We will be modeling addition with positive and negative numbers by talking about balloons and sand bags that are tied to a basket. Hot air balloons tied to a basket raise the basket. Sand bags tied to the basket lower the basket. Each problem begins with the basket hovering in the clouds. We do not care how high the clouds, in which the basket is hovering. We only care how much higher or lower than the clouds the basket ends up when the problem is finished.
While using balloons and sand bags to model addition of positives and negative numbers, we will assume each balloon raises the basket one foot. Each sand bag lowers the basket by one foot.
When modeling addition, we need to imagine that the basket begins in the clouds and then items, balloons or sand bags, are tied to the basket. Where the basket ends up relative to the clouds at the end of the problem is the answer.
Example 1
(+3) + (+5) means three balloons are tied to the basket which raises the basket three feet above the clouds. Then five more balloons are tied to the basket, which raises the basket five more feet. In all, the basket is now eight feet above the clouds. Therefore, the answer to (+3) + (+5) is +8.
Example 2
What does the addition problem (-3) + (+5) mean?
Example 3
(+3) + (-5) =
Example 4
(-3) + (-5) =
Example 5
(-6) + (+6) =
This last problem shows a total change of zero. This is called the additive inverse.
Directions: Use the blue and purple sliders to create a problem. Notice how the arrows change. Use the "hint" slider to show (or hide) the balloons and sandbags. |
Duane Habecker, Created with GeoGebra |
Using balloons and sandbags to solve addition problems will always work. However, it might be considered a little childish if you are still using balloons and sandbags to answer questions while in high school! Let's look at the results we have gathered so far to find two rules that will make adding negative numbers (especially ones with large absolute values) easier.
(+3) + (+5) = +8 |
(-3) + (+5) = +2 |
(-3) + (-5) = -8 |
(+3) + (-5) = -2 |
If you look closely, you will see the two rules for adding negative numbers:
When adding two numbers with the same sign: |
When adding two numbers with different signs: |
Add the absolute values of the numbers and attach the sign of the numbers to the sum. |
Subtract the absolute values of the two numbers and attach the sign of the number with the largest absolute |
Let's practice using the rules.
Example 1 | Example 2 | Example 3 |
(-3) + (-8) = The signs are the same, so we just add the two absolute values (3 + 8) and then attach the negative sign to the sum. So the answer is –11. |
(-5) + (+9) = The signs are different, so we subtract the absolute values (9 – 5 = 4) and then attach the positive sign since +9 has a larger absolute value than –5. So the answer is +4. |
(-52) + (+35) = The signs are different, so we subtract the absolute values (52 – 35 = 17) and then attach the negative sign since -52 has a larger absolute value than +35. So the answer is -17. |
Q1: $\,\,\,\,\,$-5 + -4 = $\,\,\,\,\,$ [show answer]
Q2: $\,\,\,\,\,$-8 + (+6) = $\,\,\,\,\,$ [show answer]
Q3: $\,\,\,\,\,$-15 + (+19) = $\,\,\,\,\,$ [show answer]