Lesson Study Final Report
Mathematics Lesson Plan
Sixth Grade
Comparing Polyhedrons
Prepared by:
Lisa Armstrong
Jose E. Carrillo
Margaret Kesler
Connie Perez
Amanda Sanchez
April 25, 2007
Central Elementary School
Santa Clara, New Mexico
Lesson Study Final Report
Central Elementary School
Santa Clara, NM
Fourth and fifth grade teachers at Central Elementary School in Santa Clara, NM were introduced to Lesson Study through the TIA-RETA Program in the fall of 2003. In conjunction with NMSU and WNMU professional development was provided to this group of teachers integrating technology and the lesson study concepts. At that time the focus of the research lessons was science due to low achievement test scores. Central Elementary School was afforded the opportunity to take Lesson Study as a class, receiving three university credit hours, as well as technology equipment to use in the classroom.
In the fall of 2004, lesson study became the school-wide professional development plan. Science continued to be the area of focus for the research lessons. Participation in lesson study increased because of the incentives that teachers received through the TIA-RETA Program and through the encouragement of the building administrator. The lesson study process was beginning to be realized by participants as a valuable teacher-led professional development process.
For the following two school years, the subject area of the research lessons changed from science to math. The staff decided to continue with lesson study although the program was not fully funded. The technology incentives that the participants received in the past were no longer available. The intrinsic value of the lesson study process was appreciated by participants. It was realized that lesson study impacts teaching and learning.
In October of 2006 Central Elementary School received information about a lesson study project through New Mexico State University. The building administrator and four teachers felt it was a great opportunity to receive professional development from an expert on lesson study, Dr. Akihiko Takahashi. The lesson study group goal was developed and emphasized during the development of the lesson. The goal was, “Students will actively construct, utilize, and communicate mathematical concepts.” During the lesson the students connected their prior knowledge to the new knowledge presented. They utilized this new information to solve problems and use mathematical concepts. The students also communicated orally and in written form their knowledge of polyhedrons and their attributes. The following is a synopsis of the process of the research lesson project.
Central Elementary Lesson Study Group Activities 2006-2007
October 2006-April 2007; regular meetings and public research lessons
· 1st Lesson Study Conference, October 27-28, 2006, 8:30-3:30 at NMSU, Las Cruces, NM
· 1st Meeting, October 30, 2006, 2:00-3:00
· 2nd Meeting, November 13, 2006, 2:00-3:00
· 3rd Meeting, November 14, 2006, 2:00-3:00
· 2nd Lesson Study Conference, November 17-18, 2006, 8:30-3:30 at NMSU, Las Cruces, NM
· 4th Meeting, November 20, 2006, 2:00-3:00
· 5th Meeting, November 29, 2006, 2:00-3:00
· 6th Meeting, December 11, 2006, 2:00-3:00
· 7th Meeting, December 21, 2006, 2:00-3:00
· 8th Meeting, January 8, 2007, 2:00-3:00
· 9th Meeting, January 9, 2007, 2:00-3:00
· 10th Meeting, January 11, 2007, 8:00-3:30
· 11th Meeting, January 12, 2007, 8:00-11:30
· 12th Meeting, January 16, 2007, 2:00-3:00
· 13th Meeting, January 17, 2007, 2:00-3:00
· 1st and 2nd Public Lessons and Debriefings, January 18, 2007, Las Cruces High School, Vista Middle School, Las Cruces, NM 8:30-3:30
· 14th Meeting, January 23, 2007, 2:00-3:00
· 15th Meeting, January 30, 2007, 2:00-3:00
· Research Lesson and Debriefing, February 1, 2007, Central Elementary School
· 16th Meeting, April 13, 2007, 2:00-3:00
· 17th Meeting, April 16, 2007, 2:00-3:00
· 18th Meeting, April 18, 2007, 8:00-11:30
· 19th Meeting, April 19, 2007, 3:00-5:00
· 20th Meeting, April 20, 2007, 12:00-3:30
· 21th Meeting, April 23, 2007, 2:00-4:00
6th Grade Public Research Lesson on February 1, 2007,
At Central Elementary School
Mathematics Lesson Plan for Sixth Grade
For the Lesson on Friday, January 19, 2007
At Central Elementary School, Santa Clara, New Mexico
Lesson Study Group: Lisa Armstrong, Jose Carrillo,
Margaret Kesler, Connie Perez, Amanda Sanchez
1. Title of the Lesson: Comparing Polyhedrons
2. Goals:
a. To identify solid figures
b. To name and count the faces, edges, and vertices of prisms and pyramids
c. To learn how different views of a solid figure compare
d. To compare and contrast polyhedrons and explain the relationships
3. Relationship of the Lesson to the New Mexico Grade-level Standards, Mathematics
4. Instruction of the Lesson
New Mexico sixth grade students are expected to make generalizations based on observed patterns and relationships. They should be able to communicate these mathematical generalizations verbally and in writing. After investigation of geometric figures, students will conceptually understand proportional relationships and develop a formula such as E=F + V – 2, where E=the number of edges of a polyhedron, F=the number of faces of a polyhedron, and V=the number of vertices of a polyhedron.
