Week 1:

1. Bring in three examples of planar patterns to share with the class. Planar patterns can be found on fabrics, wrapping paper, wallpaper, tiled floors, and many other places.
2. Write a one page outline of a lesson on symmetry. On a separate page, discuss your lesson's objectives, format (e.g. lecture vs. group work), assessment, grade level, and whether computers could be used to enhance your lesson.

Week 2:

1. Look for simple translation, rotation, reflection, and glide-reflection symmetries in the patters your classmates brought in. Indicate two translational symmetries by arrows. If you find mirror symmetries, draw one or two lines of reflective symmetry. If you find a rotational symmetry, put a colored dot at a few of the centers of rotational symmetry. If you find a glide reflective symmetry, indicate it by drawing a dotted arrow from one point in the pattern to the point it is sent to by the glide reflection. Try to find the smallest possible region (motif) which can be repeated to make up the entire pattern. Circle it.

   HINT: you might want to copy your pattern before you start writing on it!

2. Write approximately one and one half pages comparing the computer programs Kali, Tangram, Geometer's Sketchpad, and TesselMania! Your discussion should address the following points: How are they similar? How are they different? How and when would you use them to teach? If you would not use one of them in your teaching, why not?

Week 3:

1. Work the exercise on quadrilaterals taken from the Discovering Geometry textbook: define the five terms, create a tree diagram for quadrilaterals, and create a Venn diagram for quadrilaterals. Be prepared to construct examples of your figures using Sketchpad next period.
2. Write a preliminary proposal for your final project. The proposal should include who you will be working with and what subject you will address.

Week 4:

1. It is true that any transformation of the plane can be described as the product of at most three reflections. (We looked at products of transformations when we made a "multiplication table" of transformations during week two.) Given two congruent triangles in the plane, describe how you could find three (or fewer) reflections which, when combined, send the first triangle to the second. Assuming that a transformation is completely determined by where it sends three points (for instance, the points (0,0), (1,0), and (0,1)) explain why this proves that any transformation can be written as a product of at most three reflections.

   HINT: If triangle ABC is congruent to triangle A'B'C', find a way to send A to A', B to B', and C to C' in three steps, each step being a reflection.

2. The Mallard "interactive learning environment" developed at UIUC allows instructors to post quizzes on the World Wide Web which are graded automatically and give instant feedback to students. Instructors can provide several different versions of the same question, so that the quiz appears differently to different students. Each student may have a secure personal account which no other student can access without their permission. However, there does not appear to be any way to prevent one student from inviting another student to take their quizzes for them.

   If it were easy for you to create quizzes using Mallard and easy for your students to take them, how would you use Mallard in your teaching? When would you not use Mallard? What subject matter would it be most useful for teaching? In the future, many students will be able to use Mallard from their home computers. How do you think programs like Mallard will change the way mathematics is taught (if at all)?

Week 5:

If we cut the outside rectangle along the bold line shown in the (figure, i.e. cut the blue squares out of the corners), we get a fattened plus shape. By folding up the "legs" of the plus shape we get a lidless box.

1. Write a formula for the volume of this box as a function of s.
2. Substitute the values l=11 and w=5 into your volume function. Graph the volume function. What shape is your graph? Why? (I.e. what features of the volume function cause it to have that shape?)
3. For what value of s do you get the greatest box volume when l=11 and w=5? (Extra credit (+2 points): Find the volume maximizing value of s for arbitrary l and w. Show your work!)
4. Write one or two pages outlining a lesson based on this experiment. Your outline should answer the following questions: What grade level is this intended for? What topic are you investigating: functions, volume, graphing, finding maxima and minima, or something else? What materials do the students need for the exercise? Would you use a computer in preparing for the lesson? How? Would students use computers or calculators during the lesson? How?

Week 6:

Note: this is essentially "clock arithmetic" or "modulo arithmetic". If you get stuck, look up these terms in a math text.

1. Divide a sheet of paper into 12 categories labeled by remainder. Place each the following numbers in the appropriate category:
   • The numbers 0 through 24.
   • 25, 38, 42, 93, 98, 145, and 365.
   • The multiples of six up to 96.
   • The multiples of 36 up to 108.

2. Add a number from category 1 to a number from category 2. In what category is the answer? Add a different number from category 1 to a different number from category 2. Is your answer in the same category as before? Add a number from category 5 to a number in category 7. In what category is the answer? Add a number from category 9 to a number in category 11. In what category is the answer?

3. Add a number in category 0 to a number in each of the categories 0-11. In what category are the answers? Complete the sentence: "The sum of a number m in category 0 and a number n in category A will be in category ___." Why should this be true?

4. Add a number in category 3 to a number in each of the categories 0-11. In what category are the answers? Make a conjecture of the form: "The sum of a number m in category 3 and a number n in category A is ..."

5. Suppose you are given a number n in category A and a number m in category B. What is the category of n+m? Extra credit (+2 points): Prove your answer is correct.

