Module 9

    

Hello again everyone!

 

   This week covered a lot of information but it was all very helpful! I have included by thoughts and ideas about each part of the module in this blog posting. The required blog area are headed in bold red.

Nets with Pentominoes

 

    This was an interesting activity mainly because I have never worked or even heard of pentomino nets! I must say that it was a little difficult to determine which pentamino pieces would fold into a net without physically doing it. Nevertheless, I felt comfortable with my selection because I was able to find 7 other figures (in addition to the example one) that I believed would work. Here is picture of what my groups looked like before folding the pieces. The "X" on each figure indicates where I think the bottom of the net will be:

 

   Next, I followed the instructions and attempted to fold each shape to figure out if my predictions were accurate.



• For every pentomino that cannot form a net for an open box, explain why this happened. There were only four pentominoes that could not form a net for a box. After working with them, it is clear why they wouldn't have been able to form a box.
   
                            
    This piece did not work because the shape did not have enough squares to fold into a box. One square even overlapped another, giving it only 4 sides instead of the needed 4.
   
               
     I knew that just by looking at this piece, it would not fold into a box. There are too many squares clumped together, making it impossible to fold over to join sides with others.
   
  
     This figure would clearly not fold into a box. It does not have squares that extend outward, so you would only be folding the shape in one direction. This gives you no bottom (or side pieces depending on which way you make it stand).
   
     
      
     By just looking at the unfolded piece, I think it's clear to see that it is not going to work. The two squares that stick out overlap one another, thus making only 4 sides instead of 5.

Were you correctly able to identify the shapes that formed into nets as well as identify the bottom of the box? Discuss what happened.

   It turns out that I was correct on all 8 pieces because these shapes did fold to become nets. I also accurately predicted what side would be at the bottom of the closed boxes on all but 1 piece:

 

     This T-shape was the only one I failed to guess correctly. I thought the middle square would be the bottom but that did not work. It would have worked if a square was placed above or below it but then that would have made it a cube where all sides are considered the bottom.

 

  I think the activity found on the Illuminations website was great for practicing what I learned about pentomino pieces. I love how the animations allow you to see the folding process! The game was a little harder than it looks but the challenge helped me further my understanding of this geometric concept. It took me a little over 5 minutes to complete the game and only answered 18 out of the 24 correctly. I missed a few of them because I misread the directions. I thought it was looking for figures that would form the 5-sided figures (as in the previous activity) and not cubes. After going through the activity, I considered the posed questions:

• What properties are common to all nets that will form a cube?  
  

      The biggest and perhaps most obvious similarity is that the nets have six squares or units. They would have to have only 6 since they fold up to be cubes (6-sided figures). I could be wrong about this next similarity but I found that of the nets that formed cubes in the game, all had side counts of either 9, 10, or 12 units. That means all of the cubes had nets were either octagons, decagons, or dodecagons. This may be something insignificant to mention but I thought it was interesting. It was hard to identify other similarities or ones that I could put into my own words. What other common properties could you find?

• What types of nets will not work? Why not?
     Nets that have one row of squares/units and no others extending outwards will not form a cube. One unit would overlap and there would be no others to fold underneath or over to form the top and bottom. Moreover, an entire row of 5 or more units would not fold to be a square since you will have 1 or more overlapping pieces that still form an open figure. The same goes for nets that don't have exactly 6 units--you simply don't have enough. I also noticed how that nets that have a cluster of units (i.e. two squares in one row and 2 in a row below that) cannot form a square because you cannot fold them over. I hope that makes sense.
    
  
• Without folding, is there a quick way to determine whether or not a net will fold into a cube? 
     I think the quickest thing to do is simply count the number of units in a net to see if there are exactly six. If not, then you know it cannot be a cube. I would also look to see if there are more than four consecutive squares that might overlap.
  

Suggested blog question: How could you use a similar activity with students in the classroom? Were you able to complete the activity without too much frustration? What are some anticipated issues while doing this activity with students?

