GCSE Mathematics Revision Topics

Four Operations with Fractions

• Addition and Subtraction of Fractions:     * To add or subtract fractions, they must have a common denominator. This is typically achieved by finding the Lowest Common Multiple (LCM) of the denominators.     * General Formula: $\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$.     * Step-by-step:         1. Find a common denominator.         2. Adjust the numerators proportionally.         3. Add or subtract the numerators while keeping the denominator the same.         4. Simplify the resulting fraction if possible.

• Multiplication of Fractions:     * This is the most straightforward operation. Simply multiply the numerators together and the denominators together.     * General Formula: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$.     * It is often easier to simplify by cancelling common factors before performing the multiplication.

• Division of Fractions:     * To divide by a fraction, you must multiply by its reciprocal (invert the second fraction).     * General Formula: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$.     * Mnemonic: "Keep, Change, Flip" (Keep the first fraction, change division to multiplication, flip the second fraction).

Laws of Indices

• Multiplication Law: When multiplying terms with the same base, you add the indices.     * Formula: $a^m \times a^n = a^{m+n}$.

• Division Law: When dividing terms with the same base, you subtract the indices.     * Formula: $a^m \div a^n = a^{m-n}$.

• Power of a Power Law: When raising a power to another power, you multiply the indices.     * Formula: $(a^m)^n = a^{m \times n}$.

• Zero Index: Any non-zero base raised to the power of zero is equal to one.     * Formula: $a^0 = 1$.

• Negative Indices: A negative index indicates the reciprocal of the base raised to the positive power.     * Formula: $a^{-n} = \frac{1}{a^n}$.

• Fractional Indices (Roots and Powers):     * The denominator of the fraction indicates the root, and the numerator indicates the power.     * Formula for Unit Fractions: $a^{\frac{1}{n}} = \sqrt[n]{a}$.     * General Formula: $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$ or $\sqrt[n]{a^m}$.

Substitution

• Definition: Substitution is the process of replacing algebraic variables (letters) with specific numerical values to evaluate an expression.
• Procedure:     1. Replace every occurrence of the variable with the given number, ideally using parentheses to manage signs.     2. Follow the order of operations ($\text{BODMAS/BIDMAS}$).
• Example Case: If $x = 3$ and $y = -2$, evaluate $5x + y^2$.     * Calculation: $5(3) + (-2)^2 = 15 + 4 = 19$.

Angles in Shapes

• Triangles: The sum of interior angles in any triangle is always $180^{\circ}$.
• Quadrilaterals: The sum of interior angles in any quadrilateral is always $360^{\circ}$.
• General Polygons:     * Sum of Interior Angles: For a polygon with $n$ sides, the sum is given by $(n-2) \times 180^{\circ}$.     * Sum of Exterior Angles: The sum of exterior angles for any convex polygon is always $360^{\circ}$.     * Regular Polygons: In a regular polygon where all sides and angles are equal, a single exterior angle is calculated as $\frac{360^{\circ}}{n}$. A single interior angle is $180^{\circ} - \text{exterior angle}$.

Percentages - All Topics

• Percentage of an Amount: To find $x\%$ of $y$, calculate $\frac{x}{100} \times y$.
• Percentage Change: Used to find the increase or decrease relative to the original value.     * Formula: $\text{Percentage Change} = \frac{\text{Change}}{\text{Original Value}} \times 100$.
• Reverse Percentages: Finding the original value after a percentage increase or decrease has occurred.     * Method: Use a multiplier. If a value increased by $20\%$, the multiplier is $1.20$. If it decreased by $20\%$, the multiplier is $0.80$.     * Calculation: $\text{Original Value} = \frac{\text{New Value}}{\text{Multiplier}}$.
• Compound Interest: Calculating interest on both the initial principal and the accumulated interest from previous periods.     * Formula: $A = P(1 + \frac{r}{100})^n$, where $P$ is principal, $r$ is rate, and $n$ is number of time periods.

Transformations

• Translation: Sliding a shape without rotating or resizing it. Defined by a translation vector $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$, where $x$ is horizontal movement and $y$ is vertical movement.
• Reflection: Flipping a shape over a mirror line (line of reflection). Each point of the image is the same distance from the line as the original, but on the opposite side.
• Rotation: Turning a shape around a fixed point called the center of rotation. Requires three pieces of information: the center of rotation (coordinates), the angle (e.g., $90^{\circ}$, $180^{\circ}$), and the direction (clockwise or anticlockwise).
• Enlargement: Resizing a shape by a scale factor from a center of enlargement.     * Scale factor $k > 1$: The shape gets larger.     * Scale factor $0 < k < 1$: The shape gets smaller.     * Negative scale factor: The shape is enlarged and inverted on the opposite side of the center point.

