Essential Question

How are integers and absolute value used in real-world situations?

Integer Basics

To use integers to represent data, we need to understand that integers are positive whole numbers, their opposites, and zero. Negative integers are used to represent data less than zero, while positive integers are used to represent data greater than zero. For example, when asked to write an integer to represent 12 feet below sea level, the answer would be -12. The integer 0 represents something at sea level. Similarly, to represent winning 3 tokens in a game, the integer would be 3, while 0 represents not winning any tokens.

Graphing Integers

When we graph integers on a number line, we place positive numbers to the right and negative numbers to the left of zero. For example, when graphing -2 on a number line, it is placed to the left of 0. We can also graph a set of integers, such as (-3, 1, 0), on a number line, where -3 is to the left of 0, and 1 is to the right. Similarly, we can graph the set (-4, 1, 3) on a vertical number line.

Representing Absolute Value

To represent absolute value on a number line, we need to understand that absolute value is always positive or zero. For example, when finding the opposite of -12, we can use a number line to graph -12, which is 12 units to the left of zero. Therefore, the opposite of -12 is 12. Another method to find the opposite of a negative integer is to use symbols. The opposite of a negative is always a positive, so the opposite of -12 is +12 or simply 12.

Examples of Absolute Value

When evaluating the absolute value of -18, the absolute value is 18, as distance cannot be negative. Similarly, the absolute value of 5 is 5.

Comparing and Ordering Integers

To compare integers, we can compare the signs as well as the magnitude of the numbers. Greater numbers are graphed to the right on a number line. For example, when comparing 2 and 6, we graph both numbers and then compare their positions on the number line.

Ordering Integers

To order integers, we can use a number line and graph the numbers from least to greatest, or we can compare the signs and values of the numbers to determine their order.

Understanding Terminating and Repeating Decimals

Terminating decimals are the decimal form of rational numbers, which have a group of one or more digits that do not repeat. On the other hand, repeating decimals are the decimals form of rational numbers, which have a group of one or more digits that repeat indefinitely.

Examples and Small Exercises

Examples of terminating and repeating decimals are provided to understand the concept better. These examples include writing rational numbers as decimals and using bar notation to indicate repeating patterns in decimals.

Comparing and Ordering Decimals and Fractions

In this section, we learn how to compare decimals and fractions and how to compare and order rational numbers. We can represent decimals as fractions and then compare the fractions to determine their order. Graphs and exercises are used to illustrate this process.

Order Rational Numbers

To order a set of rational numbers, we can write fractions as decimals, graph all decimals on a number line, and then determine their order from least to greatest. This helps in better understanding how to compare and order rational numbers in different forms.

By learning how to use integers to represent data and how to represent absolute value on a number line, we can better understand and apply these concepts in real-world situations. These skills are essential in various aspects of life, including financial transactions, sports, and daily measurements.Integers can also be found in jobs such as accounting, mathematics, and physics, where accurate representation and calculation of data are essential. Additionally, the concept of graphing integers on a number line is vital in understanding number relationships and making comparisons. Overall, understanding how to use integers correctly and where they are used in daily life can greatly improve our ability to analyze and interpret data.