  I think this would be a fun activity to use with students. I can't say that my experience was completely stress-free because I did get frustrated when I inaccurately guessed on the nets. It was a little difficult to "fold" the nets in my head because I got confused on where on the figures I would like to start. I think there would be some similar issues to arise for students. I can see them getting a little overwhelmed by the visual folding of the nests since there so many ways to fold the pieces to make a cube. Another thing to consider before doing this activity with students is their familiarity with cubes. I think it's important for them to understand the important properties a cube has. Just a thought. 

 

Hiding in the Nets--NCTM Article

    I think this article was a great extension to the previous activity. I enjoyed reading through most of the article because it involved future teachers like me exploring geometric concepts. I think that nets can be a difficult thing for students to understand so that is why I was surprised to read about third-grader Morgan's fantastic discovery that all cube nets have six units and perimeter of 14 units. I never even considered or even came close to thinking about this fact! I like how the article described the teacher candidates' struggles with their net activities. It makes me feel better to know that other future educators make mistakes with certain math activities. I recall encountering a similar problem that the student teachers did when they listed 2 nets that were the same, just flipped around. It was interesting to read about the problem this later posed that if two figures have the same perimeter, they would be congruent. However, once you look at all eleven possible nets for cubes, they follow these criteria but are not necessarily congruent to one another.

   I will admit that the section about the study of fold lines was little confusing to me. I understand the concept but I know I would have ever thought of this on my own. I struggled to fully understand the process of finding numbers for the icosahedrons (platonic solid with 20 face). Moreover, I have never even heard of Euler's formula before so that idea was new to me.
   After reading the article, I considered the following questions:

How could you use a similar activity with students in the classroom? Were you able to complete the activity without too much frustration? What are some anticipated issues while doing this activity with students?

  I think it would be a neat idea to use the Polydron pieces in the classroom to help students learn about geometric nets. Students can work to find the different 11 nets that make cubes and ask them to identify the similarities they see. The article suggested to challenge students to explore the perimeter-unit theory that nets wit6 unis and a perimeter of 14 will always fold into a cube. They can learn that this is not an accurate statement. As I mentioned before, this was a tough activity for me to follow. I know that young students don't learn about Euler's formula or other concepts that the student teachers explored but I still think young learners can run in some problems as they work with lower-level concepts. For instance, they might have a hard time constructing nets that are not a repeat of ones they already have.

Textbook Reading--Chapter 2

   

    This textbook chapter provided a lot of information about different things. I like how it gave a more in-depth look at the van Hiele levels we learned a few modules back. It's great that it also provides tips on how to support students as they make the transition from one level to another. It was helpful to read through the text and learn ways to help students at various van Hiele levels of thinking learn properties, transformations, location, and visualization of geometric shapes.     

   After reading the chapter, I did consider the reflection questions on page 60.  As suggested in the module guide, I focused on question #4:

4) Find one of the suggested applets, or explore GeoGebra (www.geogebra.org) and explain how it can be used. What are the advantages of using the computer instead of hands on materials or drawings?

    For this question, I worked with the Interactives: 3-D Shapes (Annenberg Foundation) https://www.learner.org/interactives/geometry/index.html applet. I love how this resource allows you to work with so many geometric concepts, especially ones that have come up in this module. Some of the things I struggled with are a little easier to understand now thanks to the applet's emphasis on things like Euler's Theorem and Platonic solids. The applet lets you control the view of a shape at any angle, unfolded net, folded, and any view in between! While drawings and hands on activities are good to use in the classroom, some students cannot fully comprehend or visualize the geometric concept. Computer/Internet programs like this can help students focus on specific things to look for in a shape. For example, the applet highlights the number of faces, vertices, and edges a shape has and the net folding process:

 

 

   Another advantage is that students can somewhat personalize their mathematical experience with the shapes they are learning. Some students might need more attention with basic concepts of 3D shapes while others need more time understanding surface area of different shapes. With this applet, they can do this and so much more!