6. Multiply a number from one category by a number from another category. In what category is the product? Repeat this experiment several times with several different numbers in several different categories and make some conjectures about multiplication of categories. (Here is a sample conjecture: The product of a number in category 11 with a number in category A is always a number in category 12-A, if A is not 0.)

   
   

7. Browse through the list of web sites selected to be of interest to Mtht480 students. Find three web pages whose content you consider to be worthwhile (please don't include search engines or lists of links to other locations.) Record the URL's of those pages and write a one paragraph description of the content of each page.

Week 7

1. (3 points) In class, you read the notes in section one of an Internet Seminar. How would you describe the internet to your students? What analogies could you use?

2. (7 points) Final project progress report: Give an outline of your final project presentation. List all books, software programs, web sites, etc. that you have identified as useful sources of information for your presentation. Students doing in-class presentations of computer software may have well developed outlines and few information sources, while those doing research projects may have many information sources and a minimal outline at the present time.

Week 8

1. How can the internet have a positive role in mathematics teaching? For what sort of tasks and purposes will it be an effective tool? How will you recognize and avoid frivolous uses of the internet in teaching?
2. How would you control your students' access to materials on the internet? How can you keep them focused on the internet project you have chosen for them?
3. List what you consider to be the four most important issues covered in the Acceptable Use Policies referenced on the District 97 web site. (Sadly, several of these links are broken. Please be patient.)
4. What will you do when an 8 or 12 year old in your classroom downloads pornography? Hate literature? Video game demos? Are you willing to teach from web sites on which there is advertising?

5. In class, you worked on a meteorology project that was completely computer based. There were no handouts and a minimum of lecturing in this lab. Compare this to the the "What is the Relationship" project we did in week five. Your comparison should be up to one page long, and should mention advantages and disadvantages of having the entire assignment on the computer, under what circumstances it is advantageous to have a "paper-free" assignment, and under what circumstances it is advisable to provide handouts to your students.

Week 9

1. Write up a proof that any transformation can be written as a product of three or fewer reflections. Your proof should take the form of a statement of hypothesis followed by a step by step logical argument proving your hypothesis. You need not number your steps or use a "two column" format. Your argument will probably consist of a proof that any triangle can be sent to any other triangle by three or fewer reflections followed by an argument that any transformation (i.e. rigid motion of the plane) is completely determined by where it sends some fixed triangle.

   You may wish to refer to the web page listing the different transformations. If you have not yet done proofs in a college level course, please make an appointment or send email to burgiel@math.uic.edu to get help.

2. Complete the web page we started in class. The instructions for posting your web page given in class were correct; we were just unable to view the posted pages. You may wish to post your web page before coming to class to ensure that there is no conflict between Macintosh and Windows file formats.

1. (5 points) In your homework for Week 1, you designed a lesson on symmetry. Design a web page that you would use to support or extend that lesson, or for some other lesson. Include a brief summary of the lesson on your web page.

   This web page may take the form of a list of links to sites with content relevant to your lesson, a gallery of pictures providing examples for your lesson, an introductory discussion to your lesson, or a series of questions about the lesson with a link to an answer key.

   Post the completed web page on your UIC account and write down and turn in the URL of the page. (Remember, office hours are from 4-5 on Tuesdays and by appointment. You can get help posting the page if you have finished it before 4PM Tuesday.)

2. (3 points) Design a Logo program poly that takes an argument numsides and draws a regular polygon with that many sides. For example, typing:

   poly 5

   at the ? prompt should cause the turtle to draw a regular pentagon.

3. (2 points) Design a Logo program star that draws a five pointed star.

1. Review the results of the Logo exercise you did in class. What is the relationship between the total turn of a Logo path and the number of times the path must be repeated before the turtle returns to its starting place?
2. If you were going to use Logo in your classroom, how would you use it? Would you design games for your students to play? Would you teach your students to program? Or would you simply let students maneuver the turtle about the screen using setxy, fd, rt and lt commands? Your answer should mention the grade level of your students and what skills you would be using Logo to teach.
3. Define a p/q star to be the figure generated by the Logo command star p q, where star is the procedure defined in the second set of Logo notes. How would you describe a p/q star to someone who doesn't know Logo? (How many points, how sharp are the points, which points are joined by segments, etc.) When q is prime, the different p/q stars each have q points. If p and q have a common divisor d, this appears to no longer be true. Why not? Would you say that a 2/8 star has four points, or four pairs of points? Why? (In the first case, you have an unpleasant exception to a simple rule. In the second case, you are saying that something that looks like a square is actually an octagon. Which do you prefer?)

1. Subscribe to a mailing list with content related to teaching. Forward one message from the list to the burgiel-mtht480 mailing list.
2. Turn in a rough draft of your final project. This should include a detailed list of the topics you will cover and a copy of any handouts you plan to use.