Questions and Answers

One of the assignments for today's math class was to find a algorithm that can be used to solve the area of any polygon. We talked about two different methods that have been found to work. They are Pick's Theorem and the Chop Strategy. I think that I found another one but I'm not completely sure if it will work in all cases.
I start out by listing all the coordinates in order and listing the first one again at the bottom so it completely closes off the polygon. Then I multiplied all the diagonal numbers slanting down and to the right and all the diagonals slanting down and to the left. I added both of these columns and subtracted the smaller one from the larger one and divided it by two.
The first example looked like this.
0 4
8 2 3 0
12 4 4 8
16 4 0 0
0 2 2 8
0 0 0 0
0 0 4 0
= =
36 16

36-16=20
20/2= 10
The area of the first polygon on the sheet is 10...

Summary and Synthesis

Today in class (Thursday 27th) we discussed area of polygons and triangles. First we learned how to find the area of a triangle by figuring out the area of the rectangle and dividing that by 2. Then we were given the task of finding the area of the polygons. At first I tried finding the area of the rectangle for each polygon and then dividing just like we did for triangles. That just gave me numbers that didn’t seem to fit. So then I tried to divide each polygon into triangles then find the area of those triangles rectangles and dividing by 2. That didn’t work either. But I never thought about taking the unused portion of the geoboard and finding the area of each of those then adding that up and subtracting that from the total geoboard to find the area. It all seems so easy now but that was one way that I would never of thought of on how to do it. The right way to solve for area is actually way easier then I was trying to make it when I was trying to come up with my algorithm.
I think it was good for the students to have to work on and try to figure out algorithms because it really got our minds thinking of all the different aspects to think about. To use this same strategy with younger students I think will benefit them just as it did us as college students. They will get a chance to look at and think about possible solutions and then will be taught the “right” solution and they will have a better understanding of why that is the “right” solution and why theirs didn’t work.

Personal Concerns and Next Steps

I wasn't sure if I should put this blog under this title, or under questions and answers. While reading the text, I completly agree that students need to be able to explore questions on their own, without the teacher giving them the answer. But where do we, as teachers draw the line? If the students aren't getting anywhere, their frustration level grows. Even mine, while working on formulas for irregular polygons, was eventually shot. And I'm a college student. Obviously they wouldn't be working on something quite so high of a level, but for them, multiplication will be as difficult as advanced algebra is to me. Hints and helpful, and very useful up to a point. But what if I have a student who just can't figure it out? Would it be better to try and have a peer explain it? Or would they feel worse because their peer understood it, but they didn't? There's a difference between giving them the answer, and explaining the process. Explaining the process is okay to do, isn't it? They still find the answer and their own. And just looking at the incredible time constraint teachers are under to help students understand math, I'm not sure that students could realistically explore too many processes on their own. I know I'm rambling a bit, but I'm trying to work it out in my own mind. Is giving an explaination as to why a formula works enough, or are the students supposed to figure out the formula themselves? I want to be a great math instructor for my students, and give them a good, solid foundation in math. Having them actually enjoy it is one of my main goals, since I never got that as an elementary student. I'm just not sure the best way to go about it. I need to balance the needs of my students against the needs that the administrators and state force me to adhere to. Thanks for any help you can give me!
Mary Fink