MATHS

1. Number and Arithmetic

1. Whole Numbers & Operations

   • Whole numbers include non-negative integers: $0, 1, 2, 3, \dots$

   • Order of Operations (BIDMAS/PEMDAS):

   • B/P: Brackets / Parentheses

   • I/E: Indices / Exponents

   • D/M: Division and Multiplication (evaluated from left to right)

   • A/S: Addition and Subtraction (evaluated from left to right)

2. Factors, Multiples, and Primes

   • Factors: Whole numbers that divide evenly into another number without a remainder (e.g., factors of $12$ are $1, 2, 3, 4, 6, 12$).

   • Multiples: Products obtained by multiplying a number by integers (e.g., multiples of $5$ are $5, 10, 15, 20, \dots$).

   • Prime Numbers: Numbers strictly greater than $1$ that have exactly two distinct factors: $1$ and the number itself ($2, 3, 5, 7, 11, 13, \dots$). Note that $2$ is the only even prime number.

   • Prime Factorisation: Breaking down a composite number into a product of its prime factors (e.g., $60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5$).

3. Square and Cube Roots

   • Square & Square Root: The square of $x$ is $x^2$. The inverse is the square root: $\sqrt{x}$. Example: $\sqrt{49} = 7$.

   • Cube & Cube Root: The cube of $x$ is $x^3$. The inverse is the cube root: $\sqrt[3]{x}$. Example: $\sqrt[3]{64} = 4$.

2. Integers

1. Definition

   • Integers consist of all positive whole numbers, negative whole numbers, and zero: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$

2. Addition and Subtraction

   • Adding a negative is equivalent to subtraction:     $a + (-b) = a - b$

   • Subtracting a negative is equivalent to addition:     $a - (-b) = a + b$

   • Same signs: Add the absolute values and keep the common sign (e.g., $-4 + (-5) = -9$).

   • Different signs: Subtract the smaller absolute value from the larger absolute value and take the sign of the larger (e.g., $-8 + 3 = -5$).

3. Multiplication and Division

   • Multiplying or dividing two numbers with the same sign yields a positive result:     $(+) \times (+) = (+)$     $(-) \times (-) = (+)$

   • Multiplying or dividing two numbers with different signs yields a negative result:     $(+) \times (-) = (-)$     $(-) \times (+) = (-)$

3. Fractions, Decimals, and Percentages

1. Conversions

   • Fraction to Decimal: Divide the numerator by the denominator (e.g., $\frac{3}{8} = 3 \div 8 = 0.375$).

   • Decimal to Percentage: Multiply by $100\%$ (e.g., $0.375 \times 100\% = 37.5\%$).

   • Percentage to Fraction: Write the percentage over $100$ and simplify to lowest terms (e.g., $45\% = \frac{45}{100} = \frac{9}{20}$).

2. Ordering Values

   • Convert all terms into the same format (typically decimals) to compare and arrange them in ascending or descending order.

3. Arithmetic Operations

   • Addition and Subtraction of Fractions: Find a common denominator before combining numerators:     $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

   • Multiplication of Fractions: Multiply numerators together and denominators together:     $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

   • Division of Fractions: Multiply by the reciprocal of the divisor:     $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

   • Multiplying Decimals: Multiply the values ignoring the decimal points, count the total number of decimal places in both factors, and insert the decimal point that many places from the right in the product.

4. Approximation and Estimation

1. Place Value and Powers of $10$

   • Each digit in a number has a value determined by its position relative to the decimal point: thousands ($10^3$), hundreds ($10^2$), tens ($10^1$), units ($10^0$), tenths ($10^{-1}$), hundredths ($10^{-2}$), thousandths ($10^{-3}$).

   • Multiplying by $10^n$ shifts the decimal point $n$ places to the right.

   • Dividing by $10^n$ shifts the decimal point $n$ places to the left.

2. Rounding Rules

   • Identify the target place value.

   • Check the decider digit immediately to the right:

   • If the decider digit is $5$ or greater ($\ge 5$), increase the target digit by $1$.

   • If the decider digit is $4$ or less ($\le 4$), keep the target digit unchanged.

   • Significant Figures (s.f.): The first non-zero digit from the left is the first significant figure.

