Developing Algebraic Reasoning

The following sequence of problems deals with a fairly well-known pattern. Similar questions have appeared on SATs, on other standardized tests and in texts. The intent here is to provide an extended activity for students of diverse math backgrounds and abilities to develop a systematic approach to analyzing patterns. Students should also be encouraged to make a table of values in which the first column is the number of 'crosses' and remaining columns are reserved for other 'dependent' variables. This function-based approach is also an essential feature of this development.

Notes: There are many ways to approach these questions. Encourage students to share theirs! These questions involve pattern-based thinking, combinatorics, recursive sequences, arithmetic sequences and algebraic reasoning. Parts (d) and (e) are more challenging for some. Based on the pattern of the first 3 or 4 terms, some students will simply develop a linear formula of the form aN+b for the perimeter (which is somewhat harder than the area). It is important for our prealgebra and algebra students to recognize that any arithmetic sequence like 12,20,28,36,… can be described this way. My experience is that if a class has 20 students, there will be at least 5 different ‘counting’ methods discussed, Students often are very creative here and not all use ‘linear’ thinking!