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!

 

Module 8

 

 

Hello again everyone!

     I hope you had a great Spring Break last week! I enjoyed having a couple of days to get caught up on some other work. It's hard to believe that there are only a few more weeks left of the semester. It will be here before we know it!
    Module 8 was filled with a lot of great information so I certainly learned a lot! As always, I have included my thoughts, answers, and ideas to what was presented in the module readings/activities. I have put the headings of required blog postings in bold red.

 

Quick Images--Video

 

 

      I thought this was an interesting activity! I actually recall reading about something similar when I took EDN 322 last year. The task was to project a group of dots for a few seconds and students write down how many dots there were. The goal was to learn about the different strategies students come up with for adding up the dots to make memorization easier. For this particular video, I enjoyed seeing how the students described what they saw that reminded them of the shape.

 

      For the first shape, I immediately saw a crescent moon shape. A student in the video commented on how shaped also reminded him of the moon and that he used it to help him redraw it from memory. Another student described seeing the shape as the letter C. Interestingly, one child mentioned that the shape looked like part of a jet ski but I can't seem to picture it. Perhaps this is because I've never been on a jet ski before! Another student discussed how the shaped looked like 1/2 of a circle. She clearly pointed out where she could see the circle. I thought it was great that other students explained different features of the shape--like that it has 2 points. By flipping the transparency sheet, other students demonstrated how the shape reminded them of sliced cantaloupe, a boat, and a banana. The child at the end of the video had similar ideas to mine considering that he just looked at the shape and remembered it and knew it was a crescent. He even went on to say that it only had two curved sides.

 

   I liked how I saw that the children were so engaged in this activity. I believe this is a great thing to do before or during a math unit on geometric shapes. I think it is a fun way to help them make a connection between Geometry and the real world. They can begin to see that math and shapes are all around us!

Case Studies--Geometric Definitions

 

There were a lot of great ideas brought forth in these case studies! For this posting, I will provide a summary of my thoughts to each of the reflection questions.

 

•Follow the thinking of Susannah throughout Andrea's case 19. What does she understand about triangles? What is she grappling with? What ideas or questions does she contribute to the class discussion? What does she figure out by the end of the case?

     I think Susannah has an inaccurate but developing understanding of triangles. She knows that triangles have 3 sides and don't necessarily have to look the same. From her comments, she seems to know that triangles can be different sizes but she cannot let go of her idea that triangles cannot be "stretched out" (in her response to the right triangle). Her input sparked a little bit of debate among the class as other students explained how it shouldn't matter if the triangle was "stretched out" or rotated, it was still going to be a triangle. Towards the end of the case study, she seemed to gain a better understanding that triangles don't have to always have the same side lengths. She demonstrates this by later explaining that triangles simply have to have two "slanty" sides without specifying that the sides had to be equal. What an interesting way to put it!

 

•Now go back and follow the thinking of Evan throughout Andrea's case 19. What does he understand about triangles? What is he grappling with? What ideas or questions does he contribute to the class discussions? What does he figure out by the end of the case?

     I thought that Evan had a much better understanding about triangles than Susannah did. He realizes that triangles have pointy, not curved points and they also have straight, not bumpy sides. He can't seem to let go of the fact that triangles can come in different sizes by "stretching" them and flipping them to look certain ways. I thought it was interesting how he helped his peers understand this by explaining that if someone stretched him out and flipped him upside down, he would still be the same person. This sparked conversation about how other students in the class viewed triangles. For example, Zachary explained his struggle with viewing triangles that he wasn't used to seeing. This also led other students to explain that you really can "stretch" sides and flip a triangle and expect that it will still be a triangle. I think the entire lesson helped Evan solidify his understanding of triangles. By the end of the case, he was able to writ his own definition that triangles have three sides and three corners and even demonstrated how the shape can be rotated or turned by drawing a variety of triangles.

 

•Consider Natalie’s case 20. What are the students learning about squares and rectangles? What do they still need to figure out? Refer to specific examples from the case to illustrate your ideas.

   The students in this case are learning that the definition for both shapes are very similar. Charlie explained how that different looking shapes have the same "rules" or characteristics. I think they still need more work on noting the differences between squares and rectangles so that they can develop an accurate definition. They have yet to realize that a square is a unique shape because all the sides are equal. The realize that squares have four sides and four corners but can't find a way to describe these being different than other four-sided shapes. Brett (line 283) seemed to make some sort of realization about this when he turned the square block and saw that it looked the same after each rotation. If only he had done this with a rectangular piece, he would see important differences in each shape.

