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Rationalization of School Curriculum

Background

The conventional teacher centred and rote learning form of education has served us well through ages. As the education system in Bhutan embraces the 21st Century education framework and principles, it warrants a paradigm shift in curriculum design and development, including the pedagogy, commensurate the competency based learning. An approach, which underscores that learning in the 21st century, is for the development of competencies through active engagement of learners in learning experiences, guided by formation and utilisation of “working knowledge”. This empowers learners to take responsibilities of their learning and develop “portable skills or soft skills,” such as critical thinking, creativity, communication and collaboration, vital for all as individuals with unique talent and competencies. The current culture of curriculum design and practices in schools, however, do not render condition to facilitate realisation of the national aspiration of nurturing “nationally rooted and globally competent” citizen. Amongst others, it has always been a concern for REC on the extent, relevancy and quality of the curriculum in all subjects. Thus, in order to facilitate quality learning for 21st Century education, REC has initiated major curriculum reform in all subjects.

Rationale

The Bhutan Education Blueprint 2014-2024 indicated that the existing curriculum was ‘heavy’. This was echoed as one of the major pointers in the National School Curriculum Conference 2016 that the curriculum was ‘vast’. These findings led to the need for curriculum “thinning” [Resolution 3.1.10 (IV)]. In response to these findings, REC started the rationalization of the existing curriculum by reviewing and screening out the obsolete and irrelevant content, and updating them with the most recent information and also rectifying errors in the textbooks. Therefore, some portions of the syllabi from several subjects, for instance, have been dropped. The rationalization or thinning of curriculum is one of the important considerations made while developing new textbooks based on new curriculum frameworks.

The curriculum rationalization process also aligns very well with Resolution 13 of the National Education Conference 2018 of “Doing away with the Saturday classes”. The para 13.4 of the resolution requires ‘REC to work on curriculum thinning and review of time allocation for each subject’. This resolution has further facilitated REC to expedite the curriculum rationalization and review the time and period allocation for each subject. Rationalization of the school curricula is based on the following strategies:

i. Review the goals and outcomes of each subject to identify topics, chapters, learning activities, exercises and assessment. ii. Develop rationalized syllabus for each subjects ensuring conceptual linkages and progression within the chapter or topic in the textbooks.

iii. Minimize lexical density in text by reducing heavy textual materials from the textbooks. iv. Remove topics, learning activities or assessment items, which are redundant, overlapping irrelevant or inappropriate. v. Delete irrelevant or inappropriate illustrations or diagrams, and examples from the text.

vi. Update and align the content width and depth with the teaching time available for each subject.

vii. The revised syllabi for each subject are categorised and compiled under four subject classifications, namely STEM, Social Sciences, Language, and TVET & Commercial Studies.

The review of the instructional time allocation is based on the following criteria:

i. Maintain the instructional time requirement at the international standard. ii. Maintain gradual increase of instructional time across most of the key stages.

iii. Reduce the instructional time for each subject across the grades based on the doing away of the Saturday classes. iv. Allocate time for personal development learning areas, such as HPE, Arts Education, Values Education, CGC, TVET Program (clubs and PVOP). v. Non-curricular activities and programmes are to be conducted outside the instructional hours.

vi. Calculate 150 actual curricular instructional days (excluding examination days in June and November months) in an academic year based on 5 working days per week.

vii. Calculation of instructional time is based on 8 periods a day of 40 minutes each.

Conclusion Instructional time refers to the actual contact time in the classroom. This is the minimum time available for the delivery of the curriculum including assessment. Instructional Time equals to number of days multiplied by number of periods per day times duration of one period (180 x 8 x 40). The rationalization of the curriculum is based on 150 days of the actual instructional time.

Instructional days are the total number of days within which the curricular activities are conducted. Within these days, a maximum of 5.33 hours (320 minutes) are available for actual classroom instruction per day. This calculation is based on 8 periods a day of 40 minutes each. The average instructional time in the OECD countries ranges from 799 to 915 hours per year. This includes all the educational activities that happen in the school in a day. However, the calculation of instructional time for the rationalized curriculum is based on the actual contact time for curriculum delivery, which has resulted in more instructional time than in OECD countries.

viii.

Lastly, it must be noted that the instructional time and days are suggested guide. Thus, it is envisaged that schools will make adjustment in instructional time as deemed applicable.

Class: VII Subject: Mathematics All the additions and omissions mentioned are recommended by the Expert Group during Pre-Conference of the National School Curriculum Conference 2016 and further agreed during the review of School Mathematics Curriculum Framework. All the games are extra opportunities to help deepen the conceptual understanding. Teachers are advised to use the games as per the understanding level of the students. If the formative assessments reveal that the students have not adequately understood certain concepts, then the games can be used to deepen the understandings. No questions are to be asked from the games. The connections lessons, although good in their own right, give a sense that it’s the only way to make connections. Students should get the sense that there are many different ways to make connections. Without these lessons also understanding of the concepts can be achieved. The connections lessons are also to be used to deepen the understanding of concepts. Some topics. Sub-topics, explore lessons, connections and games that may not the used are mentioned below. Also it is to be noted that it’s not compulsory to answer all questions in the practising and applying.

Sl No Units Time

(min) Weighting (%) Changes Reasons

1

UNIT 1 NUMBER Getting Started Chapter 1 Whole Numbers and Decimals 1.1.1 EXPLORE: Divisibility by 3 and 9 1.1.2 Divisibility Tests CONNECTIONS: Casting Out Nines GAME: Divisibility Spin 1.1.3 Lowest Common Multiple 1.1.4 Greatest Common Factor CONNECTIONS: Carrom Math 936 13

Delete connections: Casting Out Nines Delete GAME: Divisibility Spin Delete connections: Carrom Math Conceptual

understanding can be achieved without these lessons. Chapter 2 Powers

1.2.1 Introducing Powers GAME: Rolling Powers 1.2.2 Expanded, Standard, and Exponential Forms

Delete GAME: Rolling Powers

Chapter 3 Decimal Operations 1.3.1 Multiplying Decimals 1.3.2 Dividing Decimals 1.3.3 EXPLORE: Mental Math with Decimals 1.3.4 Order of Operations

Sl No Units Time


2

UNIT 2 FRACTIONS Getting Started Chapter 1 Fraction Addition and Subtraction 2.1.1 Comparing and Ordering Fractions 2.1.2 Adding Fractions Using Models 2.1.3 Adding Fractions and Mixed Numbers Symbolically GAME: A “Whole” in One 2.1.4 Subtracting Fractions and Mixed Numbers 2.1.5 Subtracting Mixed Numbers in Different Ways

864 12

Merge lessons 2.1.1 and 2.1.3 and teach using the Concrete, Representational/Pictorial and Abstract Strategy. https://notafraidtolearn.wordpress.com/mathematics-the-cra-approach/ Delete GAME: A “Whole” in One

Both lessons deal with adding fractions but using different strategies. There need not be two separate lessons.

