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MATHEMATICS STUDY GUIDE Grade 10
Grade 10
Study guide
Mathematics
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Í2*È-E-MAM-SG01SÎ 1810-E-MAM-SG01
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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
CONTENTS
1
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
Study Guide G10 ~ Mathematics
1 1 1 1 2 2 2 1 1 2 1 1 2 2
Exam paper. 2 16 32 54 63 82 109 135 140 174 220 258 296 343
Page 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
Unit 1: Number systems
CONTENTS 3 4 6 7 8 9 10 13 14
Page
Unit
1
2
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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x
x
x
x
x
x
x
x x
The objective of this theme is:
1 2 3 4 5 6 7 8
Exercise
Study Guide G10 ~ Mathematics
Unit
1
Links: Ctrl + click x Video1
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Fractions
3 is Non Real
3
Whole numbers
Irrational numbers
Negative integers Zero Positive integers (counting numbers)
Remember: 0 0 x x is undefined 0
x x x
Integers
Rational numbers
Real numbers
Summary of the real number system:
= 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.}
an int eger } non zero int eger
an int eger } non zero int eger
Q’ = {numbers that cannot be written as
= Rational numbers
Q = {numbers that can be written as
N = {1; 2; 3; 4; 5 ...} = Natural numbers N0 = {0; 1; 2; 3; 4 ...} = Whole numbers Z = {... -2; -1; 0; 1; 2; 3; ...} = Integers
Exercise 1: Different types of 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.
Unit 1: Number systems
Study Guide G10 ~ Mathematics
a , where a and b are b
has no answer and is thus a non-real number.
2 This is an integer, a rational number and a real number.
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.
- 2
2
4
3
8?
8 ?
8?
What type of number is 3 8 ?
What type of number is
What type of number is
What type of number is
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4
Not all number systems can be presented as in the example below. x Rational and real numbers cannot be tabulated. x Irrational numbers cannot be written in any of the notations. We are therefore limited in the representation of a group irrational numbers. x In the examples below, integers are used:
Link: Ctrl + click x Video 1.1 to 1.5
Is the number zero a positive or a negative number?
Exercise 2: Notations
1.5*
1.4*
1.3*
1.1* 1.2*
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.
Examples:
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.
x x
integers. The following are rational numbers: ¾ Fractions of which both the numerator and denominator are integers. ¾ Integers. ¾ Decimal numbers that ends. ¾ 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.
A rational number is any number that can be written as
Unit
x
x
x
x
x
Summary:
Study Guide G10 ~ Mathematics
1
Remember: Integers are indicated by dots.
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 ࣅ Z} Interval notation = x ࣅ >-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
Unit
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2.5**
-1
0
1
2
3
4
5
6
7
Video 2.1
Video 2.2 to 2.4
R
-2
-1
0
1
2
3
5
4
5
6
7
R Video 2.5
Study the graph below and provide the answer in interval notation.
-2
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 ࣅ >-1; 10], is 10 included or excluded? 2.3 ** Draw the graph for 30 < x d 33 if x ࣅ 5 Can you see that 30 is excluded, and 33 is included? Study the graph below and give the answer in builder notation. 2.4**
6.
4. 5.
1. 2. 3.
Examples:
Study Guide G10 ~ Mathematics 1 Unit
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3.3**
3.2**
1 van of 12 3
Video 3.2
Video 3.1
6
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.
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.
Example: 2 + 3 x 4 = 2 + 12 = 14 and not 5 x 4 = 20
The normal calculation order is: 1. Power 2. Brackets 3. Multiply and divide 4. Add and subtract
4 3 x 6 y 9 2 4 (10 8 x3)
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.
LO1 and LO2
Exercise 3: Terms, factors and multiples
Study Guide G10 ~ Mathematics
1
Divisible by 8 Divisible by 9
Divisible by 3 Divisible by 4 Divisible by 5 Divisible by 6 Divisible by 7
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1
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
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
Divisible by 2
LO1
Unit
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.
7
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 – (2 x 2) = 175 and 175 is 17 – (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.
Exercise 4: Divisibility rules
3.5**
3.4**
Study Guide G10 ~ Mathematics
Video 4.1 to 4.3
Exercise 5: Types of fractions
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a b
Non-repetitive,
Repetitive fractions
Terminating fractions
Equivalent fraction
8
The decimal representation does not terminate and repeat one or more digits (forming a pattern). The decimal representation
The decimal representation terminates or ends.
A combination of a whole number and a proper fraction Fractions that represent the same values but have different numerators and denominators.
Numerator > denominator
Improper fraction Mixed number
Numerator < denominator
Proper fraction
Denominator
Numerator
A rational number consists of a numerator, which is whole numbers, and a denominator, which is not equal to 0.
2 4
5 11 3 ........ 6
= 45, 1 2 3
45,123123123123 ...
2,3456784356178903 has 16 numbers that follow after the decimal sign.