Prior to the research lesson, students have been introduced to and explored basic geometric concepts such as the study of lines, angles, and polygons; they have measured angles, and constructed congruent line segments and angles. In further exploration the students identified line and rotational symmetry. They also identified solid figures; named and counted faces, edges, and vertices of prisms and pyramids; and modeled prisms and cylinders from nets.
If students comprehend the numerical relationship between the faces, edges, and vertices of polyhedrons and their algebraic connections, they can incorporate them to find a formula such as E=F + V – 2. A lesson from Harcourt Brace Math Advantage, Faces, Edges, and Vertices Algebra Connection, was chosen and revised. The lesson was extended to help students realize the relationships between the faces, edges, and vertices of polyhedrons. Students will use models of polyhedrons previously built to explore the physical attributes of the solid figures. They will develop a method of communication to present their observations.
5. Lesson Procedure
Learning ActivitiesTeacher’s Questions and Expected Students’ Responses / Teacher’s Support / Points of Evaluation
1. Introduction
We explored polygons and polyhedrons, we’ve built models of polyhedrons, and we’ve compared and classified pictures of polyhedrons. Now we will compare our models of polyhedrons and your group will present information about them. Use the available materials to visually present your findings to the class.
Ask the question: “What do we need to know to compare the polyhedrons?”
Now you will compare the polyhedrons and use the available materials to record your groups’ findings.
Students will present their findings. / Review prior knowledge; pose the problem to the class in written format on the overhead projector.
Provide examples of polyhedrons (see appendix).
Write students’ responses on the board.
Provide the materials for the students to use during this portion of the lesson.
Call students to the front of the class one group at a time. / Do the students know the vocabulary?
Can the students apply their knowledge to describe polyhedrons?
Can students describe attributes of polyhedrons?
Are students discussing and recording their observations? Are they working collaboratively?
Are students appropriately using vocabulary in context?
2. Posing Problem
Students’ anticipated responses:
· Organize information according to polyhedron attributes such as number of edges, faces, and vertices
· Derive the formula for the relationship between the number of edges, faces, and vertices of any polyhedron, E=F + V – 2. / If students cannot graphically organize their results according to attributes, recall prior examples of graphic organizers. / Do students recall methods of presenting information?
4. Summing up
a. Using the students’ representations of polyhedron comparisons review what students learned through their explorations.
b. Ask students to write a summary explaining the relationship between polyhedrons. / Ask open ended questions to keep the discussion going.
Ask the question “Is there a shorter way to express this relationship?”
“When letters are used to represent numbers, what is it called?” / Can students explain and represent their comparison of polyhedrons?
Do the students use variables to represent an algebraic expression?
Do students appropriately use variables in their summaries?
6. Evaluation
a. As a small group were the students able to organize and present their comparisons of polyhedrons?
b. As a whole group were students able to compile information and explain the relationship between polyhedrons?
c. Independently were students able to explain their reasoning in written form?
Post Discussion for Research Lesson-Comparing Polyhedrons
Central Elementary School
Group Participants: Lisa Armstrong, Jose Carrillo, Margaret Kesler, Connie Perez, Amanda Sanchez
Guest Observers: Daena Davis, Sonia Marrujo, Betsy Montes
Lisa Armstrong: The students understood faces, edges, and vertices of polyhedrons. They discussed this but didn’t make the connection between them. There was a lot of thinking, they used good organizational skills, but once they had to present their findings they weren’t able to communicate their thinking because they were shy and learning stops. They’re thinking it, but we didn’t know what they were thinking.
