Complete Grade 8 Mathematics Study Guide: Squares, Variables, Equations, Similarity, Circles, Probability, and Geometry

Unit 1: Squares, Square Roots, Cubes, and Cube Roots

1.1 The Square of a Number

1.1.1 Square of a Rational Number

Operations in mathematics are categorized into different levels:

• Operations of the first kind: Addition and subtraction.

• Operations of the second kind: Multiplication and division.

• Operations of the third kind: Raising to a power and extracting roots.

Definition 1.1: Squaring a Number The process of multiplying a rational number by itself is called squaring the number. If the number is $a$, the product $a \times a$ is written as $a^{2}$. This is read as "$a$ squared," "the square of $a$," or "$a$ to the power of $2$."

Examples of Square Patterns (Patterns of Squares):

• $1 \times 1 = 1$ (The $1^{st}$ square number)

• $2 \times 2 = 4$ (The $2^{nd}$ square number)

• $3 \times 3 = 9$ (The $3^{rd}$ square number)

• $4 \times 4 = 16$ (The $4^{th}$ square number)

Components of Power Form: In the expression $7^{2} = 49$:

• Base: $7$

• Exponent: $2$

• Power Form: $7^{2}$

• Standard Numeral Form: $49$

Key Distinctions: There is a fundamental difference between $a^{2}$ and $2a$:

• $a^{2} = a \times a$

• $2a = a + a$

• Example: $30^{2} = 30 \times 30 = 900$, whereas $2 \times 30 = 60$.

Definition 1.2: Perfect Square A rational number $x$ is called a perfect square if and only if $x = n^{2}$ for some $n \in Q$. A perfect square is the product of a rational number times itself, and its square root is also a rational number.

Square values of common natural numbers ($x$ to $x^{2}$):

• $x = 1, x^{2} = 1$

• $x = 2, x^{2} = 4$

• $x = 3, x^{2} = 9$

• $x = 4, x^{2} = 16$

• $x = 5, x^{2} = 25$

• $x = 10, x^{2} = 100$

• $x = 15, x^{2} = 225$

• $x = 20, x^{2} = 400$

• $x = 25, x^{2} = 625$

• $x = 35, x^{2} = 1225$

Squaring Fractions and Decimals:

• If $x = \frac{1}{3}, x^{2} = (\frac{1}{3})^{2} = \frac{1}{9}$

• If $x = 0.26, x^{2} = (0.26)^{2} = \frac{26}{100} \times \frac{26}{100} = \frac{676}{10000} = 0.0676$

Notes on Powers:

• The squares of natural numbers are always natural numbers.

• $0^{2} = 0$

• The symbol $0^{0}$ has no meaning.

• If $a \in Q$ and $a \neq 0$, then $a^{0} = 1$.

Theorem 1.1: Existence Theorem For each rational number $x$, there is a rational number $y$ ($y \geq 0$) such that $x^{2} = y$.

Approximating Squares: Rough calculations for approximation often involve rounding decimals to the nearest whole number:

• If $x = 3.4$, then $x^{2} \approx 3^{2} = 9$.

• If $x = 9.7$, then $x^{2} \approx 10^{2} = 100$.

• If $x = 0.026$, then $x^{2} \approx (0.03)^{2} = 0.0009$.

1.1.2 Use of Table of Values of Squares

To find the square of a decimal like $2.54$ using a numerical table:

1. Under the column headed by $x$, find the row starting with $2.5$.

2. Move right until reaching the column headed by the digit $4$.

3. The intersection provides the value: $(2.54)^{2} = 6.452$.

1.2 The Square Root of a Rational Number

Definition 1.3: Square Roots For any two rational numbers $a$ and $b$, if $a^{2} = b$, then $a$ is called the square root of $b$. Extracting a square root is the inverse operation of squaring.

Notation:

• Radical sign: $\sqrt{}$

• Radicand: The number $b$ in $\sqrt{b}$.

• Radical $b$: The full expression $\sqrt{b}$.

