# Module 13

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?

-When you measure something, you are trying to see how long or tall something is. You are also possibly trying to solve how heavy something is.

-When I thought of measuring, I automatically thought of a ruler. It is the first thing that comes to my mind. Also, because I just moved this weekend, I mostly thought of measuring as trying to see how big, wide, tall, etc. things are. My explanation for measuring did not work well for area, volume, or time. My explanation did not relate to time or volume at all.

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

-I believe the most important reason to use nonstandard units when measuring would be because they provide younger students a good rationale for using standard units in the future. The whole goal is to improve student’s math and measuring skills. Nonstandard units can help prepare them for math in the future which is important.

Annenberg Video Circumference and Diameter:

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

-Ms. Scrivner let students find circles around the classroom and measure and figure out the relationship of the circumference and diameter on their own. I thought giving them the hands own experience of finding their own circles around the classroom to measure was a great idea. Another way you could have students do this would be to actually give them a list of circles that are around the room. The students could focus on these things and that might possible knock out some of the incorrect information.

• 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.

-Students in this class learned the relationship of circumference and diameter by exploring own their own. I do remember something similar to this in my elementary school experience. However, we were given a list of certain circles around the classroom to measure and guided throughout the experiment. We also did not work in groups.

• How did Ms. Scrivner have students develop ownership in the mathematical task in this lesson?

-Students developed ownership of their mathematical tasks by finding their own circles and discovering the relationship between circumference and diameter own their own.

• How can student’s understanding be assessed with this task?

-Student’s understanding can be assessed by the charts that they completed throughout their activity. Also, by their explanations throughout the discussion after the activity was completed.

Annenberg Circles and Pi Module:

I had a hard time understanding problem A9…

Since Pi is an irrational number, can both the circumference and the diameter be rational numbers? Can one of them be rational? Explain using examples.

In my head, if we are multiplying by an irrational number, wouldn’t our answer be irrational? This question has me very confused.

I found problem B8 very interesting…

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.

When I first read this question I thought it was obvious that they both double. It was not until I actually worked through the problem that I realized the area was actually multiplied by 4 and the radii was doubled.

Further consideration:

Throughout this semester, we have explored many different ideas. One of the main ideas I will take with me to my classroom is having students collect data. I never realized all the different ways there are to collect data and how younger students can be involved in this as well. I also really enjoyed exploring different graphs. I think these are great and would like to help my students graphing skills as early as possible. I also learned through our children’s literature project that there are multiple book that relate to math standards that are available. I will definitely be using these with future students. I also really enjoyed learning about measurement in this last section. I would love to use different measurement techniques and relate different forms of measuring together like we have reviewed in the case studies and from the video in this module.

# Module 12:

TCM Article:

I found this article very helpful and a great resource. Throughout last week’s module, I wondered if having students begin measuring with different items effected their ability to use a ruler properly. The article discussed that this could lead to some confusion and inaccurate solutions. I think it was so neat to see how the students worked together to see that each ruler had different measurements. I think this activity proved that the students really did understand how to measure with a ruler. I would really like to complete this activity with students in my classroom to see if they can make the same connections. I think one misconception students may have when dealing with measurement would be counting the first mark as one instead of zero. I feel that this is a common mistake many people, especially children make.

Angle Video and Case Studies:

I was very impressed by the students in the video. Each student that explained seemed to have a very good idea of what an angle is. They also did a great job drawing and pointing out the angles on the chart. I believe that angles can sometimes be a difficult thing to understand. The case studies always show what the students are struggling with. I believe that many of the students were struggling with identifying the difference between obtuse and acute angles. The understood the definitions, but when it was time to draw the angle of find it, they had trouble. I feel as if this is probably very common throughout each grade level. What do you think Hope? I believe that most of the students had a great definition in their heads of what these angles are; however, just needed more practice actually using the angles and understanding the definitions.

Annenberg Angles Module:

I found section A in the Annenberg module very informative. The first part that I really enjoyed was learning that angles can be measured as a radian. I was not aware of that until this module. I just always assumed that angles were measured in degrees. Also, I really enjoyed seeing the activity of making angles out of bendy straws. I think this would be such a great and fun way to introduce angles to a class. I also liked that during the activity you cut the length of one of the sides and see that this does not affect the angle. I think this would be a great visual for the students.

I really enjoyed this module and did pretty well answering all of the questions. One question I did have a difficult time with was B7. When completing the chart by recording the angles degrees, I really wanted to measure and figure out exactly. I had a hard time just trying to use the polygons. Hope, how did you do with completing the chart? Did you have the same problem as I did or do you find it easier to measure with the polygons?

