Making Triangles

• 4.2.1 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.

Instructions

We will refer to the pegs that the rubber band is attached to as nodes. To begin, click on the rubber-band box and drag a rubber band to the geoboard. Click on a node to attach one end of the rubber band to that node. Move the mouse and click on another node to attach the other end of the rubber band to that node.

• To attach the rubber band to more than two nodes, drag the rubber band from the middle.
• To move a rubber band to a new node, click on the current node and drag the rubber band to the desired node.
• To remove a rubber band from a node, click on the node to select it (a double circle appears), then click on the Delete Node button.
• To delete a rubber band, click on it to select it, then click the Delete Band button.
• To clear the geoboard, click on the Clear All button.
• To color the interior of a shape, click on the rubber band and then click on a color.

Exploration

Using 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.

Note

Students 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 Geoboards

Working 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.

Take Time to Reflect

• What are some strategies you can use to help students focus on the properties of triangles when they make four-sided shapes that they consider triangles?
• What experiences, knowledge, and vocabulary should students have in order to identify and define triangles?
• What are some activities students of this age can engage in that involve congruence?

Objectives and Standards

NCTM Standards and Expectations
• Geometry / Measurement
• Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
• Pre-K - 2
• Geometry