In previous posts, we took a look at how dynamic geometry environments, like GeoGebra, might be used to give students a relational understanding of the construction of quadrilaterals. By manipulating various combinations of line segments, perpendicular bisectors, angles, and other geometric elements, students can understand the implications of the criteria required for a quadrilateral to be, for example, a parallelogram (opposite angles congruent, two pairs of parallel sides, etc.). Once a student creates a quadrilateral that is inherently a parallelogram, he or she can then drag and morph the structure to investigate all of the specific forms a parallelogram can take, such as a square, for instance. This exercise enables students to grasp the relationships among quadrilaterals. To help solidify these findings, a teacher might have his or her students create a flow chart with the most broadly defined quadrilaterals being funneled to the most specific. Several programs exist to aid in making digital flowcharts, but PureFlow was used to create the example below:

As delineated above, a parallelogram can be used to construct a rhombus, rectangle, or square. A kite can make a rhombus and square, while an isosceles trapezoid can make a rectangle and square. Given the debate on the exact definition of a trapezoid, it is worth noting that the definition used here does not limit an isosceles trapezoid to having only one pair of parallel sides.

Having students create a graphic with their geometric findings is a critical last step for effective use of dynamic geometry environments because it gives students a visual of what they have learned and provides them a means of communicating their discoveries. In *Principles to Action*, the National Council of Teachers of Mathematics highlights the importance of students sharing their mathematical thinking. By communicating their results, students build confidence in their mathematical ability, which fuels their desire to investigate further.

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I really like your flow chart. The bright colors make it stand out, and the use of different text bubbles is a neat idea. My chart is similar to yours except that I have an arrow connecting the kite to the square, since a square is a special case of a kite.

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I like the way your flowchart is done with the shapes. Also, I agree with your statement that mentioned students can create a visual of what they have learned because it allows them to communicate their discoveries.

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Solidifying concepts in students minds is so challenging. I agree that that exercises like this helps solidify lessons.

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I like how you setup your flowchart. I set mine up pretty much the same way you did!

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