Representations of Number Operations

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, learn how you can determine which number operation to use when you see different problems laid out in different ways. Decipher between graphic problems, picture problems, and word or verbal problems.

Number Operations

In math, there are only four basic number operations. They are addition, subtraction, multiplication, and division. Addition is when you have numbers that are grouped together and you are asked to find the total sum of the numbers. Subtraction is when you begin with a number and then are asked to take numbers away from it. Multiplication is when you have repeated addition. Division is when you are asked to split a number into a number of equal groups.

Math problems can be presented in one of four ways or a combination of these four.

  • Symbolically: Problems given using math numbers and symbols. For example, 2 + 3.
  • Graphically: Problems shown via a graphic.
  • Pictorially: Problems shown via a picture.
  • Verbally: These are your typical word problems.

Let's go over these one by one.


The first way and the most common is when math problems are presented symbolically. This is your typical worksheet of problems where students get a chance to practice their knowledge of math facts. Here are some examples of problems.

number operations

With this method, you are familiar with the symbols used to represent the four basic math operations. The cross, +, symbol is used to represent addition. The dash, -, is used to represent subtraction. The dash with a dot on top and bottom is used to denote division. Division can also be noted with a slash, as in 81 / 9, or as a fraction with a horizontal line separating the two numbers, one on top and one on bottom. And an x is used to denote multiplication. Multiplication can also be noted with an asterisk, as in 3 * 8, or with a dot in between the numbers.


When presented graphically, you'll find a graphic with your math problem. Perhaps, it might look like this:

number operations

The instructions may ask for the number of stars and cans together.

To solve this type of problem, you need to first make sense of the graphic. Then, once you can see what is going on in the graphic, you can then go ahead and figure out how to find the answer to the problem. In this case, the problem is asking for the number of stars and cans together. You can go ahead and count the stars and cans together (8) or count the stars and cans separately and then add them together (4 + 4 = 8).


Pictorial problems may picture a scenario with some given information and a piece of missing information you need to find. Pictorial pictures may use actual photos, or they may use simplified drawings of real world objects, like this one for example.

  • Farmer John is now going to build a pen for his pigs. His finished pen will measure 30 feet by 18 feet. What is the total footage contained in the finished pen?

number operations

In this problem, instead of a real-world picture, a simplified drawing is used. In order to solve this problem, you need to be able to link the numbers with various parts of the picture. For this problem, you'll look at the simplified picture and realize that the longer side is the side that measures 30 feet and the shorter side the one that measures 18 feet. Then, to find the total footage, you'll realize that this requires repeated addition of the feet in each row or column of feet. So, you can multiply the feet of one side by the feet of the other side (30 * 18 = 540 square feet).


Last, verbal problems are the ones that may be the hardest for students to solve. These are your typical word problems. They usually don't come with any pictures. You have to come up with any helpful diagrams and pictures by yourself. You also have to come up with the mathematical expression by yourself too.

Here's an example.

  • Farmer John has just finished harvesting his apples from his apple orchard. From his whole orchard, he harvested 960 apples. He packs them in boxes of 12 apples. How many boxes does Farmer John need to pack all his apples?

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