Common Perfect Squares and Roots:

• $\sqrt{100} = 10$

• $\sqrt{225} = 15$

• $\sqrt{169} = 13$

• $\sqrt{256} = 16$

• $\sqrt{625} = 25$

• $\sqrt{1600} = 40$

Properties of Square Roots ($a > 0, b > 0$):

1. $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$

2. $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ (where $b \neq 0$)

Method of Prime Factorization for Roots: Example: To find $\sqrt{324}$

• $324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3$

• Arrange into identical sets: $(2 \times 3 \times 3) \times (2 \times 3 \times 3) = 18 \times 18 = 18^{2}$

• $\sqrt{324} = 18$

Using Tables for Roots: If $\sqrt{17.89}$ is needed:

1. Find $17.89$ in the body of the square table.

2. The row to the left gives the first two digits ($4.2$).

3. The column heading at the top gives the third digit ($3$).

4. $\sqrt{17.89} \approx 4.23$

1.3 Cubes and Cube Roots

1.3.1 Cube of a Number

If the number to be cubed is $a$, the product $a \times a \times a$ is written as $a^{3}$ and read as "$a$ cubed."

Definition 1.5: Cube Number A cube number is the result of multiplying a rational number by itself, then multiplying by the number again.

• $1 \times 1 \times 1 = 1$ (The $1^{st}$ cube number)

• $2 \times 2 \times 2 = 8$ (The $2^{nd}$ cube number)

• $3 \times 3 \times 3 = 27$ (The $3^{rd}$ cube number)

Table of Integer Cubes ($x$ to $x^{3}$):

• $x = -4, x^{3} = -64$

• $x = -3, x^{3} = -27$

• $x = -2, x^{3} = -8$

• $x = -1, x^{3} = -1$

• $x = 0, x^{3} = 0$

• $x = 1, x^{3} = 1$

• $x = 2, x^{3} = 8$

• $x = 6, x^{3} = 216$

1.3.2 Cube Root of a Number

Definition 1.6: Cube Root The cube root of a given number is one of the three identical factors whose product is the given number. It is written as $\sqrt[3]{b}$.

• In $\sqrt[3]{512}$, the index is $3$ and the radicand is $512$.

• The relation is: if $a^{3} = b$, then $a = \sqrt[3]{b}$.

Properties of Cubes:

1. $(ab)^{3} = a^{3} \times b^{3}$

2. $(\frac{a}{b})^{3} = \frac{a^{3}}{b^{3}}$ (where $b \neq 0$)

Unit 2: Further on Working with Variables

2.1 Further on Algebraic Terms and Expressions

2.1.1 Use of Variables in Formula

Definition 2.1: Variable A variable is a symbol or letter (such as $x, y, z$) used to represent an unknown number. Formulas utilize variables to describe mathematical relationships between quantities.

Examples of Geometric Formulas:

• Perimeter of a Rectangle: $P = 2(l + w)$, where $l$ is length and $w$ is width.

• Volume of a Rectangular Prism: $V = l \times w \times h$.

• Area of a Triangle: $A = \frac{1}{2}bh$.

• Area of a Trapezium: $A = \frac{1}{2}h(b_{1} + b_{2})$.

2.1.2 Variables, Terms, and Expressions

Definition 2.2: Algebraic Expressions Algebraic expressions are formed using numbers, letters, and operations (addition, subtraction, multiplication, division, powers, and roots).

• Constant: An expression with no variables (e.g., $72 - 16\pi$).

• Terms: Parts of the expression connected by plus or minus signs. In $5x - 13x + 20$, the terms are $5x$, $-13x$, and $20$.

Classification of Expressions:

• Monomial: Contains one term (e.g., $8x$, $18xy$).

• Binomial: Contains two terms (e.g., $2x + 2y$, $n - 3$).

• Trinomial: Contains three terms (e.g., $4x^{2} + 3x + 10$).

Definition 2.6: Like Terms Terms that have the same variables raised to the same powers are called like terms. Only like terms can be combined (added or subtracted).

• Combining Like Terms: $18x + 27 - 6x - 2 = 12x + 25$.

2.1.3 Use of Variables to Solve Problems

Solving word problems involves translating verbal statements into mathematical equations.

• Example (Consecutive Integers): The sum of two consecutive odd integers is $-188$. Let $x$ be the first. Equation: $x + (x + 2) = -188 \rightarrow 2x = -190 \rightarrow x = -95$. The integers are $-95$ and $-93$.

• Example (Amoeba Splits): An amoeba cell splits by fission, doubling every split.

  • $0$ splits: $1$ cell ($2^{0}=1$)

  • $1$ split: $2$ cells ($2^{1}=2$)

  • $2$ splits: $4$ cells ($2^{2}=4$)

  • $10$ splits: $2^{10} = 1024$ cells.

2.2 Multiplication of Binomials

2.2.1 Multiplication of Monomial by Binomial

This utilizes the Distributive Property:

• $a(b + c) = ab + ac$

• $a(b - c) = ab - ac$

• Example: $4c^{2}(16abc - 5a^{2}bc) = 64c^{3}ab - 20c^{3}a^{2}b$.

