It is important to describe the ideas used in prose as well as formulas. This connection between ideas and equations is essential to understanding the ideas.
A problem solution is a story. In order for a story to be effective it must be constructed with at least the basic rules for effective communication.
The principal difficulty many have with problem solving is that it is taught, and therefore done, as if problem solving is somehow different from the processes used in writing an essay, drawing conclusions in history, investing, riding a bicycle, executing art, writing poetry or any other human activity. Successful completion of such activities is a result of orderliness and connectedness. Rules are used to guide the activities.
Various presentations made in the WEB include the ideas of orderliness and connectedness in mathematical problem solving. Problems are solved reliably by following rules. The ideas shown in this WEB provide these rules. (It is interesting to note that the preparation of materials for the WEB requires hierarchical, nested sets of instructions. No coincidence; that's the effective way to communicate and think!)
The cause of the difficulties which are created in problem solving can be verified by comparing the semantics of any professionally written essay or novel with the problem solution presentations shown in commonly used introductory physics and math textbooks. The author of a written work paints a picture in the reader's mind. The author involves the reader. This is in sharp contrast to the manner in which a problem solution is normally presented.
Textbooks and answer books present problem solutions in a sketchy, shortcut manner designed to occupy minimum space on the printed page. The reader is left to unravel the solution. The logical processes the expert used in solving the problem are not shown. Learning to solve problems then becomes quite difficult since the reader is presented with a defective model.
In the following solution, connectedness is apparent. Compare this presentation of a solution with those typically found in textbooks and answer books.
Using words as an integral component of a problem solution follows commonly accepted practice in many fields. For example in computer programming comments are used to document the program. The comments help others to understand what you have done. It also helps you to understand what you have done especially on looking at the program (problem solution) at some later time. Words, pictures and equations are all tools of effective communication. They work together to inform ourselves and others. Computer programmers frequently write psuedocode, that is, a discription of the program to be developed using words. A flow diagram is frequently used to supplement the words. A flow diagram provides a picture of the words.
Explanatory comments are used in many different situations. The following shows the use of verbal statements in computer programing. In the case they are referred as comments and preceded by a double slash (//).
The problem solution was developed using SureMath.