The Math Forum || Annie's Sketchpad Activites || Printable Version (No Java)

Investigating the Properties of Quadrilaterals

An example of animation, middle school math, and tables!

In this activity, we'll construct a demonstration sketch which could be used by middle (and high) school students to investigate the properties of quadrilaterals. In doing so, we'll learn some more about action buttons and will see how tables can be used to collect data and back up conjectures.

We're going to construct a sketch that consists of a random quadrilateral which can be manipulated to form various "special" quadrilaterals by use of buttons. It's not especially hard, but there are a lot of details.

Making a Movement Button

A movement button lets you control where a specific point goes very exactly. While an animation button lets you have constant movement, a movement button is a simple motion from one point to another. Let's try one.
Construct a segment AB. Put a point C somewhere else in your sketch. Select A and C and under the Edit menu, go down to Action Button and over to Movement. You'll get to choose a speed - medium is probably okay. Be careful - on a slow machine, slow is SLOW! Once you say okay, you'll get a button on your sketch. Double click on that and see what happens. Note that A is not in any way attached to C at this point. It's simply sitting right on top of it. As we use this technique to turn a random quadrilateral ABCD into different special quadrilaterals, ABCD will not become any of those quadrilaterals, but will simply look like them. So our rectangle will not be a draggable rectangle, but will simply look like a rectangle at that moment in time, and we can study it properties in a static manner. Let's draw some pictures! Sorry, this page requires a Java-compatible web browser.

Constructing the demonstration sketch

Open a new sketch and draw a quadrilateral. Label it ABCD. This is the figure that will be manipulated. Now we need to construct some "special" quadrilaterals onto which we can move ABCD. This could include rectangles, squares, rhombi, parallelograms, kites, trapezoids, and a scalene quadrilateral. You can go wild with this part, depending on what you've covered with your students or what you'd like them to learn.
Construct a rectangle. There is a script tool to do this in the Sample Scripts:Polygons folder - if you use this script, you will have to get the fourth vertex back. You can do this by choosing Display->Show All Hidden, holding the shift key down, and clicking on that fourth vertex, then choosing "Hide" again. Make it a pleasing shape - not too square, not too anything else, and make it in the area of the existing quadrilateral. You might delete any interior just to save space. We want to create a button that will move each vertex of ABCD onto a vertex of this rectangle.

Select A. Then, with the shift key down, select G, then B and H, C and F, and D and E. Each point is chosen, and then its destination point is chosen. Choose Movement from the Edit->Action Button menu. Fast is fine. You'll get a Move button (we'll rename it after we make sure it works).

Sorry, this page requires a Java-compatible web browser.

Hide all the parts of the rectangle, including the vertices, sides, and any interior. Double click on your move button and see what happens!

If that worked, use your text tool, double click on the button, and rename it something like "Rectangle".

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Create a couple more quadrilaterals and repeat the above sequence until you feel comfortable. Don't forget to include a scalene quadrilateral! You can get creative about having the vertices of ABCD go to far-away vertices of the quadrilaterals, just make sure that it always stays a quadrilateral and doesn't cross over itself.

So now we have some buttons that will give us specific quadrilaterals. Let's look at how we might explore their properties.

Exploring the Quad Properties

This might look like something your students would do. Measure the edge lengths, diagonals, and angles of ABCD.

Using the buttons, make ABCD into different quadrilaterals. For example, make ABCD a parallelogram. See if you see anything interesting about any of the measurements. Let's make a table of values to take a snapshot of the angle measures and their sum.

Select the measurements you want in the table. I'll choose <DAB, <ABC, and their sum. Go to the Measure menu and choose Tabulate.

Make sure you note that the first line in your table is a parallelogram. Now investigate those angles for the other quadrilaterals. Is the sum 180 for other quadrilaterals? Investigate other properties of the quadrilaterals. (This could go on for a while!) For each quadrilateral, be sure to look at least at the following parts: are any of the sides congruent? How about the diagonals? Do the diagonals bisect each other? Are any of the angles equal? Do any of the edges have equal slopes?

What shape(s) has diagonals that bisect each other? What shape(s) has four congruent edges? What shape(s) has consecutive angles that are supplementary? Etc... Some things to think about: You could invest a lot of time in preparing this sketch. Would it be worth it? What if you shared a set of such sketches with other people? How would you use this in your classroom? Introduction? Review? How could you use the tables to assess the students' work? Could students produce similar sketches? How does Sketchpad make this activity happen? How could you do it without a dynamic geometry tool?

What other topics lend themselves to investigation in this manner?

References: An activity similar to this, using the preSupposer, appears in Geometry in the Middle Grades, the NCTM Addenda Series Grades 5-8. Another activity appears in Geometry from Multiple Perspectives, the NCTM Addenda Series Grades 9-12.