Module 7

Hello again everyone!
   Wow! Can you believe that we are already halfway through the semester?! It has gone by fast! I must admit that I am a little relived to be moving onto a new section of the course where we will be focusing on Geometry. I learned a lot this week and hope you all did, too! I look forward to reading your posts! As always, I have listed my thoughts and comments about each activity in the module. The required items are listed in bold red.

Key Ideas in Geometry

 



What are the key ideas of geometry that you want your students to work through during the school year?

 

  Essentially, I would want my future students to work on the concepts that are given by the Common Core State Standards as well as the NCTM. It is important that they are familiar with the objectives so it will be my responsibility to give them a lot of practice. I would also want students to start using their knowledge of the shapes and other ideas they learn about and apply it to the real world. In my experience with geometry, I can say that got through by memorizing and not fully understanding some of the information. This is something I do not want my students to do. I believe that my response to this question will likely change once I gain some experience in teaching and reflecting on geometry and the standards that accompany the subject.

 

Van Hiele Levels

 

    The ideas mentioned within this part of the module were very interesting. I don't think I've heard of the van Hiele levels, so it was great to learn about them and practice applying them to a specific activity.  

 

   Before starting the activity, I  read the article that was attached in the module. I learned about the five different levels of geometry understanding that were named by Dutch educators, Pierre and Dina van Hiele. They explained that students learn and understand geometry at different hierarchical levels 1) Visualization, 2) Analysis, 3) Abstraction, 4) Deduction, and 5) Rigor. It was interesting to learn that students must master one level in order to move up to another. It made sense to read that many teachers often think on higher van Hiele levels than their students, making the interpretations very different from one another. It is super important to teach at levels students can understand. The article emphasizes the importance of including appropriate, mathematical language when helping students progress through each level because they need practice discussing and verbalizing learned concepts. It was great to read about the different methods teachers can use to assess their students' placements on the van Hiele scale. Simple observations and analyzing their work in the classroom can reveal a lot about their understandings. This will come in handy for me one day!

 

   Next, I viewed the PowerPoint video that also highlighted these important ideas. I did notice, however, that this PowerPoint did mention that it is possible for students to skip a hierarchy level but only if they are really advanced in their thinking. I like how Dr. Hargrove gave us some ideas on what kinds of activities we can use to gather student understanding and reasonings. For example, the "What's My Shape?' activity seems like a great one to use that would help identify students' understanding and use of geometrical vocabulary. Very interesting.

 

    I enjoyed working through this same activity for myself and was successful in sorting out the correct shape in the first two examples. The questions helped guide me to sort out the undesired shapes. I had a little bit of trouble with the final example and ended up guessing the wrong shape. I think I was confused on some of the questions like "Does this shape only have straight sides?" To me, this meant that you sort out the shapes that have all straight lines. However, she explained to sort out the shapes that do no have straight lines (meaning any straight lines at all). I was a little confused by this. I think I messed up on sorting the symmetrical from the asymmetrical shapes. I say this because I got confused by the final question of "Does the shape look like a kite?" The two shapes that I had left looked almost the same (except one was big the other was small). After redoing the activity again, I realized that I had a shape that I wasn't supposed to have and did not have one I needed (the kite shape). It was a little difficult but I was able to correct my mistakes afterwards. This would undoubtedly be a great activity to use in the classroom!

Polygon Properties--NCTM Article

     I thought this article gave some really great ideas on how to help build students' understandings of polygon properties and advance their geometric thinking skills. I like how the teacher used hands-on activities to help them learn different ways a shape can be made while still keeping the properties of that shape. I learned a lot of important information from this article regarding the particular NCTM standards the activities satisfy. I will certainly consider using them in my future classroom. Moreover, it was great to learn some ways to complete the lesson with some students in the class being at different Hiele levels than others. Some simple modifications within the activities would allow all students to successfully participate. In addition to shape sort activity, I like how the students became actively engaged in a "Riddle" activity where they used geoboards to help them solve the problems. It was great to actually work through some of these for myself. As prompted in the module description, I tried out the questions on pg. 528 of the article using a geoboard. I could be wrong but as I worked through the problems, I found that only the problems 2, 5, 9, and 10 could be solved. I will say that having the geoboard was helpful because I was able to manipulate the shapes in anyway I wanted. As one students in the article exclaimed, you don't have erase anything on a geoboard, you just move the rubber band. Here are the images of the shapes I made according to the questions that were asked.

 

 

 

 

 

 

 

11) Describe what you have learned about each kind of polygon.

     I have learned that some polygons can be made in many different ways, you just have to make sure that you keep in mind the particular properties of the shape. Also, there are some polygons that cannot be made in other ways.

 

Was anyone else able to solve these questions differently?