 

•Also in Natalie’s case 20, after line 250, the students are working to define the term square. Their conversation is as much about what a definition should be as it is about the particular term square. What does their discussion make clear about definitions? In particular, consider Roberto’s definition (“four sides, four corners, four angles, and it’s a square”) and the other children’s responses in the lines that follow.

     I thought It was interesting to read how the students struggled to define what a square was without including the word "square" in the definition. Thanks to comments from students like Charlie, the children learned that different shapes can carry the same definitions. The collective idea that a square simply has four sides and four corners was diminished when Charlie drew a chevron and trapezoid shape on the board. Ultimately, they learned that while there are many similarities, they need to keep working on providing a clear-cut definition for the shapes.

 

•In Dolores's case 18 (lines 25-43) and in Andrea's case 19 (lines 162-168), students are talking about what it feels like to make sense of a new idea. Describe their conversations. Refer to specific portions of the text in your discussion. What is your reaction to their comments?

     I enjoyed reading the student's comments about expanding their knowledge of triangles. Children in both cases admitted that learning new ideas about triangles was difficult because they have always been used to working with one "type" of triangle. It was surprising to read student's ideas that they have to forget about some things they know about triangles in order to "make more room" for new information. It's not too often that you hear children say things like this. Zachary's comments in case #19 were very insightful! He knew that the shapes were triangles but knew that it was hard for him to recognize them because again, he had been used to seeing certain kinds of triangles.
   I was pleasantly surprised to hear such wise words come from young children. It was great to pick up on their eagerness to learn new information about triangles. Their comments also led to think about why children often struggle to identify certain triangles. Many baby books and T.V. shows teach children about shapes but normally showcase one type of triangle--one with equal side lengths and angles. No wonder students like Susannah had a hard time letting go of her "stretched out" side theory! Just a thought.

 

•Reread the questions posed at the beginning of this task. Discuss your answers to those questions, taking into account all the cases you were asked to read.

        There were a lot of ideas to consider as I answered the questions given at the beginning of the case studies. The question about coming up with your own definition for the shapes was harder than I thought but reading each case helped m make better descriptions later on. I realize that I encountered similar problems as the students in terms of developing similar definitions for a square and a triangle. Moreover, I had trouble coming up with explanations of a definition and an attribute/property. l simply said that a definition is necessary to have in any situation because it gives meaning to a word or statement. Attributes and properties are the "rules" or characteristics that relate to that word/statement.
       The students in each struggled with describing the attributes for the shapes they were learning. They first needed to consider a number of things before fully understanding their definitions. They had many misconceptions about the shapes such as thinking a shape had to be turned or rotated a certain way, had to have a certain number of "slanty" sides, and could not be "stretched out". Their lack of ability to put words to these attributes leads me to think that they the students need to learn certain vocabulary words to describe the shapes. Maybe then the students would understand things a little more clearly.
      While many children struggled with their ideas about the shapes, I think they contributed a lot to the discussions. They even used different methods to make discoveries such as drawing pictures, using math manipulatives (square pattern block), and writing out their own definitions. It's great that the teachers encouraged the children to explore their ideas. In response to the final question about how children develop a sense of purpose for definition, I believe that they first need to have chances to explain things in their own words. I think they also need to have plenty of opportunities to "test" their ideas and make mistakes to see that accurate definitions are necessary.

  

Annenberg Module--Polygons

      I learned some new ideas from this Annenberg session. I have included my thoughts and answers to the questions given throughout.

Part AQuestion A1: Look at the shapes above that are not polygons. Explain why each of these shapes does not fit the definition of a polygon.

   These shapes are not considered polygons for a number of reasons. The first shape is a circle which obviously means that it has no straight lines, something that is one characteristic of a polygon. The second shape also does not have straight line and it is not a closed figure, another quality of a polygon. While the third shape has straight lines, it is also an open figure because one of the points do not meet. The last figure does not qualify as a polygon because it is divided up into more than two regions. There are two shapes that have an inside and outside. Polygons only require one shape with one outside and one inside.

 

Problem A2: How many polygons can you find in the following figure?

13 Polygons:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Problem A3: How many polygons can you find in the following figure?

