Secondary School Mathematics Material (All Level Compilation) Version 20260224

Title Secondary School Elementary Mathematics Materials Compilation Series (20260224A) Author AprilDolphin – Lim Wang Sheng Date 24/02/2026 Secondary Level Topic Page 1 Basic Algebra – Basic Algebraic Manipulation 2 1 Basic Algebra – Formula Manipulation 12 1 Basic Algebra – Simple Inequalities 18 1 Factors and Multiples 25 1 Percentages 31 1 Coordinate Geometry – Basics 39 1 Coordinate Geometry – Straight Lines 44 2 Direct and Inverse Proportion 47 2 Maps and Scales 50 2 Expansion & Factorisation of Algebraic Expression 54 2 Quadratic Equations and Functions 58 2 Simultaneous Equations (Linear) 70 2 Basic Statistics – Ungrouped Data 75 2 Basic Statistics – Grouped Data 81 2 Shapes of Graphs of Functions 89 3 Applications of Quadratic Equations 100 3 Laws of Indices 103 3 Set Language & Venn Diagram 107 3 Trigonometry, Congruency & Similarity of Triangles 117 3 Matrices 124 4 Financial Arithmetic – Money Exchange 130 4 Financial Arithmetic – Simple & Compound Interest 133 4 Financial Arithmetic – Hire Purchase 136 4 Vectors in Two Dimensions 138 This document is to be released into CC0 Public Domain Dedication by Lim Wang Sheng and all the other authors and editors upon creation.

Title Basic Algebra – Algebraic Manipulation Author AprilDolphin – Lim Wang Sheng Date 2/2/2026 Algebra is basically • Representing known or unknown quantities and numbers with letters and symbols • Performing mathematical operations using letters and symbols, together with numbers Without algebra • Textbook would be ridiculously thick and heavy • Certain statistical analysis will be virtually impossible to conduct as the person conducting the analysis will need to make endless amount of mathematical explanation • Certain fields of Mathematics, dependent on algebra, cannot exist as well Tips to students who are new to basic algebra • Please name your unknowns systematically, so that you won’t be confused. • Never use letters that can be easily confused for something else as shown in the below table (Unless the question specified them as part of the question or insist that you must use them): Common Letters to try Avoid Using Reasoning/Explanation 𝑄 (Upper Case Only) Upper case letter 𝑄 can be confused with “𝑂” or Zero (0) 𝑂, 𝑜 (Lower Case and Upper Case) Upper case letter 𝑂 can be confused with zero (0). Lower case letter 𝑜 can be confused with zero (0). 𝐽, 𝑗 (Lower Case and Upper case) Upper case letter 𝐽 can be confused with 𝐿 and 𝐼 Lower case letter 𝑗 can be confused with 𝑖 𝑙 (Lower Case only) Lower case letter 𝑙 can be confused with 𝑖 or one (1) 𝐼, 𝑖 (Lower case and upper case) Upper case letter 𝐼 can be confused with lower case letter 𝑙 Lower case letter 𝑖 is typically reserved for a specific use at higher level Mathematics

𝑒 (Lower Case Only) Lower case 𝑒 is typically reserved for a specific use at higher level Mathematics 𝑆, 𝑠 (Upper and Lower case) Upper case letter 𝑆 can be confused with number five (5) Lower case letter 𝑠 can be confused with the number five (5) Basic Operands in Algebra Expression English Meaning 𝑥 + 4 Add 4 to 𝑥 𝑥 − 𝑦 Subtract 𝑦 from 𝑥 6𝑚 Multiply 𝑚 by 6 𝑚 𝑛 Divide 𝑚 by 𝑛 Basic Operands in Algebra – Index Notation Expression English Meaning 𝑘5 = 𝑘 × 𝑘 × 𝑘 × 𝑘 × 𝑘 Write 𝑘 five times in a row and multiply all the written values of 𝑘 together 𝑚−4 = 1 𝑚 × 𝑚 × 𝑚 × 𝑚 1 divided by 𝑚4, where 𝑚4 is 𝑚 written four times in a row and multiplied together

Example 1. Evaluate the following expression 𝑥 + 7 + 3𝑥 + 4 = 𝑥 + 3𝑥 + 7 + 4 4𝑥 + 11 Let’s convert it to models and see how they correlate 𝑥 4 3𝑥 7 Sum up all the values represented by the models and you should get the same expression How algebra is translated into models in Example 1. 1. 1.There are four boxes of the unknown 𝑥, which equates to 4𝑥 when summed up. 2. 2.There is one box of 7 and as well as one box of 4, hence adding up everything above, we get a total value of 4𝑥 + 11