 

Knowing what you now know, what do you think about geometric instruction and how will this impact your planning in your own future classroom?
    So far, I have learned quite a few things that deal with geometric instruction. Knowing that students (and even teachers) understand geometry at different levels certainly impacts my future instruction because I have to be mindful of reaching students at each level. As I mentioned earlier, I understand that I might even be at a higher Hiele level and must make sure I teach in ways that meet the lower level needs. This greatly affects the way I will plan and teach lessons. 

 

Annenberg Module--Triangles and Quadrilaterals

     I don't know about you guys but I always learn so much from completing the modules. Some things may be really confusing for me so that's why I like to share my answers to the questions in each section.

 

Part A

 

 

 Problem A4: To make "different" triangles, you have to change some feature of the triangle. Make a list of the features of triangles that you changed.

    I mainly changed the lengths of the sides of the triangles.

Problem A5: More than one feature can be combined into a triangle. Decide which of the following combinations are possible. If the combination is possible, draw a sketch on a piece of paper. If not, explain why not.

 

Part B

 

Problem B1

 

 

 

 

 

 

 

 

 

 

 

 

Problem B2: Suppose you were asked to make a triangle with sides 4, 4, and 10 units long. Do you think you could do it? Explain your answer. Keep in mind the goal is not to try to build the triangle, but to predict the outcome.

    No, I don't think this can be done. The side with 10 units seems like it would be too long to allow the 4s to meet. The sides would be too short.

 

Problem B3: Come up with a rule that describes when three lengths will make a triangle and when they will not. Write down the rule in your own words.
   This one was a little tricky so I decided to peek at the solution. It makes sense that the longest side cannot be greater than the sum of the other two sides. When it is greater, a triangle cannot be formed.

Problem B4: Suppose you were asked to make a triangle with sides 13.2, 22.333, and 16.5 units long. Do you think you could do it? Explain your answer.

   Yes. In thinking about this question, I used the rule I just learned. Essentially, you can make a triangle with these side because when you add the shortest sides (13.2 + 16.5), you get 29.5, which is larger than the longest side of 22.333.

 

Problem B5: Can a set of three lengths make two different triangles?

   No. One set of 3 lengths can only work to form one triangle.

 

Problem B6

 

 

 Problem B7: For some of the lengths above, can you connect them in a different order to make a different quadrilateral? If so, which ones? How is this different from building triangles?
     Yes, you can connect some of the lengths in a different order to me a different quadrilateral. For example, you can see that the 4 3 2 2 lengths form one shape while rearranging the lengths to 4 2 3 2 creates a different shape:
 

             

                    4 3 2 2                                                              4 2 3 2

 

  This is different than building triangles because triangles have only 3 sides. Rearranging the lengths would result in the same shape.

Problem B8: Come up with a rule that describes when four lengths will make a quadrilateral and when they will not. Write down the rule in your own words. (You may want to try some more cases to test your rule.)
    I thought about this question in the same way I did the one with triangles. Essentially, the longest side cannot be longer than the sum of the other three sides.

Problem B9: Can a set of four lengths make two different quadrilaterals?
   Yes. This is possible. For example, the side lengths 1 2 3 4 can be arranged to make the following shapes:

 

 

Part C

Problem C1: Gather toothpicks and mini marshmallows, or other connectors. Your job is to work for 10 minutes to build the largest freestanding structure you can. "Freestanding" means the structure cannot lean against anything else to keep it up. At the end of 10 minutes, stop building, and measure your structure.
   I must say that this task was a lot harder than I thought it was going to be. I think I took a lot longer than 10 minutes due to having to stop and start over because I couldn't get anything to stand freely. And to be completely honest, I'm not sure if I did this correctly. Nevertheless, here is a picture of the biggest structure I could make.

 

Problem C2: What kinds of shapes did you use in your structure? Which shapes made the building stronger? Which shapes made the building weaker?

    On my first attempt, I tried using squares instead of triangles since I thought four sides would hold better than three. I was so wrong! My building wouldn't even hold up with the squares. These made the structure weaker while the triangles made it stronger.

 

 

Problem C3: If you had the chance to build the structure again, what would you do differently?
  After viewing the Annenberg video segment, I realize that I did not complete the task correctly I thought it wanted a structure no sides. In that case, I would certainly build a structure that has supportive sides and try incorporate more triangles throughout. I would also try to create a larger base so that my building would have more support for added height.

