Make Me a Quadrilateral
What types of quadrilaterals are possible?
The purple band is one side of a quadrilateral.
- What types of quadrilaterals can you make?
- How do you know what types of quadrilaterals you have made?
- Are there any types of quadrilaterals that you can't make? How do you know?
- What angles can you make using a second band? What does this tell you about the quadrilaterals you could make?
- How can you place a second band that doesn’t touch the purple band? How could you describe this band in relation to the purple band? What does this tell you about the quadrilaterals you could make?
- Start with a band that is connected to the center peg, like this. How does this change the quadrilaterals you can make?
- Can you adjust the position of the original purple band so that it’s possible to make all types of quadrilaterals?
This task asks students to create different types of quadrilaterals, given one side of the shape. Students use the circular geoboard configuration to complete and identify quadrilaterals; in the process, they investigate and examine properties of different types of quadrilaterals on a circular geoboard. Angle sizes, side lengths, and relationships between the line segments that make up the sides provide multiple opportunities to reason about the different quadrilaterals that can and cannot be made.
Students might use the Geoboard app in different ways for this task. They might construct a quadrilateral out of 4 separate rubber bands. They might also choose to only use the purple band, but stretch it and attach it to 4 vertices, seeing which quadrilaterals can be made as they adjust side lengths, angles, and vertices.
As students begin this problem, it may not yet be clear which quadrilaterals can and cannot be made. One way to begin could be placing 1 additional rubber band on the geoboard. From there, they could reason about the quadrilaterals that can and cannot be made from these 2 bands based on side lengths, angles, or the presence or absence of parallel and perpendicular lines. Different options for the second band include:
- Creating a right, acute, or obtuse angle.
- Creating a line parallel to the purple band, either congruent to or longer than the purple band.
Alternatively, students might choose a specific type of quadrilateral, and determine if that shape can be made, given the first purple band.
To extend student thinking, you might ask How can you know for sure that 2 or more sides of your quadrilateral are congruent? Despite not being able to measure any of the line segments, the structure of the geoboard allows students to reason about the different possible lengths between two pegs on the geoboard. You might also want to ask, How can you be sure that you have created sides that are parallel or perpendicular? The circular arrangement of the pegs may make this more challenging, but it’s possible to reason logically about the relationships between different sides on the circular geoboard.