Another question I had a difficult time figuring out was question B12 part C. When I first read to find the sum of the measures of the vertex angles in the hexagon I became confused. I began to add the degrees of the triangles together to come to my answer. After checking the solution I realized that this was correct and felt better about the information. I felt just like I was back in geometry class when completing the Annenberg module. Thankfully, I remembered a lot more than I thought I did. What was some of the difficulties you had completing the Annenberg module Hope?

TCM Article- How Wedge you Teach?

I found reading this article very interesting. I always found it difficult to measure with a protractor and had to remember how to use one throughout reading this module. One of the biggest things I will take away from reading this article will be to have students describe angles without using the words acute, right, and obtuse. I feel that based on the students in this module, and from the case studies, students just memorize these terms without having a clear understanding of what they mean. I think it is so surprising how students can know terms, but not truly understand them. That is why I would like to take this away from the article to use in my own classroom. By having students explain without these terms, you really get an understanding of what they know and do not know.

I think there are many different misconceptions that students can have about angles. For example, one of the most common I have seen throughout the case studies and readings in this module would be recognizing how many angles are in shapes and different objects. I feel as if I have this same misconception at times. For example, in the case study, one of the students points out that there is an angle on the outside of the shape, as well as on the inside. I honestly did not even realize this until I read what the student had said. I think shows how it is important to ensure that students really know and understand the concepts, not just memorize definitions.

Exploring Angles with Pattern Blocks:

Green Triangle: each angle would be 60 degrees because it is an equilateral triangle. Each angle is equal and they must add up to a total of 180 degrees.

Blue Rhombus: clockwise starting from top left- 60,120, 60, 120. I honestly have no idea how we were supposed to do this using the mirror. I tried and reread the instructions and just became even more confused. I broke the rhombus into two equilateral triangles to solve the problem.

Red Trapezoid: Clockwise starting from to left- 60, 60, 120, 120. Again, I broke this shape into triangles to solve the answer. The trapezoid can be broken into three equilateral triangles.

Tan Rhombus: I am not sure how to solve this problem. I can see that it can be broken into two acute triangles. However, I am not sure how to determine what the angles would be.

Yellow Hexagon: I broke this shape into two equal trapezoids. They would be the same as the trapezoid I solved for before.

I was very confused on how to complete this activity. Hope if you can help I would extremely appreciate it!

Further Discussion:

My husband uses nonstandard measurements when cooking all the time. I am a very literal person and I like everything to be exact. He always argues that I take too long to cook because I do measure everything! I cannot just add things in the way he does, I need it to follow the exact recipe. I do believe that nonstandard measurement is preferred when cooking. It is so much simpler than measuring everything out and adds more flavor.

# Module 11:

Coordinate Grids:

I explored the website Greg’s Grid Graphs. One of the main advantages to this website is the information it gives. The website provides great definitions and examples for how to plot points on a coordinate grid. For example, the definition of how to plot the points would be (x,y) = x how many times you move right from the number 0; y: how many times you move up from x. I thought this was a great definition for younger students and they had great examples for students to complete. It was a great refresher for me. The only disadvantage I personally saw was that it taught students to write the ordered pair in parentheses, but when it came time to type in the ordered pairs you could not put parentheses in the blank. I felt that this could cause confusion. Also, the description of this site was to help Greg and his family plan their visit to the museum. I thought this was going to be a very fun and interactive site and it was only plotting points on a blank graph each time and it was pretty time consuming for me, so it would take quite some time for students to complete. Hope, do you think that since this activity can be time consuming it would still be worth using in the classroom? I would like to but I’m just not sure how to incorporate it because I’m estimating it would take each student around 20 minutes to complete. Do you think this could be used on student’s tablets if they have them?

Miras and Reflections and Kaleidoscopes Article:

I had never worked with miras before. I loved learning how to use this! I did have some trouble at first trying to get everything to line up, but once I became more familiar with moving the mira around it became easier. I loved learning how to use this to see if shapes and letters have lines of symmetry. I think this is such a great advantage and can really help students better understand lines of symmetry. Hope, have you worked with a mira before? I don’t know that I have ever even heard of one before this module! My husband was also amazed. He completed this activity after I did!

I also really enjoyed reading the article. I think it is so amazing that you can use items such as miras to create your own kaleidoscope to teach geometry. This is definitely a project I would love to complete in my classroom someday.

Annenberg Measurement Module:

During the Annenberg module, I was completely puzzled by problem B1 and B2. First off, I can understand why they are using tinfoil because it will wrap around the rock, but it this a reliable measurement source? What if the tinfoil is overlapping in some areas, wouldn’t this make a difference in your answer? Also, I do not understand what how the measurement is in cm2. These two questions just had me very confused. Hope, do you think that using the tinfoil to measure would give you a number that is close to being correct? Also, do you understand how the answer would be in cm2? I am just very confused with this.

