Module Ten

Nets Activity

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?

While this activity was a challenge at first, once I cleared my mind and focused on one cut out at a time I was able to correctly identify the seven additional pentominoes that would make the "box." I did have a mistake in placing the bottom on just one of the shapes. It was on one of the octagon pieces, and I just miscalculated it by one square. I knew it was one or the other and I went with the wrong one. I think some students would get easily discouraged while doing this activity, at least at first. I think it would be beneficial to use the Polydons manipulatives that were used in the article so that students could not easily rip the paper. This would work well with students that could visual the changes but could pose difficulties for those that are more hands on. Ensuring that the students understand that having a tough time is okay would be important for my classroom. 

Aubrey, did you have any trouble making the connections to the creation of the "boxes"?

Textbook Reading

4. Find one of the suggests applets, or explore GeoGebra and explain how it can be used. What are the advantages of using the computer instead of hands-on materials or drawings?

I decided to look up geogebra.org, and I pretty much got sucked into creating shapes and lines and polygons. You can use this program for a variety of different applications dealing with graphing calculators and geometry. Going into classic mode allows the user to do graphing, 3D shapes, and probability to just name a few things. Using this applet to create geometrical shapes is that if you know what you are doing, you can make the shapes and measurements rather fast. You don't have to necessarily worry about the size of the shape you are drawing about the students. You don't have to hear a student say "I can't draw that shape." Aubrey, what do you think about this program? 

2. Briefly describe the nature of the content in each of the four geometric strands discussed in this chapter: Shapes and Properties, Location, Transformations, and Visualization. 

Shapes and Properties: this is the content that is associated with both 2- and 3-dimensional shapes, that they learn what they are called and the way the shapes are similar and different. The students can begin to classify the different types of shapes and how the properties of the shapes will connect and be developed. 

Location: is the analysis of paths from points on a map by way of using a coordinate system. Grid systems are used to identify locations, Level-1 thinkers the coordinates slides but do not flip or twist, like a reflection over the x- or y-axis. Level-2 thinkers can use logical reasoning, that includes the slope and Pythagorean relationships. 

Transformations: these are the changes in the position or size of the shape. This includes slides, flips, turns, and line symmetry at the Level-0 thinkers. But for Level-1 thinkers, it also includes the composition of the transformation, proportional reasoning, and the understanding of tessellations. Level-2 thinkers, use their overall understanding of symmetries and in to build a bridge between the two ideas.

Visualization: creating of mental images and visually understanding the different viewpoints that could be predicted. Level-0 thinkers think about shapes with relation to the way they look. Level-1 thinkers give way to the meaning of properties and connecting the 2- and 3- dimensional shape properties together. 

Spatial Readings/Building Plans

Did you find any of these activities challenging? If so, what about the activity made it challenging?

No, I did not find these activities challenging, I enjoyed this type of thinking and was able to get all the answers right on the first try. 

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

I almost feel that spatial visualization is almost like thinking outside the box like one cannot be closed minded to the opportunities that could happen if that makes sense at all. Not only that but it helps you view things within our world, such as driving directions, ability to put things together (furniture or puzzles), it is giving the ability to see how something could work while still having the thoughts in your head. Aubrey, does that make sense? I think I have a good bit of spatial visualization. 

At what grade level do you believe students are ready for visual/spatial activities? How can we help students become more proficient in this area?

I believe this learning should start as early as possible even before kindergarten when I think of spatial visualization it includes the ideas of putting together puzzle pieces. Encouraging younger children to put puzzles together help their minds work out the proper positions of the pieces. Though they can get frustrated with it, they will develop what they need to as time goes on. By continuing to practice and do things that involve spatial thinking and visualization they will become more open to seeing the larger pictures in their minds. 

Tangrams

B1. Start with a parallelogram. Find a way to cut your parallelogram into pieces you can rearrange to form a rectangle.

It took me a few minutes to figure this problem out, I was having trouble with just drawing the line, so I decided to cut it out. Once I did that I was able to see that I had not made a straight enough line to create the rectangle, so I made adjustments. 

B5. Start with a trapezoid. Dissect the trapezoid into pieces that will form a rectangle.


I again had trouble finding the right angle to cut the end of the trapezoid off at, I may have better luck when I am not so tired, but these problems were much more difficult for me than the other things we have done in the module. Aubrey, how did you do on this section, I think I need to revisit it to see if I can grasp a better understanding.

For Further Discussion

I decided to go online and shop at Toys R' Us and Amazon.com for this discussion, and I found the following games and toys. Each of these would promote and engage the children in learning about geometric shapes and spatial visualization without any actual teaching going on. By sorting and categorizing most of these shapes the children are learning how to make different shapes work together and how to create things with different geometric shapes. All of these toys are valuable to the children. Aubrey, what do you think? Do my toys fit into the informal recreational geometry ideas?

