Module 13

Annenberg Video Circumference and Diameter

1. Describe Ms. Scrivner's techniques for letting students explore the relationship between
circumference and diameter. What other techniques could you use?

 The children were allowed to look around the classroom for round objects to measure. Using measuring tape, students measured both the circumference and diameter of their object. They then recorded their data and discussed the data as a class. To extend the lesson a bit, you could compare the measurements from two groups  who had measured the same objects.

2. In essence, students in this lesson were learning about the ratio of the circumference to the
diameter. Compare how students in this class are learning with how you learned when you
were in school.
The students in this class were allowed to explore these concepts with a  hands on approach. This is very diffeent from my own experience. When I learned these concepts, I was juts given a formula and a lot of practice problems from the book.

3. How did Ms. Scrivner have students develop ownership in the mathematical task in this
lesson?
Ms. Scrivner allowed the students to develop ownership by allowing them to choose the objects that they wanted to measure.

4. How can student's understanding be assessed with this task?
Observation would be a strong assessment tool for this task. By observing the students during their group work, you would be able to see the way the students measure their objects, if the objects they are measuring are round, and what the students think diameter and circumference is in relation to their object. You could also assess the data that the students recorded.

Annenberg Circles and Pi Module

This module was a great refresher! It has been a while since I have had to use any of these concepts.

Problem B8    When you enlarge a circle so that the radius is twice as long (a scale factor of 2), what do you think happens to the circumference and the area? Do they double? Experiment by enlarging circles with different radii and analyzing the data.

Problem B9 Experiment by enlarging a circle by a scale factor of 3, by a scale factor of 2/3, and by a scale factor of k. Generalize your findings.

For B8, my first thought was that everything would double. After reading the solution I was a bit confused, but the more that I thought about it, the clearer it became. 
For B9 I thought I as definitely understanding the concept better until I got to the fraction and the variable. I think if I practiced this concept a bit more I would be ok, but I honestly feel a but shaky with this concept. 

Textbook Pages 1-26

1. Explain what it means to measure something. Does your explanation work equally well for length, area, weight, volume, and time?

To measure something is to attempt to assess a characteristic of that item using a specialized tool. 

3. Four reasons were offered for using nonstandard units instead of standard units in instructional activities. Which of these seem most important to you, why?

 I think that the most important reason for using non standard units is that it allows very young students to begin thinking about what it means to measure something. Learning to measure objects with nonstandard units, such as hands or feet, allows students to really play around with how to measure and observe variables that effect measurement.

For further consideration….

We have explored so many great topics this semester and I definitely will be incorporating some things that I've learned in my own classroom. I am really glad that we did the children's literature project because it allowed me to see how many rich literature resources there are for many mathematical concepts. After reading the journal articles and watching the videos of  younger students collecting and displaying data, I am confident that if I teach lower elementary I will challenge my students to collect small units of data. I would also expose my students to various types of graphs early on so that they are familiar with how to read and form their own graphs. The biggest take away I got from this class is that students are much more capable than they sometimes get credit for. You should not be afraid to introduce challenging topics. The key is to make the content meaningful.

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?

I could not get this article to load!

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.

Module 8

Quick Images Video 

I found the video this week very interesting. It really highlighted the way that everybody thinks of things differently. Most of the children connected the shape with a real life object such as a boat or banana. My first thought was that it was a picture of the shape, crescent. I always remember what a crescent looks like because I think of a crescent moon. I thought it was really interesting that one of the students said that she remembered the shape because it was part of a circle. I have never thought of a crescent as part of a circle, but it really is.

Case Studies- Shapes and Geometric Definitions 

The case studies outlined students' responses to seeing familiar shapes in unfamiliar positions and proportions. In Andre's case, she presented students with an array of triangles. All of the students agreed that the oddly shaped ones looked like triangles, but could not decide whether they were "real" triangles. The kids had become so accustomed to seeing a triangle depicted as an equilateral triangle that it did not feel comfortable to classify different triangles as being the same shape.

Annenberg Polygons Module

Problem A2
How many polygons can you find in the following figure? Problem A3    How many polygons can you find in the following figure?