Case Studies:

I really enjoy reading the case studies. I think it is so interesting how many factors go into us teaching our students about measurement. I was very impressed with how well the kindergarten students could measure using their feet and pick out that different students get different answers because their feet are various sizes. I was also a little surprised at how the older students seemed to always forget this step. It is very important, but I think it just relates back to when our brain sees a larger number we always think of something larger. I was also somewhat surprised at how the students were having trouble remembering how to use the rulers and tape measures. I think the teacher in this case study was correct in saying that we need to have students practice measuring more often.

Further Discussion:

The fifth grade teacher that believes he cannot teach his students geometry is truly mistaking. I have learned throughout these modules that students can learn different concepts of geometry at any age. By allowing students to learn a little bit at a time, the students will be able to connect different ideas of geometry and have them make more sense when they get up to geometry in high school.

After these modules, I really want to focus on teaching students more varieties of shapes so they understand that, for example, triangles can look different. I believe by just understanding this, they are on the right track to begin questioning other things and discovering how different geometry skills link together.

# Module 10:

Annenberg Symmetry Module:

In this module, I found Problem A2 very interesting.

 Problem A2 Find all the lines of symmetry for these regular polygons. Generalize a rule about the number of lines of symmetry for regular polygons.

I knew that each shape had many lines of symmetry, but I did not realize that there were as many lines of symmetry as there were sides on the shape. Hope, did you realize this to be true? I had never thought of it before reading the solution to this problem.

I also had a few problem with question B1…

Problem B1
 Each of these figures has rotation symmetry. Can you estimate the center of rotation and the angle of rotation?

After using the interactive board I could see the angle of rotation, but I would not have been able to see without that. Hope, were you able to solve this question without the interactive board? If so how?

Pentomino Activities:

I worked along with the activities from the power point. When finding the perimeter of the shapes, I was very confused. I could not figure out how the perimeter was twelve. Finally, I realized that you had to count the sides of the shape. Once I figured out that part of the activity, it was easy. I think this would be a fun activity to incorporate with students. I would show you how students are thinking at the time and if they have a grasp on perimeter.

I also thought drawing a title floor with the pentomino shapes was a fun activity. I had a pretty easy time fitting the shape I picked together. I think this would be a very fun activity for younger students to complete. Below is a picture of where I placed the pentomino shapes together.

(Sorry the picture turned out light)

Hope, what shape did you use to complete this activity? Did you have any difficulties with this or was it fairly simple for you as well?

I became very frustrated with the activity on the online website of trying to fit the pentominoes together to make a rectangle. I could not figure it out once! Not once was I able to make a rectangle! I feel like it should not be this difficult! Hope, were you able to make a rectangle? How did you do it if you did? Did you find it as difficult as I did? I am still trying to figure out this activity!

Pentomino Narrow Passage:

Below is a picture of my passage. I was only able to create a path with 13 spaces. I did find this activity much easier. Was this activity easier or harder for you than the others Hope?

Tessellating T-shirts Article:

This article was very interesting to me. I think I got the most out of this article when it discuessed how we cannot afford to wait until high school to teach students about geometry. The more this semester goes on, I find this more to be true. I have always had difficulties with math, and honestly do not remember learning many of the things discussed until high school. I think that we need to begin teaching students these conscepts, especially geometry, while they are in elementary school to help them grow further.

Tangram Discoveries:

Again, I had a really hard time trying to figure out this activity. Hope, how did you go about this? Do you have any advice to understand this better?

Ordering Rectangles Activity:

1. I believe that rectangle C has the largest perimeter because the sides are so much longer than any other rectangle. I think rectangle B has the smallest perimeter just by looking. It appears to be smaller just based on my eye. I believe the order will go from smallest to largest rectangle B, rectangle A, rectangle D, rectangle G, rectangle E, rectangle F, and rectangle C.
2. I believe it was harder to guess which rectangle had the smallest and largest area. I do believe just by looking at the rectangles, rectangle C has the smallest area and rectangle G has the largest area. Smallest to largest, I believe the area will be C, F, B, D, A, E, and G.
3. The actual order, from smallest to largest for the perimeter was D, E, B, A, G, C, F. I did not do as horrible as I believed I had done and after measuring I can see now the correct order of the shapes.
4. The actual order, from smallest to largest, for the area was C, D, E, B, F, A, and G. I also did not do as bad I as had believed estimating the area of these shapes. It is also a lot easier to see the actual order now looking at the shapes after measuring.
5. I did much better at this activity than the previous activities in the module. It was very refreshing after doing so badly on the other activities. I am still somewhat puzzled at why I could see the area and perimeter but not enough to make the shapes in to previous activities.

