The substitution process in problems solving

THE OBJECTIVE:Learning to handle the math.

Request-Response-Result Is a natural process. Making use of this in problem solving in situations in which mathematics is useful requires learning basic mathematical operations. An important operation is substitution.

The solution to Mary's Apples shown below focus on solving the problem. A common instruction given to those learning to solve problems is to collect the information about the problem. This is done in the solution shown below. The information is assembled in an ordered manner using an indented arrangement to show the logical and mathematical relationship between the various components of the information given in the problem statement. Arranging the information in an ordered manner serves to solve the problem. The solution is complete once the indented set is complete. Obtaining the result is then a mechanical process consisting of successive substitution. This is done in one step in the first image below.

There are a number of steps involved in the successive substitution done above. The steps are done automatically by a symbolic algebra program such as SureMath. The substitution process can be made visible by showing each of the successive substitution. This is shown in the solution presentation below. After the problem is solved, that is the right moving indents are complete, substitution is started by substituting the last indent into the outdented equation that requested the response. This process is continued to un-indent the solution presentation. The last equation is then necessarily at the left margin.

Sometimes is desirable to substitute at various stages of the of the solution. Indeed a useful rule is "when the going gets tough, substitute". This clears the deck and one can better see how to proceed.

This method is illustrated by the solution presentation shown below. Substitution was done at each step. This explicitly shows the Request-Response-Result character of the steps of a problem solution. At each step one has solved a problem and is presented with a new problem. Once all the little problems have been solved you have the answer to the big problem. Problems solve themselves.