Year 10 Mathematics Revision: Topics 1–8

Comprehensive vocabulary flashcards covering Year 10 algebraic techniques, equations, variation, coordinate geometry, index laws, non-linear functions, measurement, trigonometry, and probability and data.

Expanding a Single Bracket

The process of multiplying every term inside a bracket by the term outside, represented as $a(b + c) = ab + ac$.

Expanding Two Brackets

The process of multiplying every term in the first bracket by every term in the second bracket.

Expansion of $(x + a)(x + b)$

$x^2 + (a + b)x + ab$.

Difference-of-Two-Squares Identity

The algebraic identity $a^2 - b^2 = (a - b)(a + b)$, which also factorises $x^2 - a^2$ into $(x - a)(x + a)$.

Common Factor Factorisation

Finding the greatest common factor of every term and placing it outside the bracket, as shown in $ax + ay = a(x + y)$.

Factorising a Monic Quadratic ($x^2 + bx + c$)

Finding two numbers that multiply to $c$ and add to $b$.

Factorising a Non-Monic Quadratic

Finding factors that expand to the original quadratic with a leading coefficient other than $1$, checking both the $x^2$ and constant terms.

Simplifying Algebraic Fractions

Factorising the numerator and denominator, then cancelling common factors. Terms cannot be cancelled across addition or subtraction; for example, $\frac{ab}{ac} = \frac{b}{c}$, provided $a \neq 0$ and $c \neq 0$.

Perfect Square Expansion

Expanding squared binomials: $(x + a)^2 = x^2 + 2ax + a^2$ and $(x - a)^2 = x^2 - 2ax + a^2$.

First Rule of Complicated Factorisation

Checking for a common factor first before applying any other factorisation techniques.

Zero-Product Rule

The property stating that if $AB = 0$, then $A = 0$ or $B = 0$.

Solution to $x^2 = a^2$

$x = \pm a$.

Inequality Negative Operation Rule

Reversing the inequality sign whenever multiplying or dividing an inequality by a negative number.

Strict vs. Non-Strict Inequality

$x > 3$ means $x$ is greater than $3$ but does not include $3$; $x \le 5$ means $x$ is less than or equal to $5$.

Extraneous Solution

A value produced during the solving process that does not satisfy the original equation or its restrictions.

Direct Variation

A relationship where two quantities change in proportion according to $y = kx$, with a constant of variation $k = \frac{y}{x}$ (provided $x \neq 0$). When $x$ doubles, $y$ doubles.

Inverse Variation

A relationship where one quantity is inversely proportional to another according to $y = \frac{k}{x}$, with a constant of variation $k = xy$. When $x$ doubles, $y$ halves.

Speed, Distance, and Time Formulas

$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$, $\text{Distance} = \text{Speed} \times \text{Time}$, and $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$.

Percentage Change Formula

$\frac{\text{New} - \text{Original}}{\text{Original}} \times 100\%$.

Percentage Increase and Decrease Multipliers

To calculate a percentage increase, multiply the original amount by $1 + \frac{r}{100}$; for a percentage decrease, multiply by $1 - \frac{r}{100}$, where $r$ is the percentage rate.

Average Rate of Change

$\frac{\text{Change in } y}{\text{Change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$.

Distance vs. Displacement

Distance is the total ground covered, whereas displacement is the change in position, including direction.

General Equation of a Straight Line

$y = mx + c$, where $m$ represents the gradient and $c$ represents the $y$-intercept.

Gradient ($m$)

The measure of steepness calculated by $m = \frac{y_2 - y_1}{x_2 - x_1}$. A positive gradient rises from left to right, a negative gradient falls from left to right, and a gradient of zero indicates a horizontal line.

Vertical and Horizontal Line Equations

A vertical line has the equation $x = a$ (where $a$ is a constant); a horizontal line has the equation $y = b$ (where $b$ is a constant).

Axis Intercepts

The coordinates where a graph crosses the axes: find the $x$-intercept by setting $y = 0$ and solving for $x$; find the $y$-intercept by setting $x = 0$ and solving for $y$.

Midpoint Formula

$(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$.

Perpendicular and Parallel Gradients

Parallel lines share the same gradient $m$; perpendicular lines have a gradient of $-\frac{1}{m}$, provided $m \neq 0$.

Intersection of Two Lines

The point that satisfies both equations simultaneously.

Product Law of Indices

$a^m \times a^n = a^{m+n}$, requiring the same base. It applies to multiplication, not addition ($a^m + a^n \neq a^{m+n}$).

