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Abstract
This study investigates how students construct the sum of two co-varying oriented quantities (denoted by x + y=a, where a is a constant) by reorganizing their counting schemes and units-coordinating schemes. Two 9th grade students, one who reasoned with the two levels of units and one who reasoned with three levels of units, participated in a year-long teaching experiment. I found major differences in how the two students constructed sums and differences of signed quantities. Carl, the student who reasoned with two levels of units, did not construct a negatively oriented quantity as the inverse of a positively oriented quantity nor did he find the sum of two oppositely oriented quantities in a way that respected the orientation of the quantities. In contrast, Maggie, the student who reasoned with three levels of units, did construct sums and differences of oriented quantities in such a way that respected their orientations. In situations that involved two oriented quantities, denoted by x and y, that co-varied in such a way that x + y = a (a is a constant), Carl found a set of discrete points that were representative of x + y = a by experientially plotting a few points on a coordinate plane. Although he said that he could find infinitely many points, he did not envision them as belonging to a line nor did he construct the counterbalancing relation between changes in each quantity. In contrast, Maggie constructed the counterbalancing relation by additively coordinating changes in the two quantities. Her schemes were anticipatory, and she could envision a two-dimensional trace of the co-variation of x and y as a line. My findings suggest that reasoning with three levels of units and reversible reasoning are both essential in constructing graphs of two oriented quantities that co-vary in such a way that their sum is a constant.