Angles Case Study

Chad: I think that Chad understands that there are different size angles. I would like to draw some pictures with him and see if he means the lines are long and short (lines in regards to sides) or if when he says long and short he is talking about the width of the angle. We need to determine what he is referring to before we can help him proceed with his definition.

Cindy: I think that Cindy’s definition was the most vague out of all the student examples. I think that she is referring to the sides instead of angles. I would want her to show me what part she is talking about. I think having her define side and make sure that she is not thinking about that instead of angle would be helpful.

Nancy: Nancy gave a good definition. I think it would be helpful to have her draw an example of an obtuse and an acute angle to see if she really understands the difference. Also, I would like to ask her what she thinks a right angle is.

Crissy: Chrissy gives the most detail out of the students. I would like to see her prove that an angle is obtuse and/or acute. We would want to make sure that she understands the correlations that a 90 degree angle, which she speaks of, has in regards to these other two angles. 

Measurement Case Study

In Barbara’s case study her students know that when you are measuring the length of something that you must start at one edge and measure straight across to the other edge. I think that they understand that your units need to touch and that you should not leave space in between. However, they seem to think that you can use different types of units as long as you get from one edge to another. For example, they are trying to use books and baskets at one time.

Often time’s students get confused and think that bigger numbers in measurement mean bigger units. In the Rosemarie and Dolores case studies they are working through this misconception to help student’s understand why that is not correct.

In the Dolores’ case, the students were comparing a measurement at home with their parents. Students were identifying the correlation between parents and students that had about the same size feet and this caused their measurements to be close. They also noticed that one of the biggest measurements that differed came from one of the smallest students in the class. They understood why that was possible because they knew he had a small foot and that would be a big difference from his father’s foot.  

At the end of the study Les finally gathers his thoughts on this topic and proudly states “big hands and feet get low numbers and little hands and feet get high number because little things take up less room.” This is a great definition that proves to his teacher that he has established a firm understanding between a unit size and the measurement that it results in.

In the Rosemarie study, Miriam also displays knowledge of the correlation between unit size and the overall measurement. She states “I think Gita has the biggest foot because it took fewer of her foot lengths to measure the doorway. The smaller your foot is the more feet you need to measure something.” While her classmate Courtney display the misconception of the bigger the unit the bigger the measurement number. She states “I just looked at the numbers, and Miriam has the highest number of foot-lengths, so I think that her foot is the biggest.”

In Mabel’s case study the students are focusing on ways to use a ruler to measure something that is longer than a ruler. They are working hard to make sure that they do not add any extra space when moving the ruler from one spot to the next. They determine that it is easier to use two rulers because then you do not waste space in between. In Jose’s case study they have moved on and are working to make sure that they start measuring at the correct spot on the standard measuring tool. They see that you must start at the correct spot and end at the correct spot, while taking in to consideration any extra space on the measuring tool. If you do not do it correctly each time your measurements will not be the same.

Barbara’s students are going to have to understand the correlation between the size of the unit and how that effects the overall measurement. They will then work to make sure that when they are using a tool if they do not have enough to go across the entire length, they will have to figure a way to get the entire measurement without leaving spaces. They will also have to learn how to properly use standard measurements. It was interesting to see this alignment across grade levels. 

Ordering Rectangles

1.       Smallest to Largest Perimeters: D, E, G, B, A, C, F

2.       Smallest to Largest Area: C, D, E, B, F, A, G

3.       Perimeter by Measurement: D, B, G, A, E, C, F

I did a good job getting the smallest and the biggest, but I did not do to good at getting the ones in between. I don’t know that I had a good strategy. I think I just kind of eye balled it if you will and guessed.

4.       Area by Measurement: C, D, B, A,E, F, G

Again I did well with the smallest and the biggest, but I did not do well with the ones in between.

5.       It seems that a lot of them were similar in height and that caused the perimeters to be close, but did not affect the areas as much. I am not sure the best way to gauge them without measuring. I did not have a ruler at my house, so I used two colored squares that I cut off the end of Post-Its that you would use to bookmark a page.  

Mean Case Study

Trudy says : Putting it on one of the 4 foot 8’s. Because then you could even all the heights out. And you might get close to the same answer.

