Project 4: Similarity

1. Mathematics Methods and Modeling for Today's Mathematics Classroom p. 216 number 4 (Loggers). [This is on ASULearn]
   Print out your Excel work and be sure to show work for our models from class and explain which model is better from your Excel solutions. Excel Regression Reminders
   1) Enter in the data so that the x-axis is in a column before the y-axis
   2) Formulas in Excel are entered as (for example) =b2^(2/3)
   3) To insert a new column, click on the next column and Insert/Column
   4) To fill a formula down a column, click on the first box, go to bottom right, click, and fill down.
   5) To create a linear regression plot, click on the letters above the columns, then Insert/Chart/x-y scatter/Marked Scatter. Next, Click on one of the points on the Chart, so that all the points are highlighted. Under Chart choose Add Trendline, and then choose Options - Select Display r^2 value and hit OK.
2. Draw a figure in Sketchpad. Illustrate the effects of congruence using a translation, reflection, and rotation [figure out how to do these transformations yourself by looking around in the built in menus in Sketchpad]. Be sure to label your original figure and the transformed figures and to explain in Sketchpad text comments (that also include your name) which figure is which. Attach your work to the personal storage space as yourfirstnamep4n2.gsp
   
   
3. Draw a figure in Sketchpad. Illustrate the effects of similarity using magnification or dilation that is
   a) positive number >1,
   b) a positive number <1, and
   c) a negative number.
   Be sure to label your original figure and the dilated figures and to explain in Sketchpad text comments (that also include your name) which figure is which. Attach your work to the personal storage space as yourfirstnamep4n3.gsp
   
   

4. The proof of the Pythagorean Theorem that we saw in project 1 was inspired by a figure that was included in the book Vijaganita, (Root Calculations), by the Hindu mathematician Bhaskara (around 1150 A.D.). Bhaskara's only explanation of his proof was, simply, "Behold". What you see is an animated version of Bhaskara's Idea. A quote from Math Awareness Month 2000 states: "Calculus is not the time when students should be doing their first serious thinking about geometry. Rather it should be the culmination of years of consideration of increasingly sophisticated geometrical topics... It isn't necessary to wait until students have learned about square roots before they can see an illustration of the Pythagorean theorem. Children who play with geometric puzzles that illustrate decompositions will find it easier later on to appreciate formal results. "

   
   
   Bhaskara also devised a proof of the Pythagorean Theorem based upon the notion of similar triangles. On the pdf Bhaskara's Similarity Proof (see also p. 151-152 of Wallace and West Roads to Geometry), fill in the details and reasons (using the appendix A of Euclid's Book 1 from Sibley The Geometric Viewpoint and the Similarity Postulates and Definitions from class [similar to what we did for the trig identity proof in class]).
   
   
5. Examine the Euclidean proof of AAA via Theorem 4.4.5 on p. 149-150 in Wallace and West Roads to Geometry and review the solutions to problem 8 in project 1 on ASULearn. Notice that there are two sets of triangles in the project solutions that satisfy AAA (pair 1: a usual triangle and a self-intersecting triangle and pair 2: 2 triangles with angles all 180 degrees) but they are not similar or congruent. Notice that for each of the pairs the corresponding sides are not proportional. Use the counterexamples listed there and relate them to the Euclidean proof in the book to discuss what goes wrong with the proof for each pair [similar to what we did for the Euclidean SAS proof and two spherical triangles that satisfied SAS but were not congruent].