AIMS
AND OBJECTIVES OF TEACHING MATHEMATICS
Aims of Teaching Mathematics
1. Practical
Aim (Utilitarian Aim)
One cannot do without the use of
fundamental process of the subject mathematics in daily life. Any person
ignorant of mathematics will be at the mercy of others and will be easily
cheated. A person from labour class, a businessman, an industrialist, a banker
to the highest class of the society utilizes the knowledge of mathematics in
one form or the other. Whoever earns and spends uses mathematics and there
cannot be anybody who lives without earning and spending.
Counting, subtraction,
multiplication, division, weighing, selling, buying etc., will have got an
immense practical value in life. The knowledge and skill in these processes can
be provided in an effective and systematic manner only by teaching mathematics
in schools. In many occupations like accountancy, banking, tailoring,
carpentry, taxation, insurance etc., which fulfills the needs of man can be
carried out by the use of mathematics. These agencies depend on mathematics for
their successful functioning. It has become the basis for the world’s entire
business and commercial system. Ignorance of mathematics in the masses is a
formidable obstacle in the way of country’s progress. Individual resources add
up to form national resources. An individual who is ignorant of calculation is
often ruins himself and also causes for the national loss by wasting his time,
energy and money. There are family budget, school budget, factory budget,
national budget, etc., which owe the fundamentals of mathematics. Natural
phenomena like rising and setting of the moon and the suns change of seasons,
speed of rotation of planets, etc., need time specification.
Mathematics will continue to occupy
a prominent place in man’s life. In all activities of life like arranging a
party, admitting a child to school, celebrating a marriage, purchasing or
selling a property, etc, mathematical considerations are uppermost in human
mind. In order to create system in life we have to fix timings, prices, rates,
percentages, exchanges, commissions, discounts, profit and loss, areas, volumes,
etc. In the absence of these fixations life in the present complex society
revert back into confusion and chaos. The number imparts system through our
life. In this complex world passing through scientific and technological age
the practical value of mathematics is going to be increasingly felt and
recognized.
The
following are the practical aims of teaching mathematics.
1. To
enable the students to have clear ideas about number concept.
2. To
give the individual an understanding of ideas and operations in number and
quantity needed in daily life.
3. To
enable the individual to have clear comprehension of the way the number is
applied to all measures but most particularly to those frequently used concepts
such as length, volume, area, weight, temperature, speed etc.
4. To
enable the individual to become proficient in the four fundamental operations
of addition, subtraction, multiplication and divisions.
5. To
provide the basis of mathematical skills and processes which will be needed for
vocational purposes.
6. To
enable the learner to acquire and develop mathematical skills and attitude to
meet the demands of (i) daily life (ii) future mathematical work and (iii) work
in the related fields of knowledge.
7. To
enable the students to make appropriate approximations.
8. To
enable the learner to understand the concept of ratio and scale drawing, read
and interpret graphs, diagrams and tables.
9. To
enable the individual to apply his mathematics to a wide range of problems that
occur in daily life.
2.
Disciplinary Aim
The principal
value of mathematics arises from the fact that it exercises the reasoning power
more and climbs from the memory, less from any other school subjects. It
disciplines the mind and develops reasoning power. Locke is of the opinion that
‘mathematics is a way to settle in the mind a habit of reasoning. A person who
had studied mathematics is capable of using his power of reasoning in an
independent way’. Its truths are definite and exact. The learner has to argue
the correctness or incorrectness of a statement. The reasoning in the
mathematical world is of special kind processing characteristics that make it
specially suited for training the minds of the pupils. It can be studied under
following heads –
1.
Characteristics of Simplicity: In this subject
teaching and learning advances by degrees from simple to complex. It teaches
that definite facts are always expressed in a simple language and definite
facts are always easily understandable.
2.
Characteristics of Accuracy: Accurate
reasoning, thinking and judgment are essential for the study of mathematics.
The pupils learn the value and appreciation of accuracy and adopt it as a
principle of life. It is in the nature of the subject that it cannot be learn
through vagueness of thoughts and arguments. Accuracy, exactness and precision
compose the beauty of mathematics. He learns to influence and command others by
accuracy.
3.
Certainly of Results: There is no
possibility of difference of opinion between the teacher and the pupil. It is
possible for the learner to remove his difficulties by selfeffort and to be
sure of the removal. He develops faith in selfeffort, which is the secret of
success in life.
