Comprehensive Study Notes on 6th Grade Mathematics from Natural Numbers to Statistics

Natural Numbers and Operations

Natural numbers form the foundation of counting and arithmetic. A primary concept within this domain is the power of a natural number. A power is defined as the multiplication of the same number by itself a specific number of times. The notation for this is represented as $a^n$, where $a$ is defined as the base and $n$ is the power exponent. For instance, the expression $5^3$ represents the number $5$ multiplied by itself $3$ times, calculated as $5 \times 5 \times 5 = 125$. It is a fundamental rule that any number raised to the first power is always equal to itself.

Prime and Composite Numbers

Natural numbers are categorized into prime and mürəkkəb (composite) numbers based on their divisors. A prime (sadə) number is defined as a natural number that has exactly two divisors: $1$ and itself. Examples of prime numbers include $2$, $3$, $5$, $7$, and $11$. Conversely, a mürəkkəb (composite) number is an explorer that has more than two divisors. Examples of composite numbers include $4$, $6$, $8$, and $9$. It is important to note that the number $1$ is classified as neither prime nor composite. The number $2$ holds a unique status as the smallest prime number and the only existing even prime number.

Greatest Common Divisor (ƏBOB) and Least Common Multiple (ƏKOB)

The Greatest Common Divisor, referred to as ƏBOB, is the largest natural number that divides all given numbers without a remainder. To find the ƏBOB, one must follow a three-step procedure: factor the numbers into their prime factors, identify the common factors, and then multiply those common factors together. If the calculated ƏBOB of two numbers is equal to $1$, those numbers are categorized as coprime (qarşılıqlı sadə).

The Least Common Multiple, referred to as ƏKOB, is the smallest natural number that is divisible by each of the given numbers. The process for finding the ƏKOB involves factoring the numbers into prime factors, identifying all necessary factors to cover each number, and then multiplying them. A specific rule exists for cases where one number is divisible by another: the larger number is the ƏKOB, while the smaller number is the ƏBOB. As a general rule for identification, a prime number has exactly $2$ divisors, whereas a composite number possesses more than $2$ divisors.

Ratio and Ratios of Quantities

A ratio is defined as the division of two numbers, expressed either as $a : b$ or $\frac{a}{b}$. A ratio serves to illustrate how many times one number is larger than another or what part one number represents of another. For example, in the ratio $10 : 5 = 2$, it is demonstrated that the number $10$ is $2$ times larger than $5$.

When dealing with the ratios of quantities, if the quantities are of the same kind (such as $5\,m : 10\,m$), the resulting ratio is $1 : 2$ and is expressed as a unitless number. If the quantities are of different types (such as $60\,km / 2\,saat$), a new quantity is created, such as speed ($30\,km/saat$). Whenever units of measurement differ within the same kind of quantity, they must be converted to the same unit before calculating the ratio.

Dividing Quantities in Ratios and Proportions

To divide a specific quantity according to a given ratio, a three-step rule is applied: first, add the numbers in the ratio to find a total sum; second, divide the total quantity by that sum to find the value of one part; and third, multiply that part value by each number in the ratio. For example, to divide the number $60$ in a $2:3$ ratio, one calculates $2 + 3 = 5$, then $60 / 5 = 12$. The final values are $12 \times 2 = 24$ and $12 \times 3 = 36$.

A proportion is defined as the equality of two ratios, written as $a : b = c : d$. The primary property of a proportion is that the product of the extremes equals the product of the means, expressed as $a \times d = b \times c$. This property is utilized to find unknown values within the proportion.

Scale and Proportional Relationships

Scale is the ratio of a measurement on a map to the actual corresponding measurement in reality. For instance, a scale of $1 : 100,000$ signifies that $1\,cm$ on the map represents $100,000\,cm$ (which is $1\,km$) in reality.

Functional relationships between quantities include direct and inverse proportionality. Direct proportionality (düz mütənasiblik) occurs when one quantity increases and the other increases by the same factor. This is represented by the formula $y = kx$, where $k$ is the proportionality coefficient. An example is the relationship between the quantity of a product and its total price. Inverse proportionality (tərs mütənasiblik) occurs when one quantity increases while the other decreases by the same factor, represented by the formula $x \times y = k$. An example of this is the relationship where speed increases and the time required for the journey decreases.

Percentages and Percentage Change

A ratio can be expressed as a percentage by multiplying the ratio by $100\%$. For example, the ratio of $5$ to $20$ is calculated as $\frac{5}{20} \times 100\% = 25\%$. This indicates that $5$ is $25\%$ of $20$. A percentage itself shows how many parts out of $100$ a value represents.

To calculate the percentage change of a quantity, one must find the difference between the new and initial values, divide that difference by the initial value, and then multiply by $100$. The formula is expressed as: $\text{Change Percentage} = \frac{\text{Difference}}{\text{Initial Value}} \times 100\%$.