The kindergarten class that I am assigned to this semester recently began exploring shapes. Just last week I was working with some students during their Math centers and told them "I bet I can make a square out of 2 triangles." They seemed skeptical but watched me intently. They all immediately modeled what I had just done with the triangles. I challenged them to create other shapes using the manipulatives. When I came upon these series of questions I felt it was very simple. 4 triangles and a rectangle. When I read the solution I was shocked to see so many other combinations. I went back to look at the figure again and began to see more chapes jump out that I missed the first time.

For further discussion

I have never really noticed before, but my home is filled with rectangles. Tables, doors, windows, picture frames, the television, tiles, and even this computer is composed of rectangles. Most of the circles in my home are located in the kitchen. Plates, bowls, pots, pans, and a large wall clock. I went on a hunt through the house for triangular objects but, other than a few decorative patterns, I found none.

Module 6

Annenberg Probability Module 

Problem A3    What is a random event? Give an example of something that happens randomly and something that does not. When I read the solution to this question I began to feel the same confusion that has followed me through many years of working with statistics. The solution mentioned that some things are just random while others require skill. I understand that. The example they used as a random event is what card would show up on top of a freshly shuffled deck. Where I get confused is would that actually be random? You would always have a 1 in 42 chance of drawing a specific card. SO it isn't as if the probability of pulling any one card is randomly changing. Am I just over thinking this? Problem B2    Suppose you toss a fair coin three times, and the coin comes up as heads all three times. What is the probability that the fourth toss will be tails?

This question made me laugh a little bit. Logically, I know that in a coin toss, there will always be a 50/50 chance of heads or tails. It is easy to forget this in the heat of the moment, though. I know that my odds are still 50/50 but there is this nagging feeling in your mind that the next toss HAS to have this certain outcome.

A Whale of a Tale article

Create your own probability line chart that displays events that are impossible, certain, likely, or
unlikely to happen. You should have a minimum of 2 events for each area

Dice Toss 

• Based on the data, the class concluded that 7 was the most likely outcome. Some of the students did have some preconceptions, though. For example, some students thought that 12 would be unlikely because it is more difficult to roll a 6. I think that this would be a difficult concept to get out of their heads. Like I explained earlier with the coin toss, even though you know that there is an equal chance of either outcome, it is so easy to get caught up in the results. You could toss a coin 4 times and get 4 heads in a row, but that doesn't affect the probability that you could land on tails next even though you may feel otherwise. 
• I do not think that they were too young. They were able to clearly answer their teacher's questions and even expressed their surprise at some of the results. 
• The teacher asked the students to roll 36 times because it would allow the students to collect a large amount of data to be able to see the trends. One advantage to this is that the results would be more accurate. One disadvantage is that this would take up a lot of time in class. Also the large amount of data  might be overwhelming. 

Module 5

Box an Whisker Plots

1. What are some differences in the data that you can observe? 

Our class has a larger range than the German class. The German class has a larger population but their range is smaller. 

2. What does the median for each class tell us about the data?

The median for the German class is approximately 55. This means that half of the German class threw out less than around 55 pounds of trash and the other half threw out more than 55 pounds. The median for our class is 83. This means that about half of the class threw out less that 83 pounds of trash and half threw out more than 83 pounds. 

Common Core

First Impressions:

• My first thought when exploring the standards was that there are A LOT. It is a bit overwhelming, imagining how to fit in all of these standards throughout the school year.
• The second thing that I noticed was how the standards progressed as they advanced in grade level.

How do the concepts progress through the grades?

As you move through the grades, you revisit the same concept but you delve deeper into the content.

How do the concepts change and increase in rigor and complexity for the students?

Each year, students build upon the concepts that were covered in the previous year. For example, in kindergarten students are expected to master addition up to sums of 20. By the end of second grade they are expected to add to sums of 100.

Does the Common Core Standards align with what NCTM states students should be able to know 

and do within the different grade level bands?

For the most part, the Common Core Standards align with NCTM Standards.

Give examples of which standards align as well as examples of what is missing from the Common 

Core but is emphasized in the NCTM standards and vice versa.

I focused on the earlier grade levels since I am assigned to a kindergarten classroom this semester. I thought that the concepts all aligned pretty well. I noticed that one major difference is that NCTM stresses connecting materials to the students and the every day lives. This is not mentioned in the Common Core Standards.