Value‑Added Course
Closing the gap between
school mathematics
and university study.
A 30 hour value‑added course designed to rebuild core mathematical foundations — algebra, functions, trigonometry, coordinate geometry, and an introduction to matrices, calculus, & statistics — before students meet first‑year coursework.
30
Contact Hours
6
Units
15
Sessions
6
Graded Assignments
Why this course
Rebuilding the foundation, deliberately.
Many first‑year students arrive with gaps in core mathematical reasoning that quietly undermine every course that depends on it. This bridge programme is not a repeat of school syllabi — it is a focused rebuild of the ideas that recur most often in university mathematics, statistics, and the sciences.
01 · PURPOSE
Diagnose, then rebuild
Each unit opens with a short diagnostic discussion, then works forward from first principles rather than assuming prior fluency.
02 · METHOD
Problem‑first sessions
Every concept is introduced through a worked problem before its formal statement, with guided practice inside class time.
03 · OUTCOME
Readiness, not recall
Students leave able to manipulate expressions, read graphs, and reason quantitatively — the working vocabulary of their degree.
Course Structure
Syllabus at a glance
Six units, sequenced so each builds the tools the next one needs. Full detail, session plans, and problems for every unit are in the section below.
| Unit | Title | Core Topics | Hours |
|---|---|---|---|
| U1 | Algebraic Foundations | Number systems, identities, Linear & quadratic equations | 6 hrs |
| U2 | Functions & Graphs | Domain/range, standard functions, composition, inverses | 6 hrs |
| U3 | Trigonometry & Coordinate Geometry | Angle measure, ratios, identities, distance/section formulae, vectors | 4 hrs |
| U4 | Matrix Algebra | Types of matrices, Basic matrix operations. | 4 hrs |
| U5 | Differential Calculus | Limits, continuity, differentiation, differential equations | 4 hrs |
| U6 | Integral Calculus | Integration techniques, applications | 4 hrs |
| — | Consolidation & Final Assessment | Review clinic and closing evaluation | 2 hrs |
Lesson Plan
Unit‑wise plan, problems & material
Open a unit for its learning outcomes, session‑wise lesson plan, in‑class practice problems, the graded assignment, and reference links. Session notes, slides, and recordings are posted here after each class.
U1
Algebraic Foundations
6 hrs · Coordinators: Mr. Sainathan, Mr. Sampath
Algebraic Foundations
6 hrs · Coordinators: Mr. Sainathan, Mr. SampathTopics Covered
- Sets and the number system: ℕ, ℤ, ℚ, ℝ
- Laws of indices and simplification of surds
- Standard algebraic identities
- Factorization of quadratic and cubic expressions
- HCF & LCM of polynomials
- Linear equations in one and two variables
- Quadratic equations: factoring, formula, nature of roots
- Linear inequalities and their solution sets
- Remainder and factor theorems
- Proof Techniques: Induction, Converse, Inverse, Contrapositive, Contradiction, Counter Example
Learning Outcomes
- Classify a given number within the real number system
- Simplify expressions involving surds and rational exponents
- Apply identities to expand and factorize expressions
- Rationalize denominators confidently
- Solve simultaneous linear equations by substitution/elimination
- Solve quadratic equations by two independent methods
- Represent inequality solutions on the number line
- Apply the remainder theorem to evaluate polynomials
- Construct valid mathematical proofs
Session Plan
| Session | Duration | Focus | In‑class Activity |
|---|---|---|---|
| S1 | 1 hr | Number systems, indices & surds | Diagnostic quiz + guided simplification practice |
| S1 | 1 hr | Identities & factorization | Worked examples, paired problem solving |
| S2 | 2 hrs | Linear & quadratic equations | Board work + individual solving practice |
| S2 | 2 hrs | Inequalities & remainder theorem | Case discussion + short quiz |
In‑class Practice Problems
- P1.1 Simplify: (3√2 + √18) − √50
- P1.2 Rationalize the denominator of 1 / (√5 − √3)
- P1.3 Prove: (a + b)³ = a³ + b³ + 3ab(a + b)
- P1.4 Factorize: x² − 5x + 6 and 8x³ − 27
- P1.5 Solve: 2x + 3y = 12, x − y = 1
- P1.6 Solve x² − 7x + 12 = 0 using the quadratic formula
- P1.7 Solve and graph: 3x − 5 ≤ 2x + 4
- P1.8 Find the remainder when x³ − 3x² + 4 is divided by (x − 2)
Assignment 1 · Due before Session 3
- A1.1 Simplify (2⁵⁵ · 4⁴) / 8³
- A1.2 Factorize x&sup4; − y&sup4; completely
- A1.3 Find the HCF and LCM of x² − 1 and x² + 2x + 1
- A1.4 Simplify √(12) + √(27) − √(48)
- A1.5 If a + b = 7 and ab = 10, find a² + b²
- A1.6 Solve for x: 5x − 3(x + 2) = 4
- A1.7 Determine the nature of the roots of 2x² + 3x + 5 = 0
- A1.8 Solve the inequality −3 < 2x + 1 ≤ 7
- A1.9 Use the factor theorem to check if (x + 1) is a factor of x³ + 3x² − x − 3
- A1.10 Two numbers differ by 3 and their product is 40. Find the numbers.