Further Discussion:

This is actually a piece of artwork I have hanging in my house. I bought this when I was volunteering in Paraguay a few years ago. There were numerous artworks made like this. They used string and sewed these designs. All the designs used shapes and symmetry. I even had some earrings at one time from there that looked very similar but I cannot seem to find them now.

# Module 9:

Nets Activity and Mathematics Hiding in the Nets Article:

I really enjoyed reading the nets article. I was able to complete the activity with little frustration. I was amazed at how well the students did with this activity. At first I was able to tell that all the cubes would have the same area, but I did not realize that would be the same for the perimeter. It did take me awhile to see this. I would really like to complete this activity with my future students. I remember completing something similar to this in school. After we discussed the difference in two-dimensional and three-dimensional shapes, we made our own three-dimensional shapes. I think that students would really benefit from discussion different issues and coming up with their own ways to make cubes. However, I think this activity would be very difficult for younger students. I believe you would have to make the discussion much simpler for students under third-grade. Do you agree with this Hope, or do you think that grades under third would still be able to come to these conclusions?

1. Describe van Hiele’s first three levels of geometric thought.

-Level 0- Students name shapes based on their appearance.

-Level 1- Students begin to think of shapes in classes rather than individual. For example, if students see a circle on their desk, they do not just think of it as a circle, they think of the class of circles.

-Level 2- Students begin to think about properties related to shapes. When a student thinks of a square, they realize that the angles must all be right angles.

-Activities at each of these levels would differ a great deal. Students in level 0 are just starting to learn shapes. They are understanding what a circle, triangle, square, etc. looks like and are able to name them. Activities would be focused on helping them learn the shapes and being able to recall them. Activities at level 1 would be focused on helping students understand that shapes can be classified together. The activities would have to do with having students identify the shapes and which shapes should be classified together, and which should not be. Finally, at level 2, students will be focusing on the properties of shapes. Activities will be focused on helping them realize why a square is a square and a triangle is a triangle (more than students just realizing they look similar).

-Hope, can you think of any other ways that activities would differ at these levels? This is all I could think of.

1. Explore GeoGebra and explain how it can be used.

-GeoGebra was very interesting and I was amazed at how many materials were available. I believe that there are many activities for students at every van Hiele level. To incorporate this in the classroom, you could show the sight on the smartboard, or downloaded it on tablets if they are available to the class. I personally prefer using hand drawings and hands on materials when learning math. I feel as if I learn faster this way and remember longer. On advantage, however, to GeoGebra is that it is exiting and will catch student’s attention.

Spatial Readings, Annenberg, and Building Plans:

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

-I had a very easy time with the “I took a trip on a train” activity. I was able to place the pictures in order correctly on the first try. I also had a very easy time with the “Shadows” activity. I was able to get the answers correct on the first try. However, I had a very difficult time with the “Plot plans and silhouettes activity.” I am not sure why I had such a hard time, but I was never able to solve it correctly. Hope, were you able to complete this activity? How did you do it?

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

-It is very important that students become proficient at spatial visualization because it is a skill that will help them throughout the rest of their lives. I was honestly laughing so hard reading the spatial readings article. I cannot name how many times I have walked to my car from grocery shopping, only to realize I had no clue where I parked the car because I was not paying attention before I walked in. Also, my husband makes fun of me because I am horrible at directions. I would rather go a longer way to get somewhere because I know how, than try and learn a new faster way. It is so easy to use spatial visualization if we just take the time to do it. By students becoming proficient at spatial visualization, they are preparing for the future. Also, it will give them the knowledge they need to visually pick up on something to analyze it. Just as the article said Snow researched the virus and noticed that all the outbreaks were near a well and he discovered the one well was contaminated.

• At what grade level do you believe students are ready for visual/spatial activities?

-Students are ready for visual and spatial activities at the earliest of grade levels. By preparing them early and continuing to build on these activities, students will gain a great amount of knowledge that will help them throughout the rest of their lives.

• How can we help students become more proficient in this area?

-We can help students become more proficient in this area by challenging them to use their visual and spatial thinking through many different activities.

Annenberg Tangrams Module and Creation of Manipulative:

-I had no problem with question A1, creating the square. However, I did find it difficult to make a rectangle. Every option I came up with ended up being a square. Hope, were you able to make a rectangle or did you have a difficult time with this as well?

-I also had the same problem with question B1. When I cut the parallelogram, I could only form a square, not a rectangle. Again, Hope were you able to do this?

# Module 8

Quick Images Video:

I thought this video was very interesting and a great way to help students remember and draw shapes. I was not surprised when one student told the teacher the shape looked like the moon because that is the first thing I thought of when I saw the shape. It was very interesting to see how the students recognized the shape by things around them.