Module Nine

Quick Images Video

I thought the students did well making connections to what they saw and how they remembered what the figure looked liked according to them. When I saw the image at first I thought of the crescent moon shape. The students were able to notice facts about the shape that I only thought of as an after thought. When they began to move the image, I was able to think of banana or a smile, but that was only after they rotated the image. Aubrey, did think of a moon shape?

Shapes and Geometric Definitions

Triangle- Shape that has only 3 straight sides that connect.

Square- Shape that has 4 equal straight sides that connect creating 4 90 degree angles.

Rectangle- Shape that has 4 sides that connect but 2 sides that are parallel are longer than the other 2 parallel sides, also creating 4 90 degree angles.

Parallelogram- Shape that has 4 straight sides that connect where the opposite sides are parallel. Squares and Rectangles are parallelograms.

The definition tells you what the meaning of the word is, whereas the attributes tells us something that is being caused, so with my definitions above I think I may have included some attributes along with the definitions. Aubrey, what do you think about this question, are attributes an extension of  the definition?   I believe that a definition is the explanation of what the meaning of the term is. 

Through the eyes of the students...

The students need to consider not only the number of sides the geometric shape has but also the angles that are created with the shape. They also need to learn about rotations of shapes, many students struggled with the idea that a triangle was still a triangle even if it was drawn on it's side. Even with the rectangle, some students questioned the idea of the shape still being a rectangle since it was drawn differently. The students learned that as they applied the definition they had developed they were able to count the shape as that particular shape, but still would struggle in cases and would try to make sense of how it was that shape within the attributes they discovered. One student mentioned they think it is a triangle, it looks upside down, but then in the same writing says that it kind of doesn't look like a triangle. Children begin to use the definition to create the understanding that they have and while I feel it may have been shallow to start with they start to grasp a deeper understanding the more they use their minds. 

Susannah

I feel like Susannah has an understanding of triangles in general but relates them to the common equilateral triangle and the triangles that don't look the same are causes her to question what type of triangle it is. She states that those (L and R) are not real triangles. Later on Susannah mentions that the triangles are still triangles, but adding that you have to turn them and they need to have 2 slanty sides is a must for the rules. She begins to understand that L and R are triangles, they just look different. 

Evan

Evan chimes in later in the conversation, stating that triangles must have points, and that even if the triangles look stretched out and turned upside down they are still triangles. He has a good understanding that triangles have three sides, three corners and turning the shape doesn't change it. I think his explanation of the points help students grasp a better understanding and him being so sure of turning the shape, I especially liked his connection...if I turn myself, I am still me. Do you think he helped the class?

Natalie

In this case the students are trying to define what a square is and what a rectangle is. The students are struggling with how to separate the two shapes into their own definitions and use the attributes associated to the shapes correctly. Roberto states that a square has four sides, four corners, four angles, and it's a square. Clarisa chimes in with four corners and four angles is the same thing, but nothing was mentioned about the use of it's a square as part of the definition. When Charlie called out, that they are the exact same thing and drew examples it helped the students see that the definitions they we coming up with were not specific enough. Maryanne and Brett have figured out the the definition needs more differentiation to really be able to justify the difference between a square and a rectangle. 

Dolores and Andrea

For Dolores's case they speak of the difficultly they had of seeing differently shaped triangles as triangles since they were heavy exposed to the common equilateral triangle we see for shapes on posters in younger classrooms. They felt like they had to break out of her usual thinking to let new ideas in. In Andrea's case Zachary is also taking about what is in his head and how he has to use his eyes and head in connection to see that the shapes are triangles. It is like they are brain-washed into thinking shapes can only be one way, and the process of ending those misconceptions take time. Misconceptions are a topic we've discussed in Science and I feel like that is what these children are struggling with, standard shapes are introduced at home and in preschool, but these have caused an undesired effect on the students thinking process. 

I now have a new understanding that misconceptions are all around us, since I want to teach in kindergarten, I will ensure that I have many pictures of different shapes inside my classroom. Instead of having the standard shapes, I will post pictures of shapes of varying types and sizes. I will introduce children to shapes that may not be common at that age level, but breaking the misconceptions early is important to harness the definition and attributes of the shapes. Aubrey, what do you think about these case studies? Did they open your eyes to what our children are thinking and how they process definitions and attributes?

Math Activity with Color Tiles

 While creating the 5 sides I did okay, I did not find the two decagon without help from the photo on the page. Naming the shapes was easy, but I did have to remember the name for the 12 sided shape. Aubrey, how successful where you, did you know the names? The funny thing about this activity was that my children had used my colored tiles, and I couldn't find them, so I had picked up the bag  of pentominoes and thought about if I could just use those. But my children found the tiles this morning before school....I thought I was gonna have to go to the lab and borrow some. 

