Subtracting Integers - Sixth Grade Mathematics

Subtracting integers investigates the removal of signed quantities, using the universal principle that subtraction is equivalent to adding the opposite (additive inverse).

The Fundamental Equivalence: Adding the Opposite

In mathematics, subtraction is not an independent operation; it is formally defined as the inverse of addition. Subtracting any integer is mathematically identical to adding its opposite.

The Universal Subtraction Rule:
                        a - b = a + (-b)

Keep the first number (a).
Change subtraction (-) to addition (+).
Change the second number (b) to its OPPOSITE (-b).
(Commonly remembered as Keep-Change-Change: KCC).

Examples of Transforming Subtraction to Addition:
- Problem:  8 - 12      -->  Transform:  8 + (-12)     =  -4
- Problem: -5 - 9       -->  Transform: -5 + (-9)      = -14
- Problem:  6 - (-4)    -->  Transform:  6 + (+4)      =  10
- Problem: -7 - (-11)   -->  Transform: -7 + (+11)     =  +4

Why Subtracting a Negative Equals Adding a Positive

Why does subtracting a negative number produce addition? Consider financial debt. If you owe a friend $10, your financial status has a value of -$10. If that friend decides to forgive (remove or subtract) that $10 debt, you are executing: Original Balance - (-$10). Removing a debt has the exact same financial impact as giving you $10 in cash! Subtracting a negative quantity increases your total value.

The Counter Model: Introducing Zero Pairs to Subtract

When using two-color counters, subtraction means physically taking away counters.

Evaluate: 3 - (-2)  (Start with 3 positives; take away 2 negatives)
Problem: You have 3 positive counters, but NO negative counters to take away!

Resolution: Add 2 ZERO PAIRS without changing the value:
            ( + ) ( + ) ( + )   [ ( + ) ( - ) ]   [ ( + ) ( - ) ]
            |--- 3 wholes --|   |------- 2 Zero Pairs --------|

Now TAKE AWAY the 2 negative counters:
            ( + ) ( + ) ( + )   ( + )             ( + )
            Count remaining: Exactly 5 positive counters (+5)!

Conclusion: 3 - (-2) = 3 + 2 = 5.

Number Line Movement for Subtraction

On a number line, subtraction can also be viewed as finding the directed distance from the subtrahend to the minuend, or as facing left (the negative direction) and stepping forward or backward. Facing convention: Start at the first number. Subtraction instructs you to face the negative direction (left). Subtracting a positive number: Walk forward into negative territory (move left). Subtracting a negative number: Walk backward into positive territory (move right).

Evaluating 2 - 6:
Start at 2. Subtract positive 6: Move 6 units LEFT.
Landing point is -4.
Equation: 2 - 6 = -4.

Exercises

Exercise 1. Transform each subtraction expression into an equivalent addition expression and calculate the result: (a) 14 - 23; (b) (-9) - 15; (c) 12 - (-8); (d) (-6) - (-14).

Exercise 2. The summit of a mountain is 3,200 feet above sea level. A nearby deep valley floor is 180 feet below sea level. Write a subtraction expression to calculate the vertical elevation difference between the summit and the valley floor, and evaluate it.

Exercise 3. Use the two-color counter zero pair method to explain why (-2) - 3 equals -5.

Solutions

Solution to Exercise 1. For (a) 14 - 23: keep 14, change to addition, opposite of 23 is -23: 14 + (-23) = -9. For (b) (-9) - 15: keep -9, change to addition, opposite of 15 is -15: (-9) + (-15) = -24. For (c) 12 - (-8): keep 12, change to addition, opposite of -8 is +8: 12 + 8 = 20. For (d) (-6) - (-14): keep -6, change to addition, opposite of -14 is +14: (-6) + 14 = 8.

Solution to Exercise 2. The elevation of the summit is +3,200 feet. The elevation of the valley floor is -180 feet. To find the difference in elevation, subtract the valley elevation from the summit elevation: 3,200 - (-180). Applying the subtraction rule: 3,200 + 180 = 3,380 feet. The vertical elevation difference between the summit and the valley floor is 3,380 feet.

Solution to Exercise 3. Start with 2 negative counters representing -2: (-)( -). You need to take away 3 positive counters, but there are none present. Introduce 3 zero pairs: (+)(-), (+)(-), (+)(-). The collection now has 2 negatives plus 3 zero pairs, which contains 5 negatives and 3 positives total. Now take away the 3 positive counters. Exactly 5 negative counters remain. Therefore, (-2) - 3 = -5.