UNIT-1
1.1 Introduction of Teaching Materials
Instructional materials are as essential for the mathematics as spices are for the chef. They are necessary extra ingredients that make teaching and learning mathematics a pleasant, satisfying experience. Models, pamphlets, films and diagram give to a mathematics lesson breadth and depth that would be difficult to obtain on any other way.
Today a successful teacher of mathematics is able to communicate ideas, build student’s curiosity, direct independent study, pose challenging question, and plan review and reinforcement experiences. Constantly, on the lookout for ways to mathematics learning meaningful and pleasant, teacher is eager to use the best instructional materials and the best strategy for presenting each concept. To do this, teacher must not only keep abreast of new instructional materials but also understand the role of these materials and how to incorporate them in his lessons.
Since mathematics is an abstract, logical science, mathematics teachers have special need for instructional materials that lend reality toideas. At the same time, they must be aware of the danger that a concrete representation may add qualities to the student conception that are not mathematical.
Man thinks in terms of tangible or visual representations sketches models or mock-up ones. He uses these devices to solve a problem, discovers a new idea, or creates a new product. These representations link thought processes and reality; they relate past experiences to a new situation. Thus they help make transitions from one idea to another.
In the same way, the teacher uses an instructional aid to communicate a new idea, and to add meaning and interest to verbal instruction. He knows which aids will add meaning to his lesson and has them at hand when he needs them. He gives careful attention to discovering what works for him and uses it effectively.
Other teachers, however, may get little benefit from instructional aid. Some use them but fail to take students from concrete representation to the concept behind it. Other uses an inappropriate aid a model that is too small, a chart that is too complex, or a film that is too elementary. The greatest offenders of all are these who do not use any kind of aid because they are unaware of the wide range of instructional materials available to them.
1.2 Meaning of Instructional Materials
aterials use to the teaching activities by teachers and students is called teaching materials. The materials not only make mathematics attractive and effective but also by using the teaching materials, students can be skilled in the related subjects. By its application students become eager to the teaching and develop the free teaching and concepts.
1.3 Definition of the Teaching Materials
According to the Dictionary of Education ; Instructional Materials means any device with certain function that is used for teaching process, including books, textbooks, supplementary reading materials, audio-visual and other sensory materials scripts for radio and television, instructional programmed for computer managed instruction or manipulation (good and Hills 1897) this definition reveals that those materials which help in instructional process either they area audio or visual are the instructional materials.
Different educationist gives definition of instructional materials in different ways; some of them are listed below:
· Visual instruction simply means the presentation of knowledge to be gained through the “seeing experience”.
-Dorris
· Visual education is a method of imparting information which is based upon the psychological principle that one has a better conception of the thing he sees than of the thing he sees about or hears discussed. -Roberts
· All materials used in the classroom, or in other teaching situations, to facilitate the understanding of the written or spoken word.
-Dent
1.4 Importance of the Teaching Materials
To make the mathematics education effective and meaningful teaching materials plays the vital role. By applying the teaching materials, we can make the abstract concepts, facts and actions concrete that helps student to comprehend the lessons. The concepts learned by using the teaching materials are deep rooted in the student’s mind. Such concepts are permanent and meaningful than the concepts learned without using teaching materials.
Materials used in teaching learning process, are called instructional materials. They help the students for learning mathematical fact, skill, theory and concept, to make the teaching interesting and effective, the teacher should use instructional of material according to nature of the course, The instruction of technology allows wider distribution of audio visual materials and the use of information technology to introduce new knowledge, teaching skills and evaluate learners progress holds out great promise technology can help to bridge the gap between industrialized and non industrialized counties.
According to Piaget, to give abstract knowledge about mathematics to the child we should use concrete and physical object. There are many abstract mathematical concepts, which cannot be taught meaningful without use of instructional materials. We should relate these mathematical concepts skills and facts to the mental power of child with the leaf of the instructional materials. Some importance of instructional materials in mathematics teaching is as follows.
· To understand abstract mathematics concepts.
· To facilitate learning simple
· To give concepts about geometric shapes to students
· To clear mathematics ideas to the students.
· They provide interest in the study of the subject.
