Solving Integer Operations by Using The Cop-Villain Model

Integer operations or also named whole number as a fundamental concept of math seems easy as 123 but at the same time can be tricky. To conquer this topic, memorizing the formulas is a boring and outdated way. Then, how to accomplish it in a content way? Let's dive into the explanation.

Before we go any further, we need to know what integer operations are. Integer consisting of positive integers, zero and negative integers. Positive integers are on the right and are greater than zero. Whereas, negative integers are located to the left and their value is smaller than zero. Next, these numbers are usually collected in a mathematical equation where the result is definetely a whole number, neither decimal nor fraction. This operation include addition, substraction, multiplication and division. So, integer operations are mathematical operations that are specifically related to integers.

Back to the topic about a way to solve integer operations, here is the formulas. Look at the column below :

Integer Operation Formulas (personal doc.)

We don't need to memorize all of them, it is not effective way. Instead of sapping it right away, studying it through positive and negative sign is the best way to go. We can use real life situation analogy which is easier to understand. "Negative" word, things that come to many people's minds are generally not good, bad or deviant. So, let's assume it's a "criminal / villain". Then, "positive" is often interpreted as something good or according to the dictionary means firm. Therefore, "positive" can be analogous to the "cop / police". This method for completing integer operations is called "The Cop-Villain" model.

Without any further do, below is an example of a question in applying the cop-villain model :
We are going to divide it into two groups based on how the models work, that is :

1. Addition and Substraction
In these two subtopics if an equation has different signs, it means cops and villains are "battle" towards each other. However, if the equation has the same signs, the cops or villains they are companion. Here is the application of the model :
33 + (-12) = 21
If we change to the model, there are 33 cops with 12 villains. Since the equation has different sign, then the cops and villains battle and automatically their each number will diminish. Because the number of cops are bigger, so the winner is the cops that remain 21. The answer is 21
-9 - (-7) = -2               -> -9 + 7 = -2
In this case, we can simplify two adjacent negatives by combining it. It becomes 9 villains battle with 7 cops and remains 2 villains.

2. Multiplication and Division
Specifically for mulplication and division have the difference to solve it. We can't do the same way as addition and substraction. In these subtopics, thing that we have to pay attention the most is about the signs. For the numbers we just need to multiply it as common mulplication (+ × + = +) or division. How to solve it?  See the explanation below :
4 × 3 = 12
First, we know that the result is 12. Afterward, look at the sign they are all positives. You can imagine 4 cops join with other 3 cops by multiplying it to fight the villain, resulting 12 cops. Based on the explanation above, positive times positive equal positive.
8 × (-7) = -56
Next, we have 8 cops against 7 villains, leading to a negative result. It influences the positive efforts of cops. So, the answer is -56.
-6 ÷ 3 = -2
We can imagine 6 villains are divided into 3 groups of cops. So, every cop handles 2 villains.
-16 ÷ -2 = 8
This kind of equation has slightly difference. There are 16 villains divided into 2 groups. Each groups consist of 8 villains who work together. Because they work together the results are positive.
Now, we try more complex equation. Here it is :
14 - 9 {-9 + (8)} =
Note :
When working on the question above, there are three points to remember, including:
1. If there is multiplication or division in integer operations, it must be calculated at the beginning;
2. If there is an operation given in parenthesis, then it is carried out first;
3. If both are present, do the operation in parenthesis first.
Solution :
First, count operation in parenthesis. There are 9 villains battle with 8 cops, leaving 1 villain. Second, 1 villain join with other 9 villains resulting positif outcome, that is 9. Last, the 9 cops join with other 14 cops, the result is 23.

That's an explanation of integer operations using "the cop-villain" model. We can understanding this basic concept in a pleasure way without having to memorize it.