Journal Entries February 1999

Monday, February 1

I appreciated the hints you gave for several of the homework problems. Robyn and I spent a good portion of time working on 1 and 3. Your suggestions will help with the others.

Classtime spent investigating that every isometry is surjective was helpful. I still am unclear what it all means. I understand what onto means but not how it relates to what we were doing today.

I liked what we did in class revolving around the Triangle Theorem. The GSP investiagtion was useful in seeing how to go about the proof. This is another case that proves how important GSP can be in teaching a geometry class. I also was able to review how to construct a congruent triangle.

Wednesday, February 3

Today's class had several main points. A brief discussion centered around the Triangle Theorem. I do understand geometrically how to prove this theorem. But to prove theoretically is another story. I enjoyed you taking the time out for Carla's question which proved to be a teachable moment for the class; well at least for me. It helped cement the idea that every isometry is at most the product of three reflections. It also showed that you listen to your students and acknowledge their questions; something very important to model for future teachers.

I understand the concept of a fixed point. I am still fuzzy on orientation however, especially as it relates to preserving and reversing. Even after Robyn's question I still was in the fog. The table summing up the relationship between isometries and fixed and orientation was useful and tied together some important concepts. The GSP examples proved helpful in explaining the table.

As I reflect on this past homework assignment, it was challenging. It always is when working with proofs. But working with Robyn and Inchul helped. I realize though I have a long way to go in proof skills. Even though I am apprehensive about proofs, I still appreciate the fact that it is necessary to work them to get better. Practice always helps.

Friday, February 5

Today's discussion centered around representing transformations (isometries) using coordinate geometry. After reading section 5.5, Coordinate Characterization of Linear Transformations, I was completely lost. However, after today's classroom discussion and investigation, I understand several of the concepts. The book's presentation is unclear, hard to follow, and encumbered with heavy notation. Your presentation and guidance through today's lesson was clear, easy to follow, and fun, especially using GSP. I enjoyed our work using GSP. It always seems to cement the mathematical concepts and make them more clear and concrete. I do like the challenge of proofs, but using GSP to make sense and investigate the geometry is always more enjoyable to me. I seem to get more out of it. I also appreciate all of your hard work in making the ocurse web friendly.

Monday, February 8

Discussion today centered around the homework assignment and the relationship between isometries and matrices. I was confused as to both regards.

After having completed problem 1 and 2 of the homework assignment, I now understand Robyn's question. I was able to figure out what a and b are algebraically. Then the verification became easy, just time consuming. The homework assignment was extremely beneficial in seeing the relationship between the equation for the isometry and the coordinates of the transformation. This assignment was concrete and worthwhile. I am still muddy on the extra credit.

After the class discussion, I still do not see the relationship between isometries and matrices. The language and terminology was vague. I had a hard time following the path that you were taking us on. Therefore, I never really saw where we were supposed to be or what direction we were headed. I will spend time reviewing my notes and see if that helps clear the muddy waters.

Wednesday, February 10

After seeing you for office hours and getting help on the extra credit, I now understand the extra credit and what you were doing in class on Wednesday.

Today, class centered around strip patterns; a definition, how you construct them, symmetries involved in the patterns, and other important ideas concerning strip patterns. Class was interesting and enjoyable. When we do activities that are applications of many of the ideas we have discussed such as symmetry and isometry, I benefit a great deal. Both in terms of being motivated to learn the material but also the concepts are clear and more deeply embedded in my mental schema. The strip patterns are a great way to discuss ideas of symmetry. I am looking forward to the homework assignment; it sounds fun and engaging.

The idea that I walked away with today was that every strip pattern must have a translation. From your explanation and illustrations (GSP), it was clear and this point was well made. I was also amazed that the entrie world of strip patterns can only be made up on the 7 types of symmetries.

Friday, February 12

Class today was really interesting. Class discussion mainly focused on the seven types of strip patterns: L, L', V, V', E, N, and H.The discussion centered on the seven types of Hungarian needlework designs was helpful and quite motivating. It was a beautiful representation and application of the seven types of symmetries involved in strip patterns. We also began a brief discussion about why strip patterns are limited in regards to there only being seven types. This is due to the fact that there are a limited number of symmetries. Actually, there are only five types of symmetries. Since the compositions allow for repeats that explains why there are only seven types possible. Next class will elaborate on this point.