Case Studies:

I was very surprised at reading the case studies. I honestly do not remember learning about different triangles and shapes in school. I feel as if it was something I always knew and never really had a problem with. I see now how confusing learning shapes can be for students, especially triangles because there are so many different kinds. I was very impressed by Evan’s response in case study 19 that even if he is turned upside down or inside out, he is still a triangle. I think that really helped students make the connection that shapes can look different, but still be the same shape. I thought it was funny how Susannah in the same case study said there should be a rule that you might have to turn around a shape to see what it is. I think this showed that she was beginning to realize the same thing Evan had before. I believe that I will try and introduce more shapes, especially triangles, to students when I begin teaching (especially if I end up in a younger grade). I believe that this will really help students once they get older and focus more on geometry. In case study 18 many students kept saying the shapes were not triangles because they did not look like triangles they had ever seen before.

Hope, what ways do you think would be best to help students with these problems? Do you think introducing a variety of triangles to younger students would cut back on some confusion when they become older, or just confuse them more now?

Annenberg- Polygons:

In questions A2-A4 I was amazed at home many different shapes you could make. I did not realize all the different shapes. I think I just am in shock at how many there are because I have never really thought about it before. I realized that polygons were any shape with 4 sides, but I did not realize how many different polygons there were. Hope, how many different shapes were you able to come up with? Were you as shocked as I was at the many different shapes you could make?

Hope, were you able to fill out the chart for problem C6? I know that triangles make 180 degrees; however, I cannot figure out how to find the degrees of shapes with another number of sides. It seems to just have completely slipped my mind.

Further Discussion:

Geometry is everywhere around us in the world. We can see it in our homes, furniture, buildings, etc. I think you can see this used the most when looking at the structure of how things are made. We can see many triangles and many squares. It is important to understand how these shapes work to make sure we use them correctly in the world around us.

# Module 7

Key Ideas in Geometry:

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

When I first read this question, I was not exactly sure how to answer. I began by trying to remember what geometry was. I thought back to my high school class and my geometry teacher that I loved. I had struggled to pass algebra, but passed this course with an A. I was able to understand the shapes and equations so much better in geometry for some reason. I really felt as if I could have related how geometry was related to the other math subjects, especially algebra, I would have done much better. Because of this, I would want to work on how my students view geometry throughout the year. I want my students to see how geometry fits into other areas and math and in real life.

Van Hiele Levels and Polygon Properties Article:

There are four Van Hiele levels. Level 0 is visualization. In this level we think of shapes and what they look like. Students can name the shapes by how they look, but they cannot identify properties of shapes. Level 1 is analysis. In this level students are beginning to see shape properties, but cannot see how the properties of shapes go together. Level 2 is informal deduction. Students can understand some abstract definitions in this level, but the definitions should still be kept simple. Level 3 is deduction. In this level, student start to go beyond just identifying characteristics of shapes. This is usually taught at the high school level. Level 4 is rigor. In this level, students work in different geometric or axiomatic systems. This is at a college level.

I did not do so well with the interactive activity the first time; however, I completed it again and did much better. I think it just took me awhile to figure out all of the shapes. Hope, how did you do with the activity? I really enjoyed it and think it would be a great way to introduce geometry to many different grade levels. I would like to use this with my future students. I would especially like to use the activity with younger students to help them better learn shapes. Hope, how would you like to use this activity in your classroom?

I was incorrect in problem A5 from Annenberg, because I thought we were choosing one incorrect answer (I read through the directions too quickly…again). At first I could easily identify that it is not possible to have a right triangle that is also obtuse (choice D). An obtuse triangle is a triangle that has an angle larger than 90 degrees so that would not work. I was thrown off a little bit when choice C said an equilateral right triangle. At first my brain thought it made sense and was correct, but upon drawing it out I could easily see how this was not correct. Hope, did you have the same issues for this problem or were you able to know fairly quickly which choices would work or not?

I really enjoyed watching the video clip on building the towers out of toothpicks and marshmallows. It was a great example to show how strong triangles were. The groups that used triangles in their designs instead of squares had much taller and sturdier structures. I would love to use this activity with younger students. I think it would a great way for them to integrate shapes and start noticing shapes that are used in structures in the world around them.

I think that this lesson is great and with a few modifications would be perfect for elementary students. I believe that for younger students I would have the triangles already cut out for them. Then allow them to see how they fit on the geoboard and have them draw conclusions from there. Hope what do you think? Do you have any other ideas that would make the activity a little simpler?

I did remember most of the vocabulary. For instance, I knew a right triangle, acute triangle, etc.; however, I did not really remember how to explain them. Hearing the definitions in this power point helped a lot.

