Chapter 1 - Number Sequences and Integers
1.1 Number Patterns and Sequences 1.2 Integers 1.3 Addition and Subtraction of Integers 1.4 Multiplication and Division of Integers 1.5 Combined Operations of Integers Mastery Practice Chapter 2 - Fractions
2.1 Comparing Fraction 2.2 Addition and Subtraction of Fractions 2.3 Multiplication and Division of Fractions 2.4 Combined Operations of Fractions Mastery Practice Chapter 3 - Decimals
3.1 Comparing Fraction 3.2 Addition and Subtraction of Fractions 3.3 Multiplication and Division of Fractions 3.4 Combined Operations of Fractions Mastery Practice Chapter 4 - Indices
4.1 Indices 4.2 Multiplication of Numbers in Index Notation 4.3 Division of Numbers in Index Notations 4.4 Raising Numbers and Algebraic Terms in Index Notation to a Power 4.5 Negative Integral Indices 4.6 Fractional Indices 4.7 Computation Involving Laws of Indices Mastery Practice
1
2 5 12 18 22 28 31
32 34 43 50 55 59
60 60 63 70 74 76
77 79 81 82 85 88 92 96
Chapter 5 - Exponential Notation
97
5.1 Exponential Notation 5.2 Addition and Substraction in Exponential Notation 5.3 Multiplication and Division in Exponential Notation 5.4 Combined Operations Using Exponential Notation Mastery Practice
98 100 103 106 109
Chapter 6 - Geometrical Constructions
111
6.1 Constructions Mastery Practice
112 130
Chapter 7 - Solid Geometry
7.1 Cubes and Cuboids 7.2 Plan, Front Elevation and Side Elevation of 3-D Geometrical Shapes Mastery Practice Chapter 8 - Linear Equations
135 138 151
8.1 Equality 8.2 Linear Equations in One Unknown 8.3 Solutions of Linear Equations in One Unknown Mastery Practice
9.1 Relations 9.2 Coordinates 9.3 Scales of the Coordinate Axes 9.4 Line Graphs Mastery Practice
10.1 Probability Mastery Practice
155
156 158 162 171
Chapter 9 - Relations, Coordinates and Lines Graphs
Chapter 10 - Probability
134
174
175 181 186 194 199
203
204 208
Number Sequences and Integers
1 1. Find the next number in each number sequence. (a) 32, 30, 28, 26, … (b) 8, 23, 38, 53, … (c) 13, 26, 52, 104, … (d) 800, 400, 200, 100, … 2. Arrange these numbers. (a) 372, 327, 237, 273 (in ascending order) (b) 658, 568, 668, 865 (in descending order) 3. Calculate the following. (a) 434 + 635 + 12 = (b) 143 – 31 – 69 = (c) 812 ÷ 14 = (d) 567 × 32 = 4 Evaluate (a) 40 ÷ 10 × 2 + 5 = (b) 214 × 2 – (676 ÷ 26) = (c) 516 + 310 – 759 = (d) 608 ÷ 16 – 812 ÷ 28 =
By the end of this chapter, you should be able to analyse and explain relations of a given pattern. specify or give examples and compare added integral numbers, subtracted integral numbers and 0. add, subtract, multiple and divide integral numbers for the purpose of problem-solving; be aware of validity of the answers. explain the results obtained from the addition, subtraction, multiplication and division, and explain the relationship between addition and subtraction and between multiplication and division of integral numbers. apply knowledge and properties of integral numbers for problem-solving. use estimation appropriately in various situations, as well as for considering validity of answers reached through calculation. Math Online Visit these websites to know more about this chapter: http://www.mathsisfun.com/numberpatterns. html
Chapter 1 Number Sequences and Integers
1
4. (a) 13
(b) 402 (c) 67
3. (a) 1081 (c) 58
(d) 9
(b) 43 (d) 18144
2. (a) 237, 273, 327, 372 (b) 865, 668, 658, 568 1. (a) 24 Answers:
(b) 68
(c) 208 (d) 50
1.1
Number Patterns and Sequences
A Recognising some simple number patterns A sequence is a set of numbers written in an order according to a certain pattern or rule. The pattern of a number sequence is the method of obtaining numbers in the number sequence. The numbers in a sequence are called terms. The even numbers make a sequence. 2 4 6 +2
+2
8
+2
+2
The first term in the above sequence is 2. To get the next term, we add 2 to its previous term. The odd numbers make a sequence by adding 2 every time. 1 3 5 +2
+2
7
+2
+2
The square numbers are 12, 22, 32, 42, … = 1, 4, 9, 16 … The cube numbers are 13, 23, 33, 43, … = 1, 8, 27, 64, …
The triangular numbers are
1 3 6 +2
+3
+4
10 +5
The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, …. To get the next term, we add the last two terms. 1+1=2 1+2=3 2 + 3 = 5 3 + 5 = 8 5 + 8 = 13
2
Mathematics Mathayom 1
1
Describe each of the following number sequences. List the 6th term of each sequence. (a) 113, 115, 117, 119, … (b) 27, 64, 125, 216, … (c) 25, 36, 49, 64, …
(a) 113, 115, 117, 119, 121, 123 +2 +2 +2 +2 +2
Even numbers. The 6th term is 123.