For Further Discussion

Inside my home, my work space is in my bedroom, the room itself is a rectangular shape, the tray ceiling however is an octagon shape. My desk is a rectangle as are my computer screens. I have many things that are circular in shape around me as well, like the CD's, coins and speakers. Shapes are all around us, I have picture frames that are squares and rectangles. When driving my children to school we see circles in the traffic lights, octagons in the stop signs, rectangles in the speed limit signs and so much more.  Triangles are not as common in my home, but I see the yield pedestrian signs while driving. Our world is full of geometric shapes, inviting the students to see this inside a classroom would be a bit of a challenge at first, but I think this concept would open their minds to the shapes all around us. What do you think Aubrey?  

Module Eight

Key Ideas in Geometry

What are the key ideas of geometry that you want your students to work through during the school year?
I would think that the key ideas that the students should work through are knowing the shapes, finding the perimeter and area, knowing the different types of angles and how to identify them. But it would depend on the grade level I am teaching, I am still drawn too kindergarten, so I think geometry for them would be knowing shapes for the most part.

Van Hiele Levels and Polygon Properties

For the PowerPoint activity I did well, I was prepared and had the shapes printed and cut out. I already knew the many of the terms used and was successful in removing the specific groupings that needed to be removed. I was able to get 3/3 on the activity. How about you Aubrey, was this a hard task?

What I didn't know was the classification of levels using the Van Hiele levels, I have never heard of him before this lesson. I think I would rate myself at a Level 2 or 3, I get confused with the reflections and rotations of shapes about creating a certain number of triangles, but that was from the Triangle activity. 

Understanding the levels are essential so that we as teachers are not over teaching before our students can relate what they already know. It also helps having school-aged children at home.

Thinking about Triangles

I wasn't able to come up with many words with "tri", I have tripod and tricycle. I feel silly not knowing more, but the words were just not coming to me last night. 

Is it possible to make a three-sided polygon that is not a triangle? 

No, because the shape must have three lines that intersect the only possible polygon with three-sides is a triangle. Does this reasoning make sense to you?

Is it possible for a triangle to have two right angles?

No, the triangle is three-sided with three-angles, the total size of an angle is 180 degrees. If two angles were inside the triangle that would equal 180 degrees and would not create a triangle. How about this one?

How many different right triangles can be made on the geoboard? 

I wasn't sure on this one, I created 11 or 12 myself and then got lost. So when I went forward on the PowerPoint and Dr. Higgins said 14, I decided to look them up. However, when I looked it up they said 17 could be made. I am interested in finding the 14 hat were mentioned to know where the others came from on the website I found. I decided to add the picture I found with the 17 right angles. Dr. Higgins and Aubrey, can you see any mistakes on this picture? Did you get 14 Aubrey? Dr. Higgins, will you show use the 14 that we should have gotten?

How many different types of triangles can you find?

I was able to create 5 types: right angle, obtuse, acute, isosceles, and the scalene. My children might have helped me a little :) The scalene was the difficult one for me, I created it but couldn't remember what it was called. I almost put equilateral, but I had my ruler and measured it, what about you Aubrey, did you find the different types?

Follow-Up

How would you structure this lesson for students in an elementary classroom?

I think one thing I would do is have partners, so that it is a little easier to keep track of the different triangles being made, it was difficult to come up with the right angles myself and ensure that I didn't just rotate one I already had. I'm still not sure I didn't, LOL. I think that having the geoboards for each students is great too, and maybe more different colored rubber bands, that way many bands can be on the board at once. What age level would you introduce this activity to Aubrey? 

Common Core:

Kindergarten:

CCSS.MATH.CONTENT.K.G.B.4
Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/"corners") and other attributes (e.g., having sides of equal length).

CCSS.MATH.CONTENT.K.G.B.5
Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.

First Grade:

CCSS.MATH.CONTENT.1.G.A.1
Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

Second Grade:

CCSS.MATH.CONTENT.2.G.A.1
Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.1 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Third Grade:

CCSS.MATH.CONTENT.3.G.A.1
Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

Fourth Grade: 

CCSS.MATH.CONTENT.4.G.A.1
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

CCSS.MATH.CONTENT.4.G.A.2
Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

Fifth Grade:

CCSS.MATH.CONTENT.5.G.B.4
Classify two-dimensional figures in a hierarchy based on properties.

What parts did you have issues with? 

I had to remember the different types of triangles at first, I knew the right, obtuse, acute, equilateral, and isosceles, but I had to read about the scalene. After the review I knew all the shapes and could make them.