Chapter 2 Fraction Multiplication and Division 2.2.1 Multiplying a Fraction by a Whole Number 2.2.2 Dividing a Fraction by a Whole Number

Chapter 3 Relating Fractions and Decimals 2.3.1 Naming Fractions and Mixed Numbers as Decimals 2.3.2 EXPLORE: Relating Repeating Decimals and Fractions CONNECTIONS: Repeating-Decimal Graphs

3

UNIT 3 RATIO, RATE, AND PERCENT Getting Started Chapter 1 Ratio and Rate 3.1.1 Solving Ratio Problems 3.1.2 Solving Rate Problems

792 11

Chapter 2 Percent 3.2.1 Percent as a Special Ratio 3.2.2 Relating Percents, Fractions, and Decimals CONNECTIONS: The Golden Ratio GAME: Ratio Concentration 3.2.3 Estimating and Calculating Percents 3.2.4 EXPLORE: Representing Numbers Using Percents

Delete GAME: Ratio Concentration

4

UNIT 4 GEOMETRY AND MEASUREMENT Getting Started Chapter 1 Angle Relationships 4.1.1 EXPLORE: Angles in a Triangle CONNECTIONS: Angle Measurement Units 4.1.2 Drawing and Classifying Triangles 4.1.3 Constructing and Bisecting Angles

https://notafraidtolearn.wordpress.com/mathematics-the-cra-approach/




Sl No Units Time


4

Chapter 2 Transformations 4.2.1 Translations 4.2.2 Reflections GAME: Reflection Archery 4.2.3 Rotations

1152 16

Delete GAME: Reflection Archery Not required

Chapter 3 3-D and 2-D Measurement 4.3.1 Measurement Units 4.3.2 Volume of a Rectangular Prism 4.3.3 Area of a Composite Shape 4.3.4 Area of a Trapezoid 4.3.5 Circumference of a Circle

5

UNIT 5 INTEGERS Getting Started Chapter 1 Representing Integers 5.1.1 Integer Models 5.1.2 Comparing and Ordering Integers CONNECTIONS: Time Zones 5.1.3 The Zero Property

864 12

Delete CONNECTIONS: Time Zones Merge 5.1.3 The Zero Property with 5.2.1 Adding Integers Using the Zero Property

Time Zones is covered in Geography Class 6 and Class 7 Zero Property is must while adding Integers, so before actually starting/doing lessons 5.2.1 and 5.2.2 or 5.2.3 and 5.2.4 Zero Property must be known to make appropriate connection.

Chapter 2 Adding and Subtracting Integers 5.2.1 Adding Integers Using the Zero Property 5.2.2 Adding Integers that are Far from Zero GAME: Target Sum –50 5.2.3 Subtracting Integers Using Counters 5.2.4 Subtracting Integers Using a Number Line 5.2.5 EXPLORE: Integer Representations

Delete GAME: Target –50 Merge 5.2.5 EXPLORE: Integer Representation with 5.1.1

6

UNIT 6 ALGEBRA Getting Started Chapter 1 Patterns and Relationships 6.1.1 Using Variables to Describe Pattern Rules 6.1.2 Creating and Evaluating Expressions 6.1.3 Simplifying Expressions CONNECTIONS: Using Variables to Solve Number Tricks

1296 18

Sl No Units Time


6

Chapter 2 Solving Equations 6.2.1 Solving Equations Using Models 6.2.2 Solving Equations Using Guess and Test 6.2.3 Solving Equations Using Inverse Operations GAME: Equations, Equations 6.2.4 EXPLORE: Solving Equations Using Reasoning

Delete GAME: Equations, Equations

Not really appropriate

Chapter 3 Graphical Representations 6.3.1 Graphing a Relationship 6.3.2 Examining a Straight Line Graph 6.3.3 Describing Change on a Graph 6.3.4 EXPLORE: Are all Relationship Graphs Straight Lines?

7

UNIT 7 PROBABILITY AND DATA Getting Started Chapter 1 Probability 7.1.1 Determining Theoretical Probability 7.1.2 EXPLORE: Experimental Probability 7.1.3 Matching Events and Probabilities GAME: No Tashi Ta-gye!

1296 18 Status Quo

Chapter 2 Collecting Data 7.2.1 Formulating Questions to Collect Data 7.2.2 Sampling and Bias CONNECTIONS: Estimating a Fish Population 7.2.3 EXPLORE: Conducting a Survey

Chapter 3 Graphing Data 7.3.1 Circle Graphs 7.3.2 Histograms

Chapter 4 Describing and Analysing Data 7.4.1 Mean, Median, Mode, and Range 7.4.2 Outliers and Measures of Central Tendency

Total 7200 100%

Class: VIII Subject: Mathematics All the additions and omissions mentioned are recommended by the Expert Group during Pre-Conference of the National School Curriculum Conference 2016 and further agreed during the review of School Mathematics Curriculum Framework. All the games are extra opportunities to help deepen the conceptual understanding. Teachers are advised to use the games as per the understanding level of the students. If the formative assessments reveal that the students have not adequately understood certain concepts, then the games can be used to deepen the understandings. No questions are to be asked from the games. The connections lessons, although good in their own right, give a sense that it’s the only way to make connections. Students should get the sense that there are many different ways to make connections. Without these lessons also understanding of the concepts can be achieved. The connections lessons are also to be used to deepen the understanding of concepts. Some topics. Sub-topics, explore lessons, connections and games that may not the used are mentioned below. Also it is to be noted that it’s not compulsory to answer all questions in the practising and applying. Sl No Units Time


