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MATHEMATICS STUDY GUIDE Grade 10
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Mathematics Study guide
1810-E-MAM-SG01
Í2*È-E-MAM-SG01SÎ
Grade 10
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DM Oost
Study Guide G10 ~ Mathematics
CONTENTS
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5.1 Unit 5.2 Unit 6 Unit 7 Unit 8 Unit 9 Unit 10 Unit 11 Unit 12 Unit 13
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Number systems Exponents Algebra Number patterns and relations Geometry: measurement, space and form Geometry: Euclidian geometry Transformation geometry Ratio and rate Finances Statistics Probability Functions Trigonometry Analytical geometry
1
Exam paper.
Page
1 1 1 1 2 2 2 1 1 2 1 1 2 2
2 16 32 54 63 82 109 135 140 174 220 258 296 343
Study Guide G10 ~ Mathematics
Unit
1
CONTENTS Exercise 1 2 3 4 5 6 7 8
Unit 1: Number systems Different types of number systems Notations Terms, factors and multiples Divisibility rules Types of fractions Approximating decimals by rounding off Roots, squares and irrational numbers Mixed exercises Bibliography
Page 3 4 6 7 8 9 10 13 14
The objective of this theme is: • • • •
• • • •
•
Recognising different types of number systems and numbers. Most of these number systems have already been discussed in Grade 8 and 9. Different notations such as builder notation, interval notation, tabulation and graphs. The difference between multiples, factors and terms. Divisibility rules. It is essential in order to answer questions where calculators are prohibited. Different types of fractions – decimal and normal fractions, as well as the conversion form one type to another. The handling of the addition, subtraction, multiplication and division of fractions. Approximating decimals by rounding off. At the end of this theme, you should be able to round off any decimal number (fraction) to as many digits as asked. Roots, squares and irrational numbers. In previous grades, these questions were not discussed in full. You should be able to answer these questions with ease. The two major questions here are always: 1. Simplify 2. Solve for x (equations).
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Study Guide G10 ~ Mathematics
Unit
1
Unit 1: Number systems LO1 This unit is a summary of the work discussed in Grade 8 and 9. The idea is to support you with examples rather exercises that must be done. Should you have trouble in completing this section, you should go back to the work discussed in Grade 8 and 9. Use the provided links.
Exercise 1: Different types of number systems 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 = {numbers that can be written as
Links: Ctrl + click • Video1
an int eger } non − zero int eger
= Rational numbers Q’ = {numbers that cannot be written as
an int eger } non − zero int eger
= Irrational numbers (non-terminating and non-repetitive decimal numbers.) R = Real numbers = {rational and irrational numbers.} R’ = Non-real numbers = {numbers that do not exist.} Summary of the real number system: Real numbers Rational numbers Integers • • •
Fractions
Negative integers Zero Positive integers (counting numbers)
Remember: 0 =0 x x is undefined 0
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Irrational numbers
− 3 is Non − Real
3
Whole numbers
Study Guide G10 ~ Mathematics
Unit
Summary: • •
• •
A rational number is any number that can be written as
a , where a and b are b
integers. The following are rational numbers: Fractions of which both the numerator and denominator are integers. Integers. Decimal numbers that ends. Decimal numbers that repeat. Irrational numbers are not rational. They cannot be written with an integer numerator and denominator. If the nth root of a number cannot be written as a rational value, this nth number is called a surd.
•
If a and b are positive integers, with a < b, then n a < n b .
• •
A binomial is a mathematic expression with two terms. The product of two identical binomials is known as the square of the binomial.
Examples:
4 = 2 This is an integer, a rational number and a real number. − 2 has no answer and is thus a non-real number. - 2
does exist, but is an irrational number. answer is not known. We always have to round it off 3 − − 64 = +4 is also an integer, a rational number and a real number.
Always try to simplify the number first. If there is an answer, the number is real, but if there is no answer, the number is non-real. 1.1* 1.2*
Is the number zero a positive or a negative number? What type of number is
8?
1.3*
What type of number is
−8 ?
1.4*
What type of number is
1.5*
What type of number is 3 − 8 ?
3
Link: Ctrl + click • Video 1.1 to 1.5
8?
Exercise 2: Notations Not all number systems can be presented as in the example below. • Rational and real numbers cannot be tabulated. • Irrational numbers cannot be written in any of the notations. We are therefore limited in the representation of a group irrational numbers. • In the examples below, integers are used:
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Study Guide G10 ~ Mathematics
Unit
Examples: 1. 2. 3. 4. 5. 6.
