Grade 8 Mathematics Study Notes

Number Sequences

• A number sequence is an ordered list of numbers governed by a specific mathematical pattern or rule.

• Fibonacci Sequence: A sequence where each term after the initial two is formed by adding the two preceding terms:

$x_n = x_{n-1} + x_{n-2}$

• The standard Fibonacci sequence begins: $1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$

• Sequences of Ordered Pairs: Expressed in the coordinate format $(x, y)$, where the sequence of all first values ($x$) and the sequence of all second values ($y$) each independently follow their own distinct rules.

Indices

• Negative Index Law: For any non-zero base $a$ and positive integer $n$:

$a^{-n} = \frac{1}{a^n}$

$a^n = \frac{1}{a^{-n}}$

• Power of a Product Rule:

$(ab)^n = a^n \times b^n$

• Power of a Quotient Rule:

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

where $b \neq 0$.

• Unit Fractional Indices and Radicals:

$a^{\frac{1}{n}} = \sqrt[n]{a}$

• Square root representation: $a^{\frac{1}{2}} = \sqrt{a}$

• Cube root representation: $a^{\frac{1}{3}} = \sqrt[3]{a}$

• Radical Properties for Products and Quotients:

$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

where $b \neq 0$.

Real Numbers & Decimal Approximations

• Classification of Real Numbers ($\mathbb{R}$):

• Natural Numbers ($\mathbb{N}$): Counting numbers $\{1, 2, 3, \dots\}$.

• Whole Numbers: Natural numbers including zero $\{0, 1, 2, 3, \dots\}$.

• Integers ($\mathbb{Z}$): Whole numbers and their negatives $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.

• Rational Numbers ($\mathbb{Q}$): Numbers expressible as $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$.

• Irrational Numbers: Non-terminating and non-recurring decimals that cannot be expressed as a fraction of integers (e.g., $\pi$, $\sqrt{2}$, $\sqrt{3}$).

• Decimal Forms:

• Terminating Decimals: Decimals with a finite number of digits after the decimal point.

• Recurring Decimals: Non-terminating decimals featuring a repeating pattern of digits, denoted with dot notation (e.g., $0.\dot{6} = 0.666\dots$, $0.\dot{7}\dot{2} = 0.7272\dots$, $0.\dot{7}1428\dot{5} = 0.714285714285\dots$).

• Midpoint of Two Rational Numbers: The rational number halfway between $a$ and $b$ is:

$\text{Midpoint} = \frac{a + b}{2}$

• Rounding Rules:

• Inspect the critical digit immediately following the desired place value.

• If the critical digit is $< 5$, keep the target digit unchanged.

• If the critical digit is $\ge 5$, increase the target digit by $1$.

Rate and Proportion

• Rate: A comparison of two quantities with different units, expressed using "per" or "$/$".

• Unit Rate: A rate comparing a quantity to a single unit of another quantity.

• Compound Unit Conversions:

$1\,\text{km/h} = \frac{1\,000\,\text{m}}{3\,600\,\text{s}} = \frac{5}{18}\,\text{m/s}$

$1\,\text{m/s} = \frac{3\,600}{1\,000}\,\text{km/h} = \frac{18}{5}\,\text{km/h}$

• Map Scales: Given as a ratio $1 : n$, meaning $1\,\text{cm}$ on the map represents an actual ground distance of $n\,\text{cm}$.

• Inverse (Indirect) Proportion: Occurs when an increase in one quantity results in a proportional decrease in the second, maintaining a constant product:

$x_1 \times y_1 = x_2 \times y_2$

Personal and Household Finance

• Value Added Tax (VAT):

$\text{VAT Amount} = 15\% \times \text{Price without VAT}$

$\text{Price with VAT} = 1.15 \times \text{Price without VAT}$

$\text{Price without VAT} = \frac{\text{Price with VAT}}{1.15}$

• Percentage Change:

$\text{Percentage Increase} = \frac{\text{Increase}}{\text{Original Value}} \times 100\%$

$\text{Percentage Decrease} = \frac{\text{Decrease}}{\text{Original Value}} \times 100\%$

• Commercial Profit and Loss:

$\text{Profit} = \text{Selling Price} - \text{Cost Price}$

$\text{Loss} = \text{Cost Price} - \text{Selling Price}$

$\text{Percentage Profit} = \frac{\text{Profit}}{\text{Cost Price}} \times 100\%$