Quotient Law of Indices

$\frac{a^m}{a^n} = a^{m-n}$, for $a \neq 0$.

Power Laws of Indices

Power-of-a-power: $(a^m)^n = a^{mn}$; power-of-a-product: $(ab)^n = a^n b^n$; power-of-a-quotient: $(\frac{a}{b})^n = \frac{a^n}{b^n}$, for $b \neq 0$.

Zero and Negative Index Laws

$a^0 = 1$ for $a \neq 0$; $a^{-n} = \frac{1}{a^n}$ for $a \neq 0$; and $\frac{1}{a^{-n}} = a^n$.

Fractional Index Laws

$a^{1/n} = \sqrt[n]{a}$ and $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$, where the root is defined.

Quadratic Forms and Features

General form is $y = ax^2 + bx + c$ ($a \neq 0$), forming a parabola opening upwards if $a > 0$ and downwards if $a < 0$. Factorised form is $y = a(x - p)(x - q)$ with $x$-intercepts at $x = p$ and $x = q$. Turning-point form is $y = a(x - h)^2 + k$ with turning point $(h, k)$.

Exponential Graph

General equation $y = ab^x$, typically with $a \neq 0$, $b > 0$, and $b \neq 1$. For a positive starting value, $b > 1$ represents exponential growth, and $0 < b < 1$ represents exponential decay towards zero as $x$ increases.

Reciprocal Graph (Hyperbola)

General equation $y = \frac{a}{x}$ ($a \neq 0$), with asymptotes at $x = 0$ and $y = 0$. The translated form $y = \frac{a}{x - h} + k$ has asymptotes at $x = h$ and $y = k$.

Cubic Graph

An equation of the form $y = ax^3 + bx^2 + cx + d$ ($a \neq 0$), which often displays a characteristic S-like shape depending on its coefficients.

Plane Area Formulas

Rectangle: $A = lw$; Triangle: $A = \frac{1}{2}bh$; Parallelogram: $A = bh$; Trapezium: $A = \frac{1}{2}(a + b)h$; Circle: $A = \pi r^2$ (with circumference $C = 2\pi r = \pi d$).

Prism and Cylinder Volumes

Rectangular prism: $V = lwh$; General prism: $V = \text{cross-sectional area} \times \text{length}$; Cylinder: $V = \pi r^2 h$.

Pyramid, Cone, and Sphere Volumes

Pyramid: $V = \frac{1}{3}Bh$ ($B$ is base area); Cone: $V = \frac{1}{3}\pi r^2 h$; Sphere: $V = \frac{4}{3}\pi r^3$.

Surface Area Formulas

Rectangular prism: $SA = 2(lw + lh + wh)$; Cylinder curved surface: $CSA = 2\pi rh$; Closed cylinder total: $SA = 2\pi r^2 + 2\pi rh$; Sphere: $SA = 4\pi r^2$.

Vertical Height vs. Slant Height

Vertical height is perpendicular to the base, while slant height runs along the sloping surface.

Composite Figures

Shapes or solids formed by multiple components; solved by splitting into simpler shapes or solids, calculating their individual areas or volumes, and adding or subtracting as required.

Trigonometric Ratios (SOH CAH TOA)

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$, where the hypotenuse is the longest side opposite the right angle.

Inverse Trigonometric Functions

Functions ($\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$) used to calculate an unknown angle in a right-angled triangle.

Pythagorean Theorem

$a^2 + b^2 = c^2$, where $c$ is the hypotenuse; applied strictly within right-angled triangles.

Theoretical Probability

$P(A) = \frac{\text{favourable outcomes}}{\text{total equally likely outcomes}}$, bounded by $0 \le P(A) \le 1$, where an impossible event has probability $0$ and a certain event has probability $1$.

Probability Rules

Complement rule: $P(\text{not } A) = 1 - P(A)$; Addition rule for mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$; Multiplication rule for independent events: $P(A \text{ and } B) = P(A) \times P(B)$.

Independent vs. Mutually Exclusive Events

Independent events do not affect each other's probabilities; mutually exclusive events cannot occur simultaneously.

Measures of Centre and Spread

Mean: sum of values divided by count; Median: middle value when ordered (or average of two middle values); Mode: most frequent value; Range: maximum minus minimum.

Outlier and its Statistical Effects

A value unusually far from the rest of the data. An extreme outlier pulls the mean towards it, whereas the median is usually less affected.

Reliability vs. Validity

Reliability is the consistency of results upon repetition (improved by repeating trials, using consistent methods, and calculating a mean); validity is whether the method fairly tests what it is intended to test (improved by controlling relevant variables).