The Hexagon Pattern Train Task was interesting to me. The first task of computing the perimeter for the first four trains was easy. I went about determining the perimeter by counting the sides on the first three and then by the fourth train, I saw the pattern that we were adding four to our previous units for each hexagon added. On the second task, I found the perimeter to the fifth train by adding 4 to my answer from the fourth train or 18 units plus 4 units equals 22 units. To find the 10^{th} train perimeter, I took my answer from the fifth train of 22 units and added 20 more units (which is 4 units for each additional hexagon added or 4 x 5). The third task asked us to write a description that could be used to compute the perimeter of any train in the pattern. The description that our group came up with is that you multiply 4 times the number of hexagons in the train pattern and then add 2 units for the front of the train and the back of the train. This description helped us complete the fourth task which asked for an equation in terms of the number of hexagons. Our group came up with the formula of 4n+2=Perimeter of the hexagon train with n representing the number of hexagons in the train. I understand the reasoning but figuring out how to help students understand is a challege.

Our second discussion of Thinking Through a Lesson Supporting Students’ Exploration of the Task was a little more difficult for me. I had to look at each student’s response and figure out questions to ask the student regarding their response. Student 1 has the right idea and actually looked at it the same way that I first attempted the task when I had to do it. He/she is counted the sides on the first hexagon and then adds 4 for each additional hexagon added to the train. The student sees a pattern developing of adding 4 each time a hexagon is added. However, the student would have to do a lot of adding in order to find out the perimeter for a 40 hexagon train. Student 2’s response is a little more advanced than Student 1 but still basic. The student is multiplying the number of sides in a hexagon (6) times the number of hexagons in the pattern and then subtracting 2 for the centers of the touching hexagon pieces. This strategy works but I wonder how long it would take the student to answer the question for a 20 hexagon pattern train. The student is very close to coming up with an equation that could help solve longer pattern trains easier. Student 3’s response took me a while to understand. He/she saw the pattern of adding four for each additional hexagon added to the train but took it a step further to come up with a formula to use if you didn’t have a previous number of units to build upon. This student took the hexagon train and split it into parts. He/she realized that 2 sides of each hexagon comprise the upper section of the train. The student says to take the number of hexagons and multiply by 2 for the top portion. Then double the number to account for the bottom and top of the train. He then adds 2 for the front and end of the car. This is definitely a more advanced thought process thatn Students 1 and 2. Student 4’s response is very similar to Student 3 as far as looking at the train in sections and developing a formula. He/she sees 2 units at the top of the train for each hexagon so he/she multiplies the number of hexagons by 2. Knowing that there is also a bottom of the train with the same number of units, he takes the answer from the top and multiplies it by 2. Finally, 2 is added to the answer to account for the front and back sides of the train. It appears to me that Students 1 and 2 are at a basic level while Students 3 and 4 are a little more advanced. Students 1 and 2 will be challenged in their thinking process by having to solve the problems with larger number of trains. Their current way of computing the answer would require a lot of counting. Certainly, the teacher could help guide them into finding a formula to help solve the problem by asking them questions about a smaller train pattern and working to see if the formula holds true for the larger train patterns. I think each student is on their way to understanding and can come up with an equation to solve the problem with some guidance from the teacher.

I completey agree with you when you say that students need some guidance from the teacher when approaching a task like the hexagon task. I needed guidance from Dr. Polly when it came to figuring out the equation to use. As much as I like the indirect approach, a task like this sort of scares me. I’m afraid I will do to much to lead a student to the answer instead of just guiding them in the right direction. I had a hard time understanding some of the students’ explanations for solving the problems. This also frightened me a little! It was difficult for me to wrap my brain around their explanations and directions. I did think the task was interesting and fun to do. I think students would enjoy it and it is a great way to show students how there are many different ways to solve most problems.

I was very impressed by Cyndi’s analyses of the hexagon train problems. A teacher would need to get into a “psychologist” state of mind when discerning each student’s thought process, especially in math. Of course, it would not be feasible to spend a great amount of time doing this with each lesson, but would be valuable information for lesson planning. As new teachers, we will learn how to allocate our time wisely across the many facets of curriculum management. Constantly readjusting the scope and depth of our lessons based on students’ logical reasoning skills, while keeping the class engaged and excited about the content, will prove to be a true test of assessment ability, as well as time management and organization skills.

It is always interesting to see how people think through problems differently. This very helpful to us as teachers. If adults solve problems and think differently what do you expect a group of 2o students to do. This is why I think open ended questions and explanations are important. Sometimes we may just not see it the way the student saw it that is why explanations are important.

The knowledge of this is very beneficial to me because I am a math person and it didn’t take long for me to figure out the equation but some students are going to need guidance and some are just going to need a little more time.