4.
Originality of Thinking: Most work in
mathematics demands original thinking. Reproduction and cramming of ideas of
others is not very much of appreciated. The pupil can safely depend on the
memory in other subjects but in the mathematics without original thinking and
intelligent reasoning there cannot be satisfactory progress. This practice in
originality enables the situation with confidence in his future life.
5.
Similarity
to Reasoning of Life: Clear and exact thinking is as important in daily
life as in mathematical study. Before starting with the solution of a problem
the pupil has to grasp the whole meaning similarly in daily life while
undertaking a task, one must have firm grip on the situation. This habit of
thinking will get transferred to the problem of daily life.
6.
Verification of Results: This gives a
sense a achievement, confidence and pleasure. This verification of results also
likely to inculcate the habit of selfcriticism and selfevaluation.
7. Concentration of Mind: Every problem of
education and life demandsconcentration.
Mathematics
cannot be learn without wholehearted concentration. Hamilton says, ‘the study of mathematics
cures the vice of mental distraction and cultivates the habit of continuous
attention’.
In
every field new ideas and new methods are being introduced rapidly. In this
every advancing society the important thinking is not only to learn facts but
also to know how to learn facts. This is the discipline of the mind.
Mathematics has a vast scope of application. Mathematics has the ability to
apply knowledge to new situation and acquire the power to think effectively so
that intellectual power of the learner is strengthened.
The teaching of
mathematics intends to realise the following disciplinary aims,
1.
To provide opportunities that enable
the learners to exercise and discipline mental faculties.
2.
To help the learner in the
intelligent use of reasoning power.
3.
To develop constructive imagination
and inventive faculties.
4.
To develop the character through
systematic and orderly habits.
5.
To help the learner to be original
and creative in thinking.
6.
To help the individual to become
selfreliant and independent.
3.
Cultural
Aim
Cultural aim helps the pupils to
grow in cultured situation. The greatness of Indian culture is once reflected
through the glory of Indian mathematics of olden days. Similarly having come to
know the progress of Egyptians and Greeks in mathematics, one can be aware of
their progress in culture and civilization. Mathematics does not only acquaint
us with the culture and civilization but it also helps in its preservation,
promotion and transmission to the coming generation. Further after knowing what
our ancestors have done we as the students of mathematics bring new idea in the
body of mathematical knowledge. And thus increase our cultural heritage.
Culture is not possible unless there is a proper development of the power of
reasoning and judgment. The aim of teaching of mathematics is to develop
cultured citizen who can discharge their obligations to the society effectively
and successfully. Our entire present civilization depends on intellectual
penetration and utilization of nature has its real foundation in the
mathematical science. As the education commission report (196466) was
conscious of this need when it wrote ‘one of the outstanding characteristics of
scientific culture is quantification of mathematics. Therefore it assures a
prominent position in modern education…. Proper foundation in the knowledge of
the subject should be laid at the school’. Mathematics played a major role in
bringing man to the advanced stage of development. The prosperity and the
cultural advancement of man depend on the advancement in mathematics. That is
why Hogben says, ‘mathematics is the mirror of civilization’. Different laws of
science and scientific instruments are based on the exact mathematical concept.
For example astronomy and physics are the most exact science and their
exactness is the outcome of the usefulness of mathematics.
Mathematics is
the backbone of our civilization. What we have in our modern culture and
civilization owes its depth to science and technology, which in turn, depends
upon the progress in mathematics. Moreover in various cultural arts like
poetry, drawing, painting, music, architecture and design making, mathematics
is playing a vital role and therefore it can be safely said that mathematics is
intimately linked with the culture and civilization.
The cultural
aims can be summarized as follows,
1.
To
enable the student to appreciate the part played by mathematics in the culture
of the past and that it continues to play in the present world.
2.
To
enable the student to appreciate the role played by mathematics in preserving
and transmitting our cultural traditions.
3.
To
enable him to appreciate various cultural arts like drawing, design making,
painting, poetry, music, sculpture and architecture.
4.
To
provide through mathematics ideas, aesthetic and intellectual enjoyment and
satisfaction and to give an opportunity for creative expression.
5.
To
help the student explore creative fields such as art and architecture.
6.
To
make the learner aware of the strength and virtues of the culture he has
inherited.
7.
To
develop in the individual an aesthetic awareness of mathematical shapes and
patterns in nature as well as the products of our civilization.