· They promote functional knowledge
· They supplement classroom lessons
· They develop a sense of reality and visionless
· They provide a kind of convenient and motivation environment
· They arouse curiosity among the students
· They provided opportunity for useful mental experience (right
things and imagining, comparing, analyzing and drawing inferences.)
· Enhancing the learning environment for all students.
1.5 Function and use of teaching materials
There have been several fine lists summarizing uses and functions of teaching aids. Many such lists apply specifically to manipulate materials. Among the most common uses of manipulative materials are the following;
1. To vary instruction activities.
2. To provide experiences in actual problem solving situations.
3. To prove concrete representations of abstracts ideas. To provide a basis for analyzing sensory data, so necessary in concept formation.
4. To provide an opportunity for students to discover relationships and formulate generalization
5. To provide active participation by pupils.
6. To provide for individual differences.
7. To increase motivation related, not to a single mathematics topic, but to learning in general.
1.6 Types of instructional materials
Mathematics teaching aids, which number in hundreds, generally falls into three broad categories:
Literature
A large selection of readable and interesting books, pamphlets, references, journals, curriculum materials and others in mathematics are available for students and teachers. These materials add enrichment, broaden the mathematical background of students, and stimulate curiosity in new ideas. Internet also provides varieties of literatures in mathematics.
Audiovisual Aids
Mathematics teachers can make good use of films, filmstrips, power point, videotapes and overhead projectors as well as traditional equipments such as flannel boards and chalkboard. The overhead projector has become increasingly popular. With it, the prepared transparencies, or use a variety of the commercial transparencies now available. To facilitate accurate board walking, the teacher can use stencils for chalkboard drawings and graphs. These audiovisual aids add a new dimension to learning.
Models and Manipulative Materials
Several types of charts, models and manipulative materials are available for mathematics teaching .Some is demonstration aids. Others are laboratory manipulation. Many of the manipulative materials for teaching mathematics at school levels can be made locally by students and teachers for classroom use.
1.7 Criteria of Selecting Instructional Materials
1. Pedagogical criteria for selection of teaching materials
-The material should provide a true embodiment of the mathematical concept or ideas being explored.
-The material should clearly represent the mathematical concept.
-The materials should be motivating.
-The materials should be multipurpose if possible.
-The materials should provide a basis for abstraction
-The materials should provide for individual manipulation.
2. Physical criteria for selection of manipulative materials
-Durability
-Attractiveness
-Simplicity
-Appropriate size
-Low cost
The cube is the Platonic solid (also called the regular hexahedron). It is composed of six square faces that meet each other at right angles and has eight vertices and 12 edges. It is also the uniform polyhedron and Wenninger model . It is described by the schlafli symbol and Wythoff symbol .
The cube is illustrated above, together with a wireframe version and a net (top figures). The bottom figures show three symmetric projections of the cube.
There are a total of 11 distinct nets for the cube ( Turney 1989-85, Buekenhout and Parker 1998, Malkevitch),of the cube can be addressed using the polya enumeration theorem.
A cube with unit edge lengths is called a unit cube. The surface area and volume of a cube with edge length are .Because the volume of a cube of edge length is given by, a number of the form is called a cubic number (or sometimes simply “a cube’). Similarly, the operation of taking a number to the third power is called cubing.
The cube has a dihedral angle of
In terms of the inradius
of a cube, its surface area 
and volume 
are given bySo the volume, in-radius, and surface area are related by
2.2 History of Cube
Mathematical puzzles vary from the simple to deep problems which are still unsolved. The whole history of mathematics is interwoven with mathematical games which have led to the study of many areas of mathematics. Number games, geometrical puzzles, network problems and combinatorial problems are among the best known types of puzzles.
The Rhind papyrus shows that early Egyptian mathematics was largely based on puzzle type problems. For example the papyrus, written in around 1850 BC, contains a rather familiar type of puzzle.
Seven houses contain seven cats. Each cat kills seven mice. Each mouse had eaten seven ears of grain. Each ear of grain would have produced seven hekats of wheat. What is the total of all of these?
Similar problems appear in Fibonacci's Liber Abaci written in 1202 and the familiar St Ives Riddle of the 18th Century based on the same idea (and on the number 7).
