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This two-part example describes activities that use interactive geoboards to help students identify simple geometric shapes, describe their properties, and develop spatial sense. This first part, Making Triangles, focuses attention on the concept of triangle, helping students understand the mathematical meaning of a triangle and the idea of congruence, or sameness, in geometry. In the next part, Creating Polygons, students make and compare a variety of polygons, describing the salient properties of the shapes they create. Both parts help students reach the goals described in the Geometry Standard. TaskUsing one band for each triangle, make as many different sizes and shapes of triangles, as you can on the computer geoboard. Explain to a friend the ways in which these triangles are different and how they are alike. [How to Use the Interactive Figure]
Talking about Triangles in the ClassroomStudents enjoy working with geoboards, whether they are interactive computer geoboards or physical ones. As with any manipulative, students need ample time to explore the material before specific tasks are presented. Most students in prekindergarten through grade 2 know the word triangle and have some idea of what it means. However, their definition is often not the conventional one. To encourage students to focus on the properties of a triangle, teachers can ask them to make many different triangles on a geoboard and then to pick one to show to the class. Students can compare the triangles on their geoboards and discuss whether or not each shape is actually a triangle. Some students may feel that a triangle with a vertex oriented toward the bottom of the geoboard is not really a triangle. Teachers can challenge students to justify their thinking, inviting quiet students into the conversation with comments such as "So are you saying that Marc is still Marc even if he is upside down, so this is still a triangle?"
Others may make four-sided shapes that they consider to be triangles.
Teachers might summarize the discussion by explaining that mathematicians agree that all closed figures with three straight sides are triangles. The teacher could ask the students to check once more the shapes they made and decide how many are triangles according to this definition. They thus give students an opportunity to revise their earlier choices. Students at this level can check congruence in two dimensions by moving one shape to exactly cover another. Geoboard shapes can be described with a simple system of coordinate geometry; thus two shapes on a geoboard are also congruent if their constructions can be described in the same way. If designs are made on two different geoboards, one geoboard can be moved so that eventually the constructions can be viewed in the same way (perhaps by a flip, top to bottom, or a rotation of 90 degrees). Students might copy their triangles onto geoboard dot paper and cut them out so they can physically lay one shape over another to check for congruence. Students' Experiences As They Work with Interactive Computer GeoboardsWorking with an interactive computer geoboard allows students to shade their figures and to make a greater variety of triangles than they can create on a traditional geoboard with a five-by-five array of nails. Teachers can assess students' understanding of the properties of a triangle by asking them to explain how they know that all the shapes are triangles.
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