Dividing Integers - Sixth Grade Mathematics

Dividing integers represents the inverse of multiplication, sharing identical sign patterns and establishing rules for distributing signed quantities equally.

The Inverse Relationship with Multiplication

Because division is the inverse operation of multiplication, every division equation corresponds directly to a related multiplication equation.

Division Equation:           Related Multiplication Equation:
       12 / 3 = 4            because   4 * 3 = 12
      -12 / 3 = -4           because  -4 * 3 = -12
      -12 / (-3) = +4        because  +4 * (-3) = -12
       12 / (-3) = -4        because  -4 * (-3) = 12

The Sign Rules for Division

Because division mirrors multiplication, the sign rules for integer division are identical to the multiplication rules.

+-------------------------------------------------------------------------+
|                   RULES FOR DIVIDING TWO INTEGERS                       |
+-------------------+-----------------------+-----------------------------+
| Operation Signs   | Sign of Quotient      | Concrete Examples           |
+-------------------+-----------------------+-----------------------------+
| Positive / Pos    | ALWAYS POSITIVE (+)   | 36 / 4 = +9                 |
+-------------------+-----------------------+-----------------------------+
| Negative / Neg    | ALWAYS POSITIVE (+)   | -36 / (-4) = +9             |
+-------------------+-----------------------+-----------------------------+
| Positive / Neg    | ALWAYS NEGATIVE (-)   | 36 / (-4) = -9              |
+-------------------+-----------------------+-----------------------------+
| Negative / Pos    | ALWAYS NEGATIVE (-)   | -36 / 4 = -9                |
+-------------------+-----------------------+-----------------------------+

Division Rule Summary:
- Dividing two numbers with the SAME SIGNS yields a POSITIVE quotient (+).
- Dividing two numbers with DIFFERENT SIGNS yields a NEGATIVE quotient (-).

Special Cases Involving Zero

Zero exhibits two strict mathematical behaviors under division:

Zero in the Dividend:          0 / a = 0   (where a is not zero)
                               Example: 0 / (-8) = 0 because 0 * (-8) = 0.

Zero in the Divisor:           a / 0 is UNDEFINED
                               Example: -8 / 0 cannot be solved!

Why Division by Zero is Undefined

If -8 / 0 equaled some number q, then by definition of division, q * 0 would have to equal -8. However, any number multiplied by zero equals zero (q * 0 = 0). It is impossible for q * 0 to equal -8. Therefore, dividing any number by zero is strictly undefined in mathematics.

Real-World Rate Applications

Integer division models the average rate of change over time: Over a 4-hour period, the temperature in Alaska fell by a total of 28 degrees (-28 degrees). The average hourly temperature change is (-28) / 4 = -7 degrees per hour. A company recorded a cumulative loss of $45,000 over 5 months. The average monthly performance is (-45,000) / 5 = -$9,000 per month.

Exercises

Exercise 1. Calculate each quotient: (a) (-48) / 6; (b) (-72) / (-8); (c) 100 / (-25); (d) 0 / (-14); (e) (-35) / 0.

Exercise 2. Evaluate the multi-step expression: [ (-8) * (-9) ] / (-6).

Exercise 3. A scuba diver descends 135 feet in 9 equal stages. Write a division equation to determine the change in elevation during each stage.

Solutions

Solution to Exercise 1. For (a) (-48) / 6: different signs produce a negative quotient: -8. For (b) (-72) / (-8): same signs produce a positive quotient: +9. For (c) 100 / (-25): different signs produce a negative quotient: -4. For (d) 0 / (-14): zero divided by any non-zero number is 0. For (e) (-35) / 0: division by zero is strictly undefined.

Solution to Exercise 2. In [ (-8) * (-9) ] / (-6), perform the multiplication in the brackets first: (-8) * (-9) = +72 (same signs yield positive). Now divide: 72 / (-6). Different signs produce a negative quotient: 72 / (-6) = -12.

Solution to Exercise 3. The total descent is represented by the integer -135 feet. Dividing this distance across 9 equal stages gives the equation: (-135) / 9 = -15 feet. During each stage, the diver's elevation changes by -15 feet (a descent of 15 feet).