In Investigating an Ordering of Quadrilaterals published in ZDM, Gunter Graumann shared a good activity for developing students mathematical thinking. The activity is about ordering quadrilaterals based on its characteristics. He gave the following list of different aspects of quadrilaterals as possible basis for investigation.
- Sides with equal length (two neighbouring or two opposite or three or four sides)
- Sum of the length of two sides are equal (two neighbouring or two opposite sides)
- Parallel sides (one pair of opposite or two pairs of opposite sides)
- Angles with equal measure (one pair or two pairs of neighbouring or opposite angles, three angles or four angles)
- Special angle measures (90° – perhaps 60° and 120° with one, two, three or four angles)
- Special sum of angle measures (two neighbouring or opposite angles lead to 180°)
- Diagonals with equal length
- Orthogonal diagonals (diagonals at right angles)
- One diagonal bisects the other one or each diagonal bisects the other one,
- Symmetry (one, two or four axis’ of symmetry where an axis connects two vertices or two side-midpoints, one or three rotation symmetry, one or two axis’ of sloping symmetry). With a sloping-symmetry there exists a reflection – notabsolutely necessary orthogonal to the axis – which maps the quadrilateral onto itself. For such a sloping reflection the connection of one point and its picture is bisected by the axis and all connections lines point-picture are parallel to each other.
The house of quadrilaterals based on analysis of the different characteristics of its diagonals is shown below. Knowledge of these comes in handy in problem solving.
I made the following applet for students to explore. Technically the three given quadrilaterals could be formed into the shape beneath it. Mind the arrows. Happy exploration. Click here to save the applet.
Read my post Problem Solving with Quadrilaterals. You will like it.:-)