4. Recreational Value
Mathematics not
only give pleasure through its application to various arts, it also entertains
through its own riddles, games and puzzles. While developing and subject, its
dedicated students have been playing with its numbers, figures, shapes and
problems. We can also have magic squares though which one can derive pleasure
by getting an equal sum every time after adding horizontally, vertically or
diagonally.
Entertainment
in mathematics
1)
‘9’ is a wonder number. In the
multiplication table of mathematics the sum of digits of every product is 9:
9X1=9 9+0=9 9X6=54 5+4=9
9X2=18 1+8=9 9X7=63 6+3=9
9X3=27 2+7=9 9X8=72 7+2=9
9X4=36 3+6=9 9X9=81 8+1=9
9X5=45 4+5=9 9X10=90 9+0=9
·
Similarly
√999999999= 999999.999999 to six decimal places the characteristic is
wonderful.
·
9X9+7=88
98X9+6=888
987X9+5=8888
9876X9+4=88888, etc.
2)
37
is a prime number, but it divides the following numbers completely,
111,222,
333, 444, 555, 666, 777, 888, 999, etc,.
3)
Using
‘8’ 8 times to get 1000?
888+88+8+8+8=1000
4)
(12)^{2}=144
(144)^{2}=441
5) 1X8+1=9
12X8+2=98
123X8+3=987
1234X8+4=9876
12345X8+5=98765
123456X8+6=987654
1234567X8+7=9876543
12345678X8+8=98765432
123456789X8+9=987654321
6)
Statement: A door is half opened, that means that door is half closed!
i.e,. Door is half opened = ½ closed
X 2
½ X
2 opened = ½ X2 closed
à
1 Door opened = 1 Door closed!!!
i.e., one
door fully opened, it is fully closed!!!!
7) 1 Re = 100 paise  (1)
2
Re = 200 paise  (2)
½ a
Re = 50 paise  (3)
Add
(2) and (3)
2 ½
Re = 250 paise
(2) –
(3) à
1 ½ Re = 150 paise
(2) X
(3) à 2 ½ Re = 200 X 50
1 Re = 10000 paise!!!!
Divide (2 ) by (3) à 2/1/2 Re = 200/50 paise
4 Re= 4 paise!!!
8) By using
English alphabets except first four (ABCD),is it possible to write 100 words in
1 minute?
Ans: Yes!! Starting from ZERO TO NINETY NINE, We get 100
words without using ABCD!!!
9) KEPREKAR CONSTANT
Born:
1905.Jan.17. Place: Mumbai
Steps:
1. Any 4 different numbers.
2. Write the number in ascending
and descending orders. (43211234)
3. Take difference of 1 and 2.
4. Again ascending and descending
order and difference.
5. Repeat maximum 7 times you will
get constant 6174 (before also we can get this number).
Example: 8640 – 0468=8172
8721
– 1278=3996
9963 3699= 6264
6642 – 2466 =4176
7641 – 1467=6164
* Note: For two digits we get 9
as constant.
For
three digits we get 495 as
constant.
For
five digits we get 63954 as a
constant.
10) Easy multiplication, the number end with 5 (two digits
only)
25X25 à * 5x5=25àkeep as it is.
625 * add 1 to second digit,
(1+2=3).
* 3x2=6
5.
Social Aim:
The important social aims of
teaching mathematics are as under,
1.
To
develop in the individual an awareness of the mathematical principles and
operations which will enable the individual to understand and participate in
the general, social and economic life of his community.
2.
To
enable the student to understand how the methods of mathematics such as
scientific, intuitive, deductive and inventive are used to investigate,
interpret and to make decision in human affairs.
3.
To
help the pupil acquire social and moral values to lead a fruitful life in the
society.
4.
To
help the pupil in the formation of social laws and social order needed for
social harmony.
5.
To
provide the pupils scientific and technological knowledge necessary for
adjusting to the rapidly changing society and social life.
6.
To
help the learner appreciate how mathematics contributes to his understanding of
natural phenomena.
7.
To
help the pupil interpret social and economic phenomena.
Objectives
of Teaching Mathematics – National Policy of Education (1986)
At
the end of high school stage, a pupil should be able to –
·
Acquire knowledge and understanding of
the terms, concepts, principles, processes, symbols and mastery of
computational and other fundamental processes that are required in daily like
and for higher learning in mathematics.