Problem C4: Get another set of building materials and take an additional 10 minutes to create a new freestanding structure. Your goal this time is to build a structure taller than the one you made before.
  For this round, I decided to use pipe cleaners in place of the tooth picks. I felt like these would be a little lighter, allowing me to stack more. I could not find any other materials to act as the "connectors" so I had to use the marshmallows. I would like to have used something like Play-Doh or sticky tack instead but did not have any. It was a little difficult trying to poke the pipe cleaner pieces into the marshmallows, something that took up a lot of the 10 minutes. However, I was able to build a taller structure with the items I used. Here is what it looked like:

               Front view                                                                       Top view

 

 

Thinking About Triangles

 

This was an interesting activity! Before hearing the answers Dr. Hargrove provide, I came up with my own answers.
   As the PowerPoint prompted, I looked around me to take note of any triangles I saw around me. Surprisingly, I could not identify very many objects but among the things I did see included:

   

       

What do these shapes have in common?
   Not only do these shapes look similar, they each have three sides. The points come together to form a triangle-like shape.

 Where do you think the word triangle comes from? What other words start with tri?
     The word triangle means 3 angles. Tri=3 I can't really say that I know where the root word itself comes from but I'm going to guess it's a Greek or Latin word. A lot of other mathematical words come from these languages. I can think of many words that start with tri:
  
   tricycle    triathlon  trilogy   triplets   tripod   triceps

These are similar to the word triangle in that they each have three of something. They are different because they each describe what there are three of: 3 angles, 3 wheels, 3 books, etc.

Polygon definition: Polygons are closed figures that have three or more straight sides.

As I worked through each question, I used my tools to solve them:

1) Is it possible to make a three-sided polygon that is not a triangle?  No. It has to be a triangle if it has 3 sides.

2) Is it possible to make a right triangle that has two right triangles? No. A triangle has to add up to 180 degrees. If there are 2 right angles, that equals 180 for just the 2. You would need another angle for it to be a triangle.

3) How many different right triangles can be made on the geoboard? I came up with 10 different triangles. I started at a corner of the geoboard and made the smallest triangle possible. I then extended the rubber band  down the side--one peg  time. On my geodot paper, I drew the triangle as seen on my board and wrote the dimensions (in terms of pegs) so I would not record duplicate triangles. This is a strategy I might teach my future students. Here is a picture of my work:

 

 

4) How many different types of triangles can you find? I came up with 19 different triangles (not including the right triangles). I started with the 2nd peg of one side of the geoboard and extended the rubber band outwards to make a variety of sizes. I fist completed the possible triangles on the 1st row and then moved down. Again, I wrote the "dimensions" so I would not repeat triangles. He are triangles I drew on the geodot paper.

5) Cut out the shapes your drew on the geodot paper and sort them into different piles.
   I sorted my triangles into 3 different piles: 1) right triangles, 2) triangles with 2 different side lengths, and 3) triangles with 3 different side lengths. Here are pictures of my sortings:

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

     First of all, I would break this lesson up into multiple lessons for it can take a while to complete. I would begin by making the idea of triangles relevant to students by having them explore objects around them that are shaped like triangles. They can then compare and contrast the objects and other “tri” words to gain a better understanding of with what kind of shape a triangle is. In a separate lesson, I would have students work with the geoboard riddles. They can work in pairs/groups to determine certain characteristics of triangles. The final part of the lesson set would involve students using the geoboards and geodot paper to make different triangles and sort them into groups.

 

• What parts did you have issues with? Did you need to revisit some vocabulary words to remind yourself of their meanings? If so, which ones? 
     I had the most trouble with creating different triangles on the geoboard. I don't think I repeated any triangles but I'm not sure if I created all possible triangles. I would like to know of specific strategies that help make the process easier. What kind of method did you use? 
   As far as revisiting vocabulary words, I will admit that this is something I really need to do. It's been a while since I have taken a geometry class, so I have forgotten a lot of important terms and properties. I need to visit the different types of triangles like Isosceles and Scalene. I tend to confuse the two.

•Don’t forget to check in with your blog partner….
   I look forward to reading your responses and feedback!

 

For Further Discussion

 

The van Hiele levels can be applied to geometric thinking concepts by concept or to one’s overall thinking about geometry. How would you rate your own thinking? Does the level vary or remain fairly consistent across the subject matter? For example, are you at different levels working with the concepts of two-dimensional or plane geometry than working with three dimensional or solid geometry? Explain

        I've never been the best at geometry, even though I think it's rather interesting. Perhaps this is because I have never really experienced significant growth in my geometric understanding (I pretty much know the basics I was taught in school). Having read the descriptions of each stage in the van Hiele article, I would place myself between level 3 (Abstraction) and level 4 (Deduction) on the scale. I can understand certain properties and understand meaningful definitions or figures. However, I am not 100% confident in being able to successfully construct shapes using these ideas.

   In terms of being at a different level across the subject of geometry, I do find that 2D shapes are a little easier to work with than 3D shapes. Adding in the curved edges and extra dimension makes it a little harder for me to visualize, especially if the figure is shown on a piece of paper. As a result, I may be at a lower van Hiele level with 3D shapes (around 2 or 3).