Mathematics on SOH-CAH-TOA

M16 - Scatter Diagrams and Correlation

Sun, 05 Jul 2026 23:20:40 +0700

Scatter Diagrams and Correlation

1. Bivariate Data

Bivariate data consists of pairs of values for two different variables recorded for the same subject.

Example: Height and weight of students.
The goal is to determine if there is a relationship (correlation) between the two variables.

2. Scatter Diagrams

A scatter diagram plots bivariate data on a Cartesian plane.

X-axis: Independent variable.
Y-axis: Dependent variable.
Each pair $(x, y)$ is plotted as a single point.

M15 - Scale drawings, Bearings and Trigonometry

Sun, 05 Jul 2026 23:16:02 +0700

Scale Drawings, Bearings and Trigonometry

1. Scale Drawings

Scale drawings represent real-life objects or areas at a reduced size while maintaining proportions.

Scale Factors

A scale is written as $\text{Drawing Distance} : \text{Real Distance}$ (e.g., $1 : 50,000$).

Finding Real Distance: $\text{Real Distance} = \text{Drawing Distance} \times \text{Scale Factor}$.
Finding Drawing Distance: $\text{Drawing Distance} = \text{Real Distance} \div \text{Scale Factor}$.

2. Bearings

Bearings are used for navigation to specify direction.

M14 - Further Solving of Equations and Inequalities

Sun, 05 Jul 2026 22:58:42 +0700

Further Solving of Equations and Inequalities

1. Simultaneous Equations

Two linear equations with two unknowns (usually $x$ and $y$) that must be solved simultaneously.

Methods of Solution

Elimination: Multiply equations so that coefficients of one variable are the same (or opposites), then add or subtract the equations to eliminate that variable.

Example: Solve $2x + 3y = 13$ and $x - 2y = -4$
1. Multiply the second equation by 2 to match $x$ coefficients $\rightarrow 2x - 4y = -8$
2. Subtract the new equation from the first $\rightarrow (2x + 3y) - (2x - 4y) = 13 - (-8)$
3. Simplify and solve for $y \rightarrow 7y = 21 \rightarrow y = 3$
4. Substitute $y = 3$ back into $x - 2y = -4 \rightarrow x - 2(3) = -4 \rightarrow x = 2$
5. Result: $(2, 3)$
Substitution: Rearrange one equation to express one variable in terms of the other (e.g., $x = \dots$), then substitute this expression into the second equation.

Example: Solve $y = 2x + 1$ and $3x + 2y = 16$
Steps:

M13 - Understanding Measurement

Sun, 05 Jul 2026 22:32:56 +0700

Understanding Measurement

1. Standard Metric Units and Conversions

Metric units follow a base-10 system. Conversion involves multiplying or dividing by powers of 10.

Common Units

Length: millimeter (mm), centimeter (cm), meter (m), kilometer (km)
Mass: milligram (mg), gram (g), kilogram (kg), tonne (t)
Capacity/Volume: milliliter (ml), liter (l), cubic centimeter ($\text{cm}^3$), cubic meter ($\text{m}^3$)

Conversion Factors

$1\text{ cm} = 10\text{ mm}$
$1\text{ m} = 100\text{ cm} = 1000\text{ mm}$
$1\text{ km} = 1000\text{ m}$
$1\text{ kg} = 1000\text{ g}$
$1\text{ l} = 1000\text{ ml} = 1000\text{ cm}^3$

Tip: To convert from a larger unit to a smaller unit, multiply. To convert from a smaller unit to a larger unit, divide.

M12 - Averages and Measures of Spread

Sun, 05 Jul 2026 22:30:44 +0700

Averages and Measures of Spread

1. Types of Averages (Central Tendency)

Averages provide a single value that represents a data set.

Mean

The sum of all values divided by the total number of values.

Non-grouped data: $\text{Mean} = \frac{\sum x}{n}$.
Frequency data: $\text{Mean} = \frac{\sum (f \cdot x)}{\sum f}$ (where $f$ is frequency and $x$ is the value/midpoint).

Median

The middle value when data is ordered from least to greatest.

M11 - Pythagoras Theorem and Similar Shapes

Sun, 05 Jul 2026 22:28:09 +0700

Pythagoras’ Theorem and Similar Shapes

1. Pythagoras’ Theorem

Applies only to right-angled triangles.

The Formula

In a right-angled triangle with legs $a, b$ and hypotenuse $c$ (the side opposite the right angle): $$a^2 + b^2 = c^2$$

Calculations

Finding the Hypotenuse: $c = \sqrt{a^2 + b^2}$.
Finding a Shorter Side: $a = \sqrt{c^2 - b^2}$.

2. Similarity

Two shapes are similar if they have the same shape but different sizes.