English: All integers greater than and equal to -4 and less than 2. Normal notation -4 ≤ x < 2 Table notation = {-4; -3; -2; -1; 0; 1} Builder notation = = {x / -4 ≤ x < 2, x ϵ Z} Interval notation = x ϵ [-4; 2] NB: The “)” brackets mean NB: This notation is only used for x ∈R . excluded and the “]” brackets mean included. Graphical (number line) -4
-3
-2
-1
0
1
2 Remember: Integers are indicated by dots.
Use the above examples and answer the following questions. 2.1** Give five different notations for the following numbers: “Real numbers between -3 and 6, with -3 included, and 6 excluded.” 2.2** If x ϵ [-1; 10], is 10 included or excluded? 2.3 ** Draw the graph for 30 < x ≤ 33 if x ϵ R Can you see that 30 is excluded, and 33 is included? Study the graph below and give the answer in builder notation. 2.4** Video 2.1 -2
-1
0
1
2
3
4
5
6
7
R
Video 2.2 to 2.4 2.5**
Study the graph below and provide the answer in interval notation. -2
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-1
0
1
2
3
4
5
5
6
7
R
Video 2.5
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Study Guide G10 ~ Mathematics
Unit
Exercise 3: Terms, factors and multiples LO1 and LO2 Terms are separated by a “+” or “–” symbol. Terms 7n Example: 2m + 4 – + y consists of four terms. 8 Factors are separated by an “x” symbol. Factors Example: 2 x 23 x 3 consists of three factors. If 12 = 2.2.3, 12 is factorised in its three prime factors. If 12 = 22 .3, 12 is factorised in its prime factors and written in exponent format. Multiples of any number is when that number is multiplied by 2, then by 3, then by 4, etc. Multiples Example: Multiples of 12 = {12; 24; 36; 48; ...} Multiples of 100 = {100; 200; 300; 400; ...} More examples: 2(x + 3) has one term if it is not simplified, but two terms if it is simplified. = 2x + 6 The question must specify whether you must simplify or not. xy xy is one term with four factors. = 4 2 .2 3.1** Simplify the following without the use of a calculator and prove that you know the calculation order by doing only one calculation per step.
4 − 3 x 6 ÷ 9 + 2 4 − (10 − 8 x3) −
1 van of 12 3
Video 3.1
The normal calculation order is: 1. Power 2. Brackets 3. Multiply and divide 4. Add and subtract Example: 2 + 3 x 4 = 2 + 12 = 14 and not 5 x 4 = 20 3.2**
3.3**
How many terms does (x – 1)(4x) have 3.2.1 before it was simplified? 3.2.2 after it was simplified? Write down the first 10 prime numbers.
Video 3.2
Do you know the definition of a prime number? Example: 5 is a prime number, as it can only be divided by 1 and 5 without a remainder. The number 1 is not a prime number.
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Study Guide G10 ~ Mathematics
3.4**
Unit
1
Take the numbers 24 and 36 and: 3.4.1 draw a Venn diagram to illustrate the factors of these two numbers. 3.4.2 Give the common factors of 24 and 36 from the Venn diagram. 3.4.3 Give from these common factors the biggest one. In other words, the HCF (highest common factor) or HCD (highest common denominator) NB: a denominator and a factor is the same concept. Do you remember what a Venn-diagram looks like? The objective is to find the common factors to put in the intersection of the following two circles. Video 3.3 and 3.4
3.5**
Take the numbers 12 and 8 and: 3.5.1 draw a Venn diagram to illustrate the first 10 multiples of these two numbers. 3.5.2 give the common multiples of 12 and 8 from your Venn diagram. 3.5.3 give the lowest common multiple of 12 and 8. In other words, the LCM (lowest common multiple) Video 3.5
Exercise 4: Divisibility rules LO1 Divisible by 2 Divisible by 3 Divisible by 4 Divisible by 5 Divisible by 6 Divisible by 7
Divisible by 8 Divisible by 9
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All numbers ending in 2; 4; 6; 8; or 0. I.e., even numbers. If all the numbers are added and the sum is a multiple of 3. If the last two numbers are a multiple of 4. All numbers ending in 5 or 0. If the number is divisible by 2 and 3. Take the last number, multiply it with 2, and subtract it from the rest of the number. If the answer is a multiple of 7, the number is a multiple. Repeat this until you evaluated the whole number. e.g. 1792 is divisible by 7, because 179 â&#x20AC;&#x201C; (2 x 2) = 175 and 175 is 17 â&#x20AC;&#x201C; (5 x 2) = 7, which is divisible by 7. If the last three numbers are a multiple of 8. If the sum of all the numbers is a multiple of 9.