 

Spatial Reasoning

 

     The "A Plea for Spatial Reasoning" article was an interesting read. I never really thought about spatial reasoning being a mathematical concept. I agree that many people don't take the idea of spatial reasoning as serious as other subjects and I will admit that I have had these thoughts before. I had always assumed that things like putting a crib together, packing a trunk, or installing a car seat were things that simply involved learning effective strategies---not exactly spatial reasoning. Interestingly, there is so much more associated with spatial reasoning. I did not know that the use of maps and Venn diagrams involve spatial reasoning. Moreover, so do abstract situations--like debates! Having learned such interesting information about this mathematical concept, I would say that many of us lack spatial reasoning skills. I probably do! While many children today might gain practice with these skills by playing video games and other forms of technology, it is important that they have opportunities to practice them in the classroom. Parentingscience.com lists some great evidence-based ideas for parents (and teachers!) to use with kids to develop skills like intruding them to spatial language, encourage spatial thinking by asking questions about everyday challenges (i.e. Will the groceries fit in one bag?), encourage gesturing, playing construction games, putting together jigsaw puzzles, and even working with tangrams! These are things to consider in future instruction. I'm interested in seeing what future research will tell us about spatial reasoning and the strategies that foster the skill.

 

Space and Shape

      I will admit that the activities in this section were really hard--a lot harder than I thought they would be! It makes sense now why visualization is an important skill to have in the area of geometry. There are times where you have to think about shapes at different angles and visualize them in your head without really seeing them. Before doing the activities, I read through the introduction page and the examples of visualization examples: telling someone how to make a paper airplane over the phone without even doing it, using your imagination to pack presents, etc. From these scenarios, I would say that I'm not that great at visualization. I'm good with it as long as I am actually doing or experiencing the activity but I know this is cannot always be the case.

I Took a Trip on a Train

  This was super hard! As hard as I tried, I simply could not put them in order in my head because I kept losing track of the order. I tried numerous times before drawing a sketch of map's outline and going through each picture and placing them in the appropriate order as I saw them. This helped me reach the answer on the first try!

 

Plot Plans and Silhouettes

 

   This activity was extremely hard! I worked for about 25 minutes on one problem and got really frustrated. I had a difficult time switching "views" or perspectives for the shape but was more confused of what the numbers meant. I know you were supposed to consider the side of the shape (3 blocks wide and 4 blocks long) which makes sense why there are 12 boxes to enter in numbers. I tried following the example but I was still a little confused. I think I became more confused at what the numbers themselves represented. I don't like how the activity doesn't give you the exact answer as a guide once you think you have it. It gives you an outline of the blocks but I still didn't know if I got the numbers right. I would feel better about it if I had the solutions after submitting my answers.

 

Shadows

  This was a much, much easier activity! I only got one of the questions wrong about the shadow of a cube but it helped that it gave you the correct answer and even showed you using a visual.

 

Building Plans

      

    I was a little hesitant wanting to continue with this section since it mentioned a similar activity to the Plot Plans and Silhouette activity I really struggled with. Nevertheless, I’m glad helped me gain hands on experience working with the plot plans.

   The first part of the PowerPoint was easy to understand. I didn’t even have a problem working through the first problem with identifying the different angles of the given base. Here is what my representation looked like:

 

    However, I had a really hard time completing the second problem where we were only given the front view of a building. I was really confused on how the right side looked the way it did. I was a little lost on how to know how many cubes go in the spaces that were not shown. It may inaccurate but here is what my structure looked like:

 

   Surprisingly, I was able to complete the final part of the PowerPoint activity with hardly any problems. I wish I knew how to work with spatial abilities that come up in activities like this but it was relieving to hear Dr. Hargrove mention that things will get better with practice.