# Module 6:

The textbook readings for this module begins the overview of probability. I believe the easiest way to introduce students to the idea of probability would be to use the coin example given in the text (pg. 87). If you flip a coin, there are two choices that the coin can land on (heads or tails). However, the coin can only land on one of the choices at a time. Therefore the coin can either land on heads or tails one time out of two possible outcomes (1/2).

Annenberg: Probability:

For problem A5 part C, I believe that it would be possible to increase your skills at the game push penny that was used for an example. I understand that you will never be able to exactly know where your penny will land; however, in my opinion you would be able to determine how much force to use when pushing the penny which would increase your chances. Hope, do you agree or disagree that you could acquire a greater game-playing skill for this push penny?

I thought problem B2 was very interesting and brought up a misconception many people have. “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?” I believe our heads want to say that there is a small chance that the coin will come up tails because the coin has landed on heads all three times, when in reality, the coin still has a ½ chance of coming up tails.

I also thought that the tree diagram in part C was an interesting way to map your data. I don’t think that I’ve ever seen this done before. I thought it was a great way to see your outcome of tossing coins.

A Whale of a Tale Article:

Below is my example of a probability line chart.

Dice Toss:

• Ms. Kincaid wanted the students to make predictions about their experiment on the basis of mathematical probability. Discuss preconceptions that students exhibited about tossing dice even after discussing the mathematical probability. Discuss the instructional implications of dealing with these preconceptions.

-Ms. Kincaid started the video by asking if students remembered what mathematical probability was. After she made sure the students understood and had no misconceptions, she asked students what were some possible outcomes they could have if they rolled 2 dice. Ms. Kincaid let students raise their hands and tell her numbers as she wrote the numbers down on the blackboard. Once the students had all of the possible outcomes written down, they discussed different ways they could roll the dice. The students write down all the possible outcomes that could lead to the numbers they had already written down. This helps the students become very aware and understanding of the experiment.

• Were these students too young to discuss mathematical probability? What evidence did you observe that leads you to believe that students did or did not grasp the difference between mathematical probability and experimental probability? At what age should probability be discussed?

-The students were not too young to discuss mathematical probability. They were very smart and completely understood what was happening to their experiment. I loved the comment one student made after observing the data collected that 7 would be the most likely outcome “because it has the most ways you can make it.” I believe that it is important to start discussing data with every age, even in kindergarten. Obviously you will not be able to go very far with the topic, but all children can really catch on the things that are more likely than others.

• The teacher asked the students, “What can you say about the data we collected as a group?” and “What can you say mathematically?” How did the phrasing of these two questions affect the students’ reasoning?

-The phrasings changed the student’s answers by a great deal. The first student that was asked to say something about the data as a group described how the graph reminded him of a rocket. When the students answered what he could say about the data mathematically, his answer changed to 7 would be more likely based off of the chart.

• Why did Ms. Kincaid require each group of students to roll the dice thirty-six times? What are the advantages and disadvantages of rolling this number of times?

-I am honestly not sure why the number 36 was picked for the number of rolls. (Hope did you catch why?) I think that maybe the number corresponded with the groups and added to a nice even number that would be easy to view.

• Comment on the collaboration among the students as they conducted the experiment. Give evidence that students either worked together as a group or worked as individual.

-The students worked together in groups based on their seating arrangements. One person was in charge of the dice, one person was in charge of recording, etc.

• Why do you think Ms. Kincaid assigned roles to each group member? What effect did this practice have on the students? How does assigning roles facilitate collaboration among the group members?

-I believe Ms. Kincaid assigned roles to each group member to help keep the experiment organized and help each student pay attention to detail on their specific job. By students having a certain job, they will be required to keep track of their job, but also to explain their information to their group members which helps with communication skills and working together.

• Describe the types of questions that Ms. Kincaid asked the students in the individual groups. How did this questioning further student understanding and learning?

-I liked that Ms. Kincaid asked students questions to figure out the possible outcomes and which number they believed would be more likely before they completed the experiment. This really helped the students to understand what would be happening during the experiment and let them test whether or not their predictions were correct.

• Why did Ms. Kincaid let each group decide how to record the data rather than giving groups a recording sheet that was already organized? When would it be appropriate to give students an organized recording sheet? Discuss the advantages and disadvantages of allowing students to create their own recording plans.

-I believe that Ms. Kincaid let each group decided how to record their own data so she could see the different ways students knew how to record data already. I believe it would be more appropriate to give students an organized recording sheet when they are older and collecting data on their own and not in groups. I think it was an advantage to have students make their own because you could really see how they were thinking and quickly observe the students that might need more help on recording data. One of the disadvantages would be that the student’s data could become very unorganized and hard to read.