Greek mathematics produced many classic puzzles. Perhaps the most famous are from Archimedes in his book The Sandreckoner where he gives the Cattle Problem.
If thou art diligent and wise, O Stranger, compute the number of cattle of the Sun...
In some interpretations of the problem the number of cattle turns out to be a number with 206545 digits!Archimedes also invented a division of a square into 14 pieces leading to a game similar to Tangrams involving making figures from the 14 pieces. Tangrams are of Chinese origin and require little mathematical skill. It is interesting however to see how many convex figures you can make from the 7 tangram pieces. Note again the number 7 which seems to have been associated with magical properties. They were to have a new popularity when Dodgson, writing as Lewis Carroll, introduced Alice type characters.
Fibonacci, already mentioned above, is famed for his invention of the sequence 1, 1, 2, 3, 5, 8, 13, ... where each number is the sum of the previous two. In fact a wealth of mathematics has arisen from this sequence and today a Journal is devoted to topics related to the sequence. Here is the famous Rabbit Problem.
A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begins a new pair which from the second month on becomes productive?
Fibonacci writes out the first 13 terms of the sequence but does not give the recurrence relation which generates it. One of the earliest mentions of Chess in puzzles is by the Arabic mathematician Ibn Kallikan who, in 1256, poses the problem of the grains of wheat, 1 on the first square of the chess board, 2 on the second, 4 on the third, 8 on the fourth etc. One of the earliest problem involving chess pieces is due to Guarini di Forli who in 1512 asked how two white and two black knights could be interchanged if they are placed at the corners of a 3 × 3 board (using normal knight's moves).
Magic squares involve using all the numbers 1, 2, 3, ..., n2 to fill the squares of an n × n board so that each row, each column and both main diagonals sum to the same number. They are claimed to go back as far as 2200 BC when the Chinese called them lo-shu. In the early 16th Century Cornelius Agrippa constructed squares for n = 3, 4, 5, 6, 7, 8, 9 which he associated with the seven planets then known (including the Sun and the Moon). Dürer's famous engraving of Melancholia made in 1514 includes a picture of a magic square.
The number of magic squares of a given order is still an unsolved problem. There are 880 squares of size 4 and 275305224 squares of size 5, but the number of larger squares is still unknown. Durer's square shown above is symmetrical and other conditions were also studied such as the condition that all the diagonals (traced as if the square was on a torus) added to the same number as the row and column sum.Euler studied this type of square known as a pandiagonal square. No pandiagonal square of order 2(2n + 1) can exist but they do for all other orders. For n = 4 there are 880 magic squares of which 48 are pandiagonal. Veblen in 1908 used matrix methods to study magic squares.
The dual polyhedron of a unit cube is an octahedron with edge lengths . The cube has the octahedral group of symmetries, and is an equilateral zonohedron and a rhombohedron. It has 13 axes of symmetry: (axes joining midpoints of opposite edges), (space diagonals), and (axes joining opposite face centroids).
The connectivity of the vertices of the cube is given by the cubical graph.
Using so-called "wallet hinges," a ring of six cubes can be rotated continuously (Wells 1975; Wells 1991, pp. 218-219).
lllThe illustrations above show the cross sections obtained by cutting a unit cube centered at the origin with various planes. The following table summarizes the metrical properties of these slices.
REFERENCES
Atomium. "Atomium: The Most Astonishing Building in the World." http://www.atomium.be/.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, 1987.
Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 127 and 228, 1987.
Brückner, M. Vielecke under Vielflache. Leipzig, Germany: Teubner, 1900.
Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension
." Disc. Math. 186, 69-94, 1998.Cardot C. and Wolinski F. "Récréations scientifiques." La jaune et la rouge, No. 594, 41-46, 2004.
Cundy, H. and Rollett, A. "Cube.
" and "Hexagonal Section of a Cube." §3.5.2 and 3.15.1 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 85 and 157, 1989.Davie, T. "The Cube (Hexahedron)." http://www.dcs.st-and.ac.uk/~ad/mathrecs/polyhedra/cube.html.
Dorff, M. and Hall, L. "Solids in
Whose Area is the Derivative of the Volume." College Math. J. 34, 350-358, 2003.
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