M10 - Straight Lines and Quadratic Equations

Sun, 05 Jul 2026 22:05:59 +0700

Straight Lines and Quadratic Equations

1. Straight Line Graphs

Represented on a 2D Cartesian coordinate system $(x, y)$.

The Equation of a Line

General form: $y = mx + c$

$m$: Gradient (slope).
$c$: $y$-intercept (where the line crosses the $y$-axis).

Gradient ($m$)

Gradient measures the steepness: $\text{gradient} = \frac{\text{vertical change}}{\text{horizontal change}} = \frac{\text{rise}}{\text{run}}$.

Calculating Gradient from Two Points: Given $(x_1, y_1)$ and $(x_2, y_2)$: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

M9 - Sequences, Surds and Sets

Sun, 05 Jul 2026 21:13:32 +0700

Sequences, Surds, and Sets

1. Sequences

A sequence is an ordered list of numbers following a specific rule.

Term-to-Term Rules

Describes how to get from one term to the next.

Arithmetic Sequence: Constant difference added or subtracted (e.g., $+3, +3, +3$).
Geometric Sequence: Constant ratio multiplied or divided (e.g., $\times 2, \times 2, \times 2$).

The $n$-th Term ($u_n$)

The formula to find any term in the sequence based on its position $n$.

M8 - Introduction to Probability

Sun, 05 Jul 2026 21:04:08 +0700

Introduction to Probability

1. Basic Probability

Probability Scale

Probability is measured on a scale from $0$ to $1$:

$0$: Impossible event.
$1$: Certain event.
$0.5$: Equally likely as not.

Fundamental Rules

Complementary Events: The probability of an event NOT happening is $P(\text{not } A) = 1 - P(A)$.
Relative Frequency: An estimate of probability based on experimental data. $$\text{Relative Frequency} = \frac{\text{Number of successful trials}}{\text{Total number of trials}}$$
Expected Frequency: The number of times an event is expected to occur over a number of trials. $$\text{Expected Frequency} = P(A) \times \text{Total Trials}$$

2. Sample Space

A Sample Space is a list of all possible outcomes of an experiment.

M7 - Perimeter, Area and Volume

Sun, 05 Jul 2026 20:50:06 +0700

Perimeter, Area, and Volume

1. 2D Shapes

Area and Perimeter

Shape	Perimeter	Area
Rectangle	$2(l + w)$	$l \times w$
Triangle	-	$\frac{1}{2} \times \text{base} \times \text{height}$
Parallelogram	-	$\text{base} \times \text{height}$
Trapezium	-	$\frac{1}{2}(a + b)h$

Circles

Circumference: $C = 2\pi r$ or $\pi d$.
Area: $A = \pi r^2$.
Arc Length: $L = \frac{\theta}{360} \times 2\pi r$.
Sector Area: $A = \frac{\theta}{360} \times \pi r^2$.
- For major sectors, $\theta$ is the reflex angle.

M6 - Equations Factors and Formulae

Sun, 05 Jul 2026 20:38:53 +0700

Equations, Factors, and Formulae

1. Equations

Linear Equations in One Unknown

Solving for a variable by isolating it using inverse operations:

Addition $\leftrightarrow$ Subtraction
Multiplication $\leftrightarrow$ Division
Squaring $\leftrightarrow$ Square Root

Simultaneous Linear Equations

Finding a common solution $(x, y)$ for two linear equations.

Substitution Method: Express one variable in terms of the other from one equation, then substitute into the second.
Elimination Method: Multiply equations to make coefficients of one variable equal (or opposite), then add or subtract the equations to eliminate that variable.

M5 - Fractions Percentages and Standard Form

Sun, 05 Jul 2026 20:04:51 +0700

Fractions, Percentages, and Standard Form

1. Fractions

Definitions

Proper Fraction: Numerator is smaller than denominator (e.g., $1/2, 3/4$).
Improper Fraction: Numerator is greater than or equal to denominator (e.g., $5/4, 7/3$).
Mixed Number: Combination of a whole number and a proper fraction (e.g., $1 \frac{1}{4}$).

Conversions

Mixed Number $\rightarrow$ Improper Fraction: $\text{Whole} \times \text{Denominator} + \text{Numerator} / \text{Denominator}$.
Improper Fraction $\rightarrow$ Mixed Number: Divide numerator by denominator; quotient is whole number, remainder is numerator.