The above also, in a way, demonstrate the concept of “like” and “unlike” terms. In any algebraic addition and subtraction, only “like terms” can be added or subtracted in this way. 2𝑥 + 3𝑥 = 5𝑥 5𝑥 − 4𝑥 = 𝑥 Arithmetic demonstration of concept: 2𝑥 + 3𝑥 = 5𝑥 Since 2𝑥 = 2 × 𝑥, We can pretend the value of 𝑥 is 5. In this case: 2𝑥 = 2 × 5 and 3𝑥 = 3 × 5 10 + 15 = 25 The other way is true as well and valid: 2𝑥 + 3𝑥 = 5𝑥 = 5 × 5 = 25

More Complicated Subtraction and Addition and Demonstration by Purely Algebra Approach and How to Check Your Answers. Example 2: 3𝑥 + 5 − 3𝑥 + 4 = 3𝑥 − 3𝑥 + 5 + 4 (Group “Like Terms” together) = 0 + 9 = 9 (Final Answer) To verify your answer is correct, what you need to do. Pretend 𝑥 = 2 (3 × 2) + 5 − (3 × 2) + 4 = 6 + 5 − 6 + 4 11 − 6 + 4 = 5 + 4 = 9 Example 3: 15𝑥𝑦 − 5 + 2 − 13𝑥𝑦 = 15𝑥𝑦 − 13𝑥𝑦 − 5 + 2 (Group “Like Terms” Together) = 2𝑥𝑦 − 5 + 2 = 2𝑥𝑦 − 3 (Final Answer) To verify the correctness of your answer, you need to Pretend 𝑥 = 2 and 𝑦 = 3 in your original question. (15 × 2 × 3) − 5 + 2 − (13 × 2 × 3) (90) − 5 + 2 − (78) = 90 − 5 + 2 − 78 = 9 Then pretend 𝑥 = 2 and 𝑦 = 3 in your answer. (2 × 2 × 3) − 3 = 12 − 3 = 9 Some teachers also call this type of answer checking as “Checking Answers by Value Substitution” as you are indeed “Substituting the Value” of the letters by another value.

Problem Solving (Addition and Subtraction of Algebra): To solve a problem sum using algebra, while it varies slightly in different questions, we observed the following steps are necessary. 1. Find out what are the unknowns, needed to solve the question. 2. Define the known and unknown variable properly using algebra letters and numbers. 3. Construct the equation or expression needed to solve the question. 4. Solve the equation and find the value of unknown variable. The easiest example I’ve seen on books that can demonstrate this: Example 4: The sum of three consecutive numbers is 84. What is the smallest of the three numbers. 1 Find out the unknown. The smallest number out of the three consecutive numbers. 2 Define the unknown and known properly In this case, the unknown value (smallest number) is defined as 𝑥. Since they said the numbers are in consecutive order, we can define the subsequent two numbers as 𝑥 + 1 and 𝑥 + 2 3 Construct an Equation Sum means, add the three numbers together, so we have: 𝑥 + (𝑥 + 1) + (𝑥 + 2) = 84 4 Solve the Equation and find out what is the unknown value, in this case, it is 𝑥. 3𝑥 + 1 + 2 = 84 3𝑥 + 3 = 84 3𝑥 = 84 − 3 3𝑥 = 81 𝑥 = 27 [Final Answer] Thus, the smallest number is 27.

Performing Algebra Addition and Subtraction (Using Brackets) Demonstration and Concept: The below two example seems extremely straightforward: 𝑎 + 𝑏 = 𝑎 + 𝑏 𝑎 − 𝑏 = 𝑎 − 𝑏 It is the below few example students often get confused about and are careless enough to lose a total of 7 marks in exams. 𝑎 + (−𝑏) = 𝑎 − 𝑏 𝑎 − (−𝑏) = 𝑎 + 𝑏 (−𝑎) + 𝑏 = 𝑏 − 𝑎 (−𝑎) + (−𝑏) = (−𝑏) − 𝑎 Performing Algebra Multiplication and Division (Using Brackets) Demonstration and Concept: The below two examples are straightforward. 𝑎(𝑏) = 𝑎𝑏 𝑎 ÷ 𝑏 = 𝑎 𝑏 Once again students can get really confused when negative signs are involved. 𝑎(−𝑏) = −𝑎𝑏 −𝑎(𝑏) = −𝑎𝑏 (−𝑎)(−𝑏) = 𝑎𝑏 𝑎 ÷ (−𝑏) = − 𝑎 𝑏 (−𝑎) ÷ 𝑏 = − 𝑎 𝑏 (−𝑎) ÷ (−𝑏) = 𝑎 𝑏

Example 5 Simplify −2(3𝑥 − 4 + 6𝑥) We must take note of 2 things in the above example, firstly, the bracket needs to be removed, secondly, that the exp