Addition and Subtraction

An example is 1-3+7+4.  It is possible to group the terms of this sum.  Specifically, 1 minus 3 equals minus 2, and 7+4 equals 11.  Then one can add these two results and obtain the final answer.  In other words, -2 and 11 will add to 9.

To demonstrate this on a number line, one can start at the number 1.  It is important to keep in mind that as one moves toward the right, numbers become increasingly positive, and as one goes toward the left, numbers are more negative than they were before.  Therefore, with a start at 1, it is possible to move 3 integers to the left.  This will lead to a result of -2.

Then one can add 7 to this result.  In other words, one will move 7 integers or whole numbers along the number line and towards the right.  The result here is 5.

The final step to solve this sum of 4 numbers is to add 4 because it is the last term.  Hence, one will move 4 integers to the right, and the answer is 9.

Obviously, there are several approaches to reach the solution, but the final answer will always be the same in this instance.  It does not matter what order one uses for addition or subtraction, and it is acceptable to rearrange the integers in any way that one would like.  This is because the process of addition is commutative [1, 2].

When students work with number lines, they will encounter number opposites [2].  These are two numbers which are the same distance from the origin of the number line but are on opposite sides.  For example, -3 is the opposite of 3.  The two have the same absolute value, but they are different numbers.

Similarly, the opposite of -7 is 7.  The signs of two opposite numbers will therefore be opposites of each other.

It is also possible to take the opposite of a negative number or the negative of a negative value.  The negative of a negative number is the opposite of the opposite of that number.  In other words, - (-5) is the opposite of the opposite of 5, and it equals 5. 

The number - (-7) equals 7.  The opposite of - (-7) is the same as - (- (-7)).  Each time one adds a negative sign, the result will be on the other side of the number line’s origin.  In this case, the result will be -7 because - (-7) equals 7, and one must take the negative of that.

There is much that students can learn about addition and subtraction of numbers, and the use of a number line helps to make it easy.

1. Khan Academy.  (2016).  Adding and subtracting negative numbers.
2. PBS Math Club.  (2013).  Adding negative numbers/Mean girls and Darth Vader.
3. The photo is from Massachusetts Institute of Technology.