Section 1: Rational Numbers

Gold Price Analysis and Statistical Evaluation

The text provides data regarding the annual fluctuations in gold prices to illustrate the use of rational numbers in world markets. In world exchanges, all transactions are processed using rational numbers. A specific case study examines the change in gold prices from 2017 to 2022. In 2022, the price of gold decreased by $1.20$ manat. The table of price changes (in manat) for the preceding years is as follows:

For the year 2017, the change was $+8$ manat. In 2018, there was a decrease of $-2$ manat. In 2019, the price rose by $+5.5$ manat. In 2020, the increase was $+1.5$ manat. In 2021, the price decreased by $-3$ manat. To find the arithmetic mean of the changes over the period from 2017 to 2022, one must sum all the annual changes, including the $-1.20$ manat decrease from 2022, and divide by the total number of years ($6$ years).

Initial Assessment of Rational Number Fundamentals

The material includes several exercises designed to test the comprehension of equivalent fractions and basic operations. The first exercise requires determining the appropriate numbers for blank cells in fractional equalities: a) $\frac{14}{\text{blank}} = \frac{7}{4}$, b) $\frac{\text{blank}}{32} = \frac{8}{\text{blank}} = \frac{44}{33}$, c) $\frac{24}{\text{blank}} = \frac{8}{3}$, and d) $3 \frac{3}{12} = \frac{\text{blank}}{4}$.

Rounding and digit identification are also explored. Students are tasked with rounding the following rational numbers to specified places: $3.273$, $0.165$, and $12.996$. Furthermore, missing digits in decimal representations of fractions must be identified: a) $\frac{1}{8} = 0.1\text{blank}5$, b) $\frac{2}{3} = 0.6\text{blank}$, c) $\text{blank} = 0.35$, and d) $0.6 = \frac{\text{blank}}{20}$.

Comparison and Ordering of Rational Numbers

Comparing and ordering rational numbers is a core competency presented in the text. Comparison exercises involve pairs such as: a) $7.534$ and $8.785$, b) $0.548$ and $0.55$, c) $\frac{3}{8}$ and $0.41$, and d) $\frac{18}{25}$ and $0.72$.

Ordering exercises are divided into ascending and descending sequences. In ascending order, students must sort: a) $0.64$, $0.48$, and b) $-1.5$, $0$, $-6.8$. In descending order, the sequences provided are: a) $0.6$, $0.4$ and b) $3$, $-9$, $0.7$, $-4$.

Evaluation of Mathematical Expressions and Equations

The text outlines procedures for calculating the value of complex expressions involving rational numbers. Exercises include: a) $2.4 - (-\frac{3}{5})$, b) $(-\frac{2}{3}) \div 1.3$, c) $(2 - 0.6) \times (4.2 + 3)$, and d) $\frac{3.8 - 5}{1.5}$. Calculations with variables are also detailed for specific expressions: a) For $2x + 6$, find values when $x = -3$, $-7$, and $1.5$. b) For $7 - 3x$, find values when $x = -4$ and $5$. c) For $4a - b$, find the value when $a = -2$ and $b = -9$.

Equation solving is a necessary skill for working with rational numbers. The textbook provides the following equations for practice: a) $2x + 1.4 = 31.2$, b) $3(6x + 7) = -15$, and c) $4x + 17 = 7x - 25$.

Physical Quantities and Applied Problems

Problem 8 focuses on finding portions of given quantities: a) Finding $\frac{4}{5}$ of $4.5\,cm$, b) finding $0.3$ of $5.6\,kg$, c) finding the total mass of flour when $\frac{3}{5}$ equals $1.8\,kg$, and d) finding the capacity of a container when $0.8$ of its volume is $1\,L$.

The environmental and physical application of rational numbers is demonstrated through a temperature change problem. At $12:00$, the air temperature was $3\,^\circ C$. Starting from this time, the temperature began to decrease by $2\,^\circ C$ every hour. Students are asked to calculate the temperature at $14:00$ and determine at what time the air temperature reached $-5\,^\circ C$.

Automotive efficiency is another practical application mentioned. A hybrid car has a fuel tank capacity of $50\,L$. It consumes $4.8\,L$ of fuel for every $100\,km$ traveled. If the tank is $\frac{3}{5}$ full, the objective is to determine how much distance the car can cover with the current amount of fuel.

Introduction to Rational Numbers

Section 1.1 formally introduces the concept of rational numbers. Key vocabulary terms highlighted include "negative fractions" (mənfi kəsrlər) and "rational numbers" (rasional ədədlər). A conceptual scenario involves the positions of a market, a museum, and a school bus stop, which are located at equal distances from each other. The lesson poses a question on how to determine the coordinate of one object based on the coordinate of another, emphasizing the spatial application of rational numbers on a number line.