Concept Map Narrative Linear relationships contain or sometimes overlap with many other concepts such as symbolic relationships, mathematical modeling, and the skill of recognizing patterns and connections. Specifically, recognizing connections such as the relationship of the symbolic representations of algebra to the spatial representations of geometry is an important skill. Science for All Americans suggests three phases of mathematical inquiry, which seem to me especially helpful in understanding the overall intent of the subject throughout a K-12 education: abstraction and symbolic representation, manipulating mathematical statements, and application. These phases are much more complex and meaningful than students sometimes recognize. Solving systems of linear equations really requires algebraic reasoning. Vital to this process is the “crucial understanding that the variables stand for numbers and hence manipulations of them must be governed by the same arithmetic rules that govern numbers” (Beyond Numeracy, pg. 9). Without this foundation, generalizing the basic principles of algebra would be nearly impossible. Essentially linear equations represent constant rates of change using the common variables x and y. The algebraic linear equations are, however, only one avenue of representation. Graphically, the idea of a constant rate of change is much more intuitively seen. Thus, as intended within this unit, students will first explore systems of linear equations by graphing. Reaching back one step and at the same time moving forward, this new knowledge must be tagged to a student's ability to manipulate linear equations into more appropriate forms for graphing. If students are facing difficulty understanding the variables, x and y, this may be a difficult leap for them. As a teacher, you may be able to better bridge student understanding of variables by using a word problem. Suppose students are comparing the number of twixes to the number of snickers bought. You can represent that manipulation using the candy bars while also keeping both sides balanced and equal. Eventually, students should be able to move into more abstract representations of linear relationships. The Benchmark lessons are taught initially to allow students to explore the content. The investigative lessons provide a link of that content to an application. The field trip is essentially a form of an investigative lesson. Finally, the entire final product is meant to exhibit those concepts, skills, and applications learned within the unit.