13 Polygons:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   As you can see, my work is kind of all over the place with this one. Once I got to the end, I learned that there was a much easier way to figure this out this problem. I realized that you can take one size shape and multiply it by the number of size since you have that many possible combinations. For example, the largest triangle MNOL can be listed 4 different times. So, 1x4= 4 points for ONE polygon. Continue this process until you have all possible polygons then add up the total values for points.

 

Problem A4: How may polygons can you find in the following figure?

13 Polygons

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   The same goes for this problem. I took the short route and didn't write out all combinations. I simply wrote the different kinds of polygons and multiplying them by the number of sides since it tells what possible combinations thee are for that particular shape. I hope that make sense.

 

 

Part B

 

Problem B1: Make a diagram to show how these same four polygons can be grouped into the categories Line Symmetry and Not Concave. Use a circle to represent each category.

I had to look at the answer to this problem because I wasn't sure how of what kind of diagram they were looking for. Now it makes sense.

 

 

Problem B2: As a warm-up for the game, put each of the labels Regular, Concave, and Triangle next to one of the circles on the diagram. Place all the polygons in the correct regions of the diagram. Problem B3: See if you can determine the correct labels for each of the Venn diagrams in the following interactive activity: I included images of 3 different groupings of polygons:   Problem B4: Use the picture of a Venn diagram below: a.  Determine what the labels on this diagram must be.     Before checking the solution, I worked on deciding what the labels would be. I first came up with Label 1=Irregular, Label 2=Regular, and Label 3=Pentagons. I'm glad that this was the correct solution!  b.  Explain why there are no polygons in the overlap of the Label 1 circle and the Label 2 circle.      Because label 1 is full of irregular polygons while label 2 has regular ones. These shapes cannot be similar in terms of symmetry or angles.   c.  Explain why there are no polygons in the Label 3 circle that are not also in one of the other circles.          The polygons in the overlapped areas are either irregular or regular. You cannot have both. In order to have like shapes for Category 3, it cannot have either quality but with the given shapes, this is not possible. Problem B5: Create a diagram in which no polygons are placed in an overlapping region (that is, no polygon belongs to more than one category).   I first decided to separate three different categories of polygons by number of sides. This shows a diagram of Triangles, Quadrilaterals, and Pentagons: Problem B6: Create a diagram in which all of the polygons are placed either in the overlapping regions or outside the circles (that is, no polygon belongs to just one category).    I was  super confused by this question. I had to look at the solution to get an idea of what it was looking for but it didn't help much. However, I attempted to go along with what it suggested and it made a little more sense. I tried to group them as best as I could given the sizes of the shapes: Part C  This section focused on the importance of using definitions in math. This has focused on in previous modules. For example, it's crucial to make sure that a survey question is clear and doesn't confuse others. From the Geometry aspect, it is important to make sure you understand the meaning of words like dimension, length, and size. Problem C1: Use the definition above to make sense of the notion of "convex figures." What do they look like? Can you describe what they look like in your own words? Take whatever steps are necessary for you to understand the mathematical definition. Describe the steps you took to understand the definition. How did you make sense of it for yourself? Definition:  A figure is convex if, for every pair of points within the figure, the segment connecting the two points lies entirely within the figure.     Since I already know what a convex figure is and looks like, this made some sense. However, if I was a child reading this and had no idea what a convex figure was, I think I would be lost and confused. I would need to have an example of what this type of figure looks like. I did try and use the definition to understand what it was describing.    First, I broke apart the definition and considered what each piece meant. For example, I practiced drawing a pair of points. Essentially, I made angles--acute, obtuse, and right. Then, I looked at the segment connecting the two points which is really the point of the angle. Finally, I looked at the final part of the definition saying that this segment lies entirely within the figure. This makes a little bit of sense but to me, I think this sounds like more of a definition for concave figures simply because the mention that the points are "inside" the figure. Moreover, I believe children would think this means that the point has to be inside of a closed figure. Just a thought. After some time looking at convex figures, the given definition makes a little bit more sense but I wish it was a little more clear. Problem C2: Which of these definitions work for convex polygons? A polygon is convex if and only if... a.  all diagonals lie in the interior of the polygon.  YES. At first, I said no to this question because I thought it meant something different. I found a great website about convex and concave polygons that gives examples to support this idea. b.  the perimeter is larger than the length of the longest diagonal.   YES. This one was tricky but having read the solution, it makes sense that the diagonals cannot be larger than the perimeter. Otherwise, the diagonal would extend out past the figure or even have the points come short of meeting, causing it to not be a polygon. c.  every diagonal is longer than every side. YES. In order for a diagonal to reach from one point to another, it has to be longer than all sides. d.  the perimeter of the polygon is the shortest path that encloses the entire shape.  YES. I don't really understand this one but the solution says that this is an accurate statement. It makes sense that the shapes have to be enclosed but I'm confused by the perimeter being the "shortest path". The shortest path to what? *Note*--Having watched the video clip later on in the session, I understand the question. It makes sense that adding points to a shape would increase the overall path around the perimeter. e.  the largest interior angle is adjacent to the longest side.  NO. The length AND  location of the angles does not matter. The longest angle can be across from the shortest and it can still be characterized as a convex polygon. The solution gives an example of this--an obtuse triangle. f.  none of the lines that contain the sides of the polygon pass through its interior.  YES. This makes sense. Lines that pass through the sides would make it a concave shape.   g.  every interior angle is less than 180°.  YES. I had to experiment with this idea on paper but I got it! The interior angle (the point inside the shape) has to be less than 180 degrees because this would make it a straight line. Any point beyond 180 would cause the point to "cave" into the shape, making it a concave polygon.    h.  the polygon is not concave.     YES. Concave shapes are essentially the exact opposite of convex shapes because concave polygons have points that "cave" in while convex shapes have points that point outward. Problem C3: Draw several other examples of polygons divided into triangles for polygons of varying numbers of sides. Be sure not to use just regular polygons, and be sure not to use just convex polygons.    For this question, I used the polygon cutouts and simply drew lines to make triangles within the shapes: Problem C4: How would you divide the polygons below into triangles?    I divided these shapes using red lines. The intersections create a number of triangles:

   Problem C5: Describe a method so that, given any polygon, you are able to divide it into triangles.

    I would say to just start at one point and extend a straight line to meet another inside point. Oftentimes, you will have to keep making lines that will divide to make other triangles. This is a lot different to what the solution says but I think it is the same concept.

 

Problem C6: Use the method above or your own method, and fill in the table below. Remember that we are assuming that there are 180° in a triangle.

  This was a little confusing but having looked at the answers and worked backwards through the problem, I can see an interesting pattern. The completed table does make it easy to see this.

 

    

 

 

 

 

 

 

 

 

 

 

 

Problem C7: Write a convincing mathematical argument to explain why your result for the sum of the angles in an n-gon is correct.

   I think it is a little hard putting this idea into words. Taking n-2 and multiplying by 180 would result in a correct sum of angles because when you split the shape into triangles, the number of triangles is two less than the number of sides. Since a triangle is 180 degrees, you multiply 180 by the number of triangles to reach the sum of all angles in the figure. I hope that makes sense!

 

Math Activity with Color Tiles

 

This was an interesting activity! The first part of the video was easy and was able to come up with the correct number of shapes for the given number of tiles.

 

Here are images of my tiles as I completed the activity. I thought I had the right amount for the second one until the answers that were given showed that I had two of one shape. That is why I have an "X" through that figure. It's easy to overlook similar shapes! I thought this task would be easy but it was more difficult than I had expected.

 

  Next, I practiced making figures with 5 tiles. After thinking that I had all of the possible shapes, I checked them using the pentomino pieces. I was surprised that I had only half of the shapes! These are the ones that I created on my own:

 As the PowerPoint instructed, I examined each of the 12 shapes and put a name to each in terms of how many sides they had. My groupings match what was shown in the video:

 

1 Quadrilateral (4 sides)

3 Hexagons (6 sides)

5 Octagons (8 sides)

1 Decagon (10 sides)

2 Dodecons (2 sides)

 

For Further Discussion

 

If geometry is the mathematics that describes the world we live in, that means geometry is everywhere around us. Use the language and concepts of geometry to describe your own world – your home, your workplace, your possessions, your daily commute or other travels. 

 

    Geometry is all around us! The objects we see and use every day take on shapes that we learn about in geometry. The car I drive has round, circular wheels that make it possible for the car to roll. I live in a house that has a roof resembling a triangle and doors shaped like big rectangles. As I scan my room, I see many picture frames, all of different shapes but mainly squares and rectangles. The basketballs that are being used in the NCAA tournament are shapes themselves, spheres--3D shapes. The paper towel roll I placed in the recycling big yesterday also took on a 3D shape, a cylinder. Not only do objects we use take on different shapes, geometry is in nature, too! For example, bees have hives that look like hexagons, mountains often resemble triangles or pyramids, and the earth itself is a type of sphere. How amazing!