1 2
2
2 7 9 4
Examples
1
2,3456784356178903 ...
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?
LO 1
Unit
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
Divisible by 10 Divisible by 11
Study Guide G10 ~ Mathematics
Ordinary fractions Decimal fractions
Unit
23000 downwards 2, 986 downwards 2,99 upwards 3,0 upwards 0,0754 .upwards 0,075 downwards 0,08 upwards
23456 to the nearest 1000
2,9864537 to 3 decimal places
2,9864537 to 2 decimal places
2,9864537 to 1 decimal place
0,07538 to 3 significant numbers
0,07538 to 2 significant numbers
0,07538 to 1 significant number
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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.
23500 upwards
23456 to the nearest 100
Answer 23460 upwards
Examples
23456 to the nearest 10
LO1 (Seeliger graad 10) bl. 4
Exercise 6: Approximating decimals by rounding off
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 to a decimal number. Change 8 Video 5.1 and 5.2 5.2* Change 0, 246 to a common fraction. Video 5.4 5.3* 1 to a decimal number. Change 11 Video 5.5 5.4* Change 0 ,1 4 to a common fraction. Video exercise 6 5.5* Change 2 ,3 1 4 to a common fraction.
non-terminating fractions
Study Guide G10 ~ Mathematics
To the nearest 10multiple number
To decimal places
Significant figures
1 Unit
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Cube numbers
16
9
4
3
Square root
= {1; 4; 9; 16; 25; 36; …}
16n
9n
2 4n
3 2n
nth power
10
Irrational numbers are like variables/unknowns 3m m 4m 3m m 2m ?3 2 2 4 2 ?3 2 2 2 2
= {1; 8; 27; 64; 125; 216 ...}
= {13; 23; 33; 43; 53; 63 ...}
16 = 24
9 = 32
Exponent
= {12; 22; 32; 42; 52; 62 ...}
(16)2 = 256
(9)2 = 81
Square
Square numbers
16
9
Number
Examples
9
n 16
n
1
2n
4
(2 4 ) n
1
1
16 n
3n
2
(3 2 ) n
9n
1
Root of nth degree
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.
LO1
Exercise 7: Roots, squares and irrational numbers
Study Guide G10 ~ Mathematics
1
3m . m
2
3.( 2 ) 3.2 6
3m 2 2
3 2 y
y
2
3m y m
4 3
4 k4
4050
147
7.2*
7.3**
7.4**
108a
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18k 4
7.11****
2 8 3 32
7.9***
7.10***
( 19 ) 2 2 50 (
7.8***
121a .
3
121 2k 4
200
8 )2
9m 2 48m 2 75 m 2
27 36.3. 16
2
7.7***
7.6**
12
k2 9
2
3 18 4 2
4 8
7.1*
7.5**
4 12
2.
2 3
6
12
? 4 3 Examples: 1. 2 3
3
The reverse also counts: x y x.y
2
3m m 3 2 3
3
Unit
11
Video 7.13
Video 7.11
Video 7.9
Video 7.7
Video 7.5
Video 7.3
Video 7.1
Video 7.14
Video 7.12
Video 7.10
Video 7.8
Video 7.6
Video 7.4
Video 7.2
Calculate the following without the use of a calculator. Give the answers in the simplest surd form.
2 3 Choose squared numbers in order for the root to be determined without a calculator.
? 12
Example: Remember the following: x.y x y
How to determine the square root of combined numbers.
x
?3 2 x
Study Guide G10 ~ Mathematics 1
25
5
Prove that 3 100 64 144 25
100
441
Show all calculations and use the correct method without the use of a calculator. Do the left-hand 7 side and the g2 Prove that g2 g3 right-hand side separately. Prove that 25 16 4 9 z ( 25 16 ) 4 9
Do not make the following mistake: 16 9 16 9 … (it is not true)
Unit
7 3
Use square numbers
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? 1 37 2
?31 3 7 3 8
Answer: 1 < 7 < 8
12
Use cube numbers Video examples
? 3 7 2 3. Question: Between which two integers does 3 7 lie?
? 2 ! 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
? 4 7 9
7 lie?
3 4 or 12 12345678 12 12345679
1. Question: Between which two integers do Answer: 4 < 7 < 9
Examples:
Examples:
We use the following principle: If a < b with a and b > 0, then n a n b
Determine between which two integers each of the following irrational numbers (roots) lie.
7.14
7.13
7.12
1
b2
16 9
b
More examples:
Study Guide G10 ~ Mathematics
1
3 25
7.18***
5 3000
7.21****
Video 7.19
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8.8**
8.7*
Simplify the following without the use of a calculator.
8.6**
12,58
32,45 0,3 1 , 8
13
Approximate the following number truncated to four significant numbers. 5, 00769 Use a calculator to determine the following to two decimal places.
48 3 75
Change 0 , 3 4 to a common fraction.
