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Calculation:

1 m = 1000 mm

1 m² = 1 m × 1 m = 1000 mm × 1000 mm = 10,00,000 mm² 

∴ 1 m² = 1,000,000 mm²

Given:

15m² is the area.

Concept used: 

Area of square = side²

Calculation:

Let side of square = a m

a² = 15

If we take 15 square of 1cm each,then

Area = 15 × Area of one square

Area = 15 × (1)²

Area = 15m²

∴ 15 m2 is the area of 15 squares having side 1 m each.

Given:

Length of the cuboid = 3y

Breadth of the cuboid = y

Height of the cuboid = 2y

Formula Used:

The volume of the cuboid = Length × Breadth × Height

Calculation:

The volume of the cuboid = 3y × y × 2y

⇒ The volume of the cuboid = 6y³

∴The volume of the cuboid is 6y³

Mathematics is the study of numbers, shapes, quantities, and patterns. Mathematics is the ‘queen of all sciences' and its presence is there in all the subjects. It acts as the basis and structure of other subjects. 

• Values in mathematics reflect the nature of mathematical content and knowledge. It is a field that has various values in it, that must be considered seriously and taught implicitly rather than explicitly in mathematics classes when compared to others.

Key Points

• There are mainly ten values in learning mathematics.
• Practical value
• Intellectual value
• Social value
• Moral values
• Disciplinary values
• Cultural values
• International values
• Artistic values
• Vocational values
• Psychological values
• ​“Music is a modern secret exercise of the subconscious mind of a human being related to the numbers of arithmetic.” It is a function of the artistic value of mathematics as it reflects the use of artistic knowledge.
• Mathematics has aesthetic, artistic, or pleasure value. Concepts like symmetry, order, similarity, form, and size form the basis of all works of art and beauty. All poetry and music utilize mathematics.

Thus, it is concluded that option 3 is the correct answer.

Given:

RICE is a rhombus.

Concept Used:

(Hypotenuse)2 = (Perpendicular)2 + (Base)2

In a rhombus, diagonals bisect each other at right angles.

All sides of the rhombus are equal.

Calculation:

Base = Diagonal/2

Perpendicular = Diagonal/2

(Hypotenuse)2 = (Perpendicular)2 + (Base)2   

⇒ (Hypotenuse)2 = (12)2 + (5)2

⇒ (Hypotenuse)2 = 144 + 25

⇒ Hypotenuse = \(\sqrt{169}{} \)

⇒ Hypotenuse = 13 cm

⇒ The side of the rhombus = 13 cm 

⇒ The perimeter of the rhombus = the Sum of all sides

⇒ The perimeter of rhombus = 13 + 13 +  13 + 13 = 52 cm

∴ The perimeter of the rhombus is 52 cm.

Mathematics teaches us how to analyze a situation, how to come to a decision, check thinking and its results, perceive relationships, concentrate, be accurate, and be systematic in our work habits. 

 Key PointsHogben says, “Mathematics is the mirror of civilization”.

• The history of mathematics is the story of the progress of civilizations and cultures.
• By the culture of a nation or society, we mean the mode of living of its inhabitants.
• Helps in promoting cultural heritage and transmitting it to the future generation.
• A country’s civilization and culture is reflected in the knowledge of mathematics it possesses.
• Mathematics helps in the preservation, promotion, and transmission of cultures.
• Various cultural arts like poetry, painting, drawing, and sculpture utilize mathematical knowledge.

Thus, it is concluded that Mathematics is a mirror of civilization and culture.” Hogben gave this statement.

Hint

• Locke said, “Mathematics is a way to settle in the mind a habit of reasoning.”
• Roger Bacon said "Mathematics is the gate and key of the sciences.''
• Galileo said, "Mathematics is the alphabet with which God has written the universe."

Learning from observation is a common and natural method of human learning. Learning through observation is also known as Observational learning.