2.2.2 Multiplication of Binomial by Binomial

To multiply $(a + b)(c + d)$, each term in the first set of brackets must be multiplied by every term in the second set:

• $(a + b)(c + d) = ac + ad + bc + bd$

• Cross Products: In $(2x - 3)(4x - 12)$, the products $-12 \times 2x$ and $-3 \times 4x$ are cross products.

• End Products: $2x \times 4x$ and $-3 \times -12$.

2.3 Highest Common Factors (HCF)

To factorize an expression means to write it as a product of two or more factors.

• HCF of Integers: The highest whole number factor common to two or more integers (e.g., HCF of $24$ and $36$ is $12$).

• HCF of Monomials: Determine the common variables with the lowest powers and the HCF of the coefficients. Example: HCF of $7x^{3}$ and $14x^{2}$ is $7x^{2}$.

• Factoring out a Binomial: $2x(5x + 3) - 5(5x + 3) = (5x + 3)(2x - 5)$.

Unit 3: Linear Equations and Inequalities

3.1 Solution of Linear Equations

3.1.1 Equations Involving Brackets

To solve equations like $x - 2(x - 1) = 1 - 4(x + 1)$, follow these rules based on distributive properties:

• Remove brackets: $x - 2x + 2 = 1 - 4x - 4$

• Collect like terms: $x - 2x + 4x = 1 - 4 - 2$

• Solve: $3x = -5 \rightarrow x = -\frac{5}{3}$.

3.1.2 Equations Involving Fractions

To solve equations with fractions, multiply both sides by the Least Common Multiple (LCM) of the denominators.

• Example: Solving $\frac{x+1}{3} + \frac{x-1}{10} = 12$

  • LCM of $3$ and $10$ is $30$.

  • $30(\frac{x+1}{3}) + 30(\frac{x-1}{10}) = 30 \times 12 \rightarrow 10(x+1) + 3(x-1) = 360 \rightarrow 13x + 7 = 360 \rightarrow 13x = 353 \rightarrow x = \frac{353}{13}$.

3.2 Further on Linear Inequalities

Transformation Rules for Inequalities:

1. If a value is added or subtracted from both sides, the inequality sign remains the same.

2. If multiplying or dividing by a positive number, the sign remains the same.

3. If multiplying or dividing by a negative number, the direction of the inequality sign is reversed.

• Example: $-3x + 8 < 22 \rightarrow -3x < 14 \rightarrow x > -\frac{14}{3}$.

3.3 Cartesian Coordinate System

The system consists of two perpendicular number lines: the horizontal X-axis and the vertical Y-axis, intersecting at the Origin $(0,0)$.

The Four Quadrants:

• Quadrant I: $x > 0, y > 0$

• Quadrant II: $x < 0, y > 0$

• Quadrant III: $x < 0, y < 0$

• Quadrant IV: $x > 0, y < 0$

Equations of Lines:

• Vertical Line: $x = a$ (Parallel to Y-axis).

• Horizontal Line: $y = b$ (Parallel to X-axis).

• Line through Origin: $y = mx$.

Definition 3.1: Slope ($m$) The slope is the ratio of the change in $y$ to the change in $x$: $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

• Horizontal lines have a slope of $0$.

• Vertical lines have an undefined slope.

Unit 4: Similar Figures

4.1 Similar Plane Figures

Similar figures have the same shape but not necessarily the same size. Definition 4.1: Similar Polygons Two polygons are similar if:

1. Their corresponding sides are proportional.

2. Their corresponding angles are congruent.

Enlargements and Scale Factors:

• Scale Factor ($K$): Ratio of the image length to the original length. $K = \frac{\text{Image Length}}{\text{Original Length}}$.

• If $K > 1$, the figure is enlarged.

• If $0 < K < 1$, the figure is reduced.

• If $K = 1$, the figures are congruent.

4.2 Similar Triangles

Similarity Theorems:

• AA Similarity (Angle-Angle): If two angles of one triangle are congruent to two angles of another.

• SAS Similarity (Side-Angle-Side): If two sides are proportional and the included angle is equal.

• SSS Similarity (Side-Side-Side): If all three sides are proportional.

Perimeters and Areas of Similar Polygons:

• Theorem 4.4: If the ratio of sides is $k$, the ratio of perimeters is also $k$.

