We will use hops on a number line to model addition of positive and negative numbers. When adding a positive number we move to the right on the number line. Adding negative numbers moves us to the left on the number line.
Why do we move to the left when adding negative numbers? Read on…
Example 1: Simplify 5 + 3
To model this problem we use two hops to the right. The first hop is 5 units to the right. Then hop 3 units to the right. Since we end up on 8, the answer is 8.
5 + 3 = 8
Example 2: Simplify 5 + 1
To model this expression, we first hop 5 units to the right. Then we hop 1 unit to the right. Since we end up on 6, the answer is 6.
5 + 1 = 6
Example 3: Simplify 5 + (-2)
To model this problem, we hop 5 units to the right. The second hop is 2 units to the left (please see note below). Since we end up on 3, the answer is 3.
5 + (-2) = 3
Example 4: Simplify 5 + (-8)
To model this expression, we begin with a 5-unit hop to the right. The we hop 8 units to the left. Since 8 is bigger than 5, we cross over the 0 into the negative side of the number line. We end up at -3, so the answer is -3.
5 + (-8) = -3
Example 5: Simplify (-8) + 5
Since addition is commutative, (-8) + 5 should result in the same answer as the previous example 5 + (-8). The model of (-8) + 5 means a hop of 8 units to the left followed by a hop of 5 units to the right resulting in the same sum as the previous example.
(-8) + 5 = -3
Example 6: Simplify (-6) + 10
To model this expression, we begin by hopping 6 units to the left. Then we hop 10 units to the right. Since we end up at 4, the answer is 4.
(-6) + 10 = 4
Simplify: (-6) + (-2)
To model this problem, we make two hops to the left. The first hop is 6 units to the left. Then we hop 2 units to the left. We end up at -8.
(-6) + (-2) = -8
Simplify: (-10) + (-5)
To model this expression, we make a 10-unit hop to the left, followed by a 5-unit hop to the left.
In the example 5 + (-2) we saw that it was modeled with a 5 unit-hop to the right and then a 2 unit-hop to the left. How can we be certain that (-2) represents a hop to the left?
Let’s look at this table of problems and look for a pattern.
5 + 3 = 8
As the second number goes down, so does the sum. The pattern verifies that 5 + (-2) is equal to 3. This means (-2) is accurately represented by a hop to the left.