·
Develop skills of drawing, measuring,
estimating and demonstrating.
·
Apply mathematical knowledge and skills
to solve problems that occur in daily life as well as problems related to
higher learning in mathematics or allied areas.
·
Develop the ability to think, reason,
analyze and articulate logically.
·
Appreciate the power and beauty of
mathematics.
·
Show an interest in mathematics by
participation in mathematical competitions, and engaging in its learning, etc.
·
Develop reverence and respect towards
great mathematicians, particularly towards great Indian mathematicians for
their contributions to the field of mathematical knowledge.
·
Develop necessary skills to work with
modern technological devices such as calculations, computers, etc.
Objectives
of Teaching Mathematics – New Curriculum Document (2000)
The
learners
·
Consolidate the mathematical knowledge
and skills acquired at the upper primary stage.
·
Acquire knowledge and understanding of
the terms, symbols, concepts, principles, process, proofs, etc.
·
Develop mastery of basic algebraic
skills.
·
Develop drawing skills.
·
Apply mathematical knowledge and skills
to solve real mathematical problems by developing abilities to analyze, to see
interrelationship involved, to think and reason.
·
Develop the ability to articulate
logically.
·
Develop awareness of the need for
national unity, national integration, protection of the environment, observance
of small family, norms, removal of social barriers, and elimination of sex
biases.
·
Develop necessary skills to work with
modern technological devices such as calculators, computers, etc.
·
Develop interest in mathematics and
participate in mathematical competitions and other mathematical club activities
in the school.
·
Develop appreciation for mathematics as
a problemsolving tool in various fields for its beautiful structures and
patterns, etc.
·
Develop reverence and respect towards
great mathematicians, particularly towards the Indian mathematicians for their
contributions to the field of mathematics.
Objectives
of Teaching Mathematics
The objectives
of teaching mathematics at the secondary state may be classified as under:
A. Knowledge
and Understanding objectives
B. Skill
objectives
C. Application
objectives
D. Attitude
objectives
E. Appreciation
and Interest objectives
A.
Knowledge and Understanding Objectives
The
student acquires knowledge and understanding of:
1. Language
of mathematics i.e., the language of its technical terms, symbols, statements,
formulae, definitions, logic, etc.
2. Various
concepts i.e., concept of number, concept of direction, concept measurement.
3. Mathematical
Ideas, like facts, principles, processes and relationships.
4. The
development of the subject over the centuries and contributions mathematicians.
5. Interrelationship
between different branches and topics of mathematics etc.
6. The
nature of the subject of mathematics.
B. Skill Objectives
The
subject helps the student to develop the following skills:
1. He
acquires and develops skill in the use and understanding of mathematical
language.
2. He
acquires and develops speed, neatness, accuracy, brevity and precision in
mathematical calculations.
3. He
learns and develops technique of problemsolving.
4. He
develops and ability to estimate, check and verify results.
5. He
develops and ability to perform calculations orally and mentally.
6. He
develops and ability to think correctly, to draw conclusions, generalizations
and inferences.
7. He
develops skills to use mathematical tools, and apparatus.
8. He
develops essential skill in drawing geometrical figures.
9. He
develops skill in drawing, reading, interpreting graphs and statistical tables.
10. He
develops skill in measuring, weighing and surveying.
11. He
develops skill in the use of mathematical tables and ready references.
C.
Application Objectives:
The subject
helps the student to apply the abovementioned knowledge and skills in the
following way:
1. He
is able to solve mathematical problems independently.
2. He
makes use of mathematical concepts and processes in everyday life.
3. He
develops ability to analyze, to draw inferences, and to generalize from the
collected data and evidence.
4. He
can think and express precisely, exactly, and systematically by making proper
use of mathematical language.
5. He
develops the ability to use mathematical knowledge in the learning of other
subjects especially sciences.
6. He
develops the students’ ability to apply mathematical in his future vocational
life.
D. Attitude Objectives:
The subject helps to develop the following
attitudes:
1. The
student learns to analyze the problems.
2.
Develops the habit of systematic thinking and objective reasoning.
3. He
develops heuristic attitude and tries to discover solutions and proofs with his
own independent efforts.
4. He
tries to collect enough evidence for drawing inferences, conclusions and
generalizations.
5. He
recognizes the adequacy or inadequacy of given data in relation to any problem.
6. He
verifies his results.