Why is 3 125 a rational number? Write the following in interval notation. {x / 0 < x 3, x ࣅ 5} Write the following in set builder notation. x ࣅ - ; 67]
Give an example of an irrational and a non-real number.
Video 7.20 and 7.21
Video 7.18
Video 7.16
8.5**
8.4**
8.3**
8.1* 8.2*
LO1
Video 7.17
Video 7.15
Exercise 8: Mixed exercises
4 3000
7.20****
10000
3 25
7.17**
-
- 19
7.16**
7.19**
10
7.15*
Between which two integers do the following numbers lie? Use the examples on the previous page.
Study Guide G10 ~ Mathematics
(3)
(2)
(3)
(4)
(2)
(2)
(2) (2)
Total 20
Unit
1
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14
Laridon. (1994). Wiskunde vir die Klaskamer st 8. Phillips, M. Mathematics 10 Textbook & Workbook. Seeliger gr 10. (n.d.). PracMaths graad 10. Johan, P. (2006). Wiskunde Plus Leerderbroek. Oxford University Press. Laridon. (1994). Wiskunde vir die Klaskamer st 8. Laridon, e. (1996). Classroom Mathematics std 10/ grade 12. Heinemann. Phillips, M. (n.d.). Mathematics 10 Textbook & Workbook. Seeliger. (n.d.). Prac-Maths graad 10. Siyavula. (2011). Graad 10 Wiskunde. Creative Commons. Tapson, F. (2006). Oxford Mathematics Study Dictionary. Oxford University Press. Van der Lith, D. (n.d.). Study& Master Mathematics Learner's Book grade 11. Cambridge University Press.
Bibliography
Study Guide G10 ~ Mathematics Unit
1
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1 2 3 4 5 6 7
Exercise
Unit 2: Exponents
CONTENTS
16
Basic laws and definitions Combined bases Exponential equations without the use of a calculator Exponential equations with the use of a calculator Word problems and exponents Graphs of exponential functions Mixed exercises Bibliography
Study Guide G10 ~ Mathematics
18 22 25 26 27 28 30 31
Page
Unit
2
25% 25% 25%
Measuring in several contexts Data handling
25%
2
Patterns and functions
Numbers, equations and relations
Unit
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1 2 3 4 5 6
17
Understand the basic laws and definitions of exponents. Apply the rules when combined and compound bases are given. Solve exponential equations without the use of a calculator. Solve exponential equations with the use of a calculator. Understand and apply this knowledge on word problems. Draw exponential graphs.
After you have studied this unit, you should have reached the following goals:
The object of this unit is to work with compound bases when exponential variables must be simplified or solved. The basic knowledge of exponents are revision of Grade 8 and 9 work.
In Grade 10, the study of exponents will consist of the following:
Learning outcome 1 LO 1 Learning outcome 2 LO 2 Learning outcome 3 LO 3 Learning outcome 4 LO 4
Study Guide G10 ~ Mathematics
base
exponent Video
Unit
2
7
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2
21 3
4 1
2 . 2 3.2 4.2 1
Answer:
Examples : 3 4 1 1. 2. 2 .2 .2
35
3
18
4 6 2 1
3 4.3 6.3 2.3
Answer:
2. 3 4.3 6.3 2.3
Example: 22.33 cannot be changed, as the bases are not the same. Remember that 2.24 is not equal to 44, but 21+4.
It is important to realise that the exponent laws can only be applied if the bases are equal.
Simplify the following exponents and write down the answers with positive exponents. No calculator may be used. Indicate the exponent rule to prove that you did not use a calculator. Accept that the value of all variables are > 0
Definitions: an = a.a.a.a.a.a ... n times a1 = a a0 = 1 1 a-1 = a 00 is undefined
a
n
LO1 & LO2
Exercise 1: Basic laws and definitions
Study Guide G10 ~ Mathematics
5
312
3
-2 2
1 y4
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6. 3 p2 k3 37 p5 k-3 Answer: = 31 + 7 k3 + -3 p 2 + 5 = 38 k0 p7 = 38 (1) p7 = 38p7
2x 6 y4
2 x6
19
More examples: 3(a2.b3)5 = 3.a10 b15 (3a2b3)5 = 35 a10 b15
In (2x3 y-2)2 the 2 is inside the brackets. The 2 should therefore be squared. Therefore, (2x3 y-2)2 = 22x6y-4
In 2(x3 y-2)2 the square at the back of the brackets is only applicable to the bracket and not the 2 in front of the bracket. Do not therefore first multiply by 2 and then square.
Remember:
22
2 1 3
3 37
2 3
2 1
Answer:
2 1 2 3
2 1 3
4.
12 5
5. 2(x .y ) Answer: 2(x3.y-2)2 = 2(x3x2 y-2x2) = 2(x6 y-4) = 2 x6y-4
3
5
312
3 Answer:
3.
Study Guide G10 ~ Mathematics Unit
2