In this activity, students will use the TI-73 Geoboard APP to create equilateral triangles to make sense of the first four van Hiele levels of geometric reasoning.
Before the Activity
Copy student worksheet page "vanHieleLevelsRevisited"
Install the Geoboard APP on a set of TI-73 calculators, one for each student
Optional: Download the geoboard paper (see below) for recording purposes
During the Activity
The attached document contains the steps of an investigative lesson using the Geoboard APP on the TI-73. Given the position of the points on the rectangular geoboard, students should determine that it is impossible to make an equilateral triangle on any size board. Using informal reasoning, students may comment that they cannot construct three congruent sides; the best they can make is an isosceles triangle. One side may have a whole number length but the remaining one will not. Using the Pythagorean Theorem, students should be able to logically argue that they cannot construct all 3 congruent sides.
During the second part of the investigation, students will find that they are able to construct an equilateral triangle on the circular board. As students work through the investigation, they will gain a better understanding of the first four van Hiele levels as they pertain to the figures they create.
After the Activity
Review the van Hiele levels with respect to the students' discoveries using the Geoboard APP. After completing this activity, students should have increased their geometric understanding with respect to triangles and their pedagogical understanding of the van Hiele levels with respect to teaching geometry to children.