Curricular Goals(Learning Objectives) & Competencies
Curricular Goals, Competencies and
Illustrative Learning Outcomes (LOs) Issued by CBSE & NCF
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PREPARATORY
STAGE (Nursery to 5nd) |
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CG-1 Understands numbers (counting
numbers and fractions), represents whole numbers using the Indian place value
system, understands and carries out the four basic operations with whole
numbers, and discovers and recognizes patterns in number sequences. |
C-1.1 Represents numbers using the
place-value structure of the Indian number system, appreciates the key role
of zero in this system, compares the sizes of whole numbers, and knows and
can read the names of very large numbers. C-1.2 Represents and compares commonly used
fractions in daily life (such as 1 ⁄ 2, 1 ⁄ 4, etc.) as parts of unit wholes,
as locations on number lines, and as divisions of whole numbers. C-1.3 Identifies relationships amongst
operations and applies the four basic operations on whole numbers to solve
daily life problems. C-1.4 Discovers, recognises, describes, and
extends simple number patterns such as odd numbers, even numbers, square
numbers, cubes, powers of 2, powers of 10, and Virahanka--Fibonacci numbers. |
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CG-2 Analyses the characteristics
and properties of two- and three-dimensional geometric shapes, specifies
locations and describes spatial relationships, and recognises and creates
shapes that have symmetry. |
C-2.1 Identifies, compares, and analyses
attributes of two- and three-dimensional shapes and develops vocabulary to
describe their attributes/properties. C-2.2 Identifies and builds a
three-dimensional object from two-dimensional representations of that object. C-2.3 Describes location and movement using
both common language and mathematical vocabulary; understands the notion of
map (najri naksha). C-2.4 Recognises and creates symmetry
(reflection, rotation) in familiar 2D and 3D shapes. C-2.5 Discovers, recognizes, describes, and
extends patterns in 2D and 3D shapes. |
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CG-3 Understands measurable
attributes of objects and the units, systems, and processes of such
measurement, including those related to distance, length, mass, weight, area,
volume, and time, using non-standard and standard units. |
C-3.1 Measures using non-standard and
standard units and recognises and appreciates the need for standard units. C-3.2 Uses an appropriate unit and tool for
the attribute being measured. C-3.3 Carries out simple unit conversions,
such as from centimetres to metres, within a system of measurement, and
solves daily life problems. C-3.4 Devises strategies for estimating the
distance, length, time, perimeter (for regular and irregular shapes), area
(for regular and irregular shapes), weight and volume. C-3.5 Deduces that shapes having equal areas
can have different perimeters and shapes having equal perimeters can have
different areas. C-3.6 Measures distance, length, perimeter,
time, weight, area, and volume and to solve daily life problems. |
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CG-4 Develops problem solving
skills with procedural fluency, to solve mathematical puzzles as well as daily life problems,
and as a step towards developing computational thinking. |
C-4.1 Solves puzzles and daily life problems
involving one or more operations on whole numbers. C-4.2 Selects appropriate methods and tools
for computing with whole numbers such as mental computation, estimation, or
paper and pencil calculation, in accordance with the context. |
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CG-5 Knows and appreciates the development of numeration
through human history including the major contributions of India. |
C-5.1 Understands the development of the
representation of numbers through human history, from tallying (e.g., on the
Lebombo bones), to Roman numerals, to the Mayan and Babylonian systems,
leading up to the development of zero in India and the modern Indian system
of writing numerals (from Yajurveda, story of Buddha, Bakshali Manuscript,
Vasavadatta, Aryabhatiya, Brahmasphutasiddanta, Gwalior inscription, etc.)
