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!