7. He
understands and appreciates logical, critical and independent thinking in
others.
8. He
expresses his opinions precisely, accurately, logically and objectively without
any biases and prejudices.
9. He
develops selfconfidence for solving mathematical problems.
10. He
develops personal qualities namely, regularity, honesty, objectivity, neatness
and truthfulness.
11. He
develops mathematical perspective and outlook for observing the realm of nature
and society.
E.
Appreciation and Interest Objectives:
The student is
helped in the acquisition of appreciation and interest in the following way:
1. He appreciates the role of mathematics in
everyday life.
2.
He appreciates the role of mathematics in understanding his environment.
3.
He appreciates mathematics as the science of all sciences and art of all arts.
4.
He appreciates the contribution made by mathematics in the development of civilization
and culture.
5. He appreciates the contribution of
mathematics with field and other branches.
6. He develops the interest in the learning of
the subject.
7. He feels enter by mathematical recreations.
8. He develops act interest in the activities of
mathematics clues.
9. He develops act interest inactive library
reading, mathematical projector.
10.
He appreciates the aesthetic nature of mathematics by observing symmetry,
Similarity, order and arrangement in
mathematical facts, principles and processes.
11.
He appreciates the contribution of mathematics in the development of other
branches
of knowledge.
12. He
appreciates the recreational values of the subject and learn to utilize it in
his
leisure
time.
13.
He appreciates the vocational value of mathematics.
14.
He appreciates the role of mathematical language, graphs and tables in giving
Precision and accuracy to his
expression.
15.
He appreciates the power of computation developed through the subject.
16.
He appreciates the role of mathematics in developing his power of acquiring
Knowledge.
17.
He appreciates mathematical problems, their intricacies and difficulties.
18.
He develops interest in the learning of the subject.
19.
He feels entertained by mathematical recreations.
20.
He takes an active interest in the activities of mathematics club.
21.
He takes an active interest inactive library reading, mathematical projects,
and doing practical work in
mathematics laboratory.
Objectives
of Teaching Arithmetic:
Arithmetic
is the science of numbers and art of computation. It is the oldest branch of
the subject mathematics. Historically arithmetic developed out of a need for a
system of counting. It is considered to be essential for efficient and
successful living. That is why arithmetic is divided as the science that deals
with numbers with relations between numbers, numbers in term, or abstraction
arising from such concrete situations as counting measuring and ordering the
various quantities and objects that we encounter in everyday life. The need of
a good command of arithmetic by a housewife, by a farmer, by a successful
merchant, by a skilled worker is too obvious to need any discussion. Also its
utilitarian, cultural and disciplinary values are too obvious to need any
argument at this stage.
The teaching of arithmetic has to
fulfill two responsibilities.
1. The
inculcation of an appreciative understanding of number system and an
intelligent proficiency in its fundamental process.
2. The
socialization of number experiences.
The
following are the objectives of teaching arithmetic –
1. To
teach the learner mathematical type of thought, to understand the statement to
analyze them and to arrive at right conclusions.
2. To
arose pupil’s interest in the quantitative side of the world around him and its
use as a simple tool in business.
3. To
develop fundamental arithmetic concepts like the concept of number, order,
units of measurement, size and shape etc.
4. To
give accuracy and facility in simple computation of the fundamental process.
5. To
develop speed and accuracy in arithmetical calculation and computation.
6. To
impart a working knowledge of practical arithmetical applications which are
useful in life.
7. To
appreciate the use of arithmetic in daily life.
8. To
help in the learning of other branches and higher studies in mathematics.
Objectives of Teaching Algebra
Algebra
is called the science of letter. It refers to the methods of reasoning about
numbers by employing letters to represent their relationship. Algebra is
concerned largely with structure of number system, operations with numbers and
statements involving numbers as well as the solution of problems. Algebra is a
language used to develop and express much of the scientific data. Algebra
comprehended a more general treatment of numbers and number relation than thus
arithmetic. It is concerned with the general statement about numerical
situation. Algebra refers to the operation of taking a quantity from one side
of the equation to another by changing its signs. It presents a radically new
and different approach to the study of quantitative relationships characterized
by a new symbolism, new concepts, and a new language much higher degree of
generalization and abstraction than has been encountered in arithmetic. But it
is primarily taught for manipulative skill. Solutions of problems by equations
give a power of generalization and use of formulae and idea of functionality.