(b) 27, 64, 125, 216, 343, 512 ↓ ↓ ↓ ↓ ↓ ↓ 33 43 53 63 73 83 Cube numbers. The 6th term is 512. (c) 25, 36, 49, 64, ↓ ↓ ↓ ↓ 52 62 72 82 Square numbers. The
81, 100 ↓ ↓ 92 102 6th term is 100
Try Question 1 in Test Yourself 1.1
B Recognising number pattern Consider the number sequence below. 2, 5, 8, 11, 14, …
We can get the next term by adding 3 to its previous term. 2, 5, 8, 11, 14, …
+3 +3 +3 +3
1st term, T1 = 2 2nd term, T2 = T1 + 3 = 2 + 3 3rd term, T3 = T2 + 3 = 2 + 3 + 3 = 2 + (2 × 3) 4th term, T4 = T3 + 3 = 2 + (3 × 3) nth term, Tn = 2 + [(n – 1) × 3] The number pattern for this number sequence is Tn = 2 + [(n – 1) × 3]
Chapter 1 Number Sequences and Integers
3
15
Simplify each of the following. (a) 4 – (–3) (b) – 6 – 2 (a) 4 – (–3) = 7
–3 –2 –1
+7 0
(b) – 6 – 2 = –8
–6
–5 –4
1
2
3
4
–1
0
1
2
Simplifying addition and subtraction of integers: • a + (+b) = a + b • a + (–b) = a – b • a – (+b) = a – b • a – (–b) = a + b
Move 7 steps to the right.
–8
–3
–2
Move 8 steps to the left.
16
–17 + (– 4) = –17 + (– 4) = –17 – 4 = –21
+ (– 4) = – 4
Try Question 7 in Test Yourself 1.3
To perform a subtraction involving three integers, always work out from left to right. 17
Simplify 8 – (– 4) – 3. 8 – (– 4) – 3 = 8 + 4 – 3 = 12 – 3 =9
Work out from left to right.
Try Question 8 in Test Yourself 1.3
Chapter 1 Number Sequences and Integers
15
13
In the morning, the temperature of a city was –3°C. Its temperature then dropped by 5°C in the afternoon. At night, its temperature dropped by another 4°C. Find the temperature of the city at night. –3 – 5 – 4 = –8 – 4 = –12 –8 –3
–1
1
–12 3
–8
5
–4
0
4
Therefore, the temperature of the city at night was –12°C. Try Questions 9 – 11 in Test Yourself 1.3
1.3 1. Calculate each of the following. (a) –4 + 7 (b) –9 + 3 (c) 5 + (–13) (d) –7 + (–2) (e) 6 + (–6) (f) –4 + (–8) 2. Simplify each of the following. (a) 5 + (–7) + 4 (b) 3 + 6 + (–10) (c) –7 + 1 + 2 (d) –4 + 9 + (–5) (e) –6 + (–4) + (–3) (f) –8 + (–7) + 10 3. A submarine was 40 m below sea level. Three hours later, it rose by 15 m. What was the new position of the submarine? 4. The temperature of a town was –3°C in the morning. Its temperature rose by 7°C at noon. What was the temperature of the town at noon?
16
Mathematics Mathayom 1
Example 1 Arrange –5, 4, –2, 3 and 1 in increasing order.
Increasing order: 1, –2, 3, 4, –5
–5
• Did not consider the negative sign. • Positive integers are always greater than negative integers.
