The definition of square can determine whether it is a special case of a rectangle. Using the Cabri Jr. application, students can get a feel for why its definition makes sense. Along the way, they get experience with perpendiculars, parallels, measuring lengths, and an informal look at the intermediate value theorem.
Before the Activity
Prior to beginning this activity, students need to be familiar with how Cabri Jr. operates.
During the Activity
Details are in the attachment. Start with a new sketch. Draw a line near the bottom of the screen. Construct the line perpendicular to the line previously drawn, passing through the control point on the left.
Place a point on the perpendicular line.
Construct the line parallel to the first line drawn, passing through the point just put on the perpendicular line.
Place a point on the parallel line.
Construct the line perpendicular to the last line constructed, passing through the point put on it. Alternately, students can use the Parallel tool to construct a line parallel to the vertical line, passing through the point put on the line previously constructed.
Place a point where the last line constructed intersects the first line drawn.
Hide the lines in the drawing. 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 rectangle and explain why it is that kind of quadrilateral.
Measure the lengths of the base and height of the rectangle. You may need to remind the students that they will have to wait for a short time in order to measure a single edge of a rectangle and not the rectangle?s perimeter. After getting the length of an edge, depress the + key to get a second decimal place in the length?s value. Move those lengths to convenient places on the screen. You may want to have them discuss what other lengths they know, even if they are not overtly shown.
Select the upper right hand point of the rectangle. Use it to move left and right. As you carry out this investigation, you will find a place at which the rectangle becomes a line segment instead of a polygon. At another place the two displayed lengths become equal (or very close to it).
After the Activity
If they are not visibly equal, they will change from the height being greater than the base to the height being less than the base. The question to pose in that situation s ?What is the situation somewhere in between those two points??
If you want to avoid this discussion, 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 rectangle become a square?? ?Did any of the rectangle?s fundamental properties change, i.e., number of edges, straightness of its edges, etc.?? Do you think a square is a special case of a rectangle??