The
following are the objectives of teaching Algebra
1.
To give compact formulae of
generalization to be used in all cases.
2.
To provide an effective way for
expressing complicated relations.
3.
To correct the weaknesses and supplement
the deficiency of language as an instrument of abstract investigation and exact
statement.
4.
To inculcate the power of analysis.
5.
To verify the results in simpler and
more satisfactory way.
6.
To develop confidence among the pupils.
7.
To provide new and refined approach in
the study of abstract mathematical relationship though the use of new
symbolism.
8.
To enable the pupils to use it for
solving more difficult problem.
Objectives
of Teaching Geometry:
The
word geometry originally means measurement of earth. Geometry has two value:
a) the
knowledge and
b) as
a method of logical thinking.
It
is the science of lines and figures it is the science of space and extent. It
deals with the position, space and size of bodies but nothing to do with their
material properties. Geometry has two important aspects –
Demonstrative
Geometry – It deals with the shape, size and position of figures by pure
reasoning based on definitions, selfevident truths and assumptions. Euclid, a
great Greek Mathematician was the father of demonstrative Geometry. His methods
arte intuitional, observational, intentional, constructive, informal, creative,
experimental and so on.
Practical
Geometry – It covers the constructional work of the subject. Most of the work
directly or indirectly based on demonstrative Geometry.
The
following are the objective of teaching Geometry
1.
To enable the learner to acquire a mass
of geometrical facts.
2.
To implement geometrical principles like
equality, symmetry similarity in every nature of things.
3.
To develop the ability to draw accurate
figures.
4.
To demonstrate the nature and the power
of pure reason.
5.
To systematize the information received
by the pupils in the preschool stage.
6.
To aid the pupils in becoming familiar
with the basic geometrical concepts and space perceptions and in understanding
the fundamental techniques such as the use of set square, protractor, compass,
etc.
7.
To acquaint with the pupil the good
geometrical notation.
Instructional Objectives:
I.
Remembering:
It is cognitive level.
The
pupil –
1. Recalls the mathematical laws,
principle, rule formulae, etc.
2. Recognizes the mathematical laws,
principle, formulae, etc.
II. Understanding:
Goes deep into the content
The pupil –
1. Cites
or gives examples
2. Gives
reasons
3. Identifies
4. Compares
5. Finds
relationship
6. On
the basis of observation draws conclusion
7. Draws
inference or the result
8. Converts
verbal form to symbolic form or vice versa
9. Classifies
mathematical data
III. Applying:
The Pupil –
1. Analyses
the problem into its components
2. Judges
the adequacy of the given data
3. Suggests
the alternate methods
4. Suggest
the most appropriate method
5. Generalizes
IV. Skill:
The Pupil –
1. Reads
mathematical figures, statements, problems, charts, tables, etc.
2. Labels
the geometrical figure.
3. Draws
the most appropriate, neat and proportionate geometrical figures
4. Solves
oral problems quickly and accurately
5. Solves
written problems quickly and accurately.
Trigonometry Objectives
 Measurement
of Angles, Arcs, and Sectors
 Using Radians, Degrees, or Grads
to Measure Angles
 Length of an Arc and Area of a
Sector of a Circle
 Circular Motion
 The
Trigonometric Functions
 Definition of the Six
Trigonometric Functions
 Values of the Trigonometric
Functions at some multiples of 15 degrees.
 Trigonometric Functions for
right triangles
 Solving Right Triangles
 Applications of Right Triangle
Trigonometry
 Circular Functions
 Graphs
of Trigonometric Functions
 Graphing Generic Sine and Cosine
Functions
 Shifting Generic Curves
Right/Left or Up/Down
 Using the Graphing Calculator to
Graph Functions by Addition of Ordinates
 Graphing the Tangent and
Cotangent Functions
 Graphing the Secant and Cosecant
Functions
 Qualitative Analysis of
Trigonometric Functions
 Inverse
Trigonometric Functions
 Relations, Functions, and Their
Inverses
 Inverses of Trigonometric
Functions
 Finding Inverses of
Trigonometric Functions Using a Calculator
 Basic
Trigonometric Identities
 Fundamental Identities
 Opposite Angle Identities
 Additional Techniques to Prove
Identities
 Sum
and Difference Identities
 Sum and Difference Identities
for Cosine
 Some Identities Useful in
Calculus
 Sum and Difference Identity for
Tangent
 Identities Involving Sums and
Differences of Pi and Pi/2.
