Grade 8 Mathematics - Units 1-4 Vocabulary Review

Unit 1: Rational Numbers and Their Properties

A rational number is formally defined as any number that can be expressed in the form of $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. In this mathematical classification, the set of rational numbers is denoted by the symbol $\mathbb{Q}$. This unit explores how these numbers relate to other sets, their representation on the number line, and their operations in real-world contexts.

Rational numbers are visually represented on a number line based on specific positional facts. Positive rational numbers are placed to the right of zero, while negative rational numbers are placed to the left. Positive proper fractions, where the numerator is less than the denominator, always reside between $0$ and $1$. Conversely, improper fractions, where the numerator exceeds or equals the denominator, are typically converted into mixed fractions (a whole number plus a proper fraction) to determine their placement between specific integers. For example, to locate $\frac{3}{2}$, one converts it to $1\frac{1}{2}$, placing it exactly midway between $1$ and $2$. Two rational numbers are considered opposites if they are equidistant from zero on opposite sides, such as $\frac{3}{2}$ and $-\frac{3}{2}$.

The relationship between number sets is hierarchical and often illustrated via Venn diagrams. The set of Natural numbers ($\mathbb{N}$) is a subset of Whole numbers ($W$), which is a subset of Integers ($\mathbb{Z}$), which is ultimately a subset of Rational numbers ($\mathbb{Q}$). Thus, every integer is a rational number because it can be written as $\frac{a}{1}$, but not every rational number is an integer.

Absolute value, denoted by $|x|$, represents the distance of a rational number from zero on a number line, ignoring direction. It is defined piecewise: $|x| = x$ if $x \ge 0$ and $|x| = -x$ if $x < 0$. This concept leads to absolute value equations of the form $|x| = a$. If $a > 0$, there are two solutions ($x = a$ and $x = -a$); if $a = 0$, there is one solution ($x = 0$); if $a < 0$, there is no solution, as distance cannot be negative.

Operations and Comparisons of Rational Numbers

Comparing rational numbers involves ensuring they are in a comparable format. For decimals, one compares place values from left to right, starting with the integer part. For fractions with the same denominator, the fraction with the larger numerator is greater (e.g., $\frac{15}{7} > \frac{13}{7}$). For fractions with different denominators, two primary methods are used: the Least Common Multiple (LCM) method, where fractions are converted to a common denominator to compare numerators, and the Cross-Product method. In the latter, for two fractions $\frac{a}{b}$ and $\frac{c}{d}$, $\frac{a}{b} < \frac{c}{d}$ if and only if $ad < bc$. General rules state that any positive rational number is greater than zero and any negative number, and among negative numbers, the one with the larger absolute value is smaller (e.g., $-45 < -23$ because $|-45| > |-23|$).

Addition of rational numbers requires common denominators. When signs are identical, one adds the absolute values and applies the common sign. When signs differ, one subtracts the smaller absolute value from the larger and applies the sign of the number with the greater absolute value. Subtraction is treated as the addition of the additive inverse: $\frac{a}{b} - \frac{c}{d} = \frac{a}{b} + (-\frac{c}{d})$. Addition possesses key properties: Commutative ($a + b = b + a$), Associative ($a + (b + c) = (a + b) + c$), Identity ($a + 0 = a$), and Inverse ($a + (-a) = 0$).

Multiplication involves multiplying numerators together and denominators together: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$. The sign of the product depends on the factors: two signs that are the same yield a positive result, while opposite signs yield a negative result. Division is performed by multiplying the dividend by the reciprocal of the divisor: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$, provided $c \neq 0$.

Real-Life Applications and Simple Interest

Rational numbers are applied in daily activities such as sharing items and financial management. A key application is Simple Interest, calculated using the formula $I = PRT$. In this formula, $I$ represents the interest, $P$ is the principal (the initial amount borrowed or invested), $R$ is the annual interest rate (expressed as a decimal or fraction), and $T$ is the time the money is used (expressed in years). For example, if Abebe borrows Birr $21100$ at a $15\%$ simple interest rate for $5$ months, the time $T$ is $\frac{5}{12}$ years. The interest is calculated as $21100 \times 0.15 \times \frac{5}{12} = 1318.75\,\text{Birr}$.

