Module 10

Hello!

 

   I hope everyone has enjoyed their Easter break and hopefully you had some time to spend with friends and family. This module was very informative so I learned a lot about other geometric concepts. I have provided my thoughts and ideas about each part of the module but have indicated the required content in bold red.

More Pentomino Explorations-PowerPoint Video

This activity allowed me to explore different geometric concepts using pentominoes:

 Line Symmetry

   I thought this activity was pretty simple. However, I kind of second-guessed myself when I found that only half of the pieces were symmetrical. Here is what my pentominoes looked like after adding my lines of symmetry but before checking my answer through the PowerPoint. You can't see it that well but I included how many lines of symmetry I found for each piece: The "W" shape has 1 line of symmetry, the "L" piece has 1, the "upside down T" has 1, the "U" (or example piece used in the video) has 1, the rectangle has 2, and the cross piece has 4 lines of symmetry.

After going through the activity I was glad to know for a fact that there were only 6 pieces that were symmetrical. I even had the correct number of lines for symmetry for each shape!

Line and Rotational Symmetry

   This was a little tricky to begin with because I wasn't exactly sure what it was asking. But now, I see that it was asking me to identify the pieces that would rotate to look the same (at 90 degree intervals) before making it around the entire rotation (360 degrees). Due to my confusion, I had to look at the answer to get an idea of what to do and then played around with each shape. Here is a picture of my traced shapes. I managed to fit them all on one sheet of paper as I worked.

Area and Perimeter

  This was an was an easy task but it did help that I had my previous outlines of the pieces to figure out the perimeter of each. I simply drew lines for the squares on each and counted the perimeter. As the PowerPoint mentioned, I found that all but one of the pieces had a perimeter of 12. I did find that the remaining one had a perimeter of 10.

 

 

Tessellations

  I have honestly never worked with tessellations. In fact, I don't recall even hearing about them as a young student so this concept is relatively new to me. I am a little familiar with M.C. Escher's unique and mind-bending work so it helped to have an example of the tessellations he formed. As instructed, I used one of the pentomino pieces to form my own tessellation. I used one of the hexagon pieces that looks like an L. I used the graph paper provided in a previous module but had to add a bit of extra paper to the top to complete the image. Here is my finished product:

 

    I did not do a very good coloring job (my colored pencils kept breaking) but I think it shows how one shape can make a cool looking pattern. It would be interesting to see what students would come up with using one or more pentomino pieces!

  

Pentominoes for Spatial Sense Activities

 

    I actually enjoyed working through these activities, even though I thought they were are a little challenging. I started out doing the easy version of the game on the Scholastic website. I tried to find at two different ways to make the a solid rectangle before moving on to the next level:

 

Easy Level 

 

 At times, this level was hard, especially after I realized I was using similar pieces each time. I played around with it some and then got the hang of it.

 

Medium Level

   This level was a lot harder! It took me a long time to reach one solution, let alone two! 

Hard Level

 Unsurprisingly, this puzzle was super hard to solve because you had to use all of the pentomino pieces. Because of this, I was unable to find more than one way to piece them together. 

 

 

Use the pentomino pieces to make different size rectangles

  For these problems, I drew out the different sized rectangles and placed the pentomino pieces inside to make a complete shape. I spaced some of the pieces out a little so that you can see which ones I used.

3 x 10 rectangle:

I used 6 pentomino pieces to form this rectangle.

 

5x5 rectangle:
I used 5 pieces to form this rectangle.

 

5 x 8 rectangle:

I used 8 pieces to form this rectangle.

 

4 x 10 rectangle:

This one was super hard! It took me really long time to find one way to form this rectangle. I finally was able to use 8 pentomino pieces to complete it.

 

6 x 10 rectangle:

I was surprised that I found this puzzle to be the hardest to piece together. I thought it would be a little easier since I knew I had to use all 12 pieces and that we were told there were 1,339 different ways to make this size rectangle! I was wrong! After about an hour of rearranging the shapes, I admit that I had look online for a solution.

 

Talk about which websites you explored and which ones you’d like to try in your own classroom. Also, summarize how you progressed with the activities and if they caused any frustration.

  The Play Pentominoes game was an interesting activity. I did have some difficulty but was eventually able to find solutions on my own. Putting the pieces together with the pentominoes from my math kit was easy at first but then I experienced problems with the last two. I could tell that as the rectangles got a little bigger, the task of arranging the pentominoes became increasingly harder. It was certainly frustrating to almost have the puzzle solved only to find out that I needed a piece that I already used. I had to basically start over each time this happened. I would like to know if there are any specific strategies out there that help people determine what shapes might or might not work or which ones will work the best. There has to be an easier way!
    Aside from this website, I did search for other websites that might help students practice pentominoes. It was hard to find websites that didn't have pages of information about pentominoes--stuff that would not really appeal to young children. In an effort to look for fun activities or games, I did find a similar one to the Play Pentominoes game. A website called Wall of Game has a ton of fun games to play from logic puzzles and Sudoku to board games and cards. Under the "puzzles" category, there is a game called Pentominoes that allows players to choose from 10 puzzles to create only using pentominoes. Link: http://wallofgame.com/free-online-games/arcade/396/Pentominoes.html. I will say that it is rather challenging but I like how it gives you hints when you need help. Here is an image of the camel puzzle I completed using all 12 pentomino pieces:

 

      I didn't find and other websites like this that strictly use pentominoes. However, since starting these modules about this geometric concept, I can't help but think about the classic game Tetris and how some of the shapes used in the game look like pentominoes--each piece just has 4 cubes and not 5. As the article from last week's module mentioned, playing video games can help children develop spatial skills and learn strategies for overcoming problems. Perhaps playing a game like Tetris would all them to also visualize pentomino-like shapes in different ways since the point of the game is to match up the shapes to make lines. Even though the game shapes are not pentominoes, maybe this can be used as an introduction activity before teaching students about the actual geometric concept. Just a thought. There are a lot of different websites that feature Tetris like games (and even phone/tablet apps). Here one you can play for free: http://www.tetrisfriends.com/games/Marathon/game.php

 

Pentomino Narrow Passage

 

I thought this was a difficult task but I kind of enjoyed the attempt to build a narrow passage greater than Dr. Hargrove's. However, I was not able to construct a passage with more than 22 units. I came close, though! Here is an image of my passage with 19 units.

 

 

 

Tessellating T-Shirts (NCTM Article)

 

How has this article furthered your understanding of transformational geometry? What does it mean to tessellate? Look online for different examples of tessellations and share what you’ve found

    
    I really enjoyed reading this article! Who knew that learning about tessellations could be so fun? I like how the pre-service teachers had an opportunity to work with different patterns and create their own unique designs using one shape. I have learned new ideas about tessellations from this article, especially since I have little experience with the concept. Essentially, I learned that tessellations can help students dig deeper into the idea of transforming and rearranging shapes. I even gained some insight on how to introduce children to repeating patterns like this—by using physical motions. It’s a great idea to use sideways steps, mirror reflections, and twirls to help kids learn about transforming shapes. It was interesting to see how some students (and even student-teachers) can make mistakes when creating tessellation patterns. I admit that when I did my pentomino tessellation, I messed up the design.

   I must say that I love the idea of creating t-shirt designs when teaching kids about tessellations. It sounds like a really fun activity that students would get excited about. Not only are the subjects of art and math being integrated, the children are using are enhancing their creativity to complete their shirts. I agree with the authors of this article in saying that this can help young learners see that math can be fun!!

 

    I searched the Internet for examples of tessellations and found a lot of interesting ideas. I came across a lot of Escher’s work, which is pretty amazing. I love this watercolor tessellation he did of a sea horse (my favorite sea creature!).

 

Other images like these caught my eye during my search:

 

 

 

I even found some examples of tessellations we see in nature or in man-made objects:

 

 

 

 

Tangram Discoveries-Area

 

I followed the module directions and used the three different tangram pieces to create the shapes: a triangle, square, rectangle, and trapezoid. Here are pictures of my work:

 

 

-  Which polygon has the greatest perimeter? …the least perimeter? How do you know?

     I would say that the triangle has the greatest perimeter. I can see that this shape has three sides that are longer than any of the other shapes (except for the trapezoid but this has only one of these long sides). It's easy to see that the square has the least perimeter because is has the shortest sides. Clearly, the other shapes have greater perimeters because their sides are longer.

 

-  Which polygon has the greatest area?…the least area? How do you know?  

  All of the polygons have the same area. This is because they each are made up of the same shapes.

 

Annenberg Symmetry

 

The Annenberg lesson provided a lot of information but it gave me some interesting insight about symmetry. I have provided my responses to the questions asked in each section.

 

Section A

 

Problem A1: For each figure, find all the lines of symmetry you can.

 Of the 8 shapes given, I found that only 3 did not have lines of symmetry. Here is an image of the shapes and the lines I drew. The numbers indicate how many lines of symmetry I found. For the circle, I stated that there are infinite numbers of lines. I hope this is correct because the solution did not mention this.

 

Problem A2: Find all the lines of symmetry for these regular polygons. Generalize a rule about the number of lines of symmetry for regular polygons.

 

 

Rule: I would say that the number of lines of symmetry a regular polygon has is equal to the number of sides it has.

 

Problem A3: For each figure, reflect the figure over the line shown using perpendicular bisectors. Check your work with a Mira.

  This part about perpendicular bisectors was new to me but it was pretty easy to understand. I printed out the shapes and drew their reflections on the other side of the line:

 

  

I was confused on the direction's mention of using a Mira to check my work but I have never even heard of that before. I did a little research on the Internet and found that a Mira is a math tool you can use to reflect shapes or other writings/drawings to trace on paper. The pictures of this tool helped me see that we have one of them in our manipulative kits so I practiced using it. It was very interesting!

 

  It's a little hard to see from these pictures but I was a bit off on my lines of reflection. This neat tool is really helpful in creating the perfect symmetrical shapes! I will be sure to use this in my future teaching!

 

Part B

 

Problem B1

a) Each of these figures has rotation symmetry. Can you estimate the center of rotation and the angle of rotation?

