Contents
1 Numbers and Four Operations of Integers ... 1
2 Multiples and Factors ... 7
3 Fractions and Decimals ... 13
4 Percentages ... 20
5 Simple Equations ... 27
6 Perimeters and Areas ... 34
7 Volumes ... 41
8 Statistics ... 48
9 Directed Numbers ... 57
10 Introduction to Algebra ... 64
11 Introduction to Geometry ... 71
12 Introduction to Rectangular Coordinate System ... 79
Revision Test ... 85
Preparing for the New School Year ... 91
... 94
... 95
Common Question Commands in Mathematics ... 96
Sample
Daily Life Mathematics
Peter is playing a mathematical game. There are some numbers written on a paper card as shown in the figure.
He has to add brackets and the signs +, –, × and ÷ to form a mathematical expression with an answer of 1. The order of the numbers cannot be interchanged and he does not need to use the brackets and all the signs.
Peter suggests that there is only 1 solution as shown below:
(5 – 4) × (3 – 2) × 1
Do you agree with him? If not, how many solutions can you find?
Knowledge Reminder
1. Numbers
(a) Natural numbers: All positive integers (b) Whole numbers: All natural numbers and 0
(c) Even numbers: All whole numbers which are divisible by 2 (d) Odd numbers: All whole numbers which are not divisible by 2
(e) Prime numbers: Any natural number greater than 1 which is divisible by 1 and itself only (f) Composite numbers: Any natural number which is divisible by some numbers other than
1 and itself 2. Basic Operations
(a) Addition: When two numbers are added together, the result is called the ‘sum’ of the two
numbers.
(b) Subtraction: When one number is subtracted from another, the result is called the
‘difference’ of the two numbers.
(c) Multiplication: When a number is multiplied by another, the result is called the ‘product’
of the two numbers.
(d) Division: When a number is divided by another, the result is called the ‘quotient’ and the remaining part (if the number cannot be divided completely) is called the
‘remainder’.
Unit 1
Numbers and Four Operations of Integers
5 4 3 2 1
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Pan Lloyds Mathematics Study Pack — Summer Revision (Latest Upgraded Edition) P.6 – S.1
3. Order of Operations
(a) Multiplication and division must be completed before addition or subtraction.
(b) If there are brackets in an expression, complete the parts in the brackets first.
Exercise
Level 1
1. Classify each of the following numbers into prime number or composite number.
2, 5, 8, 9, 12, 15, 17, 22, 29 Prime numbers: ____________________________
Composite numbers: _______________________________
C oncept Builder
C oncept Builder
Determine whether each of the following statements is correct or not.
1. Multiplication should be always performed before division. Correct Incorrect
2. 100 ÷ 2 × 5 Correct Incorrect
= 100 ÷ 10 = 10
3. (2 + 3)× 10 ÷ 5 Correct Incorrect
= 5 × 2 = 10
4. The smallest prime number is 2. Correct Incorrect
5. A natural number that is divisible by more than 2 numbers is not a Correct Incorrect prime number.
Common Mistakes
Calculate 12 × (4 – 3) × 9.
Solution:
12 × (4 – 3) × 9 = 12 × 4 – 3 × 9 = 48 – 27
= 21
i The operations in brackets should be completed first, that is,
12 × (4 – 3) × 9 = 12 × (1) × 9
= 108
Sample
2. Write down all the prime numbers between 1 and 50.
2, ___, ___, 7, 11, ___, ___, 19, ___, ___, 31, ___, ___, ___, 47
Calculate each of the following expressions (3 – 14).
3. 3 + 5 – 2 4. 8 ÷ 4 × 7 5. 60 – 34 + 25
6. 16 + 7 × 10 7. 35 × 8 – 9 8. 5 + 39 ÷ 13
9. (12 + 15) × 3 10. (99 – 51) ÷ 6 11. 24 − (40 − 27)
12. 6 ÷ (8 ÷ 4) 13. 7 ÷ 10 × 20 14. 32 – 80 + 49
Level 2
15. Write down all the prime numbers from 51 to 100.
16. Determine if 361 is a prime number.
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Pan Lloyds Mathematics Study Pack — Summer Revision (Latest Upgraded Edition) P.6 – S.1
Calculate each of the following expressions (17 – 22).
17. 10 × 2 – 60 + 2 × 25 18. 5 × (2 − 8 + 9) × 6
19. (4 + 12) + 50 ÷ (12 – 7) 20. 84 ÷ [3 × (19 – 15)]
21. (42 ÷ 6) × (70 ÷ 14 + 3) 22. 12 ÷ 3 − (25 ÷ 5) × 2 + 11
23. Calculate 1 – 2 + 3 – 4 + ... – 50 + 51.
24. Find the units digit of the value of the expression
(5 × 2) + (5 × 4) + (5 × 6) + (5 × 8) + ... + (5 × 998). Briefly explain your answer.
Hint
Consider the value of the expression in each pair of brackets.
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Sample
25. Find the units digit of the value of the expression 1 × 2 × 3 × ... × 2012 × 2013.
Briefly explain your answer.
26. Which of the following is NOT a prime number?
A. 1 B. 7 C. 53 D. 89
27. Which of the following is NOT a composite number?
A. 295 B. 183 C. 76 D. 2
28. Which of the following is a prime number?
A. 111 B. 113 C. 115 D. 117
29. 6 × (15 – 7) ÷ 4 = A. 3
B. 12 C. 24 D. 32
Multiple-choice Questions
30. (91 ÷ 13 + 2) × 3= A. 3
B. 9 C. 27 D. 81
31. [4 + (5 – 2 × 2) – 9 ÷ 3] × 2= A. 1
B. 2 C. 3 D. 4
32. Which of the following has the same value as 7 × 8 – 5 × 2 + 6?
A. 35 – 16 + 6 B. 7 × 3 × 2 + 6 C. 56 – 10 + 6 D. 56 – 5 × 8 33. Calculate
1 × 2 × 3 × ... × 99 ÷ (6 × 7 × 8 × ... × 98).
A. 113 B. 120 C. 490 D. 11 880 Hint
Consider the product of the first 5 integers.
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Pan Lloyds Mathematics Study Pack — Summer Revision (Latest Upgraded Edition) P.6 – S.1
G lossary G lossary
addition 加法 composite number 合成數 difference 差
division 除法 even number 偶數 multiplication 乘法
natural number 自然數 odd number 奇數 prime number 質數
product 積 quotient 商 remainder 餘數
subtraction 減法 sum 和 whole number 完整數
Enrichment Corner
Referring to the situation in Daily Life Mathematics, actually, there are many answers for the question, for example:
(5 − 4) + (3 − 2) − 1 = 1 (5 − 4) ÷ (3 − 2) ÷ 1 = 1 (5 − 4 + 3) ÷ 2 − 1 = 1
If brackets and all the signs (+, –, × and ÷) must be used in the expression, can you find an answer for the question? (Hint: You can use more than 1 pair of brackets.)
Open-ended Question
Write down three expressions that each involves at least two kinds of operations and has an answer of 20.
For example, (2 + 3) × 4 = 20.
M ake Yourself Smarter
M ake Yourself Smarter
Determine the units digit of the value of the expression
1 × 9 + 2 × 8 + 3 × 7 + … + 9 × 1 –––––– (*).
Hence, find the units digit of the value of the expression
1 × 99 + 2 × 98 + 3 × 97 + … + 99 × 1 –––––– (**).
Briefly explain all your answers.