3. Estimation

   • Approximate values to $1$ significant figure before performing operations to calculate a quick check of reasonableness.

5. Algebra

1. Algebraic Terms

   • Variable: A letter representing an unknown quantity (e.g., $x$, $y$).

   • Coefficient: A numerical factor multiplying a variable (e.g., in $7x$, the coefficient is $7$).

   • Constant: A fixed numerical value with no variable attached (e.g., $5$).

   • Expression: A collection of terms joined by operators without an equals sign (e.g., $3x + 2y - 4$).

2. Simplifying and Collecting Like Terms

   • Like terms share identical variable parts raised to identical powers.

   • Combine like terms by adding or subtracting their coefficients:     $5x + 3y - 2x + 4y = 3x + 7y$

3. Substitution

   • Replace variables with given numerical values and calculate following the order of operations.

   • If $p = 4$ and $q = -3$, evaluate $2p - 3q$:     $2(4) - 3(-3) = 8 + 9 = 17$

4. Solving 2-Step Equations

   • Use inverse operations in reverse order to isolate the variable:

   • Step 1: Undo addition or subtraction.

   • Step 2: Undo multiplication or division.

   • Example: Solve $4x - 7 = 13$     $4x = 20$     $x = 5$

6. Ratio

1. Concept of Ratio

   • Compares quantities of the same kind measured in identical units ($a : b$).

2. Simplifying Ratios

   • Divide each part by the Highest Common Factor (HCF):     $18 : 24 = \frac{18}{6} : \frac{24}{6} = 3 : 4$

   • Clear fractions or decimals by multiplying all terms by an appropriate common multiple:     $1.2 : 2 = 12 : 20 = 3 : 5$

7. Geometry and Measurements

1. Angle Relationships

   • Intersecting Lines:

   • Vertically opposite angles are equal: $\angle a = \angle b$

   • Angles on a straight line add to $180^\circ$

   • Angles around a point add to $360^\circ$

   • Parallel Lines Cut by a Transversal:

   • Corresponding angles are equal (F-shape).

   • Alternate interior angles are equal (Z-shape).

   • Co-interior angles add to $180^\circ$ (C-shape or U-shape).

   • Polygons Angle Sums:

   • Interior angles of a triangle sum to $180^\circ$

   • Interior angles of a quadrilateral sum to $360^\circ$

2. 2D and 3D Shapes

   • 2D Properties: Quadrilaterals include squares, rectangles, parallelograms, rhombuses, trapezia, and kites.

   • Symmetry: Line symmetry (reflection across an axis) and rotational symmetry (order of times a shape looks identical during a full $360^\circ$ turn).

   • 3D Properties: Prisms, cubes, and cuboids defined by faces ($F$), vertices ($V$), and edges ($E$).

   • Nets: A two-dimensional unfolded pattern that can be folded to form a specific three-dimensional solid.

3. Perimeter and Area

   • Perimeter ($P$): Total distance around the outer boundary.

   • Area of a Triangle:     $A = \frac{1}{2} \times b \times h$

   • Area of a Compound Shape: Split the compound shape into non-overlapping standard shapes (e.g., rectangles, triangles), determine individual areas, and calculate their sum.

4. Volume

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   • Cube:     $V = s^3$

   • Cuboid:     $V = l \times w \times h$

8. Coordinates and Graphs

1. The Cartesian Plane

   • Defined by two perpendicular axes intersecting at the origin $(0, 0)$: horizontal $x$-axis and vertical $y$-axis.

   • Coordinates are written as an ordered pair $(x, y)$.

   • Four quadrants:

   • Quadrant I: $(+, +)$

   • Quadrant II: $(-, +)$

   • Quadrant III: $(-, -)$

   • Quadrant IV: $(+, -)$

2. Tables of Values

   • A method to generate coordinate pairs from an algebraic rule.

   • Choose integer values for $x$, substitute into the relation, compute the resulting $y$-values, and plot the points $(x, y)$.

3. Linear Functions

   • The general gradient-intercept equation of a straight line is:     $y = mx + c$

   • $y$ -intercept ($c$): The point where the line crosses the $y$-axis at $(0, c)$.