For further consideration….

Knowing what you now know about probability concepts in the elementary school, how will you ensure that your students have the background to be successful with these concepts in middle school?

-I will begin introducing probability with fun activities, such as the dice roll video, and making sure students have a good understanding of things that are more or less likely to happen.

# Module 5:

Box and Whiskers Plot:

1. Make a box and whisker plot for the data in your class and draw it under the German class’s plot using the same scale.

1. Suggest three good questions that you could ask your class in making comparisons between the two plots. Answer each of your questions.
2. Based on the box and whiskers plots, what are the minimum and maximum values of each class? What does this tell us?

-In the German class, the min. value is around 30 and the max. value is around 90. In our class, the min. value is 32 and the max. value is 114. This shows us that the range of our class is higher than the German class.

1. What are the main differences you can tell between the box and whiskers plots?

-The box on the German class starts around 41 or 42 and ends around 88-89 and the median appears to be around 53-54. In our class, the box starts at 57 and ends at 90 with a median of 83.

1. Why does our class have a higher range than the German class when they have 42 students and we only have 18?

-I am personally not sure why our class has a larger range when the German class has considerably more students. The only reason I could think of is maybe the German class emptied the same trash (like the kitchen trash) each night and since everyone was measuring the same room the results were similar. Maybe our class used different room’s trash to measure? Hope, do you have any ideas on this?

Also, Hope I was wondering if you knew what the second line through the box (after the median) on the German box and whiskers represents? I wasn’t sure of this.

Common Core Standards:

Respond to the following questions/ statements…

1. Write down two “first impressions” you have about the standards.

-One of the first impressions I had about the standards was that they were really detailed. However, when looking further, I could see that they were very broad. Each standard was an end goal and it is our job as teachers to build objectives off of these standards.

Also, I became a little overwhelmed when first looking at just how many standards are in each grade. I started looking at third grade standards since this is the grade I am working with for my field experience. I couldn’t believe all the standard and wondered how we were supposed to fit them all into a year.

1. How do the concepts progress through the grades?

– I believe that the concepts pretty much stay the same through the grades, they just become more detailed and difficult. I provided an example of this in the question below. (I believe I answered this question correctly. I thought it was very similar to the next. Hope am I missing the meaning to this question? I feel like I should have had a better answer.)

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

– One of the things I noticed from looking at different grades was that a lot of the standards were similar; however, they just became a little more detailed and more difficult. For example, Common Core Standard 2.OA.1 (2nd grade) is to use addition and subtraction within 100 to solve one-and two-step word problems. Common Core Standard 3.OA.3 (3rd grade) is to use multiplication and division within 100 to solve word problems.

1. Does the Common Core Standards align with what NCTM states students should be able to know and do within the different grade level bands? (Note that NCTM is structured in grade level bands versus individual grade levels.)

-The common core standards do align fairly well with what NCTM states students should be able to know and do. The main difference is that the common core standards really focus on what they feel each individual grade should be able to complete.

1. 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 really focused on looking at the K-2 section. The Common Core did a good job of splitting each grade up; however, it cut a lot out of the Kindergarten grade that the NCTM standards say they should be able to complete. For example, Common Core Measurement and Data K.MD 3. States that students should classify objects into given categories; count the number of the objects in each category and sort the categories by count (limit category counts to be less than 10). However, the NCTM standards say that this same grade should be able to pose questions and gather data about themselves and their surrounding; sort and classify objects according to their attributes and organize data about the objects; and represent data using concrete objects, pictures, and graphs.

Curriculum Resources:

1. When using this activity, what mathematical ideas would you want your students to work through?

-I would want my students to be coming up with different ways to collect and represent their data. The assignment is to ask each student in the class whether they would rather be an eagle or a whale. The students need to think of the best way to collect their data from the class, then think of the best way to represent the data they collected.

1. How would you work to bring that mathematics out?

-I would work to bring this out by having a class discussion before having students collect data. I believe this will help their ideas start flowing and help them discover new ways of collecting data they might not have known before.

1. How would you modify the lesson to make it more accessible or more challenging for your students?

-I believe this is a very simple lesson that I could easily make easier or harder. I could make it easier by already having pictures of eagles and whales cut out and having student use these to make a pictograph with their results. I could make this lesson more challenging by having students graph their data without using pictures.

1. What questions might you ask the students as you watch them work?

-I would like to have students paired for this activity (they would each complete their own information, but have someone to discuss information with and to ask questions). I would ask them to look at their partner’s data and graph. Do they look the same as yours? Why or why not? Did you both represent your data the same way? If you both chose different ways, which works best?