She is understanding that if you spread all the numbers out to make a height that is the same for every person that would be your average. So she is using the numbers she has and adding or subtracting a few to each one until they are all the same.

Celica says: I think so because, like, if all the fourth graders stood on each other’s shoulders, and the third graders did that too, I think that the fourth graders would be higher.

She is understanding that you need to add up all the heights to see which group is taller, but is not thinking to figure out an average for each person. She is going in the right direction, but is not quite there yet.

Laural seems to have an understanding of how to arrive at the mean, because she states “we got the average of 13.2, but we rounded it down to 13 because we can’t have a point something.

She knows how to calculate the mean, but does not fully understand what it means. She is just getting rid of the .2 and this tells us that she needs to continue working on a deeper understanding of what the mean stands for. 

Space and Shape

I thought that the shadow activity was the hardest. I thought that it was interesting to see the shadows that it casted. I was not good at visualizing all the different ways that it could make a shadow. I did not think that it would make a hexagon.  

It is important that students become proficient in spatial visualization because it helps students when they are reading a map, giving directions, and packing efficiently. Last year I went to Charlotte to the distribution warehouse where they pack up Christmas shoeboxes to be sent around the world. A man from my church and I were in charge of putting the shoeboxes into bigger boxes and it became a competition of who could pack the most in one box. His spatial reasoning was clearly sharper than mine! As we continued on I did slowly get better and figured our strategies to use to make sure that I was packing in as many boxes as possible.

I think that students can begin working on their spatial reasoning skills as early as kindergarten. When I was teaching kindergarten my student enjoyed doing pattern block puzzles without the shape lines on them. They would have to figure out which shapes would fit where. I think that they were starting to practice spatial skills through this type of activity.  

I like the pentominos activity that we did the other day and I could see how this would help students with their spatial reasoning skills. I also think that nets that we discussed in the webex meeting would be a great way to practice as well. I was not very good at these activities, but I do think that I am better at them now than I was when I was in school. I did well on the first two activities “Trip on a Train” and Plot Plans and Silhouettes.” Providing student’s time to work through activities like this I think would be very beneficial. 

Geometric Definitions Case Studies

Susannah knows that triangles have 3 sides and 3 vertices. I am thinking though that in her head she visualizes what the standard triangle looks like. If a triangle fits or is close to her vision she considers it a “real” triangle. If is too different than she places it in the triangle family, but it is not a “real” triangle. She makes clear that her definition of “real” triangles are similar to triangle B. Near the end of the discussion she is proposing the idea of turning the triangle to see the sides in different ways, leading another student to understand that no matter which way you turn a triangle it is still a triangle.

In Mrs. Rivera’s class the students are working on developing definitions of a square and a rectangle. They are using square as part of their rectangle definition. They state “two squares make a rectangle.” They have not yet figured out the distinguishing quality between a rectangle and a square. At the end Brett is becoming close because he is turning an orange square pattern block around and around in his hand. He is thinking through what makes that shape different from a rectangle. Roberto’s definition makes it clear that he is at a level one on the Van Hiele Model. He is using properties to describe a rectangle. The students are describing the properties that are visual and describe a rectangle, but they are not considering what makes it different from a square. They are engaging in productive talk though that is leading them closer and closer to discovering the difference between the two shapes. Once they can see that I think it will be easier for them to come to a clear definition of both shapes.

I was especially intrigued by Zachary’s comments as it seems to be very insightful to me. He seems to really understand the properties of a triangle and is articulating this thoughts very well. I think that his point is very valid and reflects the thinking of many of the students in that class and many other classes as well. I like how he is trying to a conscience effort to make sure his thoughts coincide when he says “we just have to make our eyes and our heads meet.” In Dolores’s study the students are making it clear that they understand what a triangle is, but because they are not use to seeing it that way it is much harder to classify irregular triangles, as triangles.

The purpose of a definition is to give us something to go by when trying to classify shapes correctly. It is gives us an understanding of if a shape meets the classifications to be called by a certain name. 

Geometry 

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

In geometry for the upcoming year a big feature that we work on in first grade is shape names and spatial reasoning skills. We want students to understand the attributes of shapes, how to compose shapes (2D & 3D), and how to partition shapes. We often use manipulatives, specifically, pattern blocks and tangrams to help students become proficient in the geometry strands.