 Additional
Identities
 Double Angle Identities
 Half Angle Identities
 Identities to Rewrite Sums and
Products
 Trigonometric
Equations
 Solving Basic Trigonometric
Equations
 Solving Trigonometric Equations
Involving Factoring
 Solving Trigonometric Equations
Where the Argument is a Function
 Using Identities to Solve
Trigonometric Equations
 Law
of Sines and Law of Cosines
 Derivation of the Law of Sines
 The Ambiguous Case
 Applications of the Law of Sines
 Derivation of the Law of Cosines
 Applications of the Law of
Cosines
 Area of a Triangle
 Vectors
 Addition of Vectors
 Geometric Resolution of Vectors
 Algebraic Resolution of Vectors
 Work, Inclined Planes, and the
Dot Product
 Complex
Numbers
 Algebraic Operations with
Complex Numbers
 Trigonometric and Polar
Representation of Complex Numbers
 DeMoivre's Formula
 Polar
Coordinates
 The Polar Coordinate System
 Parametric Equations
 Other Graphs in Polar
Coordinates
Objectives of TeachingTrigonometry
The students will be able,
1.
To impart knowledge of trigonometric ratios and
identities.
2.
To apply the knowledge of trigonometry to solve
daily life problems.
3.
To find heights and distances.
4.
To appreciate the use of trigonometry to solve
problems.
5.
To develop creative thinking and reasoning.
6.
To understand that it is an essential tool.
7.
to know how structures are built.
8.
To realise that it very useful in technology and
for engineers.
9.
To continue higher education.
10.
To understand the relationship between trigonometry
and other branches of mathematics.
11.
Find the value of trigonometric ratios of some
specific angles.
12.
Determine the trigonometric ratios of
complementary angle.
13.
Apply the trigonometric identities in proving
the given statement.
Objectives of
TeachingCoordinate Geometry
The students will be able to,
1.
Draw a plan for the given situation.
2.
Appraise the Cartesian system.
3.
Identify the coordination of a point.
4.
Locate the quadrants in the Cartesian plane.
5.
Plot the points in the Cartesian plane.
6.
Write the abscissa and ordinate of a point.
Specific
Objectives:
1. To find areas of plane rectilinear figures in the
coordinate plane.
2. To find the angle
between two straight lines.
3. To understand the normal form of a straight line equation
and to apply the knowledge in finding distances.
4. To find the equation of a circle and the points of
intersection of a straight line and a circle.
5. To find the equations of tangents to a circle.
6. To find the equations of families of straight lines and
circles.
ANDERSON’S REVISED
BLOOM’S TAXONOMY OF INSTRUCTIONAL OBJECTIVES
Taxonomies of the Cognitive Domain
Bloom’s Taxonomy 1956

Anderson and Krathwohl’s Taxonomy 2001


1. Knowledge: Remembering or retrieving
previously learned material. Examples of verbs that relate to this function
are:

1. Remembering:
Recognizing or recalling knowledge
from memory. Remembering is when memory is used to produce or retrieve
definitions, facts, or lists, or to recite previously learned information.


2. Comprehension: The ability to grasp or construct
meaning from material. Examples of verbs that relate to this function
are:

2. Understanding:
Constructing meaning from different
types of functions be they written or graphic messages or activities like
interpreting, exemplifying, classifying, summarizing, inferring, comparing,
or explaining.


3. Application: The ability to use learned material, or to
implement material in new and concrete situations. Examples of verbs that
relate to this function are:

3. Applying:
Carrying out or using a procedure
through executing, or implementing. Applying relates to or refers to situations where
learned material is used through products like models, presentations,
interviews or simulations.


4. Analysis: The ability to break down or distinguish the
parts of material into its components so that its organizational structure
may be better understood. Examples of verbs that relate to this function
are:

4. Analyzing:
Breaking materials or concepts into
parts, determining how the parts relate to one another or how they
interrelate, or how the parts relate to an overall structure or purpose.
Mental actions included in this function are differentiating, organizing, and
attributing, as well as being able to distinguish between the components or parts. When one is
analyzing, he/she can illustrate this mental function by creating
spreadsheets, surveys, charts, or diagrams, or graphic representations.


