Assessment Problem:

What are some of the possible values for the slope of a line that lies in the shaded region? How do you know?

Common Responses:

Students must both recognize that the lines must be of the form y=mx (the y-intercept value is 0), and that the slope of the line is between 0 and 1.

Mathematical Issues:

In addition to assessing students’ understanding of graphing, this question assesses students’ object conception of function. In order to characterize the properties of lines that pass through the shaded region, students must have a holistic understanding of the properties of lines as objects, rather than lines as a set of points. See also Connecting graphs to algebraic relationship and Action/Process/Object Concept of Function.

Domain:

Analysis of Change

Theme:

Linear functions

Learning Progressions:

Linear Equations

Common Core Standards:

A-CED: Create equations that describe numbers or relationships

Research References:

*Moschkovich, J., Schoenfeld, A., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T. A. Romberg, E. Fennema & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 69-100). Hillsdale, NJ: Lawrence Erlbaum Associates Publishers