• Theorem 4.5: If the ratio of sides is $k$, the ratio of areas is $k^{2}$.

Unit 5: Circles

5.1 Further On Circles

Definitions:

• Arc: Part of a circle.

  • Semi-circle: Degree measure of $180^{\circ}$.

  • Minor Arc: Measure $< 180^{\circ}$.

  • Major Arc: Measure $> 180^{\circ}$.

• Sector: Region bounded by two radii and an arc.

• Segment: Region bounded by a chord and an arc.

• Tangent: A line intersecting the circle at exactly one point (point of tangency).

• Secant: A line intersecting the circle at two points.

5.2 Angles in the Circle

Central vs. Inscribed Angles:

• Definition 5.5 (Central Angle): Vertex at the center. Measure is equal to the intercepted arc measure.

• Definition 5.6 (Inscribed Angle): Vertex on the circle.

Theorem 5.1: Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc:

• $\text{Inscribed Angle} = \frac{1}{2} \times \text{Intercepted Arc}$.

• Corollary: Inscribed angles subtended by the same arc are congruent.

Theorem 5.3: Intersecting Chords Angle formed by two chords intersecting inside a circle is half the sum of the subtended arcs:

• $\text{Angle} = \frac{1}{2}(\text{Arc}_{1} + \text{Arc}_{2})$.

Theorem 5.4: Cyclic Quadrilaterals In a quadrilateral inscribed in a circle, opposite angles are supplementary (add to $180^{\circ}$).

Unit 6: Introduction to Probability

6.1 The Concept of Probability

• Experiment: A process of observation.

• Sample Space ($S$): The set of all possible outcomes.

• Event ($E$): A subset of a sample space.

• Certain Event: Probability is $1$.

• Impossible Event: Probability is $0$.

• Probability Scale: All probabilities occur in the range $0 \leq P(E) \leq 1$.

6.2 Probability of Simple Events

Definition 6.7: Probability Formula Assuming outcomes are equally likely: $P(E) = \frac{n(E)}{n(S)} = \frac{\text{number of successful outcomes}}{\text{total number of possible outcomes}}$

Examples:

• Rolling a sum of $7$ with two dice: $n(S) = 36$, successful outcomes: $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$. $P = \frac{6}{36} = \frac{1}{6}$.

• Drawing a prime number from cards numbered $1$ to $20$: Primes are $\{2,3,5,7,11,13,17,19\}$. $P = \frac{8}{20} = \frac{2}{5}$.

Unit 7: Geometry and Measurements

7.1 Theorems on the Right Angled Triangle

7.1.1 Euclid's Theorem

Theorem 7.1: In a right triangle with altitude $h$ to the hypotenuse $c$, if the altitude divides $c$ into segments $e$ and $f$ adjacent to legs $b$ and $a$ respectively:

• $a^{2} = c \times f$

• $b^{2} = c \times e$

7.1.2 Pythagoras' Theorem

Theorem 7.3: In a right angled triangle with legs $a, b$ and hypotenuse $c$:

• $a^{2} + b^{2} = c^{2}$

• Converse (Theorem 7.4): If $a^{2} + b^{2} = c^{2}$, the triangle is right angled.

7.2 Introduction to Trigonometry

Definition 7.1: Trigonometric Ratios In a right triangle:

• $\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}}$

• $\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}}$

• $\tan(A) = \frac{\text{opposite}}{\text{adjacent}}$

Values for Special Angles:

• $30^{\circ}$: $\sin = \frac{1}{2}$, $\cos = \frac{\sqrt{3}}{2}$, $\tan = \frac{\sqrt{3}}{3}$

• $45^{\circ}$: $\sin = \frac{\sqrt{2}}{2}$, $\cos = \frac{\sqrt{2}}{2}$, $\tan = 1$

• $60^{\circ}$: $\sin = \frac{\sqrt{3}}{2}$, $\cos = \frac{1}{2}$, $\tan = \sqrt{3}$

7.3 Solids Figures

7.3.1 Pyramids

Definition 7.2: A solid figure formed by joining a polygonal base to a point (vertex) outside the plane.

• Regular Pyramid: A right pyramid with a regular polygon as its base. Lateral edges are equal.

• Slant Height ($s$): The altitude of a lateral face.

7.3.2 Cones

Definition 7.3: A solid formed by joining all points of a circular base to a vertex.

• Right Circular Cone: The vertex is directly above the center of the circular base.

• Slant Height ($s$): Distance from the vertex to a point on the circle boundary.