Sue's Homework 1

  1. Visit and report on at least three web sites.
      This site had good graphics. It had enough pictures to entertain the reader. Color was used in the images, but not in the background. There was a clear explanation of ideas, however, I still got the feeling that the students would need to work on the lab as a group or with an instructor's help. The graphics were not always clearly explained. There was a good use of links to another site which explained rainbows in a less "discovery-based" way. The lab always allowed the student to go forward and backward within the lab. This is good, since it allows the students to both look ahead and see where they have been.

      This site made excellent use of background colors and colored images. The explanations were very informative and linked to other locations allowing for further study. They also allowed the user to go back and for between sections and the index. This was definitely a good page!

      I failed to see the point of this page or making a icosahedron other than to brag about their world record until I was several pages in to the site. There was good use of images and graphics. The basic background was used. The explanation of how the icosahedron was made was thorough. Later on, it gave potential activities to do with the object, but I still felt cheated as a reader of the page. This page also made good use of the ability to go forward and backward to different parts of the explanation.

      This site had a definite need for animation. The graphics were confusing. The explanation assumed a certain level of understanding of both the subjects and the images. The ability to go forward and backward was also there in this page.

    All of these pages were easy to read in terms of background color, text sizes, etc. They all made appropriate use of titles. Some did better than others with the use of graphics, images, and entertainment value.

  2. Answer Question #9. In your own words, describe the effect of a "dilation" on a Geometer's Sketchpad object. How would you describe it to your students? What properties of an object are preserved by dilation? Which properties changed? How does a circle change when it is dilated?

    • First of all, I would use either the program itself or a chalkboard as a visual aid in demonstrating what I am about to explain. Since that is not possible on e-mail, my explanation in words will have to do. Dilation takes a point (marked as center (C for the purpose of discussion)) and an object and creates a reduced or enlarged copy of the object. This copy is reduced or enlarged by the amount that is input (call the input II for the sake of reference) and will be located at a distance away from C equivalent to II times the distance between C and the object. This means that reductions are located between the original object and C. Enlargements are on the opposite side of the object from C. The ratio of the lengths of edges, arcs, circles, line segments, etc. (say, A' and B') of the new object will be identical to those of the original object's A and B. Therefore, the relationship between lengths of items within the object will be preserved. The shape of the object is also preserved. All angles are preserved (ie. Meas. Of angle A = Meas. Of angle A'). The actual lengths of the items within the object (ie. lines, radii, ect.) will change in amounts proportional to II times the original lengths. When a circle is dilated, the radius chances, but the shape is preserved. The distance between points on the circle also change by an amount proportional to II times the original lengths.

    Given two circles, can you always (ever?) dilate one of the circles so that it coincides with the second? Why or why not?

    • A circle can always be dilated so that it coincides with a second. This is due to the fact that all circles relate to each other by a ratio of their radii (are dilations of each other). Since they are the same shape and their only variable is their radii, they are easily duplicated. In order to accomplish the coincidence of the dilation onto the second circle, the point marked as center, C, must be carefully chosen. Once the ratio of the circle's radius is determined and used for II, C can be chosen at the correct coordinates to locate the circles on top of each other. For a closer explanation of the location of C, see the explanation of a dilation given in the previous paragraph.

    Create an irregular polygon P using The Geometer's Sketchpad and construct its interior. Mark the center and ratio, and dilate the polygon interior to get P'. Mark a new center and ratio and dilate P' to get P''. How is P'' related to P'? What properties of P' are conserved by this "composition" of dilations? What properties change?

    • P" is proportional to P'. That is, the ratios of the edge lengths to each other in P" are the same as they are in P'. (The ratios are conserved.) The respective angles in the polygons are also conserved (ie. Meas. Of angle A' in P' = Meas. Of angle A" in P"). The shape is also preserved. The actual lengths of the edges and distances between points change between P" and P'.

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