Introduction to Negative Numbers (Integers): Fifth Grade Mathematics

Integers extend our number system to include positive whole numbers, zero, and their negative opposites, representing quantities below a reference baseline.

The Definition of Integers

The set of integers consists of all positive whole numbers (1, 2, 3, ...), zero (0), and negative whole numbers (-1, -2, -3, ...). Fractions and decimals are NOT integers.

+--------------------------------------------------------------------------+
|                  THE INTEGER NUMBER SYSTEM                               |
|                                                                          |
|       ... -4,   -3,   -2,   -1,    0,    +1,   +2,   +3,   +4 ...        |
|       <------------------------+---------------------------->            |
|            Negative Integers   |         Positive Integers               |
|            (Less than zero)    Zero      (Greater than zero)             |
|                              (Neither)                                   |
+--------------------------------------------------------------------------+

Real-World Contexts for Negative Numbers

Negative numbers represent values that exist below a designated zero benchmark:
- Temperature: Degrees below zero (-5°F is 5 degrees colder than freezing).

- Elevation: Feet below sea level (-200 feet is 200 feet beneath the ocean surface).

- Financial: Debt, withdrawals, or account overdrafts (-$40 indicates owing 40 dollars).

- Sports: Yards lost on a football play (-7 yards).

The Concept of Opposites

Every integer has an opposite located the exact same distance from zero on the number line, but on the opposite side.
- The opposite of +6 is -6.

- The opposite of -14 is +14.

- Zero is its own opposite (the opposite of 0 is 0).

Opposite Pair:
  -5 is 5 units to the left of 0.
  +5 is 5 units to the right of 0.
  Together, -5 and +5 are opposites.

Exercises and Solutions

Exercise 1: Real-World Integer Representations. Write an integer (including the positive or negative sign) to represent each real-world situation: Situation A: A submarine dives 350 meters below sea level. Situation B: A bank customer deposits $125 into savings. Situation C: The temperature drops to 8 degrees below zero. Situation D: A football team gains 15 yards on a play.

Solution 1: Situation A: 350 meters below sea level is represented by the integer -350. Situation B: Depositing $125 is an increase in money, represented by +125 (or 125). Situation C: 8 degrees below zero is represented by -8. Situation D: Gaining 15 yards is represented by +15 (or 15).

Exercise 2: Understanding Opposites and Absolute Position. State the opposite of each integer: -19, +45, 0, and -100. Explain why adding an integer and its opposite always equals zero.

Solution 2: The opposite of -19 is +19. The opposite of +45 is -45. The opposite of 0 is 0. The opposite of -100 is +100. Adding an integer and its opposite always equals zero because they represent equal movements in opposite directions along a number line. Moving 5 steps forward (+5) and then 5 steps backward (-5) returns you precisely to your starting point of 0.