$\text{Percentage Loss} = \frac{\text{Loss}}{\text{Cost Price}} \times 100\%$

• Discount and Commission:

$\text{Discount} = \text{Marked Price} - \text{Sale Price}$

$\text{Percentage Discount} = \frac{\text{Discount}}{\text{Marked Price}} \times 100\%$

$\text{Commission} = \text{Rate of Commission} \times \text{Total Sales}$

$\text{Total Earnings} = \text{Basic Salary} + \text{Commission}$

• Simple Interest:

$I = \frac{P \times R \times T}{100}$

$\text{Total Amount} = P + I$

where $P$ is the principal, $R$ is the annual interest rate, and $T$ is time in years.

Coordinates and Graphs of Linear Functions

• Standard Forms of Lines:

• Vertical line: $x = h$ (parallel to the $y$-axis, where $h$ is a constant).

• Horizontal line: $y = k$ (parallel to the $x$-axis, where $k$ is a constant).

• Slant / Inclined line: $y = mx + c$.

• Plotting Lines:

• Formulate a table of values by substituting values of $x$ into the equation to compute $y$.

• Plot the resulting coordinates $(x, y)$ on the Cartesian plane and draw a straight line through them with a ruler.

• Point of Intersection:

• The coordinates $(x, y)$ where two straight lines intersect provide the unique values that satisfy both equations simultaneously.

Pythagoras Theorem

• Fundamental Theorem: In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:

$c^2 = a^2 + b^2$

• The hypotenuse ($c$) is strictly the longest side, positioned directly opposite the $90^\circ$ angle.

• Pythagorean Triples: Sets of three positive integers that satisfy the relationship $a^2 + b^2 = c^2$ (e.g., $3, 4, 5$; $5, 12, 13$; $8, 15, 17$; $7, 24, 25$; $9, 40, 41$).

Algebraic Terms, Factorisation, and Fractions

• Highest Common Factor (H.C.F.):

• The product of the common numerical factors and common variable bases raised to their lowest powers.

• Lowest Common Multiple (L.C.M.):

• The product of the highest numerical multiple and all variable bases raised to their highest powers.

• Factorisation Methods:

• Common monomial factors: $ab + ac = a(b + c)$

• Compound factors: $y(x + 2) - 3(x + 2) = (x + 2)(y - 3)$

• Grouping four terms:

$ax + ay + bx + by = a(x + y) + b(x + y) = (x + y)(a + b)$

• Operations with Algebraic Fractions:

• Simplification: Divide both numerator and denominator by their H.C.F.

• Addition / Subtraction: Express all terms over their common denominator (L.C.M.):

$\frac{a}{x} + \frac{b}{y} = \frac{ay + bx}{xy}$

• Multiplication: Multiply numerators together and denominators together:

$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

• Division: Invert the divisor fraction and multiply:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

Linear Equations

• General Principle: Balance the equality by applying identical inverse operations (additive or multiplicative) to both sides.

• Equations with Variables on Both Sides: Collect like terms containing the variable on one side and numerical constants on the other.

• Cross-Multiplication for Fractional Equations:

$\frac{a}{b} = \frac{c}{d} \implies a \times d = b \times c$

• When unknown variables appear in the denominator, multiply through by the common denominator or cross-multiply to convert to a linear polynomial form.

Linear Inequalities

• Inequality Notation:

• $<$ : less than

• $\le$ : less than or equal to

• $>$ : greater than

• $\ge$ : greater than or equal to

• Number Line Representation:

• Unshaded circle ($\circ$): Boundary value is excluded ($<$ or $>$).

• Shaded circle ($\bullet$): Boundary value is included ($\le$ or $\ge$).