1

UNIT 1 NUMBER Getting Started Chapter 1 Powers 1.1.1 Negative Exponents GAME: Getting to a Half 1.1.2 Scientific Notation 792 11

Delete GAME: Getting to a Half

Chapter 2 Square Roots 1.2.1 Perfect Squares 1.2.2 EXPLORE: Squaring Numbers Ending in 5 1.2.3 Interpreting Square Roots 1.2.4 Estimating and Calculating Square Roots CONNECTIONS: The Square Root Algorithm

Merge 1.2.2 EXPLORE: Squaring Numbers Ending in 5 with 1.2.4

2

UNIT 2 PROPORTION AND PERCENT Getting Started Chapter 1 Proportions 2.1.1 Solving Proportions 2.1.2 EXPLORE: Scale Drawings and Similar Figures

792 11 Status Quo

Chapter 2 Percent 2.2.1 Percents Greater Than 100% 2.2.2 Solving Percent Problems GAME: Equivalent Concentration 2.2.3 Fractional Percents 2.2.4 Solving Percent Problems Using Familiar Percents

Sl No Units Time


2 Chapter 3 Consumer Problems 2.3.1 Mark-Up and Discount Consumer Problems 2.3.2 Simple Interest and Commission CONNECTIONS: Currency Conversion

3

UNIT 3 INTEGERS Getting Started Chapter 1 Multiplying Integers 3.1.1 Multiplying Integers Using Counters and Patterns 3.1.2 Multiplying Integers Using a Number Line 3.1.3 EXPLORE: Pattern Grids 3.1.4 Renaming Factors to Multiply Mentally GAME: Order the Integers

864 12 Status Quo

Chapter 2 Dividing Integers 3.2.1 Dividing Integers Using Models and Patterns 3.2.2 Relating Division of Integers to Multiplication CONNECTIONS: Mean Temperatures 3.2.3 Order of Operations with Integers GAME: Target

4

UNIT 4 FRACTIONS AND RATIONAL NUMBERS Getting Started Chapter 1 Adding and Subtracting Fractions 4.1.1 Adding and Subtracting Fractions Mentally 4.1.2 Adding and Subtracting Fractions Symbolically 864 12

Chapter 2 Multiplying and Dividing Fractions 4.2.1 EXPLORE: Multiplying Fractions 4.2.2 Multiplying Fractions CONNECTIONS: The Sierpinski Triangle 4.2.3 Multiplying Mixed Numbers

Merge 4.2.1 EXPLORE: Multiplying Fractions with 4.2.2 Multiplying Fractions

4

4.2.4 Dividing Fractions With a Common Denominator 4.2.5 Dividing Fractions in Other Ways 4.2.6 Dividing Mixed Numbers

Chapter 3 Rational Numbers 4.3.1 Introducing Rational Numbers 4.3.2 Operations with Rational Numbers 4.3.3 Order of Operations GAME: Target One

Delete GAME: Target One

Sl No Units Time


5

UNIT 5 MEASUREMENT Getting Started Chapter 1 The Pythagorean Theorem 5.1.1 The Pythagorean Theorem 5.1.2 Applying the Pythagorean Theorem

936 13 Status Quo

Chapter 2 Linear and Area Relationships 5.2.1 Area and Perimeter Relationships CONNECTIONS: Pentominos GAME: Pentominos 5.2.2 Scale Drawings 5.2.3 EXPLORE: Estimating the Area of a Circle 5.2.4 The Formula for the Area of a Circle CONNECTIONS: The History of Pi 5.2.5 Applying Area Formulas CONNECTIONS: Tangrams

Chapter 3 Volume and Surface Area 5.3.1 Volume of a Rectangular Prism 5.3.2 Surface Area of a Rectangular Prism

6

UNIT 6 PROBABILITY AND DATA Getting Started Chapter 1 Probability 6.1.1 Complementary Events GAME: Unlucky Ones CONNECTIONS: Simpson's Paradox 6.1.2 Simulations

936 13

Delete CONNECTIONS: Simpson's Paradox Delete 6.1.2 Simulations

Simpson’s Paradox is not required at this stage. Simulations is moved to class 9 keeping in mind the cognitive competence required

Chapter 2 One-Variable Data 6.2.1 EXPLORE: Sample Size 6.2.2 Selecting a Random Sample 6.2.3 Circle Graphs 6.2.4 Box and Whisker Plots 6.2.5 EXPLORE: The Impact of Altering a Data Set

Chapter 3 Two-Variable Data 6.3.1 EXPLORE: The Relationship Between Two Variables 6.3.2 Using a Scatter Plot to Represent a Relationship

Sl No Units Time


7

UNIT 7 ALGEBRA Getting Started Chapter 1 Describing Relationships 7.1.1 EXPLORE: Representing Relationships 7.1.2 Describing Relationships and Patterns 7.1.3 Recognizing Linear Relationships CONNECTIONS: Adding Values in a Linear Relationship 7.1.4 Slope 936 13 Delete GAME: Alge-

Scrabble Not required Chapter 2 Solving Linear Equations 7.2.1 Solving an Equation Using Inverse Operations 7.2.2 Using an Equation to Solve a Problem 7.2.3 Solving a Problem Involving Two Relationships GAME: Alge-Scrabble Chapter 3 Linear Polynomials 7.3.1 Adding Polynomials 7.3.2 Subtracting Polynomials 7.3.3 EXPLORE: Multiplying a Polynomial by an Integer

8

UNIT 8 GEOMETRY Getting Started Chapter 1 Representing Objects 8.1.1 Isometric Drawings 8.1.2 Orthographic Drawings

1080 15

Chapter 2 Transformations 8.2.1 Dilatations 8.2.2 Combining Transformations GAME: Isometry

