Sometimes pictures are easier to understand than numbers.
Sometimes pictures are easier to understand than numbers.

Solving a mathematics problem can often be frustrating, especially to students who are less comfortable with imagining abstract situations. The mathematical model can alleviate this difficulty, building a bridge between the mathematical world and the real world. A mathematical model is a simple, visual way of representing a problem so that both the situation and the solution become easier to imagine. While problems differ in what model should be used, you can apply a general strategy of model-building to many problems.

Reading: The Boring Part

Before you even choose a model to use, you need to wrap your head around the overall goings-on in the problem. For this phase, ignore the numbers and look at how changes occur in the problem. Because many basic math problems involve changes over time or contain multiple steps, you can make this process easier by mentally dividing the problem into phases. For example, consider the problem, “During a sale, a bookstore sold 1/2 of all its books in stock. On the following day the store sold 4,000 more books. Now only 1/10 of the books in stock before the sale remain in the store. How many books were in stock before the sale?” This problem has three phases: before the sale, the sale day, and the day after the sale. Your model should represent these three phases separately.

Embracing Your Artistic Genius

The choice of a model often depends on the specifics of the question, but for many problems a semi-realistic picture works well. For example, a probability problem with bags of marbles can use models that draw bags and marbles. In the above problem, drawing a square that represents the bookshelf and filling it to certain amounts can show the differences of the amounts of books in stock before, during and after the sale. In this section, your goal is to represent the phases of the problem in a simple way so that you and students can better visualize the situation. For the above case, you would have three pictures: squares shaded to various extents: On the first day, the full square is shaded; on the sale day, half of the square is shaded; on the final day, 1/10 of the square is shaded.

Connecting the Dots

Mathematics describes the change of quantities. Thus, it contains a certain fluidity. Your model should demonstrate this fluidity by connecting the phases of your problem in a logical way. For many models, simply drawing arrows from one section to another allows students to grasp the specific changes in the situation. For example, in the book-selling problem, drawing arrows from the first bookshelf to the second and then from the second to the third helps students understand the “movement” as a subtraction problem; the arrows represent the books being sold. Often, it is important to label the arrows, both to show what they mean and to label the quantity or numerical value attached to them. For example, the arrow going from the second bookshelf to the third should have a “-4,000 books” label on it, representing the sale of 4,000 books.

Filling in the Blanks

One of the main advantages of a mathematical model is that it allows you to fill in all the information you have. For most academic math problems, any information missing in the model is likely that which you wish to find in the problem -- note that for real-life math problems, you're likely to have much more missing information, sometimes to the extent that makes the problem impossible to solve with the given information; in the latter case, you will have to investigate more before making a useful model. The bookshelf example exemplifies this: Though the bookshelves are all shaded, only one of the two arrows has a number displayed. The remaining arrow represents the missing quantity and finding this quantity will allow the problem-solver to finish the problem. After the model is drawn and the missing value located, it is up to the problem solver to identify the step in solving the model. But because of the visual aspect of the model, this is often easy. For the bookshelf example, a careful observer will notice that the moving of 4,000 books changes the bookshelf’s quantity from 1/2 to 1/10. That is, 4,000 books occupy 4/10 of the bookshelf. This is enough of a clue to complete the rest of the problem.