and its transmission to the world (due to Al-Kharizmi, Al-Kindi, Fibonacci,
etc.). |
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MIDDLE
STAGE (6th to 8th) |
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CG-1 Understands numbers and sets
of numbers (Whole numbers, Fractions, Integers, and Rational numbers) looks
for patterns, and appreciates relationships between numbers. |
C-1.1 Develops a sense for and an ability to
manipulate (e.g., read, write, form, compare, estimate, and apply operations)
large whole numbers of up to 10 digits and expresses them in scientific
notation using exponents and powers. C-1.2 Discovers, identifies, and explores
patterns in numbers and describes rules for their formation (e.g., prime
numbers, powers of 3, etc.) and explain relations between different patterns. C-1.3 Explores and understands sets of
numbers such as whole numbers, fractions, integers, and rational numbers, and
their properties. C-1.4 Represents rational numbers in decimal
form as an extension of the Indian system of numeration `past the decimal
point’. C-1.5 Explores the idea of percentage and
apply it in solving problems. C-1.6 Explores and applies fractions (both
as ratios and in decimal form) in daily life situations. |
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CG-2 Understands the concepts of
variable, constant, coefficient, expression, and (one-variable) equation, and
uses these concepts to solve meaningful daily life problems with procedural
fluency. |
C-2.1 Extends the abstract representation of
a number in the form of a variable or an algebraic expression using a
variable. C-2.2 Forms algebraic expressions using
variables, coefficients, and constants, and manipulates them through
addition, subtraction, and multiplication. C-2.3 Poses and solves linear equations to
find the value of an unknown, including to solve puzzles and word problems. C-2.4 Develops own methods to solve puzzles
and problems using algebraic thinking. |
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CG-3 Understands, formulates, and
applies properties and theorems regarding simple geometric shapes (2D and
3D). |
C-3.1 Describes, classifies, and understands
relationships among different types of two and three-dimensional shapes using
their defining properties/attributes. C-3.2 Knows properties of lines, angles,
triangles, quadrilaterals, and polygons, and applies them to solve related
problems. C-3.3 Identifies attributes of
three-dimensional shapes (cubes, parallelepipeds, cylinders, cones, etc.) and
uses two-dimensional representations of three-dimensional objects to
visualise and solve problems. C-3.4 Draws and constructs geometric shapes
such as lines, parallel lines, angles, and simple triangles, with specified
properties, using compass and straightedge. |
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CG-4: Develops understanding
of perimeter and area for 2D shapes and uses them to solve day-to-day life
problems. |
C-4.1 Identifies, selects, and uses units of
appropriate size and type to measure and examine the relationship between
perimeter and area for 2D shapes (both regular and irregular shapes). C-4.2 Discovers, understands, and uses
formulas to determine the circumference of a circle and the area of a
triangle, parallelogram, and trapezium, and develops strategies to find the
areas of more complex 2D shapes. C-4.3 Explores and uses Baudhayana’s Theorem
on right triangles and other fundamental geometric theorems to solve puzzles
and everyday problems. C-4.4 Discovers and constructs tilings of
the plane using 2D shapes and identifies and appreciates their appearances in
art in India and around the world. C-4.5 Develops the notion of fractal and
identifies and appreciates the appearances of fractals in nature and art in
India and around the world. |
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CG-5 Collects, organises,
represents (graphically and in tables), and interprets data/ information from
daily life experiences. |
C-5.1 Collects, organises data, and applies
measures of central tendencies such as average/mean, mode, and median. C-5.2 Selects, creates, and uses appropriate
graphical representations of data, including pictographs, bar graphs,
histograms, line graphs, and pie charts. |
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CG-6 Develops mathematical thinking
and the ability to logically and precisely communicate mathematical ideas. |
C-6.1 Applies both inductive and deductive
logic to formulate definitions and conjectures, evaluates and produces
convincing arguments/proofs to turn these definitions and conjectures into
theorems or correct statements, particularly in the areas of algebra,
elementary number theory, and geometry. |
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CG-7 Engages with puzzles and
mathematical problems and develops own creative methods and strategies to
solve them. |
C-7.1 Applies creativity to develop one’s
own solutions to puzzles and other problems and appreciates the work of
others to develop their own solutions. C-7.2 Engages in and appreciates the
artistry and aesthetics of puzzle-making, puzzle-posing, and puzzle-solving. |
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CG-8 Knows and appreciates the
development of mathematical ideas over human history, and the contributions
of past and modern mathematicians from India and across the world. |
C-8.1 Recognises important mathematical
contributions of India (e.g., zero, Indian numerals, ideas around infinity,
concepts of algebra, etc.) as well as the contributions of specific Indian
mathematicians (such as Baudhayana, Panini, Pingala, Aryabhata,
Brahmagupta, Virahanka, Bhaskara, Madhava, and Ramanujan). C-8.2 Recognizes and appreciates how
concepts (like the notion of number, from counting numbers, to 0, to negative
numbers, to rational evolved over a period of time across different
civilizations. |
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CG-9 Develops basic skills and
capacities of computational thinking, namely, decomposition, pattern
recognition, data representation, generalization, abstraction, and
algorithms, in order to solve problems where such techniques of computational
thinking are effective. |
C-9.1 Approaches problems using programmatic
thinking techniques such as iteration, symbolic representation, and logical
operations and reformulates problems into series of ordered steps