Textbooks & Links
- NCERT Mathematics, Classes IX–X — Number Systems & Polynomials chapters
- Khan Academy — Algebra Basics (free video lessons & practice)
- Paul's Online Math Notes — Algebra (solving equations & inequalities)
- Reference: Hall & Knight, Higher Algebra, Ch. 1–6
Unit Material
U2
Functions & Graphs
6 hrs · Coordinators: Mr. Sainathan, Mr. Sampath
Functions & Graphs
6 hrs · Coordinators: Mr. Sainathan, Mr. SampathTopics Covered
- Relations vs. functions; domain and range
- Linear, quadratic, modulus, exponential and log functions
- Graph sketching and transformations
- Composition of functions and inverses
Learning Outcomes
- Determine domain and range of common functions
- Sketch and interpret basic function graphs
- Compose two functions and find an inverse
Session Plan
| Session | Duration | Focus | In‑class Activity |
|---|---|---|---|
| S5 | 3 hrs | Relations, functions, domain/range | Graphing exercise on grid paper |
| S6 | 3 hrs | Composition & inverses | Group activity with Desmos |
In‑class Practice Problems
- P2.1 Find the domain of f(x) = 1 / (x − 3)
- P2.2 Sketch y = |x − 2| and state its range
- P2.3 If f(x) = 2x + 1 and g(x) = x², find (f°g)(x)
- P2.4 Find the inverse of f(x) = 3x − 4
Assignment 2 ·
- A2.1 Find the domain and range of f(x) = √(x − 4)
- A2.2 Sketch y = 2ⁿ and describe its behaviour as x → −∞
- A2.3 If f(x) = x² − 1 and g(x) = x + 2, find (g°f)(3)
- A2.4 Show that f(x) = 5 − 2x is one‑to‑one and find f⁻¹(x)
- A2.5 State whether the relation {(1,2),(1,3),(2,4)} is a function; justify
Textbooks & Links
- NCERT Mathematics, Class XI — Relations and Functions chapter
- Khan Academy — Functions
- Desmos Graphing Calculator (for graph exploration)
Unit Material
U3
Trigonometry & Coordinate Geometry
4 hrs · Coordinators: Mr. Sriram, Mr. Karthik
Trigonometry & Coordinate Geometry
4 hrs · Coordinators: Mr. Sriram, Mr. KarthikTopics Covered
- Angle measurement: degrees and radians
- Trigonometric ratios and the unit circle
- Standard identities
- Simple trigonometric equations
- Heights and distances
- Distance and section formulae
- Equation of a straight line; slope
- Basics of the circle
- Introduction to vectors: magnitude, direction, addition
Learning Outcomes
- Convert fluently between degrees and radians
- Prove and apply basic trigonometric identities
- Solve trig equations over a given interval
- Apply trigonometry to height & distance problems
- Derive the equation of a line under given conditions
- Compute distance, midpoint and section ratios
- Perform basic vector addition and scalar multiplication
Session Plan
| Session | Duration | Focus | In‑class Activity |
|---|---|---|---|
| S7 | 1 hr | Angle measure & ratios | Unit circle construction exercise |
| S8 | 0.5 hr | Identities & equations | Identity proof workshop |
| S9 | 1 hr | Heights & distances | Applied word‑problem clinic |
| S10 | 1 hr | Straight lines & distance formulae | Coordinate plane plotting exercise |
| S11 | 0.5 hr | Circles & vector basics | Vector diagram workshop |
In‑class Practice Problems
- P3.1 Convert 150° to radians
- P3.2 Prove: sin²θ + cos²θ = 1 using the unit circle
- P3.3 Solve 2 sinθ = 1 for θ ∈ [0, 2π]
- P3.4 A tower's shadow is 30 m when the sun's elevation is 30°. Find the tower's height.