The assigned homework was fun and extremely helpful in understanding the seven types of strip patterns. The part I found most challenging was where we had to create our own strip pattern making sure to include one of each type using GSP. This part of the assignment took more time than I had anticipated. It proved to be more difficult but enjoyable. I learned alot more having had to create my own strip patterns versus simply indentifying the seven types in patterns already produced.

Monday, February 15

Today's class was extremely helpful in understanding why there are only seven types of strip patterns. The class discussion primarily revolved around answering the question, Why are there only seven types of strip patterns? First we completed the multiplication table for the five symmetries of strip patterns: translation, reflection with horizontal mirror, reflection with vertical mirror, glide reflection, and a rotation by 180 (half-turn). After completing the 5X5 table, the discussion turned to linking this multiplication table to help understand why there are only seven types of strip patterns. Since there are only five types of symmetries that occur for strip patterns and all have a translation, we are left with 2^4=16 possible symmetry combinations for a strip pattern. Once we identified the 16 possible symmetry comibinations, we knew 7 out of the 16.

We were then able to conclude that the other 9 combinations would not be possible based on an important underlying principle. The guiding principle was that the set of symmetry types of a strip pattern must be closed under multiplication. For example, TH=G. Since this possible combination is not closed it is automatically rules out as a possible symmetry type for a strip pattern. We went through each combination discussing why the combination did not hold, meaning the product was not closed. This information came directly from the multiplication table, hence the connection between the table and the possible strip patterns.

The order that you introduced the homework assignment was important. By having us first do the assignment then having a discussing about this topic after the homework assignment was complete was powerful and made an impact on my learning and understanding the reasoning behind why there exists only seven types of strip patterns. The class discussion was organized and flowed well. Very well done.

Wednesday, February 17

The review that you provided was beneficial and helpful. I appreciate any review you do in class. I thought it was well organized and you were great at answering all of our questions. THe only area that I feel worried about is the proofs that may be on the test. This is a weak area of mine. Since I must study for the test, I am making this journal rather short and signing off early.

Friday, February 19

I wanted to briefly comment on the first test. The test was extremely fair, well organized, and touched on the main concepts we had been studying since school started. Now that I look back over my mistakes, I realize several lessons. First, I now know that I must think more about the drawn figure for a proof and take more time to develop a well thought out diagram. Now that I know how to draw the diagram for the proof, it seems rather simple. Second, I also realize the importance of the direction of the vector when determining the new coodinates using the isometry equations. I thought I had done alittle better on the test. Taking into consideration my careless mistakes, I am pleased overall.

Monday, February 22

We focused on a new variation of strip patterns, called wall patterns. As compared to strip patterns, these are different in the number of independent translations they have. A wall pattern must have at least two nonparallel independent translations.

Most of class time was spent constructing a specific wall pattern and then identifying the types of symmetries that the pattern held. My pattern seemed more challenging as compared to the others. I am grateful that mine was more difficult. The challenge allowed me to work harder, take the assignment more seriously, and learn more about wall patterns as compared to if I had an easy pattern to construct. Overall, I thoroughly enjoyed the class lesson and how the content was organized and presented.

Wednesday, February 24

Today's class focused in more detail about wall patterns. We created a table that listed the different types of symmetries involved in the 24 wall patterns (handout in class). We then compared each of the wall patterns based on their symmetries. We did not make any final conclusions. That was left as an extra credit homework assignment. We did conclude, however, that there are 17 types of wall patterns. After your reading suggestion by Stevens, I was able to locate and identify the 17 Plane Groups. I thought the lesson went well. I especially liked collecting data from each student to help build the lesson. I found that useful and motivating.

Friday, February 26

Today was very interesting. After one day of hyperbolic geometry, you have me intrigued and interested. I have never seen hyperbolic geometry before. Your presentation of the lesson was excellent. You went slow, answered all student's questions, and seemed extremely patient. I admire that teaching quality in you. I do feel I have a grasp of the material that you presented and a basic understanding of this type of non-Euclidean geometry. I found it extremely useful how you continuously connected hyperbolic geometry to Euclidean geometry; that was great. You have definitely peaked my interest in a new mathematical topic. I am looking forward to investigating and exploring non-Euclidean geometry. You did GREAT today.

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