Integers and Absolute Value

Integers comprise natural numbers, their negative opposites, and the number zero. This group is represented as $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$. It is essential to recognize that zero is neither positive nor negative. When comparing integers, every positive number is greater than zero and greater than any negative number. Among negative numbers, the number with the smaller absolute value is considered the larger number. For example, $-3 > -7$ because the absolute value of $-3$ is $3$, and the absolute value of $-7$ is $7$, and $3$ is the smaller distance.

The absolute value (modulus) of a number represents its distance from zero on the number line, denoted as $|a|$. For example, $|5| = 5$, $|-5| = 5$, and $|0| = 0$. The result of a modulus is always either positive or zero; it can never be negative.

Operations with Integers

When adding integers with the same sign, their absolute values are added and the common sign is kept, such as $(-4) + (-3) = -7$. When adding integers with different signs, the smaller absolute value is subtracted from the larger absolute value, and the sign of the number with the larger absolute value is applied, such as $(+8) + (-3) = +5$. To subtract one integer from another, the opposite of the second number is added to the first, following the rule $a - b = a + (-b)$. For example, $5 - (-3) = 5 + 3 = 8$.

For multiplication and division of integers, the sign of the result depends on the signs of the inputs. The product or quotient of two numbers with the same signs results in a positive ($+$) value: $(+) \times (+) = +$ and $(-) \times (-) = +$. The product or quotient of two numbers with different signs results in a negative ($-$) value: $(+) \times (-) = -$ and $(-) \times (+) = -$. For example, $(-4) \times (-2) = 8$ while $(-4) \times 2 = -8$. The general order of operations (Əməl Sırası) must be followed: Parentheses, followed by Multiplication and Division, then finally Addition and Subtraction. Small errors in signs can change the entire result.

Rectangular Coordinate System

A rectangular coordinate system consists of two perpendicular number lines that intersect at a single point. The horizontal line is called the OX axis (absis), and the vertical line is the OY axis (ordinat). The point where these axes intersect is the origin, denoted as $O(0;0)$. The position of any point is indicated by two numbers in the form $A(x;y)$, where $x$ is the abscissa and $y$ is the ordinate. For example, $A(3;2)$ indicates a position found by moving $3$ units to the right and $2$ units up. The distance ($d$) between two points on an axis is found via the modulus of their coordinate difference: $d = |a - b|$. For points $A(2)$ and $B(7)$, the distance is $|7 - 2| = 5\,vahid$. Distance is always positive.

Symmetry in the Coordinate Plane

Symmetry involves the mirror-like reflection of a figure or point relative to the axes. When reflecting over the OX axis, the $x$ coordinate remains unchanged while the $y$ coordinate changes its sign; for example, $A(3;2) \rightarrow A'(3;-2)$. When reflecting over the OY axis, the $y$ coordinate stays the same while the $x$ coordinate changes its sign; for example, $A(3;2) \rightarrow A'(-3;2)$. When reflecting relative to the coordinate origin, both the $x$ and $y$ coordinates change their signs; for example, $A(3;2) \rightarrow A'(-3;-2)$.

Sets and Operations

A set (çoxluq) is a group of objects combined according to a specific characteristic. These objects are called elements. Sets are typically denoted by uppercase letters (A, B, C…), while elements are shown with lowercase letters. For example, if $A = \{1, 2, 3, 4\}$, then $A$ is the set and $1, 2, 3, 4$ are the elements. A set with no elements is known as an empty set (boş çoxluq), symbolized as $\emptyset$.

Operations on sets include union, intersection, and difference. Union ($\cup$) consists of all elements from both sets; if $A = \{1,2,3\}$ and $B = \{3,4,5\}$, then $A \cup B = \{1,2,3,4,5\}$. Intersection ($\cap$) consists only of the elements common to both sets; if $A = \{1,2,3\}$ and $B = \{2,3,4\}$, then $A \cap B = \{2,3\}$. Difference ($\setminus$) consists of elements that are in the first set but not in the second; if $A = \{1,2,3,4\}$ and $B = \{3,4\}$, then $A \setminus B = \{1,2\}$. The Euler-Venn diagram is a visual scheme using circles to illustrate the relationships and common elements between sets.

Algebraic Expressions and Equations

Expressions containing numbers, operation signs, and letters (variables) are called algebraic expressions. For instance, in the expression $3x + 5$, if $x = 2$, the value is calculated as $3 \times 2 + 5 = 11$. When opening parentheses, if there is a plus ($+$) sign in front, the internal signs remain the same: $a + (b - c) = a + b - c$. If there is a minus ($-$) sign in front, all internal signs change: $a - (b + c) = a - b - c$. In this scenario, plus becomes minus, and minus becomes plus. Simplification involves combining like terms, which are terms with the same variable and degree. For example, $7x - 3x + 2 = 4x + 2$.

