
x x
+
= =
+
-
More on negative numbers Study the following 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. 5 units to the left
-4
-3
-2
-1
Number line
0
1
2
3
4
5
Example 2: – 2 + 6 = 4. Start at – 2 and move 6 units to the right.
Number line
6 units to the right -4
-3
© Impaq
-2
-1
0
1
20
2
3
4
5
1
Study Guide G07 ~ Mathematics
Unit
Example 3: – 2 – 4 = – 6. Start at – 2 and move 4 units to the right.
4 units to the left -8
-7
-6
-5
Number line -4
-3
-2
-1
0
1
2
Example 4: – 8 – (– 4) = – 8 + 4.......multiply the two signs – x – = + Start at – 8 and move 4 units to the right =–4
Number line
4 units to the left -8
-7
-6
-5
-4
-3
-2
-1
0
1
2
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: Example 5: 50 + (– 10) = 50 – 10 = 40
Example 6: – 30 + (+ 2) = – 30 + 2 = – 28
Example 7: 40 –(– 3) = 40 + 3 = 43
Example 8: – 30 – – 8 = – 30 + 8 = – 22
Example 9: (– 2)(– 3) – – 5 =6+5 = 11
Two signs next to each other mean they have to multiply.
© Impaq
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1
Study Guide G07 ~ Mathematics
Unit
4.1*
–4–3
4.3*
5–4
4.5**
(10 – 12) + 0
4.2 *
–4+3
4.4*
2+3–6
4.6**
2+3–4–5 10 – 9 – 8
4.7*
–2–3–4–5–6
4.8*
4.9*
6–7–1+3
4.10** (10 – 15) +(6 – 5)
4.11**
(– 3)(+ 6)
4.12*
(+ 3)(– 2) (– 33)(–1)
4.13*
(– 2)(– 7)
4.14*
4.15**
(+ 4)(– 2) + (2)(6)
4.16 * 5 +(– 2)(3) + 0
4.17****
Study the word puzzle. + + –2 + + 6 + =–5
0 +
+ +
+ –2 +
+
9 + –2 + 6 =–5
Complete the table by filling in the blank spaces with integers. 4.18*
-4 -2
4.19*
8 -8
4.20***
© Impaq
2 -
Two signs next to each other mean they have to be multiplied. Example 2 – – 1 =2+1 =3 - 3 - 10 -2
Remember: A minus sign multiplied by a minus gives a positive. A minus sign multiplied by a plus sign will give a minus.
22
=0 =–5 =–5
1
Study Guide G07 ~ Mathematics
4.21***
26 -2
Unit
- 2 ´ ( -3 )
10 5
4.22 * * *
+
- 20 -4
-
+ 18 - 9
Exercise 5: Fractions (rational numbers) The definition of fractions ( rational numbers): 4 Integers like 4 = which is a numerator divided by a denominator, is called a 1 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.
D is all the fractions. It includes the integers.
Integers C include the counting numbers. Counting numbers C = {.....-3; -2; -1; 0; 1; 2; 3....}
Counting numbers B include the natural numbers. B = {0; 1; 2; 3; 4; 5; 6;....} The zero is added.
E
Natural numbers A = {1; 2; 3; 4; 5; 6;......} The letter for rational numbers is a Q.
Numerators
12 3
2 3 Denominators
Š Impaq
23
1
Study Guide G07 ~ Mathematics
Unit
The different types of fractions are: · Proper (common) fractions: example
1 2
...........numerator < denominator
1 · Mixed fractions: example 7 .............integer plus proper fraction 2 11 ....................numerator > denominator · Improper fractions: example 2 · 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. Change the question to improper fractions if necessary.
Complete the calculations
Change the answer back to mixed fractions.
100 Thousands
Millions
2
3
4
5
6
7
2 10
3 100
4 1000
5 10 000
6 100 000
7 1000 000
Hundreds
Tenths
Units
0,
5.1*
© Impaq
Thousands
10 Thousands
Decimal fraction
Give the numerical value of the letter in the equation. Equation Value of letter 5.1.1* a 4 5 3,45 =3+ + a 100 5.1.2* b 7 8 9 12,789 = b+ + + 10 100 1000 5.1.3* c c 328,012 = 328 + 1000
24
1
Study Guide G07 ~ Mathematics
5.2**
Unit
Write the following as a decimal. Do not use a calculator. Question Answer 5.2.1** 9 5 4+ + 10 100 5.2.2** 8 145 + 100 5.2.3** 9 5 5454 + + 10 1000
Simplify the following fractions: 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 = 2x3 x13 1 1 =
2 13
Alternative: 12 78 2x6 = 6 x13
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. 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.
2 13 Factorise and simplify the following fractions. Calculators may not be used. 5.3* 64 80 Use the 5.4** 225 ladder 150 method. 5.5*** 36 x24 15 x18 5.6*** 30 x 49 14 x21x 25 =
Š Impaq
25
1
Study Guide G07 ~ Mathematics
Unit
1
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 5x13 .........f actorise 5x20 13 = .......... ......canc el out 20 =
8
Improper fraction
1 2
1 17 8 = .......numerator > denominator 2 2 (8x2 + 1= 17) Mixed fraction 34 6 =4 7 7 (34 ¸ 7 = 4 remainder 6) Change the improper fractions to mixed fractions and mixed fractions back into improper fractions in the following tables. Improper fraction Mixed fraction Mixed fractions Improper fraction 16 4 11 9 9
34 7 5.7*
7
2
65 24
5 9
720 33
1 8
408
48 15
2 3
It is an improper fraction when the 7 numerator > denominator e.g. 3
Š Impaq
26
Study Guide G07 ~ Mathematics
5.8**
Unit
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
3
5 8
3,375
5
12 20
0,02
1,0101
2 3
0,125 7
Š Impaq
1 9
27
1