Delete GAME: Isometry Not relevant

Chapter 3 Angle Relationships 8.3.1 EXPLORE: Measuring Angles in Polygons 8.3.2 Angles in Polygons 8.3.3 Angles With Parallel Line and Intersecting Lines CONNECTIONS: Tools for Geometry

Delete CONNECTIONS: Tools for Geometry

The lesson is about carpentry tools than about Maths in Geometry

Total 7200 100

Class: IX Subject: Mathematics All the additions and omissions mentioned are recommended by the Expert Group during Pre-Conference of the National School Curriculum Conference 2016 and further agreed during the review of School Mathematics Curriculum Framework. All the games are extra opportunities to help deepen the conceptual understanding. Teachers are advised to use the games as per the understanding level of the students. If the formative assessments reveal that the students have not adequately understood certain concepts, then the games can be used to deepen the understandings. No questions are to be asked from the games. The connections lessons, although good in their own right, give a sense that it’s the only way to make connections. Students should get the sense that there are many different ways to make connections. Without these lessons also understanding of the concepts can be achieved. The connections lessons are also to be used to deepen the understanding of concepts. Some topics. Sub-topics, explore lessons, connections and games that may not the used are mentioned below. Also it is to be noted that it’s not compulsory to answer all questions in the practising and applying. Sl No Units Time


1

UNIT 1 NUMBER AND OPERATIONS Getting Started Chapter 1 Exponents 1.1.1 Introducing the Exponent Laws GAME: Rolling Powers 1.1.2 The Power Law of Exponents 1.1.3 Negative and Zero Exponents 1.1.4 Fractional Exponents Chapter 2 Scientific Notation 1.2.1 Scientific Notation with Large Numbers CONNECTIONS: The Richter Scale 1.2.2 Scientific Notation with Small Numbers Chapter 3 Rational and Real Numbers 1.3.1 Estimation with Rational Numbers 1.3.2 Order of Operations 1.3.3 Square Roots 1.3.4 EXPLORE: Representing Square Roots 1.3.5 Representing Real Numbers

900 15 Delete GAME: Rolling Powers

Sl No Units Time


2

UNIT 2 POLYNOMIALS Getting Started Chapter 1 Introducing Polynomials 2.1.1 Interpreting Polynomials 2.1.2 Adding and Subtracting Polynomials Chapter 2 Multiplying Polynomials 2.2.1 Multiplying a Polynomial by a Monomial 2.2.2 Multiplying a Binomial by a Binomial 2.2.3 Multiplying Polynomials Symbolically GAME: Polyprod Chapter 3 Dividing Polynomials 2.3.1 Dividing a Polynomial by a Monomial 2.3.2 EXPLORE: Dividing a Polynomial by a Binomial 2.3.3 Dividing a Polynomial by a Binomial 2.3.4 EXPLORE: Creating Rectangles to Factor CONNECTIONS: Using Number Patterns to Factor

840 14

Merge 2.3.2 EXPLORE: Dividing a Polynomial by a Binomial with 2.3.3 Dividing a Polynomial by a Binomial Delete GAME: Polyprod Delete CONNECTIONS: Using Number Patterns to Factor

Note: Algebra Tiles are to be used to represent the algebraic expressions and equations to reduce the degree of abstractness of algebraic expressions and equations.

3

UNIT 3 LINEAR RELATIONS AND EQUATIONS Getting Started Chapter 1 Linear and Non-Linear Relation Graphs 3.1.1 Patterns and Relations in Tables 3.1.2 Scatter Plots of Discrete and Continuous Data 3.1.3 EXPLORE: Graphs of Linear and Non-Linear Relations CONNECTIONS: Half-Life 3.1.4 Graphs of Linear and Non-Linear Relations Chapter 2 Equation of a Line 3.2.1 The Meaning of Slope and Y-Intercept 3.2.2 EXPLORE: The Equation of a Line 3.2.3 Slope and Y-Intercept Form 3.2.4 The Line of Best Fit 3.2.5 Standard Form Chapter 3 Linear Equations and Inequalities 3.3.1 Solving Linear Equations Algebraically GAME: Equation Concentration 3.3.2 Solving Linear Inequalities 3.3.3 Solving Linear Equations Graphically 3.3.4 Solving a System of Linear Equations

1200 20

Merge 3.1.3 EXPLORE: Graphs of Linear and Non-Linear Relations with 3.1.4 Graphs of Linear and Non-Linear Relations Delete CONNECTIONS: Half-Life Delete GAME: Equation Concentration

Both 3.1.3 and 3.1.4 are lessons on Graphs of Linear and Non Linear Relations Half-life is studied in Physics

Sl No Units Time


4

UNIT 4 DATA AND PROBABILITY Getting Started Chapter 1 Displaying and Analysing Data 4.1.1 Constructing Familiar Data Displays 4.1.2 Using Graphs to Compare and Organize Data 4.1.3 Using Graphs to Examine Change 4.1.4 Misleading Graphs 4.1.5 Drawing Conclusions from Graphs Chapter 2 Probability 4.2.1 Determining and Comparing Probabilities 4.2.2 Calculating Probability of Two Independent Events GAME: On a Roll 4.2.3 Randomness: Experimental Versus Theoretical Results 4.2.4 Conducting a Simulation 4.2.5 EXPLORE: Designing a Simulation CONNECTIONS: Computer Simulations

720 12

Delete GAME: On a Roll Delete 4.2.5 EXPLORE: Designing a Simulation Delete CONNECTIONS: Computer Simulations

4.2.4 is adequate for simulation

5

UNIT 5 GEOMETRY Getting Started Chapter 1 Similarity and Congruence 5.1.1 EXPLORE: Unique Triangles 5.1.2 Congruent Triangles 5.1.3 Similar Triangles 5.1.4 Solving Problems with Similarity Chapter 2 Transformations 5.2.1 Translations 5.2.2 Reflections and Rotations GAME: Shards 5.2.3 Dilatations CONNECTIONS: Making an Animated Movie 5.2.4 Combining Transformations

840 14 Delete GAME: Shards Not required

6

UNIT 6 MEASUREMENT Getting Started Chapter 1 Volume and Capacity 6.1.1 Volume of Prisms and Cylinders GAME: Bean Counting 6.1.2 EXPLORE: Comparing Pyramid and Prism Capacities