5. Synthesis: The ability to put parts together to form a
coherent or unique new whole. Examples of verbs that relate to this function
are:

5. Evaluating:
Making judgments based on criteria
and standards through checking and critiquing. Critiques, recommendations,
and reports are some of the products that can be created to demonstrate the
processes of evaluation. In the newer taxonomy, evaluating comes before
creating as it is often a necessary part of the precursory behavior before
one creates something.


6. Evaluation: The ability to judge, check, and even
critique the value of material for a given purpose. Examples of verbs that
relate to this function are:

6. Creating:
Putting elements together to form a
coherent or functional whole; reorganizing elements into a new pattern or
structure through generating, planning, or producing. Creating requires
users to put parts together in a new way, or synthesize parts into something
new and different creating a new form or product. This process is the
most difficult mental function in the new taxonomy.

Instructional Objectives:
1. Remembering:
The pupil acquires knowledge of mathematics.
Learning outcomes:
The pupil,
·
Recalls mathematical terms, facts,
processes, principles, formulae definitions, signs and symbols, relationships,
generalizations etc.,
·
Recognizes terms, instruments, process,
formulae, signs and symbols, relationships, generalisations etc.,
2. Understanding: The
pupil develops understanding in mathematics.
Learning outcomes:
The pupil,
·
Explains mathematical terms, concepts,
principles, relationships etc., in his own words.
·
Defines mathematical terms and concepts.
·
States mathematical principles,
relationships etc.
·
Gives illustrations for mathematical
concepts, principles, etc.,
·
Identifies mathematical terms, concepts,
relationships, figures, processes etc.
·
Finds similarities between mathematical
terms, concepts, relationships, figures etc.
·
Differentiates between mathematical
terms, concepts, relationships, figures etc.
·
Classifies mathematical terms, concepts,
figures etc.,
·
Verbalises symbolic relationships and
vice versa.
·
Frames mathematical formulae,
generalisations on the basis of data.
·
Uses the formula to solve problems.
·
Substitute relevant numbers, symbols and
signs in the mathematical formulae and operations.
·
Calculate the answers for given
problems.
·
Uses appropriate units to write answers.
·
Finds solutions for given problems.
3. Applying:The
pupils applies knowledge of mathematics to novel situations.
Learning outcomes:The
pupil,
·
Analyses a problem or data into
component parts.
·
Judges the adequacy, inadequacy or
superfluity of data.
·
Establishes relationships among data.
·
Gives a number of methods of solving a
problem.
·
Select the most appropriate formulae or
principles or methods or process to solve problem,
·
Reasons deductively.
·
Reasons inductively.
·
Makes a generalization.
·
Draws inferences.
·
Predicts results on the basis of data.
4. Skill:
a) The pupil acquires skills in handling mathematical instruments with
ease.
Learning outcomes:The
pupil,
·
Draws freely satisfactory freehand
figures.
·
Selects the most appropriate
mathematical instruments.
·
Takes necessary precautions in taking
measurements while constructing geometrical figures.
·
Takes measurements correctly.
b)Drawing geometrical figures and Graphs
Learning outcomes:
The pupil,
·
Draws figures to given specifications.
·
Draws figures quickly.
·
Uses appropriate marking to denote
different parts of a figure.
c) Computation:
Learning outcomes:
The pupil,
·
Does oral calculation correctly.
·
Does oral calculation quickly.
·
Does a written calculation correctly.
·
Does a written calculation quickly.
·
Uses correct notations and symbols.
·
Avoids unnecessary steps in the solution
of a problem.
·
Is systematic in working of a problem.
d) Reading of Tables, charts, Graphics etc.
Learning outcomes:The
pupil,
·
Selects appropriate mathematical tables.
·
Uses mathematical tables, charts, ready
reckoners etc., correctly.
·
Coordinates the different sections of
the graphs correctly.
·
Reads graphs correctly.
TASK
ANALYSIS:
It is a set of questions set on the
learnt materials in the class room should not be left in the school itself, but
be stored for use in future life. To help the learner to register the
information and for its longer retainment, teacher should make students
involved in certain activities related to the class room learning.
Example (Problem): A rectangle
is having 18 metre length and area 180 sq. metre. What will be breadth?
Knowledge:
1. Recall What is given? 5. What is the unit of
Area?
2. What is length given? 6. What is the formula to calculate
area of
3.
What
is the area of rectangle? rectangle?
4. What is the unit of length?
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