Basic Operations

Addition/Subtraction: Find a common denominator, then add/subtract numerators.
- Example: $\frac{1}{4} + \frac{2}{3} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12}$
- Example: $\frac{3}{4} - \frac{1}{6} = \frac{9}{12} - \frac{2}{12} = \frac{7}{12}$
Multiplication: Multiply numerators together and denominators together.
- Example: $\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$
Division: Multiply by the reciprocal of the divisor (Keep-Change-Flip).
- Example: $\frac{2}{3} \div \frac{5}{7} = \frac{2}{3} \times \frac{7}{5} = \frac{14}{15}$

2. Percentages

Percentage of a Quantity: $\frac{\text{percentage}}{100} \times \text{quantity}$.
Percentage Increase/Decrease:
- $\text{Change} = \frac{\text{percentage}}{100} \times \text{original}$.
- $\text{New Value} = \text{original} \pm \text{change}$.
Simple Interest: $I = P \times R \times T$ (Principal $\times$ Rate $\times$ Time).
Compound Interest: $A = P(1 + \frac{r}{100})^n$ (Total amount after $n$ years).
Reverse Percentages: Finding the original value after a percentage change.
- $\text{Original} = \frac{\text{New Value}}{1 \pm \frac{\text{percentage}}{100}}$.
Repeated Percentage Change: Applying a percentage change multiple times sequentially.
Exponential Growth and Decay:
- Growth: $y = a(1 + r)^x$
- Decay: $y = a(1 - r)^x$
- Where $a$ is initial value, $r$ is rate, and $x$ is time.

M4 - Collection Organising and Displaying Data

Sun, 05 Jul 2026 19:53:29 +0700

Collection, Organising, and Displaying Data

1. Collection and Classifying Data

Data can be categorized into two main types:

Qualitative (Categorical): Non-numerical data.
- Nominal: Categories with no inherent order (e.g., eye color, country).
- Ordinal: Categories with a logical order (e.g., satisfaction rating: low, medium, high).
Quantitative (Numerical): Numerical data.
- Discrete: Values that can only take specific values, often counts (e.g., number of students in a class).
- Continuous: Values that can take any value within a range, often measurements (e.g., height, weight, time).

2. Organising Data

Before displaying data, it must be organized to make analysis easier.

M3 - Lines Angles and Shapes

Sun, 05 Jul 2026 19:43:17 +0700

Lines, Angles, and Shapes

1. Lines and Angles

Basic Definitions

Parallel Lines: Lines in a plane that never meet, no matter how far they extend.
Perpendicular Lines: Lines that intersect at a right angle ($90^\circ$).
Types of Angles:
- Acute: $0^\circ < \theta < 90^\circ$
- Right: $\theta = 90^\circ$
- Obtuse: $90^\circ < \theta < 180^\circ$
- Reflex: $180^\circ < \theta < 360^\circ$

Angle Relationships

Angles on a straight line: Sum to $180^\circ$.
Angles around a point: Sum to $360^\circ$.
Vertically opposite angles: Are equal.
Parallel Line Angles:
- Corresponding Angles: Equal (F-shape).
- Alternate Angles: Equal (Z-shape).
- Co-interior Angles: Sum to $180^\circ$ (C-shape).

M2 - Making Sense of Algebra

Sun, 05 Jul 2026 18:34:52 +0700

Making Sense of Algebra

1. Using Letters for Unknowns

In algebra, letters (variables) are used to represent numbers that are either unknown or can change.

Expression: A mathematical phrase without an equals sign (e.g., $3x + 5$).
Equation: A statement that two expressions are equal (e.g., $3x + 5 = 11$).
Formula: A rule showing the relationship between different variables (e.g., $A = lw$).

2. Substitution

Substitution is the process of replacing variables with specific numerical values to evaluate an expression.

M1 - Review of Number Concepts

Sun, 05 Jul 2026 18:32:04 +0700

Review of Number Concepts

1. Types of Numbers

Classifications

Natural Numbers ($\mathbb{N}$): Counting numbers starting from 1 (1, 2, 3, …).
Integers ($\mathbb{Z}$): Whole numbers, including negatives and zero (…, -2, -1, 0, 1, 2, …).
Prime Numbers: Natural numbers greater than 1 with exactly two factors: 1 and themselves (2, 3, 5, 7, 11, …). Note: 1 is not prime; 2 is the only even prime.
Square Numbers: Result of multiplying an integer by itself ($1, 4, 9, 16, 25, \dots$).
Cube Numbers: Result of multiplying an integer by itself twice ($1, 8, 27, 64, 125, \dots$).
Rational Numbers ($\mathbb{Q}$): Numbers that can be written as a fraction $\frac{a}{b}$ where $a, b$ are integers and $b \neq 0$. Includes terminating and recurring decimals.
Irrational Numbers: Numbers that cannot be written as simple fractions (e.g., $\pi, \sqrt{2}, \sqrt{3}$). Their decimal expansions are non-terminating and non-recurring.
Reciprocals: The reciprocal of a number $x$ is $\frac{1}{x}$. The product of a number and its reciprocal is always 1.