  This activity reminds me of the pentomino rotations we did at the beginning of the module. It was a little confusing at first but I got the hang of it after working with the different shapes. The interactive tool was super helpful.

 

1) The center of rotation is located in the center of the figure. It has an angle rotation of 120 degrees.

    I could be wrong but another way to figure the rotation would be to take 360 degrees and divide it by 3 (for the 3 symmetrical parts within the shape. 360/3 = 120). I hope this makes sense.

 

2) The center of rotation is located in the center of the figure. It has an angle rotation of 180 degrees.

It's easy to see that a rotation of only 90 degrees is not enough but two of them would produce the original shape 90+90=180.

 

3) The center of rotation is located in the center of the figure. It has an angle rotation of 120 degrees.

   This was a tricky one because the swirly lines threw me off but again, I saw how the shape has 3 symmetrical parts so I took 360/3 and got 120 degrees.

 

4) The center of rotation is located in the center of the figure. It has an angle rotation of 90 degrees. This was an easy one because rotating the figure just 90 degrees would produce the original shape. Also, 360/4 (4 for the 4 symmetrical parts) is 90 degrees.

 

b) Do the regular polygons have rotation symmetry? For each polygon, what are the center and angle of rotation?

  

Equilateral Triangle: Yes, this has rotation symmetry. The center of rotation is located in the center of the figure and it has a rotation angle of 120 degrees (360/3=120).

 

Square: Yes, this has rotation symmetry. This shape has 90 degrees angles so it makes sense that 90 degrees would be the angle of rotation.

 

Regular pentagon: Yes, this has rotation symmetry. The angle of rotation is 72 degrees because 360/5=72.

 

Regular hexagon: Yes, this has rotation symmetry. The angle of rotation is 60 degrees because 360/6=60.

 

Problem B2: Does your design have reflection symmetry? If so, where is the line of symmetry?

  This concept was a little hard to understand at first but I just had to read  an practice following the directions a few times. Here is an image of my work. I did not find any reflection symmetry or line of symmetry in this figure.

 

Problem B3:Use the basic design element below and the given center of rotation to create a symmetric design with an angle of rotation of 120°. Does this design have reflection symmetry? If so, where is the line of symmetry?

   This was kind of easy figure out thanks to the practice I've had with rotation symmetry in this module. I simply took 360 and divided it by the angle rotation of 120 and got 3 rotations. The original shape itself is symmetrical so it m that the entire design has reflection symmetry. The line of symmetry goes through the middle of the original shape. My image is not near as neat as the picture on the solution page but it shows accurate placement.

 

 

 

For Further Discussion

 

 Multicultural mathematics offer rich opportunities for studying geometry. Research the art forms of Native Americans and various ethnic groups such as Mexican or African Americans. 
What kinds of symmetry or geometric designs are used in their rugs, baskets, pottery, or jewelry? Discuss ways you might use your discoveries to create multicultural learning experiences.

 

   I think it’s amazing to look at art work and objects from different cultures and notice the geometric patterns the items showcase. The fact that many of these objects’ patterns are geometric makes it so much more eye appealing. I found a great website called http://nativeamerican-art.com/index.html that explains reasons Native Americans use geometric patterns and symmetrical shapes in their work. Essentially, the repeated designs became “representative symbols that transcended tribal language barriers”. The geometric designs in the artwork spoke for themselves, thus making verbal communication unnecessary. Here are just a few examples of some Native American artwork from the website:



 

   I think the designs of these Native American items are so intricate and can tell that the artists put a lot into their works. The Cherokee basket was made with a symmetrical square pattern and the Sioux beaded moccasins were crafted using a unique pattern of triangles. The last item is a Sioux beaded saddle blanket has a repeated design of symmetrical triangles and rectangles.

 

  African Art also uses a unique pattern of geometric shapes that represents the culture. http://www.africanart.com/tongabaskets.aspx shows you a lot of different African paintings, carvings, masks, etc. It was really interesting reading abut the story behind the designs on certain objects. The zig-zag pattern on the Kuba Cloth (shown below) is said to reflect the African's belief in the after-life and that their clan ancestors are able to recognize the pattern after they have gone. This particular design uses a repeat of chevron shapes to create a work of art. Made from woven raffia palm leaves, these cloths are passed down from many generations.

   These Tonga baskets, woven by the women of Zambia, are made using llala palm. The website explains that no two baskets are made to look the same but the women put a lot of time (several days) and hard work making one. This set of baskets have a lot of patterns with squares, zig-zags, and swirls. It is a beautiful piece of work but I think it makes the object more interesting given that it was made by hand!

 

   I can think of many ways to use my findings (or similar ones) in the classroom to create multicultural experiences in math and in other content areas. Children are often fascinated by art like bright colors and new images/objects so what better way to introduce kids to other cultures than by introducing them to the art the people use? Surely students would find it interesting but I also enjoy learning the meaning behind some of these intriguing artifacts and works of art. The geometric patterns and symmetrical aspects of these items will certainly engage students in math lessons about these concepts. Just a thought.