1. What might you learn about their understanding by listening to them or by observing them?

-I believe with this activity you could tell whether they are collecting data correctly by observing their work. It is pretty self-explanatory. They are all collecting the same data, so their representations should be similar. If one is way off then they are obviously missing some concept.

1. How do the concepts taught in this lesson align to the Common Core Standards?

-This lesson aligns with the Common Core Standard 1.MD Represent and Interpret data. 4. Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

# Module 4:

Module 4:

Okay, let me just start by saying how much my mind was blown from watching the beginning video. I really do not think I ever considered it possible to find the mean without adding everything and dividing. When watching the teachers level all of the snap cubes, I honestly was lost. I had no clue where they were going. After the teachers leveled all of the snap cubes into stacks of 5, they had 10 left over and I was still confused! Once they explained that you would put 10 over 17 because there was 10 cubes left over and 17 people recorded their answers I FINALLY got it! The mean would be 5 and 10/17. I am still somewhat blown away that you can find the mean of data a different way than adding and dividing.

For my first question, I chose A2.

Problem A2
Create a second allocation of the 45 coins into 9 stacks.

 Record the number of coins in each of your 9 stacks, and determine the mean for this new allocation. Why is the mean of this allocation equal to the mean of the first allocation? Describe two things that you could do to this allocation that would change the mean number of coins in the stacks.

I chose this question just because it really caught my attention. At first, I changed the number of coins in my 9 stacks and continued to calculate the mean. When I saw that the mean was 5 as well I honestly thought to myself that was an odd chance, until I saw part b. Then I got to thinking, the mean is the same because there is the same number of coins and the same number of stacks. Just because the stacks looked completely different did not mean they would have a different mean. We would have to change the number of stacks or coins to create a different mean. I think this would be important to show students so they can see how and what effects the mean.

Also, I am confused about Part D. I understand how to get the deviations from the mean and I was answering the questions correctly; however, I’m not really sure what purpose this is used for. I don’t understand what the deviations from the mean tell us about our data.

Generating Meaning Article:

From what I remember, my time in the elementary classroom was completely different than the students in the article. I remember being shown how to find the mean, median, mode, and range, and then being handed worksheets. We were just going through the procedure. I loved how the teacher in the article really engaged the students in the lesson. Even from the beginning, she gave them clues so they could discover how to find the range. Once the students figured out that the range was the distance between the largest and smallest numbers in the set they were so excited! They had figured everything out on their own and were so proud of their work. I believe that this is a much better approach to teaching math. It really gets the students involved. I think it gives them a much better understanding of the concepts as well, instead of just memorizing procedures.

Working with the Mean-Activity:

This is my first representation of the data with my cubes. I was completely confused on how to figure out what the other 2 bags of peanuts needed to be without adding up the five bags we already knew. I thought about it and then tried to complete this with one of the examples I watched in the Annenberg videos.

I took all of the cubes and make the columns as even as I could. By making all of the columns an even 5, I only had 1 left over and 8 columns. I wasn’t really sure where to go from there because I knew I should only have 7 columns because there should only be seven bags of peanuts. Then I decided to split up the cubes from the set again, but this time to make sure I did not go over 7 columns. When I did this, I got 6 columns with 6 cubes each and 1 column with only 5 cubes. My conclusion after working with these cubes was that there should have to be around 5 or 6 cubes in the next 2 bags to give us 7 bags of peanuts with a mean of 8.

After completely going crazy working with the cubes and trying to find the mean without adding up the total values, I made my line plot graph, and totaled the values to see if I was close my conclusion from using the cubes.

First I plotted bags 1-5 and totaled the amount, giving me 41 peanuts. I knew that I needed 7 bags total and that the mean needed to be 8, so I multiplied 7 by 8 giving me 56. I knew that 56 divided by 7 would give me the mean of 8 I needed, so I subtracted the 41 peanuts I already knew I had from 56 giving me the answer that I needed a total of 15 more peanuts in 2 more bags. Because of this, I knew I needed 7 peanuts in one bag and 8 in another. This model helps represent the exact number of peanuts in each bag and gives and easy way to figure out what the mean will exactly be, versus my guesses with the cubes. For the cubes I did not count the total. Because of this I estimated that you would need between 5 and 6 peanuts in the next 2 bags when really we needed 7 and 8 peanuts in the bags. The mean shows us the amount that we could put in each bag if we were dividing them evenly. For example, if we put 8 peanuts in the 7 bags, it would give us the same total of 56. When we divide that by 7 we get our mean of 8.

How Much Taller Video and Mean Case Studies Summary:

It was very interesting to watch the How Much Taller video. I thought that all of the students did a great job drawing conclusions from their data. They used specific number answers to compare and contrast and even came to the point of where they felt they needed to find more data to answer the questions more specifically.