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Values Inculcated through mathematics: Values of Learning Mathematics Values are the outcomes or results achieved after teaching according to certain aims. They are the actual results of our teaching effort. They enable the students to acquire mathematical knowledge, and skills, develop interests and attitudes

Key Points  The various values are as follows:

1. Practical or Utilitarian values
2. Intellectual values
3. Social values
4. Moral values
5. Disciplinary values
6. Cultural values
7. International values
8. Aesthetic values
9. Vocational values
10. Psychological values

Important Points

• Napoleon in his statement “The progress and the improvement of Mathematics are linked to the prosperity of the state.” talks about the Social value of mathematics.
• Social values:- Social values reflect how we relate to society.
• Social values include:-
• Justice
• Freedom
• Respect
• Community
• Responsibility

Hence, the statement “The progress and the improvement of Mathematics are linked to the prosperity of the state” corresponds to the social value of mathematics.

Ganita Saar Sangrah is a mathematics text. It is a text written completely in Mathematics. Mahaviracharya was a Jain mathematician. He was one of the first few mathematicians in India who contributed significantly to the field of mathematics

Key Points

• Mahaviracharya has written a book called “Ganita Sara Sangraha” in 850 AD.
• In his book, he mentions everything about triangles, rhombus, circles, and semicircles.
• He also proves many of the theories of Aryabhata and Brahmagupta’s works.
• He also tells the method of Finding the LOWEST COMMON MULTIPLE(LCM) in his book.

Thus, it is concluded that the author of the book 'Ganita Saar Sangrah" is Mahaviracharya.

There are different kinds of teaching-learning methods in a fashion that makes learning a fruitful process. 

Key Points

The inductive method is a scientific method that promotes divergent thinking in learners and is suitable for:

• ​​establishing formula by leading learners from example to formula or particular to general.
• constructing rules and generalizations to facilitate meaningful learning.
• activating learners' previous knowledge to connect them with the new one.
• The inductive approach proceeds from-
• Particular to general
• Known to unknown
• Simple to complex
• Example to formula

Thus, it is concluded that In the Inductive method, by moving from ‘Particular to General’, we generate formulas in Mathematics.

Hint

• The deductive method is based on deduction. At first, the rules are given and then students are asked to apply these rules to solve more problems. The deductive method is a method that leads learners from formula to example.
• Lecture method: The teacher instructs or gives a lecture on a topic for all most the complete time in the period. The teacher provides information, concepts, facts, events, theories, laws, principles, etc.
• The project method is a child-centered and child-friendly method. A project is an educational method where students working individually or in small groups analyze and develop “real-life” problems or tackle a present-day theme within a preset time limit, working independently and with the division of tasks clearly defined.

Mathematics is a subject that finds application in every walk of our life. Knowingly or unknowingly, people use concepts of Mathematics in their daily life.

Key Points 

• ‘Mathematics’ is a broad term that encompasses many branches and components.
• Geometry - It is a branch of mathematics that studies shapes, sizes, angles, and dimensions of objects. It is most helpful in painting.
• Geometry offers the most obvious connection between the two disciplines. Both art and math involve drawing and the use of shapes and forms, as well as an understanding of spatial concepts, two and three dimensions, measurement, estimation, and pattern.

Hence, we conclude that the knowledge of Geometry is most helpful in painting.

Additional Information

• ​ Arithmetic - It is a branch of mathematics that studies numbers and number operations. It helps us to solve daily life mathematical calculations.
• Algebra -  It is a branch of mathematics that represents and solves practical problems in the form of mathematical expressions. It is usually used to find unknown values.
• Trigonometry - It is a branch of mathematics that deals with functions of angles and their applications. It is used in astronomical studies.

Evaluation is a systematic process of collecting and analyzing data in order to determine whether, and to what degree, objectives have been, or are being, achieved.

• “Evaluation is the act of examining and judging the worth, quality, significance, amount, degree or condition of something”.

Key Points

• Evaluation is done due to following purposes:
• To find the need and meaningfulness of a program/ activity.
• To identify the strengths and/or weaknesses of a program/activity.
• The following methods are generally used as methods of evaluation -
• Interview - This method involves interviewing one or more individuals to gather information.
• Questionnaire - It includes a fixed set of questions that need to be answered.
• Check-list - It consists of a list of all things that need to be evaluated.
• Case Study Method - It is an in-depth study of an individual or a case to gather complete information about the subject.

Hence, we conclude that Preparation is not an evaluation method.

Hint

• Preparation needs to be done before conducting the evaluation procedure for successfully complete the evaluation procedure. It is not a method of evaluation**.**

Evaluation is “a systematic process of collecting and analyzing data in order to determine whether, and to what degree, objectives have been, or are being, achieved.

• The purpose of the evaluation is to make a judgment about the quality or worth of something

Key Points

 The evaluation process is related to :

• Educational objectives - Evaluation is conducted to check whether the educational objectives are attained or not.
• Evaluation method - These are the methods through which evaluation is conducted.​​ The methods of evaluation are -
• Interview - This method involves interviewing one or more individuals to gather information.
• Questionnaire - It includes a fixed set of questions that need to be answered.
• Check-list - It consists of a list of all things that need to be evaluated.
• Case Study Method - It is an in-depth study of an individual or a case to gather complete information about the subject.
• Learning experience - It provides a learning experience to the students as well as checks the previous learning.

