Adding Integers - Sixth Grade Mathematics

Adding integers combines positive and negative quantities, modeled through two-color counters and number line vector movement to establish dependable operational rules.

The Two-Color Counter Model and Zero Pairs

A physical model for integer addition uses two-color counters: positive integers are represented by yellow counters (+), and negative integers are represented by red counters (-).

Zero Pair Neutralization:
       ( + )   +   ( - )   =   0  (Neutral Zero Pair!)

Case 1: Adding Integers with the Same Sign

When adding two integers with identical signs, combine the counters. The total count increases, and the sign remains unchanged.

Example A: 3 + 4
Combine 3 positive counters and 4 positive counters:
( + ) ( + ) ( + )  and  ( + ) ( + ) ( + ) ( + )  =  7 positive counters (+7).

Example B: (-3) + (-4)
Combine 3 negative counters and 4 negative counters:
( - ) ( - ) ( - )  and  ( - ) ( - ) ( - ) ( - )  =  7 negative counters (-7).

Case 2: Adding Integers with Different Signs

When adding integers with different signs, pair positive and negative counters to create zero pairs. Remove all neutral zero pairs; the remaining counters represent the final sum.

Example C: (-6) + 4
Place 6 negative counters and 4 positive counters:
( - ) ( - ) ( - ) ( - )   ( - ) ( - )   <-- 6 negatives
( + ) ( + ) ( + ) ( + )                 <-- 4 positives
|------- ZERO PAIRS -------|   REMAINDER
   4 Zero Pairs cancel!       2 Negatives remain!

Result: (-6) + 4 = -2.

The Number Line Vector Model

On a number line, integer addition is visualized as sequential vector movements: Step 1: Start at zero. Step 2: Move to the location of the first addend. Step 3: From that location, look at the second addend: if positive, move RIGHT; if negative, move LEFT. Step 4: Your final landing position is the sum.

Visualizing (-4) + 7 on a Number Line:
                     <-- start at 0, move 4 left to -4
                     From -4, move 7 units RIGHT -->
<---|---|---|---|---|---|---|---|---|---|---|--->
   -5  -4  -3  -2  -1   0   1   2   3   4   5
        ^                           ^
      Start                      Landing Point (+3)

Result: (-4) + 7 = +3.

The Formal Rules for Adding Integers

The counter and number line models condense into two definitive mathematical rules.

+-------------------------------------------------------------------------+
|                        RULES FOR ADDING INTEGERS                        |
+-------------------+-----------------------------------------------------+
| Condition         | Rule and Procedure                                  |
+-------------------+-----------------------------------------------------+
| SAME SIGNS        | 1. ADD their absolute values: |a| + |b|.            |
| (+)+(+) or (-)+(-) | 2. KEEP the common sign for the sum.               |
|                   | Examples: 5 + 8 = 13;   (-5) + (-8) = -13           |
+-------------------+-----------------------------------------------------+
| DIFFERENT SIGNS   | 1. SUBTRACT the smaller absolute value from the     |
| (+)+(-) or (-)+(+) |    larger absolute value: |larger| - |smaller|.     |
|                   | 2. ATTACH the sign of the number with the GREATER   |
|                   |    absolute value to the result.                    |
|                   | Examples: (-9) + 4 = -5;   12 + (-7) = +5           |
+-------------------+-----------------------------------------------------+

Exercises

Exercise 1. Calculate each sum: (a) (-8) + (-14); (b) 15 + (-9); (c) (-22) + 22; (d) (-17) + 6.

Exercise 2. Use the concept of zero pairs to explain why (-5) + 8 equals +3.

Exercise 3. At 6:00 AM, the temperature was -8 degrees Fahrenheit. By noon, the temperature had risen by 15 degrees. Write an addition equation and determine the temperature at noon.

Solutions

Solution to Exercise 1. For (a) (-8) + (-14): both integers have the same negative sign; add absolute values: 8 + 14 = 22; attach the common negative sign: -22. For (b) 15 + (-9): different signs; subtract absolute values: 15 - 9 = 6; since 15 has the larger absolute value and is positive, the sum is +6. For (c) (-22) + 22: opposites (additive inverses) sum to 0. For (d) (-17) + 6: different signs; subtract absolute values: 17 - 6 = 11; since |-17| > |6|, attach the negative sign: -11.

Solution to Exercise 2. Model -5 with 5 negative counters and +8 with 8 positive counters. Pair each negative counter with a positive counter: this forms 5 zero pairs, each having a value of zero (5 * 0 = 0). After removing the 5 zero pairs, there are 3 positive counters left over. Therefore, (-5) + 8 = +3.

Solution to Exercise 3. The starting temperature is -8 degrees. A rise of 15 degrees is represented by adding +15. The addition equation is: (-8) + 15 = 7 degrees Fahrenheit. Subtract absolute values: 15 - 8 = 7; since 15 is positive, the result is positive 7 degrees Fahrenheit.