- P3.5 Find the distance between (2, 3) and (−1, 7)
- P3.6 Find the equation of the line through (1, 2) and (3, 8)
- P3.7 Find the midpoint of the segment joining (−2, 5) and (4, 1)
- P3.8 Add vectors a = (2, 3) and b = (−1, 4)
Assignment 3
- A3.1 Convert 5π/6 radians to degrees
- A3.2 Prove: (1 + tan²θ) = sec²θ
- A3.3 Solve cosθ = −1/2 for θ ∈ [0, 2π]
- A3.4 Simplify sin(90° − θ) + cos(180° − θ)
- A3.5 From a point 50 m from a building's base, the angle of elevation to the top is 45°. Find the building's height.
- A3.1 Find the slope and y‑intercept of 3x − 2y + 6 = 0
- A3.2 A point divides the segment joining (1, 1) and (7, 4) in the ratio 2:1. Find the point.
- A3.3 Find the equation of the circle with centre (2, −3) and radius 5
- A3.4 Find the magnitude of vector v = (3, 4)
- A3.5 Show that (0,0), (4,0), (4,3) form a right triangle using the distance formula
Textbooks & Links
- NCERT Mathematics, Class XI — Trigonometric Functions chapter
- Khan Academy — Trigonometry
- Reference: S. L. Loney, Plane Trigonometry, Ch. 1–5
- NCERT Mathematics, Class XI — Straight Lines & Conic Sections chapters
- Khan Academy — Vectors
- Reference: S. L. Loney, Coordinate Geometry, Ch. 1–4
Unit Material
U4
Matrix Algebra
4 hrs · Coordinator: Mr. Om, Mr. Srinand
Matrix Algebra
4 hrs · Coordinator: Mr. Om, Mr. SrinandTopics Covered
- Definition of a matrix; order and types (row, column, square, diagonal, identity, zero)
- Algebra of matrices: addition, subtraction, scalar multiplication
- Matrix multiplication and its properties
- Transpose of a matrix
- Determinants of 2×2 and 3×3 matrices
- Inverse of a 2×2 matrix using the adjoint method
- Solving a system of linear equations using matrices
Learning Outcomes
- Classify matrices by order and type
- Perform matrix addition, subtraction and multiplication accurately
- Compute the determinant of 2×2 and 3×3 matrices
- Find the inverse of a 2×2 matrix using the adjoint method
- Apply matrices to solve a system of linear equations
Session Plan
| Session | Duration | Focus | In‑class Activity |
|---|---|---|---|
| S15 | 2 hrs | Matrix definitions & algebra | Matrix construction & operations worksheet |
| S16 | 2 hrs | Determinants, inverses & linear systems | Solving systems of equations using matrices |
In‑class Practice Problems
- P4.1 Given A = [[1,2],[3,4]] and B = [[0,1],[2,1]], find A + B and A − B
- P4.2 Multiply A = [[1,2],[3,4]] and B = [[2,0],[1,3]]
- P4.3 Find the transpose of A = [[1,2,3],[4,5,6]]
- P4.4 Evaluate the determinant of [[3,1],[2,4]]
- P4.5 Find the inverse of [[2,1],[5,3]] using the adjoint method
Assignment 4
- A4.1 If A = [[2,−1],[0,3]] and B = [[1,4],[2,−2]], find 2A − 3B
- A4.2 Find AB and BA for A = [[1,0],[2,1]], B = [[3,1],[0,2]]; comment on whether AB = BA
- A4.3 Evaluate the determinant of [[1,2,3],[0,1,4],[5,6,0]]
- A4.4 Find the inverse of [[4,3],[3,2]] and verify AA⁻¹ = I
- A4.5 Use matrices to solve: x + 2y = 5, 3x − y = 1
Textbooks & Links
- NCERT Mathematics, Class XII — Matrices & Determinants chapters
- Khan Academy — Matrices
- MIT OpenCourseWare — Linear Algebra (18.06)
- Reference: Gilbert Strang, Introduction to Linear Algebra, Ch. 1–2
Unit Material
U5
Differential Calculus
4 hrs · Coordinators: Mr. Dindi, Mr. Sumanth
Differential Calculus
4 hrs · Coordinators: Mr. Dindi, Mr. SumanthTopics Covered
- Limits: an intuitive introduction
- Derivative as a rate of change; basic differentiation rules
- Mean, median, mode
- Introduction to basic probability
Learning Outcomes
- Evaluate simple limits by direct substitution and factoring
- Differentiate basic polynomial functions
- Compute measures of central tendency for a data set
- Solve elementary probability problems
Session Plan
| Session | Duration | Focus | In‑class Activity |
|---|---|---|---|
| S12 | 3 hrs | Limits & differentiation basics | Tangent‑line visual exercise |
| S14 | 1 hr | Basic probability | Dice & card problem drills |
In‑class Practice Problems
- P5.1 Evaluate lim (x→2) (x² − 4)/(x − 2)
- P5.2 Differentiate f(x) = 3x³ − 5x + 7
- P5.3 Find the mean and median of: 4, 8, 6, 5, 3, 9, 7
- P5.4 A die is rolled once. Find P(number > 4)
Assignment 5
- A5.1 Evaluate lim (x→0) (sin x)/x conceptually using a table of values
- A5.2 Find dy/dx for y = 4x⁴ − 2x² + 9
- A5.3 Find the mode of the data set: 2, 3, 3, 5, 7, 3, 8, 5
- A5.4 Two coins are tossed. Find the probability of getting at least one head.