900 15 Delete GAME: Bean Counting

Sl No Units Time

(min) Weighting (%) Changes Reasons 6.1.3 Volume of Pyramids and Cones 6.1.4 Volume of Spheres and Composite Shapes CONNECTIONS: Perspective Chapter 2 Surface Area 6.2.1 Surface Area of Prisms 6.2.2 Surface Area of Pyramids 6.2.3 Surface Area of Cylinders 6.2.4 Surface Area of Cones 6.2.5 Surface Area of Spheres

7

UNIT 7 COMMERCIAL MATH Getting Started Chapter 1 Household Finances 7.1.1 Income and Expenditures 7.1.2 Budgets GAME: Lucky Shopper Chapter 2 Taxes 7.2.1 Reporting Income and Taxes 7.2.2 Income Deductions 7.2.3 EXPLORE: Income Tax Rates CONNECTIONS: Taxation around the World

600 10 Delete GAME: Lucky Shopper

TOTAL 6000 100

Class: X Subject: Mathematics All the additions and omissions mentioned are recommended by the Expert Group during Pre-Conference of the National School Curriculum Conference 2016 and further agreed during the review of School Mathematics Curriculum Framework. All the games are extra opportunities to help deepen the conceptual understanding. Teachers are advised to use the games as per the understanding level of the students. If the formative assessments reveal that the students have not adequately understood certain concepts, then the games can be used to deepen the understandings. No questions are to be asked from the games. The connections lessons, although good in their own right, give a sense that it’s the only way to make connections. Students should get the sense that there are many different ways to make connections. Without these lessons also understanding of the concepts can be achieved. The connections lessons are also to be used to deepen the understanding of concepts. Some topics. Sub-topics, explore lessons, connections and games that may not the used are mentioned below. Also it is to be noted that it’s not compulsory to answer all questions in the practising and applying. Sl No Units Time


1

UNIT 1 MATRICES AND NETWORKS Getting Started Chapter 1 Matrices 1.1.1 Introducing Matrices 1.1.2 Adding and Subtracting Matrices 1.1.3 Multiplying a Matrix by a Scalar 1.1.4 Multiplying Matrices Chapter 2 Networks 1.2.1 EXPLORE: Travelling Networks CONNECTIONS: The Seven Bridges of Konigsberg GAME: Sprouts 1.2.2 Describing a Network with a Matrix 1.2.3 Solving Network Problems

660 11 Delete GAME: Sprouts

2

UNIT 2 COMMERCIAL MATH AND NUMBER Getting Started Chapter 1 Commercial Math 2.1.1 Purchasing Decisions 2.1.2 Compound Interest CONNECTIONS: The Rule of 72 GAME: Target 200 2.1.3 Dividends and Stocks 2.1.4 Using Commercial Math

720 12

Delete 2.1.4 Using Commercial Math Delete CONNECTIONS: The Rule of 72 Delete GAME: Target 200

2.1.4 is not required separately but can be covered while doing 2.1.1, 2.1.2 and 2.1.3. Use the questions in 2.1.4 while teaching 2.1.2 and 2.1.3 Rule of 72 need not be highlighted separately

Sl No Units Time


Chapter 2 Radicals 2.2.1 EXPLORE: Representing Square Roots 2.2.2 Simplifying Radicals 2.2.3 Operations with Radicals GAME: Five Radicals

Square Delete GAME: Five Radicals

3

UNIT 3 LINEAR FUNCTIONS AND RELATIONS Getting Started

Chapter 1 Linear Functions and Relations 3.1.1 Linear Functions 3.1.2 Applications of Linear Functions 3.1.3 Graphs of Linear Inequalities 3.1.4 EXPLORE: Transforming Linear Function Graphs GAME: True or False Chapter 2 Solving Systems of Linear Equations 3.2.1 Solving Algebraically — The Comparison Strategy 3.2.2 Solving Algebraically — The Substitution Strategy 3.2.3 Solving Algebraically — The Elimination Strategy CONNECTIONS: Matrix Solution of a Linear System 3.2.4 EXPLORE: Counting Solutions for Different Systems

780 13

Delete GAME: True or False Change CONNECTIONS: Matrix Solution of a Linear System to 3.2.4 Matrix Solution of a Linear Equation Delete 3.2.4 EXPLORE: Counting Solutions for Different Systems

3.2.4 requires further and deeper exploration which is beyond the scope of class X. It is addressed in class XI.

4

UNIT 4 MEASUREMENT Getting Started Chapter 1 Precision 4.1.1 Precision and Accuracy CONNECTIONS: Precision Instruments 4.1.2 EXPLORE: Measurement Error Chapter 2 Efficient Design 4.2.1 EXPLORE: Regular Polygons with a Constant Perimeter 4.2.2 2-D Efficiency 4.2.3 3-D Efficiency CONNECTIONS: Animal Shapes and Sizes

660 11 Status Quo

Sl No Units Time


5

UNIT 5 NON-LINEAR FUNCTIONS AND EQUATIONS Getting Started Chapter 1 Graphing Functions 5.1.1 Forms of Quadratic Functions 5.1.2 Graphs of Quadratic Functions in Factored Form 5.1.3 EXPLORE: Transforming Quadratic Function Graphs 5.1.4 Relating Graphs of Quadratic Functions CONNECTIONS: Parabolas and Paper Folding 5.1.5 EXPLORE: The Absolute Value Function Chapter 2 Solving Non-Linear Equations 5.2.1 Factoring Quadratic Expressions 5.2.2 EXPLORE: Roots of Quadratic Equations 5.2.3 Solving Quadratic Equations by Factoring 5.2.4 EXPLORE: Absolute Value Equations GAME: Get the Points