An equation (tənlik) is an equality containing a variable. Solving an equation means finding the value of the variable that makes the equality true. Unknowns are moved to one side and known numbers to the other; during this transfer, the operation signs change: plus to minus and minus to plus. For example, $x + 5 = 12 \rightarrow x = 12 - 5 = 7$. To solve problems by constructing equations: choose an unknown quantity as $x$, build the equation based on the conditions, solve it, and verify the answer. For example, if adding $8$ to a number results in $15$, the equation is $x + 8 = 15 \rightarrow x = 7$.

Inequalities

Inequalities (bərabərsizlik) are expressions written with the signs $<$, $>$, $\le$, or $\ge$ to show comparison. The same number can be added to or subtracted from both sides. While multiplying or dividing by a positive number does not change the inequality sign, multiplying or dividing by a negative number reverses the direction of the inequality sign. For example, in the inequality $3 < 5$, multiplying both sides by $-1$ results in $-3 > -5$.

Triangles and Geometric Relations

A median is a line segment from a vertex to the midpoint of the opposite side, dividing that side into two equal parts. A bisector (tənbölən) is a ray that divides an angle into two equal parts. A height (hündürlük) is a perpendicular segment dropped from a vertex to the opposite side (or its extension), creating a $90^{\circ}$ angle. Every triangle contains three medians, three bisectors, and three heights, each set intersecting at a single point respectively.

Triangles are congruent if all their corresponding sides and angles are equal. The signs of congruence are: TTT (SSS) where three sides are equal, TBT (SAS) where two sides and the included angle are equal, and BTB (ASA) where one side and its two adjacent angles are equal.

Parallel lines are lines in the same plane that never intersect, written as $a \parallel b$. When a transversal cuts two parallel lines, internal alternate angles are equal, and the sum of internal one-sided angles is $180^{\circ}$. The sum of the interior angles of any triangle is always $180^{\circ}$. An exterior angle of a triangle is equal to the sum of the two interior angles not adjacent to it. To construct a triangle, the triangle inequality must be satisfied: the sum of any two sides must be greater than the third side ($a+b>c$, $b+c>a$, $a+c>b$). For example, with sides $5, 4, 3\,sm$, since $5+4>3$, $5+3>4$, and $4+3>5$, a triangle can be constructed.

Area and Volume Formulas

The area ($S$) of a triangle is calculated as $S = \frac{a \times h}{2}$, where $a$ is the side and $h$ is the height drawn to that side. For a right-angled triangle, the area is half the product of the legs ($S = \frac{a \times b}{2}$). For a parallelogram, the area is $S = a \times h$. A rhombus, being a special parallelogram, has its area calculated either as $S = a \times h$ or as half the product of its diagonals: $S = \frac{d_1 \times d_2}{2}$.

The circumference ($L$) of a circle is calculated as $L = 2\pi r$, and the area ($S$) of a disk as $S = \pi r^2$, where $r$ is the radius and $\pi ≈ 3.14$. As the radius increases, the area of the disk increases more rapidly than the radius.

The volume ($V$) represents the space inside a 3D figure, measured in units like $sm^3$, $dm^3$, or $m^3$. For a prism, the volume is the product of the base area and the height: $V = S_o \times h$. For a cylinder, the volume is $V = \pi r^2 h$. The total surface area of a prism is the sum of the areas of its two bases and its lateral surface. The total surface area of a cylinder is expressed as $S = 2\pi r^2 + 2\pi rh$.

Statistics and Probability

In statistics, the moda is the number that repeats most frequently in a data set. For the set $2, 4, 5, 5, 7, 8$, the moda is $5$. The median is the value in the center of an ordered list (ascending or descending). For an odd number of values (e.g., $3, 5, 8, 10, 12$), the median is the middle value ($8$). For an even number of values (e.g., $2, 4, 6, 8$), the median is the average of the two middle values: $\frac{4+6}{2} = 5$.

Probability ($P$) addresses random events (təsadüfi hadisə), which may or may not occur under the same conditions. A certain event (yəqin hadisə) is one that must happen, such as the sun rising in the morning. An impossible event is one that can never happen, such as rolling an $8$ on a standard die. A random event is one that might happen, such as rolling a $6$ on a die. The formula for probability is $P = \frac{m}{n}$, where $m$ is the number of favorable outcomes and $n$ is the number of all possible outcomes. For rolling an odd number on a die, $n=6$ and favorable outcomes are $1, 3, 5$ ($m=3$), so $P = \frac{3}{6} = 0.5 = 50\%$. Probability always falls in the range $0 \le P \le 1$.

Data can be visualized using different types of charts: bar diagrams (sütunlu diaqram) are for comparison and identifying differences; line graphs (xətti qrafik) are for tracking changes over time (increase/decrease); and circle graphs (dairəvi diaqram) show the division into individual parts, where the total sum equals $100\%$ or $360^{\circ}$.