Module 12

TCM Article – Rulers

I enjoyed this article. I think that the lesson outlined was a great way to introduce different units of measurement. I was very impressed that the students were able to figure out that their rulers are the same size without being told. One point the article made that I did not think about was the difficulty students can have actually using a ruler correctly. Not all the students knew to start at the very end of the ruler (before 1).

Angles Video and Case Studies

One thing that I took away from the case study was that many children focus more on the line segments than the actual angles when they are learning about angles. It was interesting to read the varied understandings that were taking place in one classroom. This held true in the video as well. Many of the students could demonstrate what an angle would look like. They made angles using their arms and imaginary lines. They had difficulty, though, explaining what exact part of their example was the angle.

Annenberg Angles Module

Use two or more polygons to illustrate the angles below, and explain how you would justify that some of the angles are congruent: a.  Vertical angles (the angles formed when two lines intersect; in the figure above, ad, cb, eh, and fg are pairs of vertical angles, and the angle measures in each pair are equal) b.  Corresponding angles (the angles formed when a transversal cuts two parallel lines; in the figure above, ae, bf, cg, and dh are pairs of corresponding angles, and the angle measures in each pair are equal) c.  Alternate interior angles (the angles formed when a transversal cuts two parallel lines; in the figure above, cf and de are pairs of alternate interior angles, and the angle measures in each pair are equal)

Problem B6    Find one or more polygons you can use to see examples of the following angles: a.  Central angles (for regular polygons, the central angle has its vertex at the center of the polygon, and its rays go through any two adjacent vertices) b.  Interior or vertex angles (an angle inside a polygon that lies between two sides) c.  Exterior angles (an angle outside a polygon that lies between one side and the extension of its adjacent side):

I had a little trouble with each of these problems. It has been a while since I have done these kinds of problems so this module was a good refresher. After playing around with the shapes a bit and reading the solutions it became clearer.

TCM Article – How Wedge you Teach?

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Exploring Angles with Pattern Blocks

Green Triangle: For this shape I know that all three angles add up to 180. I also recognize that this is an equilateral triangle so all of the angles are the same. 180/3=60

Blue Rhombus: I can remember that on a rhombus, the opposite angle are congruent so there was going to be two different angles. The smaller of the angles looked similar to the angles in the green triangle so I matched them up and they fit. I know that all the angles of a rhombus add up to 360. Two of the angles are 60 which add up to 120. That would mean that the remaining two angles would be 120 each.

Red Trapezoid: Going in I know that all of the angles would add up to 360. I used the 60 degree angle from the triangle to confirm that the smaller angles were 60 degrees. that would leave the two larger angles to be 120.

Tan Rhombus: I found this shape to be very difficult. I could see that there were two very obtuse angles and two acute. Because it is a quadrilateral, I know that all the angles add to 360. I played around with this for a bit and could not find  a way to get an answer.

Yellow Hexagon:  I know that all of the angles in a hexagon add up to 720 degrees. All of the angles looked to be the same size so I divided 720 by 6 to get 120 for each angle.

Challenge:
By connecting two triangles by their base you get two 120 degree, obtuse, angles.

Combining a triangle and trapezoid created a straight angle.

Combining tow hexagons created a 240 degree angle

I agree with Mai. Using the shapes we have I could not find a way to create an acute angle.

For further discussion

When I think of nonstandard measurements, I immediately think of cooking. You will often times see a pinch/ dash/ splash of a certain ingredient. I have found that these terms are used when it does not matter if there is an exact measure. It is usually an ingredient meant for flavor and left to your preference. You will never see a cake recipe using a non standard measurement on flour, baking power, or baking soda because those ingredients are critical to the cake and must be exact. The example "I'll be there shortly" also falls into this category. It lets the person know that you are probably on the way and will be there soon, but you don't say an exact time. If you were to say "I'll be there in five minutes" but it actually took ten, then it seems like you were late! You use nonstandard measurement when you don't need to be exact.