900 15 Delete 5.1.3 EXPLORE: Transforming Quadratic Function Graphs

5.1.4 Relating Graphs of Quadratic Functions Basically repeats and deals about 5.1.3 in detail.

6

UNIT 6 DATA, STATISTICS, AND PROBABILITY Getting Started Chapter 1 Data Involving One Variable 6.1.1 Histograms and Stem and Leaf Plots 6.1.2 EXPLORE: Investigating Bin Width in Histograms 6.1.3 Histograms and Box and Whisker Plots 6.1.4 Data Distribution CONNECTIONS: Normal Distribution and Sample Size Chapter 2 Data Involving Two Variables 6.2.1 Correlation and Lines of Best Fit 6.2.2 Non-Linear Data and Curves of Best Fit CONNECTIONS: Data Collection by Census Chapter 3 Probability 6.3.1 Dependent and Independent Events 6.3.2 Calculating Probabilities

840 14

Sl No Units Time


7

UNIT 7 TRIGONOMETRY Getting Started Chapter 1 Defining Trigonometric Ratios 7.1.1 Using Similarity Properties to Solve Problems 7.1.2 EXPLORE: Special Ratios in Similar Triangles CONNECTIONS: Using a Clinometer 7.1.3 The Sine, Cosine, and Tangent Ratios 7.1.4 Trigonometric Identities Chapter 2 Applying Trigonometric Ratios 7.2.1 Calculating Side Lengths and Angles 7.2.2 Angles of Elevation and Angles of Depression 7.2.3 Areas of Polygons GAME: Race to Five 7.2.4 Vectors and Bearings CONNECTIONS: Relating Trigonometric Ratios to Circles

840 14 Delete GAME: Race to Five

8

UNIT 8 GEOMETRY Getting Started Chapter 1 Symmetry and Reasoning 8.1.1 2-D and 3-D Reflectional Symmetry 8.1.2 2-D and 3-D Rotational Symmetry 8.1.3 Reasoning Chapter 2 Constructions 8.2.1 EXPLORE: Rigidity 8.2.2 Perpendiculars and Bisectors 8.2.3 Medians and Altitudes CONNECTIONS: Paper Folding Constructions GAME: Balancing Triangles

600 10 Delete GAME: Race to Five

TOTAL 6000 100

Class: XI Subject: Pure and Business Mathematics The intention of the class XI and XII Mathematics is to enable students to get a deeper understanding of the various areas of Mathematics. Mathematics learning should help students to:

• Apply the mathematical knowledge and skills in appropriate real world situations • Read, interpret, and draw inferences correctly from mathematical information • Communicate mathematical ideas and results effectively, both orally and in writing • Enhance their reasoning power and problem solving skills • Gain a strong foundation to pursue mathematics or mathematics related career after class 12

While a major revision of the class XI and XII Mathematics is underway some topics which can be immediately rationalised are as follows:

Sl No Units Time (min) Weighting (%) Changes Reasons

Pure Business Pure Business UNIT 1: ALGEBRA 2160 1800 30 25

1

Sequence and Series AP, GP: Their meanings and finding the nth term (Tn) and the sum of the series (Sn); Insertion of arithmetic and geometric means between two numbers; sum to infinity of of GP (|r| < 1); Special sums, i.e. Σn, Σn2, Σn3, n∈N; Explain the meaning and use of Σ (summation notation) Problems involving the above sequences

Status Quo

2

Binomial Theorem Binomial expansion for positive integral indices; use of Pascal's triangle; and the binomial theorem, i.e. (x + y)n = nC0xn + nC1xn-1y + … + nCnyn Meaning of nCr Binomial theorem for the expansion of binomial expressions having negative or fractional indices Finding the general term of the expansions Application of the theorem for approximation, e.g. (0.99)8 = (1- 0.01)8

Delete Binomial Theorem for BMT since its mainly used for higher order expansions which is not required for BMT

3

Logarithms Revise the laws of Exponents taught in class IX Relationship between Logarithmic and Exponential expressions Laws of Logarithm and their properties including the change of base

4

Remainder and Factor Theorem Meaning of Rational Integral Function Remainder Theorem Factor Theorem Factorization of cubic and quadratic polynomials


Pure Business Pure Business

5

Quadratic Equations and functions Solution of Quadratic equations by factorization and use of their graphs/sketches Solution of Quadratic equations by the Formula method Nature of roots - Real roots, Complex roots, Equal roots Introduction to the concept of imaginary and complex numbers through the square root of -1 Sum and Product of roots Forming quadratic equations with given roots and related data Graph of quadratic function ax2 + bx + c; Sign of quadratics function; Maximum and minimum value of quadratic functions Quadratic inequalities: Solution of quadratic inequalities; Use of graphs and number lines should be employed

For BMT, Delete Quadratic inequalities: Solution of quadratic inequalities; Use of graphs and number lines should be employed.

6

Partial Fractions Rational functions of the form f(x)/g(x), where f(x) and g(x) are polynomial functions in x CASE I - degree of numerator < degree of denominator Type 1 - Non repeated linear factors Type 2 - Repeated linear factor Type 3 - Quadratic factors (may not be factorable) CASE II - degree of numerator >= degree of denominator Type 1 - Non repeated linear factor Type 2 - Repeated linear factor

Status Quo

UNIT 2 : TRIGONOMETRY 1440 720 20 10

7

Angles and Arc lengths Angles: Convention of signs of angles; Magnitude of an angle; Measures of angles; Circular measures The relation S = rθ, where θ is in radians; Relation between radians and degrees Arc length and area of a sector of a circle

Status Quo

08

Trigonometric Functions Trigonometric ratios; Relationship between trigonometric ratios Proving simple trigonometric identities Signs of trigonometric ratios Limits of trigonometric ratios Trigonometric ratios of standard angles Trigonometric ratios of allied angles Periods of trigonometric functions Graphs of simple trigonometric functions (only sketches); Students should be exposed to Computer Generated Graphs through the use of computers

Status Quo



9

Compound and Multiple Angles Addition and Subtraction formulas: Sin (A ± B); Cos (A ± B); Tan (A ± B); Tan (A + B + C); etc., Double angle, triple angle, half angle and one third angle formula as special cases Sums and differences as products: e.g.