Daena Davis: Do you think that if they had stayed in their work areas they may have been more at ease, rather than being at the front of the room. I find that students are more comfortable presenting from where they are in the room.
Amanda Sanchez: Also maybe giving each group only one shape to discuss and present on would help them to compare with other groups and have more discussion.
Connie Perez: I think a lot of learning took place in working with all of the shapes. If you only focus on one then you can’t look at relationships.
Amanda Sanchez: The goal was the numbers part – in finding the relationship between them.
Betsy Montes: The students had a lot of prior knowledge. Were the students aware of what the goal was. The goal wasn’t met. Questioning while in their groups would also stimulate more thought. Did kids know questions that were going to be asked? If they had known what was expected, then they would have been ready. They could’ve written some responses.
Margaret Kesler: In one of our lesson study sessions in Las Cruces/NMSU we talked about not asking leading questions. If you ask leading questions their creativity could be stifled and limit their discoveries. We kept that in the back of our minds when planning the lesson however if we had asked questions during the presentations to stimulate their thinking without being leading, it could’ve helped. For example, questions like “What was the discussion in your group?”, “What was brought up?”, “What did you not agree on?”
Amanda Sanchez- Perhaps a focus question that they will answer during their presentations, maybe open-ended questions?
Connie Perez-We had considered some questions during planning but decided they were leading and would tell them how to think.
Betsy Montes- Did you ask questions while they were in groups?
Lisa Armstrong: I walked around more as a facilitator.
Margaret Kesler: Maybe specific questions could have been asked to stimulate their thoughts during the presentation of their findings.
Sonia Marrujo: Was your grouping of the students random? Some wee doing some of the work or most of the work and others were not as active in the activity.
Lisa Armstrong: For the most part they chose their own groups.
Betsy Montes- They were interacting, they were learning.
Mr. Carrillo: I observed that one group, Jose, Brittney, and Zita rotated the tasks. Each one was looking at the shapes, one writing, one facilitating, one cutting-that assured that they were understanding the process. They all had an equal opportunity to share their work. There were other groups that each student only had one task.
Betsy Montes: I wondered if roles were assigned.
Connie Perez: One group continued to work during the others’ presentations, they all had different ideas about how to present the information on their board.
Amanda Sanchez: They were trying to draw them when they saw everyone was using the pictures, and they were almost done. Then they started to use the pictures and they finished really fast.
Margaret: Drawing them out might have helped some students. They really got a feel for the shapes and the attributes of them.
Amanda: That is why I think each group having one shape to focus on may have been good.
Connie Perez: It was nice that you used 3-D objects. I noticed that when one student was using the picture of the shape he needed to recount.
Amanda: I think having the different types of manipulatives, they all chose something different, they all had enough materials to work with.
Jose Carrillo: As far as evidence of student learning: one of the students was asking probing questions. She kept involving other members of the group but it was interesting that she didn’t take what she was doing at face value but asked the others in her group. She would engage them “Do you think?” “Do you think?” … As far as the different groups, most of them would go back and ask questions and check their work.
Connie: I think in the area of algebra, and considering the development of the students that perhaps the goal – finding the algebraic equation was not appropriate. Finding the pattern between the numbers would have been more appropriate at this time.
Amanda: According to the standards and benchmarks students should have been able to come up with an equation. Students should already be finding the algebra part of it. They have to come up with their own formulas from a problem. They did get a lot of those types of problems on the practice tests. They had to come up with an equation from the information given.
Daena: Have the students been exposed to formulas?
Lisa: A little bit.
Betsy: I noticed some used variables. Have you used variables?
Lisa: Yes, in that F=faces E=edges V=vertices before the lesson.
Connie: Yes, using the variables in that way. In actually using the variables in equations, I think the first step would be in finding the pattern in looking at all the polyhedrons and move from there.
Daena: You could have given them only one of them but some of the kids can’t see the pattern in one, but need to compare them to other shapes to see what is going on between them.