Solving Quadratics

• Factorization: Expressing the quadratic expression $ax^2 + bx + c = 0$ as a product of two linear binomials.
• Quadratic Formula: Used when factorization is difficult or impossible.     * Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Completing the Square: Transforming the equation into the form $(x + p)^2 + q = 0$.
• Graphical Solution: Finding the points where the curve $y = ax^2 + bx + c$ intersects the x-axis ($y = 0$).

Equation of a Line

• Slope-Intercept Form: $y = mx + c$, where $m$ is the gradient (slope) and $c$ is the y-intercept.
• Calculating the Gradient: Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the gradient is:     * $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• Parallel Lines: Two lines are parallel if they have the same gradient ($m_1 = m_2$).
• Perpendicular Lines: Two lines are perpendicular if the product of their gradients is $-1$ ($m_1 \times m_2 = -1$).

Solving Equations with Unknowns on Both Sides

• Process: The goal is to isolate the variable on one side of the equation.     1. Collect all terms containing the unknown on one side (e.g., subtract the smaller coefficient term from both sides).     2. Collect all constant terms on the opposite side.     3. Divide by the coefficient of the unknown to find its value.     * Example: $7x - 4 = 3x + 12 \rightarrow 4x - 4 = 12 \rightarrow 4x = 16 \rightarrow x = 4$.

Changing the Subject

• Definition: Rearranging a formula to isolate a different variable.
• Method: Apply inverse operations in the reverse order of BIDMAS to "undo" the expression containing the target variable.     * Example: Make $r$ the subject of $A = \pi r^2$.     * Step 1: Divide by $\pi$: $\frac{A}{\pi} = r^2$.     * Step 2: Take the square root: $r = \sqrt{\frac{A}{\pi}}$.

Prime Factor Decomposition

• Definition: Breaking down a composite number into a product of prime numbers.
• Product of Primes: Every positive integer greater than 1 has a unique prime factorization.
• Method: Use a factor tree or repeated division by prime numbers ($2, 3, 5, 7, 11, \dots$).     * Example: $60 = 2 \times 30 = 2 \times 2 \times 15 = 2 \times 2 \times 3 \times 5$.     * Index notation: $60 = 2^2 \times 3 \times 5$.

Exchange Rates

• Concept: The value of one currency for the purpose of conversion to another.
• Formula for Conversion: $\text{Currency B} = \text{Currency A} \times \text{Exchange Rate}$.
• Formula for Reverting: $\text{Currency A} = \frac{\text{Currency B}}{\text{Exchange Rate}}$.
• Practical Application: Useful for calculating costs in foreign travel or international trade.

Bearings

• Three Key Rules:     1. They are always measured from North.     2. They are always measured in a clockwise direction.     3. They are always written as three digits (e.g., $045^{\circ}$ instead of $45^{\circ}$).

Form and Solve Equations

• Application: Turning a word problem or geometric scenario into an algebraic equation.
• Steps:     1. Assign a variable (e.g., $x$) to the unknown quantity.     2. Use the relationships described in the problem to write an equation.     3. Solve the equation using standard algebraic techniques.

Grouped Frequency Tables

• Definition: Used to organize data into classes or intervals when there are many different values.
• Estimating the Mean: Since exact values within groups are unknown, use the midpoint of each class ($x$).     * Formula: $\text{Estimated Mean} = \frac{\sum (f \times x)}{\sum f}$, where $f$ is the frequency and $x$ is the class midpoint.

Functions

• Function Notation: $f(x)$ denotes a rule that takes an input $x$ and produces an output.
• Inverse Functions: Denoted by $f^{-1}(x)$, this function reverses the effect of the original function.
• Composite Functions: Applying one function to the result of another.     * Notation: $fg(x)$ means perform $g(x)$ first, then apply $f$ to that result: $f(g(x))$.

Histograms

• Difference from Bar Charts: Unlike bar charts, the area of the bar in a histogram represents the frequency, not just the height. Bars are used for continuous data and often have unequal class widths.
• Frequency Density: The height of each bar on the vertical axis represents the frequency density.     * Formula: $\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}$.
• Interpretation: To find the frequency of a class from a histogram, calculate $\text{Area} = \text{Width} \times \text{Frequency Density}$.