Unit 2: Squares, Square Roots, Cubes, and Cube Roots

Squaring a number involves multiplying a number by itself, denoted as $x^2$. A rational number is a perfect square if it is the square of some rational number $m \in \mathbb{Q}$. Observations of patterns show that the sum of the first $n$ odd natural numbers is always equal to $n^2$ (e.g., $1 + 3 + 5 = 9 = 3^2$). Squaring properties include $(ab)^2 = a^2b^2$ and $(\frac{a}{b})^2 = \frac{a^2}{b^2}$. Rough calculations can be used to estimate squares by rounding decimals to the nearest whole number before squaring.

Square roots represent the inverse operation of squaring. For any rational numbers $a$ and $b$, if $b^2 = a$, then $b$ is the square root of $a$. The principal (positive) square root is denoted by $\sqrt{a}$. Finding square roots can be done via prime factorization, where the number is decomposed into a product of identical pairs. For instance, $400 = 2 \times 2 \times 2 \times 2 \times 5 \times 5$, which regrouped is $(2 \times 2 \times 5) \times (2 \times 2 \times 5) = 20 \times 20$, thus $\sqrt{400} = 20$. Numerical tables and scientific calculators are also used for high-precision or non-perfect squares.

A cube number is obtained by multiplying a number by itself three times ($x^3$). A perfect cube is the product of three identical rational factors. The cube root of a number, denoted $\sqrt[3]{a}$, is the value that, when cubed, equals $a$. Unlike square roots, every rational number has exactly one cube root, and the cube root of a negative number is negative. Applications of these concepts are found in geometry (e.g., calculating the side of a patio with area $400\,m^2$ as $\sqrt{400} = 20\,m$) and volume calculations ($V = s^3$).

Unit 3: Linear Equations and Inequalities

The Cartesian coordinate system is formed by two perpendicular number lines: the horizontal $x$-axis and the vertical $y$-axis. Their intersection is the origin $(0,0)$. This plane is divided into four quadrants (I, II, III, IV) in a counter-clockwise direction. Points are identified by an ordered pair $(x, y)$, where $x$ is the abscissa and $y$ is the ordinate. In Quadrant I ($x > 0, y > 0$), II ($x < 0, y > 0$), III ($x < 0, y < 0$), and IV ($x > 0, y < 0$).

Graphs of linear equations take several forms. A vertical line through $(a, b)$ has the equation $x = a$. A horizontal line has the equation $y = b$. Equations of the form $y = mx$ represent lines passing through the origin; the line passes through Quadrants I and III if $m > 0$, and II and IV if $m < 0$. Linear equations of the form $y = mx + c$ are graphed by plotting points (choosing $x$ values and solving for $y$) and drawing a straight line through them.

Linear inequalities involve relation signs like $<, >, \le, \ge$, or $\neq$. These are solved using transformation rules. Adding or subtracting the same number from both sides maintains the inequality direction ($a < b \rightarrow a + c < b + c$). Multiplying or dividing by a positive number also maintains direction. Crucially, multiplying or dividing both sides by a negative number reverses the direction of the inequality ($a < b$ and $c < 0 \rightarrow ac > bc$). Solution sets for inequalities can be graphed on a number line, where an open circle represents strict inequality and a closed circle represents "greater than or equal to" or "less than or equal to".

Unit 4: Similarity of Figures

In mathematics, two plane figures are similar if they have the exact same shape but not necessarily the same size. Similar figures appear as scale models of one another. Two polygons are similar if a one-to-one correspondence exists between their vertices such that all corresponding angles are congruent and the ratios of the lengths of all corresponding sides are equal. This constant ratio is known as the scale factor or the constant of proportionality ($k$).

Certain figures are inherently similar: any two squares or any two equilateral triangles will always be similar because their internal angles are fixed and their sides are always proportional. To verify similarity in other polygons, one must check the ratio of all corresponding sides. For example, if a quadrilateral has sides $2\,cm, 5\,cm, 6\,cm,$ and $8\,cm,$ a similar quadrilateral with a shortest side of $3\,cm$ would have a scale factor of $\frac{3}{2} = 1.5$, resulting in other sides of $7.5\,cm, 9\,cm,$ and $12\,cm$.