# Activities

• • • ##### Subject Area

• Math: Geometry: Quadrilaterals and Polygons

• ##### Author 6-8

45 Minutes

• ##### Device
• TI-83 Plus Family
• TI-84 Plus
• TI-84 Plus Silver Edition

## Which Trapezoids Are Special?

#### Activity Overview

Cabri Jr. allows students to look at several kinds of trapezoids in order to see which ones are special. With this activity students can see that an isosceles trapezoid is a special one. In a parallel manner, students can see why a parallelogram can be considered a special trapezoid. This activity follows the same pattern as the one showing that a square is a special rectangle.

#### Before the Activity

Prior to beginning this activity, students need to be familiar with how Cabri Jr. operates.

#### During the Activity

Start the Cabri Jr. application with a new sketch. Select the Line tool to draw a line near the bottom of the screen.

Select the Point tool, option Point, to place a point above and to the right of the left hand control point of the line.

Select the Parallel tool to construct the line parallel to the first line drawn, passing through the point just placed in the drawing. Select the Point tool, option Point on, to place a point on the parallel line to the right of the point placed in second step.

Select the Hide/Show tool, option Object, to hide the lines in the drawing. Select the Quad. tool to connect the four points (in either a clockwise or counterclockwise order) to form a quadrilateral. Have the students discuss if this quadrilateral is special in any way. They should be able to see that it is a trapezoid and explain why it is. Select the Measure tool, option D. & Length, to measure the lengths of the left and right edges of the trapezoid. Move those lengths to convenient places on the screen.

Use the ALPHA key to select the upper right hand point of the trapezoid. Use it to move left and right, noting the lengths of the edges as you do so. As you carry out this investigation, you will find a place at which the trapezoid becomes a triangle. To the left of that place the figure ceases being a quadrilateral. After you move the point to the right, there is a place at which the two displayed lengths become essentially equal. Pose the question What special kind of trapezoid is this?

After you move the point to the right, there is another place at which the two displayed lengths become equal (or very close to it). Again pose the question What special kind of trapezoid is this?

#### After the Activity

If you want to avoid some of the discussion when the two edge lengths are not visibly equal, you can keep the number of decimal places in the lengths to be one. In that way, it is quite likely that, to one decimal place, the edge lengths will be equal. In my experience, however, it is healthy to talk about the situations that often arise with two decimal place accuracy.

After the work on the calculator, you can pose the questions "At what point did the two measured edges first become congruent"? "Did any of the trapezoid's fundamental properties change, i.e., number of edges, straightness of its edges, etc."? "Do you think an isosceles trapezoid is a special case of a trapezoid"?

After considering the first situation, you can pose the questions "At what other point did the two measured edges become congruent"? "Did any of the trapezoid's fundamental properties change, i.e., number of edges, straightness of its edges, etc."? "Do you think a parallelogram is a special case of a trapezoid"?

With teachers, the first question will draw little discussion. The second question could be quite a different story. Many teachers learned a geometry in which a trapezoid was defined as a quadrilateral with exactly one set of parallel edges. Under that definition, a trapezoid could never be a parallelogram.

Students, on the other hand, have little difficulty with this kind of development. This is especially true if they have the experience of seeing why an equilateral triangle is a special case of isosceles triangle and a square is a special case of rectangle.

On a personal note, I was one of those teachers who learned the restrictive definition of trapezoid. It was work with dynamic geometry that convinced me that defining a trapezoid as a quadrilateral with at least one set of parallel edges makes more sense.