𝑆𝑆𝑆𝑆𝑆𝑆 𝐶𝐶 + 𝑆𝑆𝑆𝑆𝑆𝑆 𝐷𝐷 = 2 𝑆𝑆𝑆𝑆𝑆𝑆 (𝐶𝐶 + 𝐷𝐷)

2𝐶𝐶𝐶𝐶𝐶𝐶

(𝐶𝐶 − 𝐷𝐷)2

Product to sums or differences: e.g. 2 SinA CosB = Sin (A + B) + Sin (A - B) etc Conditional identities (involving angles of triangles)

Delete the whole part for BMT

10

Trigonometric Equations Solutions of trigonometric equations (General solution and solution in specified range) Type 1: Equations in which only one function of a single angle is involved e.g. Sin 5θ = 0 Type 2: Equations expressible in terms of one trigonometric ratio of the unknown angle Type 3: Equations involving multiple and sub-multiple angles Equations involving compound angles Linear equations of the form aCosθ + bSinθ = c, where |c| ≤ (𝑎𝑎2 + 𝑏𝑏2)

12 and a, b ≠ 0

Status Quo

11

Properties of Triangles Sine Rule (including ambiguous case for triangles) Cosine Rule Projection formula Napier's Formula for the area of a triangle (Proof and use)

Status Quo

12 Heights and Distances Practical problems based on angle of elevation and depression (in 2 - D)

Status Quo

UNIT 3: CALCULUS 1800 1440 25 20

13

Functions Concept of real valued functions; Domain and Range; Inverse functions; Classification of functions; Sketch of graphs of exponential functions, logarithmic functions, step functions, and simple trigonometric functions like Sinx, Cosx, and Tanx

Status Quo

14 Limits Notion and meaning of limits; Fundamental theorems on limits; Limits of algebraic and trigonometric functions

For BMT consider to Algebraic Functions only

Not required for BMT

15 Continuity Continuity of a function at a point x = a; Continuity of a function in a range

For BMT consider Algebraic functions only




16

Differentiation Meaning and geometrical interpretation of derivatives; Differentiation from first principle; Derivative of simple algebraic and trigonometric functions and their formulae; Derivative of sums, differences, products and quotients of functions; Application of derivatives: Equation of tangent and normal; Approximation; Rate measure;

For BMT consider derivatives of Algebraic (polynomial) functions only


17

Integration Indefinite integral: integration as the inverse of differentiation; Anti-derivatives of polynomials and functions like (ax + b)n , Sin(x), Cos(x), Sec2(x), Cosec2(x) Integration by simple substitution for simple polynomial functions and simple trigonometric functions

For BMT consider only Algebraic (polynomial) functions only


UNIT 4: CO-ORDINATE GEOMETRY 1080 1080 15 15

18

Points and their coordinates in 2-Dimensions Cartesian system of coordinates Distance formula, Section formula Centroid of a triangle, In-center of a triangle Area of a triangle using its three vertices, Area of a quadrilateral Slope or gradient of a line Angle between two lines Conditions of perpendicularity and parallelism of two lines

Status Quo

19

The Straight line Various forms of equation of lines: point slope form; two points form; intercept form; perpendicular/normal form; general equation of a line; slope/gradient; distance of a point from a line; distance between parallel lines; Angles between two lines; equations of lines bisecting the angle between the lines; Identical Lines Family of lines: Lines parallel to ax + by + c = 0 are of the form ay + bx + k = 0; Lines perpendicular to ax + by + c = 0 are of the form ay - bx + k = 0; any line through the intersection of two lines L1 and L2 is of the form L1 + KL2 = 0, where K ∈ R

Status Quo

20 Locus and its equation Definition of a locus and methods to find the equation of a locus; problems should be limited to fairly simple ones

Status Quo

21

Equations of Circles Equation of a circle in: Standard form; diameter form; general form; parametric form Given the equation of a circle, to find the centre and the radius Finding the equation of a circle, given 3 non-collinear points; and given other sufficient data

Status Quo



22

Theorems on Circles Theorems on chords of a circle Theorems on arcs and angles Theorems on angles in alternate segment Theorems on congruent arc and chords Theorems on tangent lines and circles

Status Quo

UNIT 5 : COORDINATE GEOMETRY 720 1080 10 15

23 Measures of Central Tendency Mean, Median, Mode; finding by direct methods, formulas, and graphs

Status Quo

24

Dispersion Range: Quartiles, inter quartiles Standard deviation - by direct method, short cut method and step deviation method; the meaning of Standard deviation should be emphasized

Status Quo + Mean Deviation about Mean and Combined mean and standard deviation of two groups only from class 12

UNIT 6 : COMMERCIAL MATHEMATICS 1080 15

25 Simple Interest and Compound Interest Meanings and methods of the interest calculations Problems involving the two types of interests

Status Quo

26 Discount Trade discount; problems based on it, Present value, True discount, Bill of Exchange; banker’s gain; days of grace; problems based on these.

Status Quo

TOTAL 7200 7200 100 100

Class: XII Subject: Pure and Business Mathematics

The intention of the class XI and XII Mathematics is to enable students to get a deeper understanding of the various areas of Mathematics. Mathematics learning should help students to:

• Apply the mathematical knowledge and skills in appropriate real world situations • Read, interpret, and draw inferences correctly from mathematical information • Communicate mathematical ideas and results effectively, both orally and in writing • Enhance their reasoning power and problem solving skills • Gain a strong foundation to pursue mathematics or mathematics related career after class 12

While a major revision of the class XI and XII Mathematics is underway some topics which can be immediately rationalised are as follows: Sl No Units Time (min) Weighting (%) Changes Reasons

Pure Business Pure Business UNIT 1: ALGEBRA 15 20 1080 1440

1

Permutations and Combinations Factorial Notation Concept of Permutation ( nPr ): Permutation of alike things; restricted permutation; circular permutations Concept of Combination (nCr): Restricted combinations; Distribution of different things into groups; Open selection of items from different things and from alike things Mixed problems on permutations and combinations (note: problems should be of fairly simple ones)