- A5.5 A particle's position is s(t) = t² − 4t. Find its velocity at t = 3
Textbooks & Links
- NCERT Mathematics, Class XI–XII — Limits, Derivatives & Statistics chapters
- MIT OpenCourseWare — Single Variable Calculus (18.01)
- Khan Academy — Statistics & Probability
Unit Material
U6
Integral Calculus
4 hrs · Coordinators: Mr. Akhil, Mr. Raghuram
Integral Calculus
4 hrs · Coordinators: Mr. Akhil, Mr. RaghuramTopics Covered
- Integration as the reverse process of differentiation
- Basic rules of integration: power, sum and constant‑multiple rules
- Indefinite integrals of standard functions
- Definite integrals and the Fundamental Theorem of Calculus
- Area under a curve using definite integrals
Learning Outcomes
- Evaluate indefinite integrals of polynomial and standard functions
- Apply basic integration rules confidently
- Evaluate definite integrals using the Fundamental Theorem of Calculus
- Compute the area under a curve between given limits
Session Plan
| Session | Duration | Focus | In‑class Activity |
|---|---|---|---|
| S17 | 2 hrs | Indefinite integration & standard integrals | Guided integration practice |
| S18 | 2 hrs | Definite integrals & area under a curve | Area‑computation workshop |
In‑class Practice Problems
- P6.1 Integrate ∫(3x² − 4x + 5) dx
- P6.2 Integrate ∫(1/x) dx and ∫eⁿ dx
- P6.3 Evaluate ∫₀² (x² + 1) dx
- P6.4 Find the area bounded by y = x² and the x‑axis between x = 0 and x = 3
Assignment 6 · Final take‑home · Due before consolidation session
- A6.1 Evaluate ∫(4x³ − 6x + 2) dx
- A6.2 Evaluate ∫₁³ (2x + 1) dx
- A6.3 Find the area enclosed between y = 4 − x² and the x‑axis
- A6.4 Integrate ∫(sin x + cos x) dx
- A6.5 If dy/dx = 3x² − 2 and y(0) = 5, find y(x)
Textbooks & Links
- NCERT Mathematics, Class XII — Integrals & Applications of Integrals chapters
- MIT OpenCourseWare — Single Variable Calculus (18.01), Integration unit
- Khan Academy — Integral Calculus
Unit Material
Evaluation
How the course is assessed
The course is graded formatively rather than through a single high‑stakes exam, in keeping with its purpose: building durable fluency, not testing recall under pressure.
Grading note
Students completing the programme with a passing standard receive a certificate of completion recording it as a value‑added course. Assignments are reviewed with worked solutions in the following session, and resubmission is permitted once per unit.
Reference Library
Textbooks & further reading
Core references for the programme, alongside open, freely accessible platforms students can use for extra practice outside class.
NCERT Mathematics, Classes IX–XII
Primary reference for the full syllabus; free to download.
ncert.nic.in →Higher Algebra
Classic reference for Units 1–2 — algebraic manipulation and equations.
Plane Trigonometry, The Elements of Coordinate Geometry
Standard reference for Unit 3, 4, with extensive graded problem sets.
Paul's Online Math Notes
Free, thorough notes spanning algebra through calculus.
tutorial.math.lamar.edu →MIT OpenCourseWare 18.01
Full lecture series for students who want to go further into calculus.
ocw.mit.edu →
Enrolment
Join the bridge programme
Open to all incoming and first‑year students, regardless of stream. No prior calculus is assumed — only a willingness to revisit the basics carefully. Sessions run in small groups to keep the pace responsive to the room.
- Duration28 contact hours + review
- FormatIn‑person
- EligibilityOpen to all incoming UG students
- PrerequisitesNone
- CertificationValue‑added course certificate
- CoordinatorProf. N Uday Kiran,
Dr. D Bhanu Prakash(Webpage)