Quadrilateral Pretenders

Ah geometry, how you suffer from a lack of attention in the elementary grades! Rare is the curriculum that doesn’t stuff geometry into its final chapter, waiting patiently in line behind number and operation.

But the one geometry topic that does command attention is classifying two-dimensional shapes into categories based on their properties. To quote the grade 5 Common Core Standards: “Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles, and squares are rectangles, so all squares have four right angles.” Such reasoning, however, can be difficult, especially if students encounter definitions of the various quadrilaterals without opportunities to experiment with the shapes themselves.

Dynamic geometry software like Web Sketchpad can play an important role in making these classifications of quadrilaterals more concrete. On page 1 of the sketch below (and here), there are five quadrilaterals that look like parallelograms. But appearances can be deceiving. Students drag the four vertices of each quadrilateral one at a time and consider whether the quadrilaterals stay parallelograms as their side lengths and angles change, or if some of them are just “pretending” to be parallelograms in their static state.

Through this kinesthetic experience, students gain a deeper sense of what it means to be parallelogram. In his book Shape Makers (2012), Michael Battista describes the benefits of this type of activity: “In essence, students can learn about properties and classes of shapes using the same processes they use in learning everyday concepts such as ‘chair’ or ‘book.’ That is, they can manipulate and reflect on numerous examples instead of trying to comprehend verbal definitions. Eventually, after extensive visual investigations have enabled students to understand shapes in terms of their properties, students can deal meaningfully with geometric definitions.”

There are four pages of quadrilateral pretenders in the websketch below focused on parallelograms, rhombuses, trapezoids, and squares. Use the page navigation controls in the lower-right corner to move between pages.

In the second websketch below (and here), students get to decide which quadrilaterals to make. Tap the name of a quadrilateral—a preview of the quadrilateral will appear. A vertex of the quadrilateral is glowing. Tap where you’d like to place it. Then, one by one, tap to place the other glowing vertices. As an alternative, just tap the green checkmark to place the quadrilateral all at once and then drag the vertices. (This brief video demonstrates the process.)

As part of students’ investigation, they might place several parallelograms in their websketch and then check, by dragging vertices, which other quadrilaterals (like a square and rectangle) their parallelograms can become. Students might also use the Length and Angle tools to measure their parallelograms to uncover length and angle relationships.

Interactive experiences like these can go a long way in helping students to develop a robust understanding of quadrilateral properties.

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