Group students into pairs. Give each pair markers
and something to
write on (e.g. whiteboard) that they
can later display to the whole class.
As a warm-up, display the overhead, Think of a Quadrilateral.
Ask students to draw two perpendicular line segments, connect the endpoints, and describe the quadrilateral that is formed.
[Some pairs will likely assume the
diagonals bisect each
other and will sketch squares, while other pairs will not assume the
diagonals bisect each other and will sketch non-square kites.]
Group students' quadrilaterals according to their type. This is a
great opportunity to prompt students to think
critically about the relationships between the diagonals and their
quadrilaterals. The goal is not to press for right answers, but
to encourage students to think about the
relationships they will see when manipulating the applet in the next activity.
- Describe the properties for this diagonal.
- Do the diagonals bisect each other?
- Are there other possible ways the diagonals could intersect?
Explain to students that they will be using a program that shows how
diagonals are related to their quadrilaterals. Launch the
Diagonals to Quadrilaterals I tool, either as a whole-class activity or as pairs of
students working on a computer.
Ask students to try a quadrilateral. Have them click on "Show Perpendicular
Diagonals" and practice
dragging the points on the diagonals to change the shape of the
students what types of quadrilaterals are possible, and have them
list the possibilities on their whiteboards. Students may see that the
initial shape is a general
quadrilateral; others may drag the points to create a
kite, a rhombus, or even what appears to be a square. Remind students
that their assumptions about what looks
like a kite, a rhombus, or a square, may not necessarily be correct. We
what we see without measures or constructions. Do not spend a lot of time
examining all quadrilateral shapes on the students’ list; these
possibilities come up again as students work through the
Next, distribute the Diagonals and Quadrilaterals
activity sheet. Note that the class just answered the question for
perpendicular diagonals; have students record the answer on the table.
Students should note that for perpendicular diagonals, the
quadrilateral can be a general shape or, in specific cases, it can be a
kite, rhombus, or square.
Prompt students to explore the effects of adding conditions to the diagonals. Click
on an appropriate button in the applet to examine other relationships
between perpendicular diagonals. For the last three shapes, students will need to use a second tool, Diagonals to Quadrilaterals II. For each quadrilateral, students will describe the type (or types) of quadrilateral and explain their reasoning.
Students return to a whole-group setting. As pairs give their answers, other students are
responsible for questioning the pair for their reasoning as well as for
clarity. Probe students for responses about their conclusions. For instance, you might ask:
- How do you know the quadrilateral is a rhombus?
[Students may respond that the diagonals appear to bisect
each other so we can get congruent triangles like GKM and HKM using
As students describe their findings, record their results on the Diagonals and Quadrilaterals
overhead. This lists categories of quadrilaterals (some will overlap;
for example, rhombi will fall in both the "general" and the
"parallelograms" categories). Mark each cell with an "A" (all
quadrilaterals in this category can be created given the conditions on
the diagonals) or "S" (some quadrilaterals can be created).
Questions for Students
1. Why does it make sense that knowing the diagonals of a quadrilateral are perpendicular is not sufficient to show that the quadrilateral is a rhombus?
[The diagonals can be perpendicular without bisecting each other; thus, the quadrilateral may not be a rhombus.]
2. Explain using diagonals why a square is both a rhombus and a rectangle.
[A rhombus must have diagonals that are both perpendicular and bisecting each other. A rectangle must have diagonals that are congruent and bisect each other. Since a square is both a rhombus and a rectangle, its diagonals are congruent perpendicular bisectors.]
3. Explain using diagonals why a square is always a rhombus but a rhombus is not always a square.
[A square has diagonals that are congruent perpendicular bisectors. A rhombus has diagonals that are perpendicular bisectors. Thus, the diagonals of a square fulfill the requirements for the diagonals of a rhombus: perpendicular bisectors. However, the diagonals of a rhombus need not be congruent. So, the diagonals of a rhombus do not fulfill one of the requirements for the diagonals of a square: congruent.]
does giving conditions about the diagonals and asking students to draw
conclusions about the quadrilaterals, or giving students conditions on
an altitude and the side it intersects and asking students to conclude
things about types of triangles, parallel what students must do in the
proof exercises often asked in high school geometry?
- How did students express their thinking when using words? Using the applet? Using sketches? Using hand motions?