As in all the computer labs, try to work with a partner. Do #1 and any one of #2 #3, or #4 below.

Effects of Some Transformations:

1. This problem follows up on some missed ideas from Lab 5, in which you investigated the effects of using the three forms of the Select Tool: translating, rotating, and dilating. I've asked the same questions as before, but in what I hope are better ways.

To start with, draw any old triangle. The idea is to describe the effect on the triangle of the three transformations - translating, rotating, and dilating - by filling in the following table based on an investigation.

An entry in the table should be a description of how the transformation changes (or doesn't change) the triangle with regard to the triangle's size/shape/orientation. Notice that I'm leaving out another possible column heading, Position.

Once you have the triangle drawn, select all of it, and then investigate the effect of using each of the tools (transformations) on the triangle. One possible point of confusion is that the Select Tool is the Translate Tool.

Introductions and Some Types of Numbers

2. We saw how the triangular numbers modeled the number of introductions - actually both the pairwise introductions and the cascading introductions. This investigation involves generalizing the triangular numbers (but not necessarily connecting back to introductions!).

Here is the sequence, with the triangular numbers in the "number" row:

geometry
number	1	3	6	10
pairwise intro's	2 people	3 people	4 people	5 people
cascading intro's	1 person	2 people	3 people	4 people

(As an example, for three people, there are 3 pairwise-type introductions and 6 cascading-type introductions; the number of vertices gives the triangular number.)

Notice that the first-differences form the sequence 2, 3, 4, 5, ... (3-1=2, 6-3=3, 10-6=4, etc.) which gives a way of generating each triangular number from the previous one.

The Problem: Try to continue the pattern of the triangular numbers in a numerical sense or in a geometric sense. (It's probably not a good idea to think about introductions.) Below is perhaps one obvious generalization; complete the table:

geometry
number	1	4	_	_

What are the first-differences for this sequence?

Now, use Sketchpad to help design and depict a next step in this pattern.

Variation on the Simulation of the Folding of the Tangent Envelope of a Parabola

(Sometimes you have to read mathematical English backwards: There's a Parabola, and its tangent envelope, and folding the envelope, and simulating the folding, and finally, variation on the simulation! Perhaps three prepositional phrases, all starting with "of," should be avoided?)

3. In Lab 6 we directed Sketchpad to construct the Tangent Envelope of a Parabola. To review that process: A point is allowed to be anywhere on a fixed line (or segment), and, as it moves, a constructed line traces out the tangent envelope. (In folding terms, many points on the edge of a square of paper are folded onto a fixed point and the creases form the tangent envelope.)

Generalize that process simply by replacing the fixed line (segment) by a fixed circle. Follow very exactly the construction process with this replacement. Construct the locus of the "crease" or "fold line" as before, or trace that constructed line. You should get some intersting and/or beautiful designs. Drag the fixed point around to see how the locus chnges. Note that the fixed point should not be the ceter of the fixed circle.)

Variation on the Partitioning of Segments and Angles by Folding

4. In class we discussed the following table for Folding Geoometry:

Investigate the bisection, trisection, etc. processes for segments and angles using Sketchpad tools, and complete a similar table (inserting descriptions of how to construct the appropriate geometric objects using Sketchpad) for Sketchpad Geometry.