Sheila Green Professor Dyrness ED200: Analyzing Schools Curriculum Project December 15, 2010 Relating Math to the Real World: A Study of Platonic Solids and Tessellations
Introduction The study of Platonic solids and tessellations at the middle school level allows the students to explore, analyze and question mathematics on their own. The student will be lead to the concept of duality as he analyzes patterns which emerge. The instructor will present models of all Platonic solids, point out intrinsic similarities, and provide examples of tessellations for the students to examine. Through the creation of a unique and original tessellation, the student is involved in a new design which would support symmetry. The objectives reflect guidelines presented in the State of Connecticut Frameworks. Through this curriculum unit, the student experiences mathematics through nontraditional means, using visual and hands-on manipulation. Inductive reasoning is utilized as the student compares two separate type die. Gardner s Theory of Multiple Intelligences played a part in the inception of this unit, as varied approaches to learning are used to augment the students information processing. The curriculum involves the student with research presented in a final paper, an accurate representation of Platonic solids and tessellations, and a justification on choice of die. Context This unit is targeted for seventh graders at Hartford Magnet Middle School. I observed in my seventh grade placement that although the students frequently worked in 2
pairs or three s, they did not work on a project together. I have also witnessed that the students became enlivened and engrossed in their work when they had had brief tactile and visual occurrences. This project was chosen for its hands-on manipulation, as well as its ability to encourage critical thinking and creative design. This unit is for students enrolled at the prealgebra level. A detracked environment is appropriate, and diversity of culture is not a factor. This is a classroom of twenty students, and groups of four are formed, totaling five groups. The project will last four days, with each session having a duration of sixty minutes. The middle school years is an ideal time to involve students in this type learning. During these years, students begin to develop the cognitive structures to allow them to reason within a linear deductive system of thought. Simultaneously, they continue informally to observe the boundless activity of their environment, tapping into native curiosities and intuitions that can, if nurtured, provide the foundation for inductive discoveries and reinventions of many of our fundamental mathematical concepts. (Navigating Through Geometry in Grades 6-8, p.vii) Overall Objective Students will find coincidences and patterns that can reveal underlying structures. This will take place when working on the chart which involves counting vertices, edges, and faces. Students will experience the two highest levels of Bloom s Taxonomy, synthesis and evaluation, as they think critically and create designs of their own. This will take place through the exercise involving the dice, and the session of group creations 3
of tessellations. The students, through working in the capacity of the group, will further their experience of working cooperatively and effectively with one another. The students will also enhance their ability to master mathematical vocabulary. Objectives The group will construct representations of the tetrahedron and the cube; the group will also design a tessellation. The group will determine congruence, similarity, symmetry, and tessellation through the handling of models, which are provided by the instructor. The relationship between the duals (as taught by the instructor) will be observed and discussed by the group. A drawing will be made of one set of duals, with precision in mind. All observations and determinations concerning the solids will be recorded by the group. This objective relates to the Common Core State Standards for Mathematics, Grade 7. The Connecticut State Standards to District Curriculum, CT.9-12-3.C.2a states Apply transformations to plane figures to determine congruence, similarity, symmetry, and tessellation. The students will examine each Platonic solis by carefully counting the edges, faces and vertices. The group will compare and categorize this data by formulating a chart. Relationships between the sides and perimeters (edges), the congruency and similarity will be noted on chart. Students will compare the data (the numbers counted) among each solid, and focus on equal numbers. They must also look for relationships among the numbers. Can you devise a way of calculating the number of vertices of a regular solid 4
once you know how many faces it has, how many sides each face has, and how many faces come together at each vertex? After you have devised the method, use it to check your entries for the number of vertices for each figure. (The Heart of Mathematics, p. 305). These objectives are supported by CT State Standard to District Curriculum, CT.7.3.1.1 which states, Classify two- and three- dimensional geometric figures based on their properties including relationship of sides and angles and symmetry (line and/or rotational) and apply this information to solve problems. The group will differentiate between a cube-shaped die and a tetrahedron-shaped die, and solve the problem of which die will better enhance their chances. The group will compare and contrast by actual throw of the dice, and record their findings. This objective reflects and expands upon CT State Standard to District Curriculum, CT.7.3.1.1 which states, Classify two- and three- dimensional geometric figures based on their properties including relationship of sides and angles and symmetry (line and/or rotational) and apply this information to solve problems. The student will use his/her unique skills (ie., drawing, paper construction of solids, writing skills, spelling skills, computer skills) in the group. This will work toward receiving the optimum grade for the group. This objective is supported by Slavin s cooperative learning strategy, incorporating the Jigsaw method. A key element of Jigsaw and other task-specialization methods is the assignment of a unique subtask to each group member. (An Introduction to Cooperative Learning, p.13). 5