–4
–3
–2
–1
0
1
2
3
4
Increasing order: –5, –2, 1, 3, 4
Example 2 Simplify 3 + (– 4).
–7
–4
–3
–2
–1
–4 0
1
–1
3
2
Therefore, 3 + (– 4) = –7.
0
1
2
3
Therefore, 3 + (– 4) = –1.
Start from 3, move 4 steps to the left.
Finding the difference between the two integers.
addition arranging combined operation decreasing order division greater than horizontal increasing order integer largest
26
Mathematics Mathayom 1
less than like sign missing terms multiplication negative direction negative integer negative number negative sign number line positive direction
positive integer positive number positive sign product quotient sequence smallest subtraction unlike sign vertical
1. A sequence is a set of numbers written in an order that follows certain rule or pattern. 2. A number pattern is the method of obtaining numbers in a number sequence. 3. An integer is a whole number that has a positive sign (+) or a negative sign (–), including zero. 4. A positive integer is a whole number with a positive sign or without any sign. For examples, +3, +5, 9, 24. 5. A negative integer is a whole number with a negative sign. For examples, –2, – 4, –50. 6. Integers can be represented using a number line. For examples, Decreasing –4
–3
–2
–1
Increasing 0
1
2
3
4
7. Positive and negative numbers are frequently used in real life situations involving (a) an increase in value or a decrease in value, (b) values greater than zero or less than zero, (c) opposite direction. 8. Addition of integers is a process of finding the sum of two or more integers. For example, 2 + (–5) = –3
Start from 2, move 5 steps to the left.
–5 –2
–3
0
–1
2
1
9. Subtraction of integers is a process of finding the difference between two integers. Move 7 For example, 3 – (– 4) = 7 steps to the right.
+7
–4
–3
–2
–1
0
1
2
3
10. Simplifying addition and subtraction of integers: • a + (+b) = a + b • a + (–b) = a – b • a – (+b) = a – b • a – (–b) = a + b 11. The product/quotient of two integers with the same sign is a positive integer. The product/quotient of two integers with d ifferent signs is a negative integer. • (+) × (+) = (+) • (+) ÷ (+) = (+) • (–) × (–) = (+) • (–) ÷ (–) = (+) • (+) × (–) = (–) • (+) ÷ (–) = (–) • (–) × (+) = (–) • (–) ÷ (+) = (–) 12. To perform computations involving combined operations, • calculate within the brackets first. • f o l l o w e d b y d i v i s i o n o r multiplication. • then addition or subtraction, from left to right.
Chapter 1 Number Sequences and Integers
27
Objective Questions 1. For the number sequence 4, 5, 7, 10, 14, , …, the missing number in the box is A 19 B 17
C 16 D 15
2. Given the number sequence 3, 5, 9, 15, 23, x, …, the value of x is A 28 C 35 B 33 D 41 3. If a number sequence follows the pattern of Tn = 2n + 1, what is the 21st term? A 43 C 20 B 40 D 21
2, 6, 10, 14, 18, …
4. What is the pattern of the above number sequence? A Tn = 2n + 2 B Tn = 4n – 2 C Tn = 2n – 2 D Tn = 4n + 2 5. –34 – (–16) – (–7) = A –57 B –43 C –25 D –11 6. How many negative integers are there between –5 and 2? A 6 C 4 B 5 D 3
x
–12
–6
0
y
7. The above diagram shows a number line. The value of x + y is A –21 C –15 B –18 D –9 8. 21 + (–7) + (–12) = A –40 C 2 B –16 D 8
28
Mathematics Mathayom 1
9. Which of the following is an integer? A 0 C 0.5 1 B D 1 2 2 3 10. The integers between –3 and 1 are A –2, –1, 0 B –2, –1, 0, 1 C –3, –2, –1, 0 D –3, –2, –1, 0, 1 11. If –7 m represents 7 m to the north, 5 m represents A 5 m to the north B 5 m to the south C 5 m to the east D 5 m to the west 12. –13 – (–8) = A 21 B 5 C –5 D –21 13. –20 – (–7) – 3 = A –10 B –16 C –24 D –30
–17
q
p
–15
14. The above diagram shows a number line. The value of p – q is A –24 C 2 B –2 D 24 –4 –3 –2 –1
0
1
2
15. The above number line shows A 3 – 4 – 1 B 3 + (– 4) + (–1) C 3 + (– 4) + 3 D 3 + (–7) + 3
3