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Study Guide G10 ~ Mathematics
Divisible by 10 Divisible by 11
Unit
1
If the number ends in 0. Count all the alternative numbers together, and subtract it from the sum of the other numbers. If the difference is 0 or 11, the number is divisible by 11.
Video exercise 4
Video 4.1 to 4.3
Video 4.4 and 4.5
Calculate the following without the use of a calculator. 4.1** Is 72645 divisible by 11? 4.2** Is 51249 divisible by 11? 4.3** Is 1926 divisible by 9? 4.4** Is 4208 divisible by 8? 4.5** Is 708 divisible by 6?
Exercise 5: Types of fractions LO 1 A rational number consists of a numerator, which is whole numbers, and a denominator, which is not equal to 0. Numerator
a b
Examples
Decimal fractions
Ordinary fractions
Denominator Proper fraction
Numerator < denominator
Improper fraction
Numerator > denominator
Mixed number
Equivalent fraction
Terminating fractions
Repetitive fractions Non-repetitive,
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A combination of a whole number and a proper fraction Fractions that represent the same values but have different numerators and denominators. The decimal representation terminates or ends. The decimal representation does not terminate and repeat one or more digits (forming a pattern). The decimal representation
8
2 7 9 4 2
5 11
1 2 3 = = ........ 2 4 6
2,3456784356178903 has 16 numbers that follow after the decimal sign. 45,123123123123 ... = 45, 1 2 3 2,3456784356178903 ...
Study Guide G10 ~ Mathematics
Unit
non-terminating fractions
does not terminate or repeat. These decimals are examples of irrational numbers. Calculate the following without the use of a calculator. Prove that you did not use a calculator by showing all calculations. Video exercise 5 5.1* 1 Change to a decimal number. 8 Video 5.1 and 5.2 5.2* Change 0, 246 to a common fraction. Video 5.4 5.3* 1 Change to a decimal number. 11 Video 5.5 5.4* Change 0 ,14 to a common fraction. Video exercise 6 5.5* Change 2 ,3 1 4 to a common fraction.
Exercise 6: Approximating decimals by rounding off LO1 (Seeliger graad 10) bl. 4
23460 upwards
23456 to the nearest 100
23500 upwards
23456 to the nearest 1000
23000 downwards
2,9864537 to 3 decimal places
2, 986 downwards
2,9864537 to 2 decimal places
2,99 upwards
2,9864537 to 1 decimal place
3,0 upwards
0,07538 to 3 significant numbers
0,0754 .upwards
0,07538 to 2 significant numbers
0,075 downwards
0,07538 to 1 significant number
0,08 upwards
To the nearest 10multiple number
23456 to the nearest 10
To decimal places
Answer
Significant figures
Examples
The first non-zero digit, read from left to right, is the first significant number. 6.1* Round off 1856, 005678 to the nearest 1000. 6.2* Round off 1856, 005678 to 4 decimal places. Video 6.1 to 6.4 6.3* Round off 1856, 005678 to 1 significant number. Video 6.5 6.4* Round off 527, 53 to 3 significant numbers. 6.5*** Round off 2,864537 to 2 significant numbers. 6.6**** Round off 2,000000789 to 2 significant numbers.