 

Tangrams

 

    This was an interesting section! I remember working with tangrams as a young student but have never experienced making my own, I followed the directions we were given and they were easy to understand. It's pretty neat how you can make a set of large tangrams from a single sheet of paper!

 

Next, I used the tangrams from the manipulative kit to complete the Annenberg activities for part A.

 

Problem A1: Given that the tangram puzzle is made from a square, can you recreate the square using all seven pieces?

     Yes! I actually used what I remembered from creating the paper tangrams--I just kind of worked backwards:

   

 

Problem A2: Use all seven tangram pieces to make a rectangle that is not a square.
   This one was a little tricky. I didn't think it was possible until I looked at the solution:

 

Problem A3: For each of the pairs of figures below, do the following:

•	Build the shape on the left with your tangram set.
•	Turn it into the shape on the right by reflecting, rotating, or translating one or two of the pieces. (This may take more than one step.)
•	Write a description, telling which piece or pieces you moved and how you moved them. a)

   This first problem was simple. I just took the first triangle and translated it to the right of the square to form a rectangle.

 

b)

 

  For this problem, I took the triangle on the bottom and translated it to the right of the square. Then, I rotated the top triangle 90 degrees to form the rectangle.

 

c)

   I simply took the triangle on the furthest right and reflect it to make a trapezoid. 

Spatial Readings, Annenberg, and Building Plans Reflections

After completing the entire module, I considered the questions given in the module guide: 

• Did you find any of these activities challenging? If so, what about the activity made it challenging?      
     Yes! The hardest activity was the Plot Plans and Silhouettes and the Base Plans PowerPoint activity. They were super confusing for me! I also had some trouble with the I Took a Train Trip activity but am glad that I was able to figure it out! They were certainly challenging! I think the hardest part was trying to picture different viewpoints in my head. I think the plot/base activities were hard because I really didn't understand how to figure out how many cube were in the plot--even with side views.

• Why is it important that students become proficient at spatial visualization?

        As the introduction article about spatial reasoning mentioned, we use spatial skills for many things in life. Whether it’s reading a map or making connections between mirrired objects, spatial skills are needed. I think it is super important for kids today to develop spatial skills mainly because of the advanced technology that our world has experienced in recent decades. This means that many kids will have to learn about seeing things from a different perspective in the virtual world and making mental representations.

  
  
• At what grade level do you believe students are ready for visual/spatial activities?

      I think the earlier kids learn about spatial reasoning, the better. Children of all ages should, to some extent, be presented with opportunities to grow in their abilities. Kindergarteners  can play games and do fun activities that increase their skills. Teachers of older students should be a bit more aggressive with teaching spatial reasoning. Much like the activities we did in this module, they can begin looking at geometric shapes in different perspectives and viewpoints.

  
  
• How can we help students become more proficient in this area? 

       As I mentioned earlier, there are a lot of things we can do to help students build on their spatial abilities. I think one of the best things to do is ask them questions that pose a challenging situation we might face on a daily basis. They can practice visualizing ways to get out of the problem. Students can practice spatial reasoning with puzzles, creating maps, and even playing educational video games.

For Further Discussion

 Informal recreational geometry is an important type of geometry in many childhood games and toys. Visit a toy store and make an inventory of early childhood toys and games that use geometric concepts. Discuss ways these materials might be used to teach the big ideas of early childhood geometry

    I did not get a chance to visit a toy store this week but I did visit the websites of toy stores and a quick search revealed a lot of toys that would help children of all ages develop understandings of geometric concepts! Here are just a few pictures of the toys I found:

 

 

    I think these are some great toys that allow children to work with geometric shapes and concepts. I like how many toys cater to all age levels. It’s great that geometry is even emphasized in video games that older children would love to play! Babies can play with the Puppy’s Dump Truck toy to feel the different shapes and notice differences and similarities. Older children can work with the connecting pieces to build different geometric shapes. I thought it was interesting to see a video game about tangrams on the toys store’s website. Older children can also play this fun game to learn about geometry!