Status Quo

2

Determinants and Matrices Determinants: Of order 2 and 3 Minors and Co-factors of a determinant Expansion of a determinant Properties of a determinant and their use in the evaluation of a determinant Product of determinants (without proof); Solution of simultaneous equations in 2 or 3 variables using Cramer's rule Conditions for consistency of 3 equations in two variables Matrices: Of order m x n, where m, n≤3, including case m = n; Types of matrices Operations: Addition/Subtraction (Compatibility); Multiplication by a scalar; Multiplication of two matrices (Compatibility) Adjoint and inverse of a matrix Application of Matrix multiplication Use of matrices to solve simultaneous linear equations in 2 or 3 unknowns

Status Quo


Pure Business Pure Business UNIT 2: TRIGONOMETRY 7 504

3

Inverse Trigonometric functions Meaning of inverse trigonometric functions (Sin-1x, Cos-1x, Tan-1x, Cot-1x, Cosec-1x, Sec-1x) Principal values (use of graphs in explanation) Properties of inverse trigonometric functions (without proof)

Status Quo

UNIT 3: CALCULUS 30 25 2160 1800

4

Differential Calculus Revision of the topics done in class XI Derivatives of trigonometric, logarithmic, and exponential functions Derivatives of composite, absolute value, implicit and parametric functions Interchange of independent and dependent variables Differentiating function with respect to another function Logarithmic differentiation Successive differentiation up to 2nd order Maxima and Minima Application of maxima and minima to practical problems Derivatives of inverse trigonometric functions reducible to simple form by substitution

For BMT consider derivatives of algebraic functions only

Other functions are not relevant for BMT

5

Integral Calculus Revision of formula of integration from class XI Standard method of integration of 1/x, ex, Tan x, Cot x, Sec x, Cosec x, (ax + b)n, where n∈Q Integration using substitution Integration by using partial fractions Integration by parts Integrals of the type Sin2x dx, Sin3x dx, Cos2x dx, Cos3x dx, f'(x)[f(x)]n dx Definite integral as a limit of sum

For BMT i) consider integration of algebraic functions only ii) Delete Definite integral as a limit of sum

Other functions are not relevant for BMT This is a higher concept for BMT

6 Properties of Definite Integrals Application of definite integrals - area of a curve included between x or y axis, volume of revolution about the x-axis or y-axis or about a line

Status Quo

7

Differential Equations Meaning of differential equation; order and degree of a differential equation Solution of differential equation of 1st order and 1st degree Variable separable

Status Quo


Pure Business Pure Business Homogenous equations and equations reducible to homogenous form; 𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑+ 𝑃𝑃𝑃𝑃 = 𝑄𝑄, where P and Q are functions of x only

Solution of differential equations of second order; 𝑑𝑑2𝑑𝑑𝑑𝑑𝑑𝑑2

= 𝑓𝑓(𝑥𝑥)

UNIT 4: COORDINATE GEOMETRY 25 10 1800 720 Decrease of 5 Marks for BMT

8

Pairs of Straight Lines General equation of a family of lines passing through the intersection of two lines L1 and L2: L1 + kL2 = 0, k∈R; finding k using additional condition General equation of second degree in x and y representing a pair of lines Conditions for general second degree equation to represent a pair of straight lines Conditions for two lines to be perpendicular or parallel Point of intersection and angle between two lines represented by a second degree equation in x and y Equation of the bisector of the angle between a pair of given straight lines

Status Quo

9

Conics As a section of a cone Definition and understanding of Foci, Directrix, Latus Rectum Recognition of Equation of a Parabola, Ellipse and Hyperbola in standard form Finding the equation for a conic when focus, directrix, and eccentricity or related data are given Finding basic information like foci, directrix, etc from a given equation.

Delete for BMT Not required for BMT

10

Points and their co-ordinates in 3-Dimensions Distance between two points; Section and mid-point formulas; Direction cosines and direction ratios of a line; Angle between two lines; Conditions for lines to be parallel or perpendicular

Status Quo

11

Plane General equation of a plane, as ax + by + c = 0, where a, b, c are direction ratios of the normal to the plane Equation of a plane: One-point form; Normal form; Intercept form Distance of a point from a plane Angle between two planes, and angle between a line and a plane

Status Quo


Pure Business Pure Business Equation of a plane though the intersection of two planes Finding the equation of a plane given a point and direction cosine/ratios of the normal and other sufficient data

UNIT 5: DATA & PROBABILITY 15 20 1080 1440

10 Measures of dispersion Meaning of dispersion; quartile deviation; standard deviation, coefficient of variation; Mean deviation from the mean or median Combined mean and standard deviation of two groups only

Move this portion to class XI for both PMT and BMT

Basically repetition in class XI and XII

11

Correlations Definition and meaning of correlations coefficient Use of scatter diagram and Line of best fit Calculation of coefficient of correlation by Karl Pearson's method for ungroup data Calculation of rank correlation coefficient by Spearman's method, for both repeating and non-repeating data Calculation of regression coefficient and the two lines of regression by the method of least squares; use of lines of regression for prediction

Status Quo

13

Probability Random experiment and their outcomes Events: sure events, impossible events, mutually exclusive events, independent and dependent events Definition of probability of an event Laws of probability: addition and multiplication laws; conditional probability

Status Quo

UNIT 6: COMMERCIAL MATHEMATICS 25 1800 Increase by 5 marks

14 Annuities Meaning, Present value, Annuity Certain, Contingent Annuity, Perpetual Annuity, Immediate Annuity, Annuity Due, PV of immediate and perpetual annuity

Status Quo

15

Application of Derivatives in Commerce and Economics Cost Function Average cost Marginal cost Revenue function and break-even point

Status Quo

UNIT 7: COMPLEX NUMBERS 8 576

16

Meaning and as an ordered pair of real number in the form a + ib Geometrical representation in complex plane - Argand diagram for z (a complex number), 1/z, z and z; equality of two complex numbers; absolute value (modulus) Argument and Conjugate of complex numbers

Status Quo


Pure Business Pure Business Operations: Sum/Difference, product and quotient of two complex numbers; additive and multiplicative inverse of a complex number Simple locus equation on complex numbers; proving using z.z = | z |2 and z1 + z2 = z1 ± z2 Square root of a complex number; Demoivre's theorem and its application and cube root of unity: 1, 𝜔𝜔, 𝜔𝜔2; application problems

100 100 7200 7200

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