The Varied Paths to Constructing a Square

Using dynamic geometry software, a student can draw what looks like a square by eyeballing the locations of the vertices. However, the quadrilateral will not stay a square when its vertices are dragged. Building a “real” square requires that it stay a square when any of its parts are dragged. This is only possible by baking the geometric properties of a square into the quadrilateral as it’s being built.

Constructing a square requires tools, and Web Sketchpad features a particularly innovative tool interface. When a student taps a tool icon, the entire object to be constructed appears  in a preview mode. For example, if you tap the Parallel Line tool in the websketch below, you’ll see a segment onscreen as well as a line parallel to the segment. These objects are previews of what you’re about to create. This effect provides immediate feedback regarding the entire construction being created; there’s no need for the student to remember or figure out what objects to click, in what order, to use the tool successfully. This overview of the entire tool gives the student an opportunity to consider what objects the tool is going to create and plan how to integrate these tool objects into the existing sketch.

The Parallel Line tool consist of three “given objects”—the two endpoints of the segment and a point through which the parallel line passes. These givens determine the segment and the parallel line. A highlighted given object can be attached to an existing sketch object or located in empty space  with no restriction on the order in which given objects are attached. As soon as the last given object is attached or located, the tool’s action is complete and the sketch is again fully interactive.

The Web Sketchpad tool interface was designed with student tasks in mind. By providing only tools needed for the task at hand (optionally arranged in the order of expected use), there’s less need to provide students with prescriptive directions and thus better support for open-ended tasks. And by immediately showing the user detailed visual information about the effect of the chosen tool, there’s less need to explain how to use tools with which the user isn’t already familiar. These innovations enable less prescriptive and more open-ended student tasks. Students can be encouraged to be more self-reliant and self-directed, concentrating on the mathematics of the task rather than following directions from a worksheet or from the teacher.

The best way to learn about tools is to try building a square yourself. The video below the websketch gives an overview of the variety of square constructions made possible by the tools. When you’re done, challenge yourself or your students to find multiple ways to construct a rhombus using the same tools.

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