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Study Guide G10 ~ Mathematics
1
Unit
Exercise 7: Roots, squares and irrational numbers LO1 A square number is any number multiplied by the same number. A square root is the inverse calculation of the above-mentioned. In other words, which number multiplied with the same number gives the original number. The nth power means that a number is multiplied n times with itself. The root of the nth degree means that a value searched for that is multiplied with itself n times give the original number. Examples Number
Square
Square root
Exponent
nth power
Root of nth degree 1 n
9
9 = 3
(9)2 = 81
9 = 32
9n = 3 2n
9 = 9n 1
= (3 2 ) n 2
= 3n 1 n 16 = 16 n
16
16 = 4
(16)2 = 256
16 = 24
16n = 2 4n
1 4 n = (2 ) 4 = 2n
Square numbers
= {12; 22; 32; 42; 52; 62 ...} = {1; 4; 9; 16; 25; 36; …}
Cube numbers
= {13; 23; 33; 43; 53; 63 ...} = {1; 8; 27; 64; 125; 216 ...} Irrational numbers are like variables/unknowns 3m + m = 4m 3m − m = 2m ∴3 2 + 2 = 4 2 ∴3 2 − 2 = 2 2
−
+
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Study Guide G10 ~ Mathematics
Unit
3m . m = 3m 2 ∴3 2 x
x
3m = 3 m 3 2 2 = =3 2
3m ÷ m =
2 = 3.( 2 )2 =3 . 2
3 2 ÷
= 6
÷
How to determine the square root of combined numbers. The reverse also counts: x y = x.y
Example: Remember the following: x.y = x y ∴ 12 =
∴ 4 3 = Examples: 1. 2 3
4 3
= 2 3 Choose squared numbers in order for the root to be determined without a calculator.
2.
2 3
12
= =
6 4
3
= 12
Calculate the following without the use of a calculator. Give the answers in the simplest surd form. 7.1*
4 8 +
7.2*
4 k4
7.3**
4050
7.4**
147 −
7.5** 7.6**
2 k2 9
12
3 18 + 4 2 +
2
Video 7.2
Video 7.3
Video 7.4
Video 7.5
Video 7.6
Video 7.7
Video 7.8
Video 7.9
Video 7.10
Video 7.11
Video 7.12
Video 7.13
Video 7.14
27 + 36.3. 16
7.7***
− 9m 2 48m 2 − 75 m 2
7.8***
( 19 ) 2 2 − 50 (
7.9***
2 8 + 3 32 −
8 )2
200
7.10***
18k 4 −
121 2k 4
7.11****
108a −
121a .
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Video 7.1
3
11
1
Study Guide G10 ~ Mathematics
Unit
More examples: b =b
1 2
16 + 9 = 25 = 5 Do not make the following mistake: 16 + 9 = 16 + 9 … (it is not true)
7.13
Show all calculations and use the correct method without the use of a calculator. Do the left-hand 7 side and the 2 3 Prove that g g = g 2 right-hand side separately. Prove that 25 − 16 4 9 ≠ ( 25 − 16 ) 4 9
7.14
Prove that 3 100 − 64 + 144 + 25 =
7.12
100 +
441
Determine between which two integers each of the following irrational numbers (roots) lie. We use the following principle: If a < b with a and b > 0, then n a < n b Examples:
3 < 4 or 12 12345678 < 12 12345679
Examples: 1. Question: Between which two integers do Answer: 4 < 7 < 9 ∴ 4< 7< 9 ∴ 2 <
7 lie?
Use square numbers
7 < 3
2. Question: Between which two integers do - 7 lie? Answer: 4 < 7 < 9 Remember: when ∴ 4< 7< 9 multiplying with a –, the signs are switched around. ∴ − 4> − 7 > − 9 ∴ −2 > − 7 > − 3 ∴ − 3 < − 7 < −2
3. Question: Between which two integers does 3 7 lie? Answer: 1 < 7 < 8 ∴31 < 3 7 < 3 8 ∴ 1 < 37 < 2
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Use cube numbers Video examples
12
1
Study Guide G10 ~ Mathematics
Unit
Between which two integers do the following numbers lie? Use the examples on the previous page. 7.15*
10
7.16**
- 19
7.17**
3 25
7.18***
- 3 25
7.19**
Video 7.15
Video 7.16
Video 7.17
Video 7.18
Video 7.19
Video 7.20 and 7.21
10000
7.20****
4 3000
7.21****
5 3000
Total 20
Exercise 8: Mixed exercises LO1 8.1* 8.2* 8.3** 8.4**
Give an example of an irrational and a non-real number. Why is 3 − 125 a rational number? Write the following in interval notation. {x / 0 < x ≤ 3, x ϵ R} Write the following in set builder notation. x ϵ (-∞; 67]
(2) (2) (2) (2)
8.5**
Change 0 , 3 4 to a common fraction.
(4)
8.6**
Simplify the following without the use of a calculator.
(3)
48 − 3 75 8.7*
8.8**
Approximate the following number truncated to four significant numbers. 5, 00769 Use a calculator to determine the following to two decimal places. 32,45 0,3 − 1 , 8 12,58
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(2)
(3)
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