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This unit took up the majority of my student teaching experience. As I reflect on it, indeed it is quite massive. Moreover, the correlation aspect of this unit was my second time around. At the 8th grade level, I was able to provide better means for helping these younger students understand higher level concepts. For example, treating the equation y = m(x) + b as a blue print, certainly made a difference in helping students understand the role that it would play later on. I was proud of these students. They really did some impressive things. This section will be devoted to looking at some of the sections of this unit that may have given me a difficult time. This section will also discuss some changes that may make a difference in the next time it is implemented.

The backgammon section was implemented during the first week of my student teaching. During this time, I had no idea of where, or how far I would take the game. Moreover, I had not yet thought of this unit. Only at about the middle of the section did I realize that what students were really exploring is probability and statistical tools that are used for predicting. At that point the bigger picture was coming to light. The backgammon game is an excellent way of exploring chance and theoretical versus experimental probabilities. It provides a new context for them to explore. Moreover, the idea of chance versus luck unfolds in a nice fashion. However, backgammon offers a limited context of what can be explored in probability. First off, the majority of what we had explored was actually through dice. I wanted to give them an opportunity to look at things that were unequally likely. The purpose for this is to provide more context for them to rethink their misconception of "everything has a 50/50 chance". Also I think it would be beneficial for them to practice listing sample spaces for different manipulatives as well. I have a strong belief that if students are to be taught a concept, they should be able to explore the concept across a variety of levels.

Another interesting episode that caught me a little off guard was the (1,2), (2,1) dilemma. One of the major drawbacks was that I did not have colored dice for students to play with. In other words, they were unable to "see for themselves" that both these events are not the same according to the rules of probability.

Earlier in this unit, I had mentioned that it is not necessary to necessarily have students take down the "teachers" notes when it came time for reflection. This is one of the biggest realizations I had. Unfortunately, this realization did not come to me until I had moved to the 7th grade! At any rate, for the kinds of activities that were performed in this unit, having students compile the notebook would be an ideal form of assessment. Not only would students see their own language and interpretation on the board, but I as the teacher, would be able to assess what the students comprehended.

The slope and the y-intercept went relatively well with the students. However, they needed a large amount of practice in understanding the concepts. Some were confused on the purpose for these concepts. Some student would even confuse the two. Now, that I can reflect on where the misunderstanding came from I realize that there is an exercise that can enhance the understanding of these concepts. Slope and y-intercept are utilized to draw a line. Both these concepts give a location and a rate of movement on a graph. Therefore, students should be asked to draw the graph given the slope and the y-intercept. In other words, they should be able to work backwards in their thinking. I think this would have given the students not only the purpose for such concepts, but moreover a concrete understanding of what they mean according to the graph. I think this would also help to clarify confusion between how slope is found and how the y-intercept is found when using the equation of a line. As I had mentioned in section V, the students were able to pick up on the fact that slope never changes, unless the line changes. However, not every student was able to see this quickly. Therefore, when students are asked to find the slope for various graphs (Appendix Q), it is a good idea to ask the students how it is that people can choose different points on the line, but yet come up with very close values of slope? In other words, this should be a natural discovery so that the whole class can realize this together.

One of the toughest areas the students faced was understanding how the equation y = m(x) + b can be used to find the y-intercept as well as predict the missing variable. They could not understand how one equation could be used twice. This misunderstanding does not surprise me. They have not had enough mathematical experience to appreciate the fact that equations can be manipulated to solve for a variety of values. However, due to the nature of the equation, I really could not think of a way to help them straighten out this issue. Here is where the argument stands to say that it is a higher level of thinking that young students will have a tough time with. Yet, I really can not think of a method to try to conquer this problem. However, I must mention that one of the most important things I did with the y-intercept was to super impose the graph. When the y-intercept is part of a broken graph ( i.e. the graph does not start at 0), it is crucial to have students view the missing piece of the graph. This really helped them understand the hole point of why one would need to use the equation of the line to get the y-intercept.

The final project provided a great culminating activity in the sense that students would have a final opportunity to bring closure to the correlation section, from beginning to end. Unfortunately I have not been able to review the final projects that the students turned in. However, when the project was assigned I was there to assist them in understanding what was required. The students took to the project pretty well. They seemed very engaged and anxious to get started. Some students wanted to turn in an exemplary project and thus began seriously searching to create their own correlation ( in other words not using the list of suggestions). Among the choices, the more popular were miles in car and the age, number of baskets attempted vs. made in a basketball game, and height of children and their parents. One student decided to find data on the depth of a river and how fast it flows. Another chose to measure the circumference of trees and the height of their trunks. So as one can see, there were some impressive thinkers out there.

The assessment had to echo what my cooperating teacher wanted. Therefore, although I had created the project and the assessment there are some things that I would change. First and for most, I would highly encourage students to make up their own correlation that they would like to explore. This includes encouraging them to use outside resources. I also feel that a short presentation from each student would allow the class to see what other students had accomplished. I think that students should also have an opportunity to assess their own work, thus giving them an opportunity to discuss their short comings as well as their achievements.

In conclusion, I am very impressed with the caliber of this unit. Moreover, I'm glad to know that I am teacher who does not under estimates the ability that students have. The packets and the thought provoking questions I posed, allowed students to quickly understand what my expectations were in their understanding of the content. They knew, that I was not so focused on the answers they were getting, but rather the explanations they were providing. Moreover, I have no doubt that due to the nature of these activities, students will remember what they have learned when it creeps upon them next year!!!

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