• Sign Reversal Rule: The inequality sign must be flipped ($< \leftrightarrow >$, $\le \leftrightarrow \ge$) whenever:

1. Multiplying or dividing both sides by a negative number.

2. Taking the reciprocal of both sides.

Surface Area and Volume of 3D Solids

• Cube (edge length $l$):

• Area of one face = $l^2$

• Total Surface Area (TSA) = $6l^2$

• Volume ($V$) = $l^3$

• Cuboid (length $l$, width $b$, height $h$):

• Total Surface Area (TSA) = $2(lb + lh + bh)$

• Volume ($V$) = $l \times b \times h$

• Packing Rule: The maximum number of small cubes of edge $x$ that can fit inside a box of dimensions $l \times b \times h$ without melting or reshaping is:

$\text{Number of cubes} = \left\lfloor\frac{l}{x}\right\rfloor \times \left\lfloor\frac{b}{x}\right\rfloor \times \left\lfloor\frac{h}{x}\right\rfloor$

Circle Geometry

• Basic Elements:

• Diameter and radius: $d = 2r$

• Chord: A straight line segment joining two points on the circumference.

• Perpendicular Chord Theorem: A line drawn from the centre of a circle perpendicular to a chord bisects the chord.

• Circumference and Arc Length:

• Circumference of a circle:

$C = 2\pi r = \pi d$

• Length of an arc subtending angle $\theta$ at the centre:

$\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r$

• Perimeter of a sector:

$\text{Perimeter} = 2r + \left(\frac{\theta}{360^\circ} \times 2\pi r\right)$

• Area Calculations:

• Area of a circle:

$A = \pi r^2$

• Area of a sector subtending angle $\theta$ at the centre:

$\text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2$

Polygons

• Terminology:

• Regular Polygon: A closed 2D shape with all sides equal in length and all interior angles equal in measure.

• Irregular Polygon: A polygon with unequal sides or unequal angles.

• Interior Angles Formulae (for an $n$-sided polygon):

• Number of triangles formed from one vertex = $n - 2$

• Sum of interior angles:

$\text{Sum} = (n - 2) \times 180^\circ$

• Each interior angle of a regular polygon:

$\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}$

• Exterior Angles Formulae:

• Sum of exterior angles for any convex polygon = $360^\circ$

• Each exterior angle of a regular polygon:

$\text{Exterior Angle} = \frac{360^\circ}{n}$

• Angle relationship at any vertex:

$\text{Interior Angle} + \text{Exterior Angle} = 180^\circ$

Statistics

• Pie Chart Interpretation and Construction:

• The entire circle represents the total frequency corresponding to $360^\circ$.

• Sector angle formula:

$\text{Angle of Sector} = \frac{\text{Frequency}}{\text{Total Frequency}} \times 360^\circ$

• Measures of Central Tendency:

• Mean:

$\text{Mean} = \frac{\text{Sum of all observations}}{\text{Total number of observations}}$

• Mode: The observation value that occurs most frequently in the dataset.

• Median: The middle value when the data is arranged in ascending order:

• If the number of items ($N$) is odd: the median is the value at position $\frac{N + 1}{2}$.

• If the number of items ($N$) is even: the median is the mean of the two middle values at positions $\frac{N}{2}$ and $\frac{N}{2} + 1$.

Sets and Venn Diagrams

• Two-Set Venn Diagram Notations:

• $n(\xi)$: Cardinal number of the universal set.

• Both sets: $n(A \cap B)$

• Only set $A$: $n(A \cap B') = n(A) - n(A \cap B)$

• Only set $B$: $n(A' \cap B) = n(B) - n(A \cap B)$

• Neither set: $n((A \cup B)')$

• Set Cardinality Formulae:

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$

$n(\xi) = n(A \cap B') + n(A \cap B) + n(A' \cap B) + n((A \cup B)')$

Geometric Constructions

• Key Instruments: Straightedge / ruler, protractor, and pair of compasses.

• Equilateral Triangle Construction (Given side $l$):

• Draw a base line segment of length $l$.

• Open compass to radius $l$, place the pointer at each endpoint of the base, and strike intersecting arcs above the line to locate the third vertex.

• Isosceles Triangle Construction (Given base $b$ and legs $a$):

• Draw the base of length $b$.

• Set compass width to length $a$, place pointer at each base endpoint, and draw arcs that intersect at the apex.

• Construction Given Three Sides (SSS):

• Draw the longest length as the baseline.

• From the respective endpoints, draw arcs with radii equal to the lengths of the remaining two sides. Connect the point of intersection to both base ends.

• Construction Given Angles and Side (ASA / AAS):

• Draw the given line segment as the base.

• Measure the required angles from the endpoints using a protractor and extend the rays until they intersect at the final vertex.