(algorithmic thinking). C-9.2 Identifies, analyses, and implements
possible solutions to problems, with the goal of achieving the most efficient
and effective combination of steps and resources and generalizes this process
to a wide variety of problems. |
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SECONDARY
STAGE (9th & 10th) |
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CG-1 Understands numbers, ways of
representing numbers, relationships among numbers, and number sets. |
C-1.1 Develops a deeper understanding of
numbers, including the set of real numbers and its properties. C-1.2 Uses deductive logic to prove theorems
such as ‘√2 is an irrational number’ and `there are infinitely many prime
numbers’. C-1.3 Uses inductive logic to prove theorems
such as the recursion relation for Virahanka numbers, `the sum of consecutive
odd numbers starting with 1 is a square number’, `the sum of consecutive
cubes starting with 1 is the square of a triangular number’, etc. C-1.4 Explores that every counting number
has a unique factorisation into prime numbers (fundamental theorem of
arithmetic). C-1.5 Recognises and appropriately uses
powers and exponents. C-1.6 Computes powers and roots and applies
them to solve problems. C-1.7 Computes simple and compound interest
and solve real-life problems. |
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CG-2 Discovers and proves algebraic
identities and uses such identities to solve equations. |
C-2.1 Learns the art of factoring
polynomials. C-2.2 Applies the division algorithm to both
integers and polynomials in order to solve problems such as those involving
GCDs and LCMs. C-2.3 Models and solves contextualised
problems using equations (e.g., simultaneous linear equations in two
variables or single polynomial equations) and draws conclusions about a
situation being modelled. |
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CG-3 Analyses characteristics and
properties of two-dimensional geometric shapes and develops mathematical
arguments to explain geometric relationships. |
C-3.1 Describes relationships including
congruence of two-dimensional geometric shapes (such as lines, angles,
triangles) to make and test conjectures and solve problems. C-3.2 Proves theorems using Euclid’s axioms
and postulates – for triangles, quadrilaterals, and circles and applies them
to solve geometric problems. C-3.3 Specifies locations and describes
spatial relationships using coordinate geometry, e.g., plotting a pair of
linear equations and graphically finding solution, or finding the area of
triangle with given coordinates as vertices. |
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CG-4 Derives and uses formulas to
calculate areas of plane figures, and surface areas and volumes of solid
objects. |
C-4.1 Visualises, represents, and calculates
the area of a triangle using Heron’s formula. C-4.2 Visualises and uses mathematical
thinking to discover formulas to calculate surface areas and volumes of solid
objects (cubes, cuboids, spheres, hemispheres, right circular
cylinders/cones, and their combinations). |
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CG-5 Analyses and interprets data
using statistical concepts (such as measures of central tendency, standard
deviations) and probability. |
C-5.1 Applies measures of central tendencies
such as mean, median, and mode. C-5.2 Applies concepts from probability to
solve problems on the likelihood of everyday events. |
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CG-6 Begins to perceive and
appreciate the axiomatic and deductive structure of mathematics. Uses stated
assumptions, axioms, postulates, definitions, and mathematics vocabulary to
prove mathematical statements and carry out geometric constructions. |
C-6.1 Uses deductive and inductive logic to
prove theorems about numbers, measurements such as areas and shapes. C-6.2 Visualises and appreciates geometric
proofs for algebraic identities and other `proofs without words’. C-6.3 Proves theorems using Euclid’s axioms
and postulates– for angles, triangles, quadrilaterals, circles, area-related
theorems for triangles and parallelograms. C-6.4 Constructs different geometrical
shapes like bisectors of line segments, angles and their bisectors,
triangles, and other polygons, satisfying given constraints. |
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CG-7 Appreciates important
contributions of mathematicians from India and around the world. |
C-7.1 Recognises the important contributions
made by Indian mathematicians in the field of mathematics. C-7.2 Recognizes how concepts (like
evolution of numbers, geometry, etc.) evolved over a period of time across
different civilizations. |
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CG-8 Sharpens skills such as
visualisation, optimisation, representation, and mathematical modelling, and
their application in daily life. |
C-8.1 Models daily life phenomena and uses
representations such as graphs, tables, and equations to draw conclusions. C-8.2 Uses two-dimensional representations
of three-dimensional objects to visualise and solve problems such as those
involving surface area and volume. C-8.3 Employs optimisation strategies to
maximise desired quantities (such as area, volume, or other output) under
given constraints. |
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CG-9 Develops
computational thinking, i.e., deals with complex problems and is able to
break them down into a series of simple problems that can then be solved by
suitable procedures/algorithms. |
C-9.1 Decomposes a problem into sub
problems. C-9.2 Describes and analyses a sequence of
instructions being followed. C-9.3 Analyses similarities and differences
among problems to make one solution or procedure work for multiple problems. C-9.4 Engages in algorithmic problem solving
to design such solutions. |
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CG-10 Explores
connections of mathematics with other subjects. |
C-10.1 Applies mathematical knowledge and
tools to analyse problems/situations in multiple subjects across science,
social science, visual arts, music, and sports. |
Introduction:
The learning outcomes of mathematics play a
crucial role in education and have broad implications for individuals and
society. Here are some key reasons highlighting the importance of learning
outcomes in mathematics.