However, in this curriculum, students are not assigned tasks, rather they assume or head tasks as they see fit, due to their abilities. This unit involves lectures by the instructor, and each group will present one final paper (two pages) and hand-made representations of a cube, a tetrahedron, and a tessellation. There will also be an additional page turned in, justifying which die was chosen. Activities Day 1. The instructor will begin b defining five Platonic solids and present several examples of these solids made of wood, plastic, sugar cube, miniature pyramid, dice. The instructor will encourage discussion of additional examples which exist in life, perhaps even world reknown structures. The instructor will give a short (10 min.) lecture on the five solids, informing the students to research why we say, Platonic solids, and will introduce two dice into the talk. The instructor will ask the students to consider two Platonic solid die shapes. The question presented to the class is Which die would better increase your chances (or would it make a difference), a cube-shaped die of a tetrahedron-shaped die? The instructor will then divide the students into groups of four, and go over instructions about what would be expected of them in order to complete their project. They will find out that they are to carry on independent research via the computer s in the classroom, and they would be given a few suggestions for relevant websites. They would take notes on observations of the solids which they handle 6
(models) and be responsible to complete the handout which compares the number of vertices, sides, and faces of the solids. The instructor also mentions that they would be covering a set of new concepts the next day, but that they would be related to the topic. It is at this time that the class is informed that a two- page paper would be expected from each group, which would show an understanding of the topics covered represented by excellent choices of expression. The instructor informs the class that they will have the opportunity to show their design and precision skills when they construct two Platonic solids, and what is known as tessellations. The students will be told by the instructor that all these tasks would be completed within four days, and would be presented as a group, and the group would share in a common grade for the project. The groups now formed, they get started on filling out their charts which have been handed out, and they mark the edges, faces, and vertices with erasable markers which have been handed out (to avoid mistakes in counting). They also discuss a strategy in figuring out the answer to the die question, studying each die in detail (faces, roll possibilities, etc.) Day 2. In addition to the group having models that they are studying closely, there are additional models on display in the classroom to which all groups may refer. The instructor lectures on congruence, similarity, symmetry and tessellation. The instructor shows floor tile, wallpaper, and fabric designs, and wall tiles as example showing tessellation. Again, the instructor encourages the class to offer their own examples of tessellations that they notice in their environment. The instructor points out the materials which are set up in order that the group may create their own tessellations. The students are informed that each new word introduced in the unit should be made familiar through 7
its meaning and its spelling. The students would be provided a list of the key words, which would be used when writing their final paper. After the lecture (which has lasted 20 min.) the students are free to begin making their cubes, tetrahedrons, and tessellations. Materials include various colorful construction type paper, colored markers rulers for precise measurements, invisible tape, scissors. Mindful of the ancient pyramids, the students are instructed to be as precise as possible. Day3. The instructor discusses duality of: the octahedron/cube, the dodecahedron/icosahedron, and the tetrahedron/tetrahedron. The group will now focus on any two duals, and recreate on paper what they have learned about duality. Along with the drawing will be the group interpretation of how duality manifests itself (resulting from research). Again, precision is the key to the drawing. The group members will designate who will se the computer to begin search for information which the instructor provided on all topics. Those gathering research information will note their sources, as the instructor has informed them to do. Day 4. This day is used for whatever purpose the group deems necessary in order to complete their project to perfection. Perhaps they have decided to remake their solids, or change their tessellation. They may need additional justification fr answering the question on the dice, or they may need to use the computer again. Their final paper is typed and printed, and they present it along with their two solids, tessellation example, and duality rendition/writing exercise. 8
Evaluation The teaching strategy is designed to open the door for learning the unit material by the student. The student plays the greater pert in thinking critically: he has compared data, he has evaluated two die, he has group-discussed, and he has created an actual example of what he has gained knowledge, therefore applying his understanding. The student has reached the highest goals (Bloom s Taxonomy) of learning in that he has evaluated, criticized, justified, and made a choice. Rubric The group has full understanding of material covered in the unit, have presented precise renditions of two Platonic solids, tessellations, and dual solids. Final paper shows excellent for expression of information. Sound justification for the die choice expressed. 5 Points Group shows an understanding of the material covered, and two other categories show strength. But one remaining category is lacking. 4 Points Group shows an understanding of material covered, and one other category shows strength. 3 Points 9
Group shows no understanding of the material covered. Remaining categories may show strength. 2 Points Group shows no understanding of the material covered. Remaining categories show little strength. 1 Point 10
Bibliography Robert Slavin, Cooperative Learning : Theory, Research Practice 2 nd edition (Boston: Allyn and Bacon, 1995) chapters 1 and 2. Navigating Through Geometry in Grades 6-8. Reston VA: National Council of Teachers in Mathematics, 2002. Edward Burger and Michael Starbird, The Heart of Mathematics (John Wiley & Sons, 2010) chap.4.5 YouTube, Platonic Solids by Peter Wetherall Tessellations.org Bloom s Taxonomy http://www.trincoll.edu/depts/educ/resources/bloom.htm Howard Gardner s theory of Multiple Intelligences http://www.ibiblio.org/edweb/edref.mi.th.html 11
Key Words Platonic Solids Cube Tetrahedron Octahedron Dodecahedron Icosahedron Vertex (Vertices) Tessellations Congruence Similarity Symmetry Duals 12