Key Points

Laboratory method:

• This method is based on the principle of "To observe and learn".
• It involves hands-on experiments, observations, and analysis in a controlled environment (e.g., science labs).
• Students actively engage in exploring concepts through experimentation, enhancing their understanding through direct observation and practice.

Hint

• Inductive method: This method focuses on moving from specific observations to general principles. While observation is involved, it is not specifically designed for the "observe and learn" approach like the laboratory method.
• Programmed Instruction method: This is a self-paced learning method where content is broken into small units, and learners progress by answering questions. It does not emphasize observation-based learning.

Thus, from the above-mentioned points, it is clear that Laboratory method is based on the principle of “To observe and learn”.

All children love to play. Play is a natural instinct of children. It is the natural expression of their needs.

• A method of teaching in which curricular concepts can be integrated into the game so that the students can enjoy playing the game and can simultaneously learn the concepts is known as “the play way method”.

Key Points

 Play way methods of teaching are : 

• Montessori - Developed by Maria Montessori, it gave emphasis on hands-on activity and collaborative play to develop the natural interests of the child.
• Kindergarten - It is a preschool approach given by Friedrich Fröbel, to impart learning through games, painting, singing, etc.
• Dalton - Propounded by Helen Parkhurst, it is a method of education by which pupils work at their own pace, and receive individual help from the teacher when necessary.

Important Points

• ​ Lecture Method- In this method, the teacher instructs or gives a lecture on a topic for most of the time in the period. She provides information, concepts, facts, events, theories, laws, principles, etc.
• It is an instructional method and not a play way method of teaching. It is usually employed at the senior level.

Hence, it is concluded that  Lecture method is not applicable for mathematics teaching through play way at the primary level.

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

Important Points 

Steps of Diagnostic Testing:

• Identifying students who are having trouble.
• Locating the errors and learning difficulties/gaps
• Discovering the causal factors of slow learning.

Key Points​ 

The Purpose of Diagnostic Testing in Mathematics:

• This test helps to know the gaps in children’s understanding.
• It is specially conducted for removing the learning difficulties of learners.
• The diagnostic test enables the educator and students to recognize issues that they have with the subject. It helps in improving the teaching and learning process in mathematics subjects.
• It is a comprehensive test that provides feedback to teachers and students on their strengths and weaknesses.
• This test provides a way to identify the part which makes the child slow down in the learning process and provide appropriate feedback to the child.

Hence, from the above-mentioned points, it becomes clear that the purpose of diagnostic testing is to improve the teaching and learning process in mathematics subject.

Teaching Methods: Different methods of teaching mathematics have been proposed by different educators. Knowledge of these methods may help in working out a teaching-learning strategy. A teacher should adopt an approach considering the nature of the children, their interests and maturity, and the resources available.

Key Points Laboratory method:- 

• This method is based on the maxim “learning by doing.”
• This is an activity method and it leads the students to discover mathematics facts.
• In it we proceed from concrete to abstract.
• The laboratory method is a procedure for stimulating the activities of the students and encouraging them to make discoveries.
• This method needs a laboratory in which equipment and other useful teaching aids related to mathematics are available.
• For example, equipment related to geometry, mensuration, mathematical model, chart, balance, various figures, and shapes made up of wood or hardboards, graph paper, etc.
• To teach the area of a circle, the laboratory method is used. The teacher can show them various instruments that are needed for finding out the area of a circle and explain it.
• This method presents mathematics as a practical subject.
• It stimulates the interest of the students to work with concrete material.

Thus, it is concluded that the best appropriate method for teaching the area of a circle is Laboratory method.

Hint

• Inductive method: The establishment of a formula or rule and application of that rule to solve the problems by using example is known inductive method. It involves the learners detecting, or noticing, patterns and working out a 'rule' for themselves before they practice the lesson.
• Deductive Method:  It involves the learners being given a general rule, which is then applied to specific language examples and honed through practice exercises.
• Lecture method:- The teacher-centered method in which students are passive learners.