Module 11

Coordinate Grids 

I explored the websites Billy Bug (1 &2), Stock the Shelf, and Greg's Grid Graphics. I think all three of these websites would be perfect for classroom use. I think that Billy Bug and Stock the Shelf would be a great choice for use in math centers in classrooms with an interactive whiteboard.  They were pretty straight forward and set up for easy turn taking. You could also use these websites in a whole class setting but it would be more difficult to allow every student to have a turn. I think that Greg's Grid Graphics would be more appropriate for whole class instruction. You would still have the opportunity to hear from every student but I feel that it would go more quickly since the teacher could just type the students' answers rather than the student coming forward to manipulate the game.

Miras and Reflections and Kaleidoscopes Article 

I have never used a mira before, I remember being perplexed by that tool when I received my kit at the beginning of the semester. These activities were good practice for me to learn how to properly use the tool. I feel like the only part of the activity that I had significant trouble with the alphabet symmetry. 

Annenberg Measurement Module 

I did not particularly have any questions for this module.The only question that really made me pause was measuring the surface area of the rock with tin foil. I figured that you would measure the flattened foil. It never occurred to me that the foil would have an irregular shape. After the module broke down how to measure an irregular shape, I understood.

Case Studies 

 It was very surprising to see that students struggle with very similar measurement concepts throughout elementary school. The idea that A larger unit of measurement would have a smaller number and that a smaller unit of measurement would result in a larger number seemed to  confuse students for many years. It was interesting to see the younger students begin to question the accuracy of measurements based on techniques used to gather data. This becomes a big focus in the upper grades as they begin using actual measurement tools.

For further discussion

A fellow teacher says that he cannot start to teach any geometry until the students know all the
terms and definitions and that his fifth graders just cannot learn them. What misconceptions
about teaching geometry does this teacher hold? Now that you've had some time to explore the world of geometry, how has your view of the key ideas of geometry that you want your students to work though changed

This teacher seems to be under the impression that geometry is only a series of theorems and complex proofs. In reality, most of his students have probably already begun exploring  geometric concepts. Coming into this course, I also associated geometry with more complex concepts such as the Pythagorean Theorem. I felt that geometric concepts would not be introduced until closer to middle school aged. Throughout the modules I have been reminded that through use of manipulatives, students can begin to understand geometric concepts such as symmetry, reflections, and spatial reasoning.

Module 10

Annenberg Symmetry Module

This module was so strange to me. While I was reading the definitions, it all felt terribly abstract and I had to re-read explanations multiple times. When I looked at the actual examples and problems, however, I knew how to solve them. 

Problem A3

For each figure, reflect the figure over the line shown using perpendicular bisectors. Check your work with a Mira.

Printable PDF of figures

For these types of problems is there a way to ensure a more accurate reflection? While I was solving the problem, I knew to reflect the shapes across the line but I felt like I was missing an important step. 

Pentomino Activities

I tried the pentomino activity on the Scholastic website. I decided to start on the easy mode as a warm up. I think I must have a different definition of easy! I filled up the square pretty quickly until I was left with three lone spaces which could not be filled by any of the remaining shapes. I reached this dead end three more times before I finally solved the puzzle. At that point I was simultaneously happy and still a little annoyed so I did not go for the medium mode!



Pentomino Narrow Passage

My passage ended up being 15 spaces long and closed on both ends, It took me a few tries to find a way to fit all of the pieces together without having wide areas in the passage.

Tessellating T-shirts Article

I really enjoyed this article. I always like the idea of integrating Art into Math activities, I like that the teacher gave students the option of creating their own shape to use in their activity. I think that will allow students to make a deeper connection to the assignment. Tessellate means repeating a shape over and over again.
When I think of tessellation, I immediately think of moroccan tiles and patterns.

I also thought of a naturally occurring tessellation: honeycombs

Tangram Discoveries

 Which polygon has the greatest perimeter? …the least perimeter? How do you know?
I think that the trapezoid had the greatest perimeter and the square had the smallest.

Which polygon has the greatest area? …the least area? How do you know?
I think that all of the polygons had the same area because I could build each polygon using the same three triangles.

For Further Discussion
Many cultures use beautiful patterns and tessellations in their art and clothing designs. These artifacts could be a great way to connect Math and Social Studies. You could challenge students to come up with their own shape-centric patterns to use in their own stylized artwork.