Topics in Level 4

Here is an interesting topic on descriptive geometry, also called construction geometry, which finds numerous applications in Architecture and Civil Engineering. Basic constructions of triangles with specified angles, based on ratios of sides are derived, by combining simple standard right and equilateral triangles. This also leads to ratios of sides and diagonals of polygons.

Outcomes: Effective use of Pythagoras theorem, similarity and dissection of triangles into standard right triangles.

From games of chance, theory of probability evolved. The fundamental rules of probability and its applications are introduced. Conditional probability and Bayes’ theorem is visited. These find multitude of applications in Finance, Medicine and Business. Discrete and continuous probabilities are also introduced. Interesting problems on geometric probability are solved.

Outcomes: Effective use of probability in problem solving. Good command of Bayes’ theorem and its applications

Basic two-dimensional transformations like translation, rotation, reflection, and scaling are presented. Effective use of this in solving Euclidean Geometry and co-ordinate geometry, is exemplified through problems. These are used in complex numbers also.

Outcomes: Manipulating geometric objects using transformations.

This is very essential for Calculus. The graphs of functions, their characteristics like periodicity, continuity, smoothness, curvature, symmetry about a line, symmetry about a point are emphasized. The features of a one-to-one function, bijective function is presented. Problems testing these concepts are solved and assigned.

Outcomes: Recognizing the characteristics of a function from the graph and vice versa.

Here, many problems where points in a geometrical object or a plane, are colored using two or three colors and questions regarding certain geometrical objects are raised. Many questions in combinatorics can be solved using coloring. Covering planar objects, problems using polyominoes can be solved using coloring.

Outcomes: Mapping from one domain to another as a means for solving problems

Inverse trigonometric ratios and properties are introduced. Evaluating trig ratios for some non-standard angles presented in multiple methods.

Outcomes: Alternate ways of finding trig ratios, Use of Geometry.

Continuity, Differentiability, Integration, and properties thereof are presented. Mean value theorems in Differentiation and Integration are explained through problems.

Outcomes: Knowledge of mean value theorems.

Series different from sequence emphasized. Convergence concepts, telescopic series, interesting problems from Olympiads where these techniques are used, are discussed.

Outcomes: Effective use of collapsing of series.

We enter the world of conics – circle, parabola, ellipse and hyperbola. Common techniques are clubbed together to reduce the load. Different approach to looking at these objects is presented.

Outcomes: Conics conquered.

A new way of dealing with Geometry is vector algebra, where 3-D is easily realized. Three dimensional objects can be represented and manipulated conveniently using the arrow algebra. This scores over the other geometries in this respect. Simpler use of 2-D transformations can be seen here.

Outcomes: Alternate solution methods to geometric problems, use of a powerful tool in problem solving.

Co-ordinate geometry in all its power is dealt here. We discuss pre-conic geometry – lines, pairs of straight lines and properties thereof. How are co- ordinates useful in solving pure geometry questions is addressed here.

Outcomes: Solving problems in co-ordinate geometry with ease.

Existential questions are discussed here. Some configurations in geometry as well some arithmetic problems may not have a solution. These questions are addressed. Proof is essential in these problems, not just a yes or no answer.

Outcomes: Comparing opinions with facts leading to better assessment skill.

Number line is extended to number plane by way of complex numbers. This extension of real numbers solves algebraic equations. It is of great help in transformation geometry as, translation, rotation, scaling is easily represented. Essential tool of higher mathematics.

Outcomes: Familiarity with the use and applications of complex numbers.

Mathematical induction proves that we can climb as high as we like on a ladder, if we can step onto the bottom rung (the basis) and from there to each successive rung. As a technique it is powerful as it can be applied in multiple domains. This is shown by way of various problems.

Outcomes: Effective use of various induction techniques.

Continuing from basic counting, we learn a wider range of techniques like, arrangement, selection, distribution of objects. Counting involves bijection technique also, wherein we map a given problem into a known domain and count. An important component of Olympiads.

Outcomes: A strong foundation in error free counting

Pigeon hole principle, principle of inclusion exclusion are the additional topics studied here. Interesting Olympiad problems will be discussed.

Outcomes: Effectively Solving entry level Olympiad problems

It focuses on effective use of algebraic inequalities through math Olympiad problems.

Outcomes: Develops boundary estimation skill

This topic helps us to learn recursions, recurrence relations and its useful applications

Outcomes: Understanding that recursion is the reverse of induction. Recursion as a programming tool.

This topic is about tracing and collecting special numbers given their properties. It involves effective use of algebra and number theory.

Outcomes: Stimulates the curiosity for such solving process.

Miscellaneous problems are dealt from various topics here.

Outcomes: Enhances multiple skills.