I have just approved a comment that has been sitting in the queue for a long time. The person was looking for an “official term” for an algorithm. Assuming the commenter was referring to the “best” technique for factoring trinomials, I propose “The Immediate Factoring Method” for factoring trinomials.
I delayed because the question evoked a larger issue. There are common patterns used in arithmetic and algebra. If it is important to be good at mental and/or paper-and-pencil calculations, then organizing by common patterns is a good idea. This has been done in the book Vedic Mathematics by Bharati Krishna Tirthaji. The book apparently is not based on ancient knowledge (read the wikipedia article ). It is however an attempt to organize the various algorithms categorized into sixteen sutras – short statements of pattern or principle. I found that the book takes quite a bit of work to extract and understand its methods. I also own Vertically and Cross-Wise by A.P. Nicholas, K.R. Williams and J. Pickles. The authors interpret and explain the sutra: vertically and cross-wise. The book is easy to read and understand. I have read it several times and now use some of its methods preferentially.
I have addressed the commenter’s query but am left this question: Is it worthwhile to master all of the sutras, that is, all of the algorithms for arithmetic and algebra? I don’t know. Certainly standardized tests and current standardized curricula assume mastery. Yet I don’t think there is a necessary link between mastery and understanding. We have calculators, actually internet sites, that will do the algebra for us now. From now on important problems, those that have real-world consequences, will be done or at least checked by such computer algebra systems. Should we be asking more questions that test understanding and application and require less calculation? It is easier in a calculus class for example to make tests and homework that overweight algebra. I have done it myself. I intend to give this more thought before the term starts. What kind of questions should I be asking in a precalculus course?