In doing my passages I tried multiple times to try to make mine as large as Dr. Higgins but I was nowhere close to being able to make mine as large. I was only able to make my passage using 13 pieces. Each time that I tried to add more in I would wind up having branches coming off. I am not able to upload a picture at the time because I am using a family members computer and for some reason the picture will not pull into the blog. I will try an upload a picture on my own computer later to see if it will work. I thought I would be able to form a pentomino narrow passage as Dr. Higgins did because I thought of it as almost like a puzzle (which I am generally good at) and was surprised to see how hard it was to actually use a lot of the pieces.
Tangram Discoveries:
Which polygon has the greatest perimeter?...the least perimeter? How do you know?
When forming my polygons in this activity I had trouble at first with myself in thinking I would not be able to figure out perimeters without a ruler but once I started forming them it made sense to me. When forming the shapes and looking at the perimeters it appeared to me that the rectangle had the largest perimeter and that the square had the least perimeter. How do I know? Well, when looking at the shapes it appeared that the rectangle had the greatest distance around where as the square had the least distance around which is what perimeter is.
Which polygon had the greatest area?...the least area? How do you know?
I believed my rectangle again to have the largest area and my square to have the least area. My reasoning for seeing this (which could be wrong) is in comparing area to perimeter. If my sides on my rectangle made the greatest perimeter it only made since to me that they would combine to make the greatest area as well. The same concept I applied to my square. Also, when looking at my square in comparison to my other shapes the area inside did appear to be the smallest.
Pentomino Activities:
When going through the PowerPoint and completing the Pentomino activities I had a fun time. I really enjoyed the first activity of the line symmetry. I feel like this would be a simple activity to use in the classroom and also one that students would enjoy. This is a good activity for students to do in order to learn hands on. The rotational symmetry activity was another hands on activity that I thought would be fun for students to complete. It is one that I found simple and enjoyed so I feel like my students would as well. Having students use the pentomino pieces to fine perimeter and area is also another activity I enjoyed. It gives the students to work with the pieces to find the dimensions instead of having to look at numbers on a paper to configure answers. The tessellation activity I did not like as much when I had to follow the instructions on the PowerPoint. I found myself growing increasingly aggravated and not being able to do it. When using the website I was however able to form the tessellation's and highly enjoyed the activities.Annenberg Symmetry Module:
When completing Part A of the Annenberg Symmetry Module I found question A1 easy to complete and I completed it with 100% accuracy. Problem A2 (although asking pretty much the same thing) I had a little more difficulty with. On the regular pentagon and the regular hexagon I was able to find half the number of lines of symmetry than what really appeared. I feel the reason I struggled on the was because of the way I was viewing them. I felt like 5-6 lines of symmetry was just too much so I did not look into finding that many. Question A3 I did not do well on. I was confused on the directions so I tried and when I did not feel like I could do it I just looked at the solution and then it began to make more sense to me.When completing Part B of the Annenberg Symmetry Module I found myself struggling with what the actual degree measurements needed to be. When I changed the degree measurement each time to rotate the object I was able to see if it was rotationally symmetrical but as for the actual degree measurement and knowing how to determine which ones would work; I need improving on.
For Further Discussion:
I found an article that talked about how the Ancient Greeks used geometry in their art forms. The Greeks were the first believed to understand "the laws of perspective" which meant they knew how to make a 3-D figure onto a 2-D surface. Artist of the Renaissance Period also understood the "laws of perspective" and they believed in forming parallel lines in their work as a symbol of forever. The artist of this time believed that parallel lines went on forever so that's what the parallel lines in their work symbolized. Also, I looked at a lot of Mexican paintings just on the Google Images page and it stood out to me that a lot of the paintings that used geometric shapes used a lot of triangles as well as a lot of circles. In researching I could not find any reasoning behind it but I did find it to be a neat attribute.
I had the same thoughts about the passage as you. I thought I would be good at it because I enjoy puzzles and I am usually pretty good at them. When I first did it, I made a huge passage but then later realized that it had to be a single passage. I was not able to come up with a large one either. I even tried to base it off of Dr. Higgins but still make it different and just could not make it too long. How do you think we could use this with our students in a geometry class? What age do you think would be good for this kind of activity?
ReplyDeleteI thought the same about the rectangle. It seemed to have the most space around the outside and that is what perimeter is, so I chose the rectangle too. It is hard to determine based on just looking at it. I feel like using a ruler would make it easier even though I have to admit I do not really remember some of the formulas for some shapes like the triangle and parallelogram. Do you remember these things?
I also enjoyed the activities in the PowerPoint. I think they are great hands on activities for students. I feel like a lot of geometry activities are better hands on for students because it is not necessarily a mathematical concept that can be based on definitions. It is really working through problems to determine the answers. Line of symmetry was always something I thought was fun in geometry for some reason. Would you change these activities in any way if you were to use them in your classroom some day?
The article you found sounds really cool. I never really thought about geometry being an art form until doing this module. Would you ever have students use this as a art project or have them research something similar to the article you found for a writing project? I think that is a good way to combine reading, writing, math and research skills all into one.
I think we could use these passages in our classrooms as a fun hands on activity just so see how the students went about making larger passages and as a way to assess whether they could make larger ones or not. I feel like you could have students make small passages in the 3rd grade and probably challenge them with larger ones in grades 4th and 5th. I do not remember forming passages in school so I can not really say for sure a grade. What do you think? I as well do not remember the ways to find the areas for those figures. I would have to look up the formulas on the internet and then try and figure it out from there. I actually enjoyed the ways that the activities were set up in the PowerPoint. I found them easy to follow and probably would not change anything about them and use the same format in my own classroom. I think an activity using art and geometry in the classroom would be a great integrated lesson. You could teach students different shapes, teach them background information from a culture, and then have them to form an art piece that would correlate with all of the information that they have learned. I love integrated lessons especially ones that incorporate the arts because students seem to really enjoy them.
ReplyDeleteI think that would be fun for students but I also think some students would get really frustrated because they could not make the passages. For this reason, I think that that would be an activity we should allow our students to do in pairs or in a small group. That way they can talk about it. 3rd grade is a good idea and harder passages in fourth or fifth grade.
ReplyDeleteI do not think I would change the format of the activities in the PowerPoint either. I think they were great activities and would really help students.
I also think integrating art would be fun for students. I always loved doing art in school and I think tessellation is one way to incorporate art into math. Most students would love it.
Teachers often shy away from things that kids will struggle through but that's really a part of the learning process. If it's easy, little learning actually occurs. However, I do like that you consider putting children in groups to work through the puzzle.
ReplyDeleteIn terms of the tangram's area and perimeter. We are often over-powered by the visual aspect of things and as adults, it's very difficult for us not to rely on the strategies and tools that we've used for so long. This means using a nonstandard form for measurement (without a ruler) can be difficult. First, when you created the different shapes using your tangrams, you used the same tangrams for each shape which means the area will be the same regardless of the shape that you made. With the tangrams that you used to make the shapes, notice that all three triangles are isosceles right triangles. In isosceles right triangles, the two legs are congruent and the hypotenuse is longer than the legs. There are some convenient relationships between these pieces in the tangram puzzle: one side of the medium triangle is the same length as the hypotenuse of the small triangle, and one side of the small triangle is equal in length to half of the hypotenuse of the medium triangle. These relationships enable us to measure the perimeters of the five polygons in terms of two units. This means that the triangle, trapezoid, and parallelogram all have the greatest perimeter and the square has the smallest.