learning outcomes of mathematics are essential
for individual development, societal progress, and global competitiveness. They
guide the teaching and assessment of mathematical concepts, ensuring that
students acquire the skills and knowledge necessary for success in various
aspects of life.
Here are some key reasons why learning
outcomes in mathematics are important:
Curricular Goal
Understands numbers (counting numbers and fractions), represents
whole numbers using the Indian place value system, understands and carries out
the four basic operations with whole numbers, and discovers and recognizes
patterns in number sequences.
Competency
Represents numbers using the place-value structure of the Indian
number system, appreciates the key role of zero in this system, compares the
sizes of whole numbers, and knows and can read the names of very large numbers.
Represents numbers using the place-value structure of the Indian number system, appreciates the key role of zero in this system, compares the sizes of whole numbers, and knows and can read the names of very large numbers.
GRADE 3 (LOs)
Recognises, reads, and writes number names and numerals up to 999 using place value concept.
Compares and forms the greatest and smallest three-digit number (with and without repetition of given digits) using the place value concept.
GRADE 4 (LOs)
Recognises,
reads, and writes number names and numerals up to 9999 using place value
concept.
Compares and forms the greatest and smallest four-digit number (with and without repetition of given digits) using the place value concept.
GRADE 5 (LOs)
Reads, writes, and compares numbers bigger than 9999 (being used in her/his surroundings) using Indian numeration system.
Poses and solves linear equations to find the value of an unknown, including to solve puzzles and word problems.
Curricular Goal
Understands the concepts of variable, constant, coefficient,
expression, and (one-variable) equation, and uses these concepts to solve
meaningful daily life problems with procedural fluency.
Competency
Poses and solves linear equations to find the value of an unknown,
including to solve puzzles and word problems.
GRADE 6 (LOs)
Uses
variable(s) to write down formulas and equation.
GRADE 7 (LOs)
Uses
number and variable with different operations and expresses a real-life
situation in the form of a simple linear equation and vice versa. Uses trial
and error method and determines the solution of a simple equation.
GRADE 8 (LOs)
Reads, writes, and compares numbers bigger than 9999 (being used in her/his surroundings) using Indian numeration system. Writes simple contextual problems as linear equations in one variable, finds its solution, and verifies.
Curricular Goal
Begins to perceive and appreciate the axiomatic and deductive
structure of mathematics. Uses stated assumptions, axioms, postulates,
definitions, and mathematics vocabulary to prove mathematical statements and
carry out geometric constructions.
Competency
Uses deductive and inductive logic to prove theorems about numbers, measurements (such as areas), and shapes.
Uses deductive and inductive logic to prove theorems about numbers,
measurements (such as areas), and shapes.
GRADE 9 (LOs)
Applies
deductive logic to prove theorems related to parallel lines. Applies deductive
logic to proves theorems related to triangles, congruence of triangles.
GRADE 10 (LOs)
Applies deductive logic to prove statements like - √2 is an irrational number, sum of two odds is even etc. Applies deductive logic to prove theorems related to properties of quadrilaterals, areas of parallelograms and triangles, mid-point theorem and theorems related to circles.
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