​Hence, we conclude that the evaluation process is related to educational objectives, evaluation methods, and learning experiences.

Diagnostic evaluation/test refers to the test which helps the teachers to know the learning difficulties or gaps in learner's understanding and identify learner's strengths, weaknesses, skills, etc.

Key Points The purpose of diagnostic testing is :

• to analyze the exact nature of the progress made by the learner in a particular topic/unit and to know the particular area of weakness/error which requires a series of carefully graded tests.
• to guide learning methods.
• classification of students according to the difficulty level.
• detecting lack of achievement in a particular subject.
• to determine the causes of learning problems and to formulate a plan for remedial action.

Thus, it is concluded that all the above options are correct regarding diagnostic testing.

Different teaching methods are used by teachers as per the requirements of students. For choosing the one, the teacher may classify students based on their abilities (ability grouping) or he can choose it based on the compatibility of the topic.

Key PointsSynthetic Method: The word “synthetic’ is derived from the word ‘synthesis which means to combine. In this method, we combine several facts, perform cer­tain mathematical operations, and arrive at the solution. In this method, we start with the known data and connect it with the unknown part.

• It is the process of putting together known bits of information to reach the point where unknown information becomes obvious and true.

It is lead from:

• Known to Unknown (what is given to providing what is required)
• Hypothesis to Conclusion
• Concrete to Abstract
• Simple to Complex

Thus, it is concluded that Synthetic method follows the principle of a process of combining different statements or parts to get something new.

Hint

• Analytic Method: In this, first the problem is broken into pieces and then each piece is analyzed first. For example, to solve a problem of geometry, first, it is necessary to understand statements, then given conditions are noted down, and it is decided which process (formula) to apply to solve problems.
• Heuristic method:- In this method, the student is put in the place of an independent discoverer. Heuristic method makes the process of problem-solving and decision-making quick and efficient.
• Lecture method: The teacher instructs or gives a lecture on a topic for all most the complete time in the period. The teacher provides information, concepts, facts, events, theories, laws, principles, etc.

Shortcut Trick

Convert all bases to 3 and compare the powers directly.

LHS: (√3)⁵ × 9² = 3^2.5 × 3⁴ = 3^6.5

RHS: 3^x × 3√3 = 3^x × 3^1.5

Equate the powers: 6.5 = x + 1.5 ⇒ x = 5

∴ The correct answer is 5.

Alternate Method

Given:

(√3)⁵ × 9² = 3^x × 3√3

Formula Used:

Law of Exponents: a^m × aⁿ = a^m+n and (a^m)ⁿ = a^m×n

Calculation:

Convert all terms to the common base of 3:

⇒ (√3)⁵ = (3^1/2)⁵ = 3^5/2

⇒ 9² = (3²)² = 3⁴

⇒ 3√3 = 3¹ × 3^1/2 = 3^3/2

Now substitute these values back into the equation:

⇒ 3^5/2 × 3⁴ = 3^x × 3^3/2

Apply the multiplication rule of exponents on both sides:

⇒ 3^{(5/2 + 4)} = 3^{(x + 3/2)}

⇒ 3^13/2 = 3^{(x + 3/2)}

Since the bases are the same, equate the exponents:

⇒ 13/2 = x + 3/2

⇒ x = 13/2 − 3/2

⇒ x = 10/2

⇒ x = 5

∴ The correct answer is 5.

Additional Information

Product Rule of Exponents

When multiplying terms with the same base, add their exponents: a^m × aⁿ = a^m+n.

Power of a Power Rule

When raising a power to another power, multiply the exponents: (a^m)ⁿ = a^m×n.

Fractional Exponents

Roots can be written as fractional powers: √a = a^1/2 and ⁿ√a = a^1/n.

Equality of Exponents

If two exponential expressions with the same base are equal, their exponents must be equal. If a^x = a^y, then x = y (provided a ≠ 0, 1, or −1).

Shortcut Trick

If x − 1/x = k, then x² + 1/x² = k² + 2.

In this problem, k = 8.

⇒ x² + 1/x² = 8² + 2 = 64 + 2 = 66.

∴ The correct answer is 66.

Alternate Method

Given:

x − 1/x = 8

Formula Used:

(a − b)² = a² + b² − 2ab

Calculations:

Squaring both sides of the given equation:

⇒ (x − 1/x)² = 8²

⇒ x² + (1/x)² − 2 × x × (1/x) = 64

⇒ x² + 1/x² − 2 = 64

⇒ x² + 1/x² = 64 + 2

⇒ x² + 1/x² = 66

∴ The correct answer is 66.

Additional Information

Positive Sign Identity

If x + 1/x = k, then x² + 1/x² = k² − 2.

Cubic Relation

If x − 1/x = k, then x³ − 1/x³ = k³ + 3k.

Square Conversion

(x + 1/x)² = (x − 1/x)² + 4.

Power Four Relation

If x² + 1/x² = m, then x⁴ + 1/x⁴ = m² − 2.