## Out of the Box Thinking Through Mathematics (OOBT)

### Course Topics & Details

## Level 1 - School Class 5 & Above

### Total number of Topics: 13 | Lecture Duration: 20 hours.

Puzzles involve two aspects: definite and suspense. The challenge is to break the suspense using the definite paths. The order of using the definite paths, enhances logical thinking and concentration. The procedure for solving the puzzles is broken into small exercises which focus on the important steps in the solution. This helps in solving any complex puzzle of the same genre.

**Outcomes**: Concentration, Topic-specific knowledge, Problem Solving skills, Memory, Enhancement of self-esteem.

Everyone is familiar with the ordered arrangement of date numbers in rows and columns of a month in a calendar. The arrangement has many interesting properties based on the repetition of the week-days. Based on these many interesting problems and exercises are discussed.

**Outcomes**: Pattern-recognition, Periodicity, Identifying properties thereof.

As in computers where binary logic is used, in arithmetic we have “odd and even” as a powerful concept, effectively used to solve several mathematical problems. Here we focus on simple but powerful application of “odd and even” theory.

**Outcomes**: Proof technique in mathematics introduced!

To watch a sports event, we first switch on the TV, go to sports channels, pick the channel broadcasting the specific sports event and then finetune the brightness, volume to make it most enjoyable. Here we see that the required conditions conducive to our enjoyment are satisfied one by one. Similarly, certain class of mathematics problems can be solved, by satisfying the required conditions one by one in a chosen order. Such techniques can be termed as Tuning Techniques.

**Outcomes**: Stepwise logical procedure, Logical flow plan.

We have a word game wherein we travel from a source word to destination word of the same length, by changing one letter at a time with the intermediate steps being valid words. Here we have a number game where we travel from a source number to a destination number through intermediary numbers satisfying some conditions. The travel might involve factors or multiples or some other mathematical operations, where the intermediate numbers may be condition specific.

**Outcomes**: Decision making, Familiarity and speed of Math operations.

It is a simple ladder technique for solving certain algebraic equations, popularized by the legendary Math Educator (late) Sri P. K. Srinivasan. From unknown variable to the destination, the operations involved in the forward direction, have to be retraced in the reverse order, to reach the variable from the destination. Hence the name “doing and undoing”.

**Outcomes**: Avoidance of the use of variables for solving competition questions, Developing arithmetic manipulation skill over algebra.

Here we visualize a given problem situation and interpret in different ways to arrive at a solution. For example, think of proof without words, which effectively uses visual medium to prove identities or solve problems. We use combinatorial ideas of counting in different ways also.

**Outcomes**: Visual interpretation, Counting in different ways, Pattern recognition, Dimensional enhancement, and generalization.

An object, may be a number or geometrical figure, gives its characteristics and properties, and queries who am I? The process of assessing and analyzing the properties to arrive at the answer, is what makes this an interesting technique.

**Outcomes**: Analysis of the properties, Use of characteristics to aid in solution procedure.

This involves problems typically with large number of digits like hundreds and thousands of digits. We solve for such numbers with certain required properties. This is typically solved, by using blocks of small number of digits and manipulating the blocks, to arrive at the properties required in the large number. The manipulations used, depend on the properties required.

**Outcomes**: Arithmetic properties, Divisibility rules, suitable blocking.

Numbers with some digits missing or hidden will be given. Need to solve so that the completed number satisfies the properties required. Known rules of arithmetic must be used to arrive at the solution.

**Outcomes**: Arithmetic properties, Divisibility rules.

A sequence of numbers is provided, which is generated following certain rules which are given. We emphasise the generating rule, as a given finite sequence can be extended in many different ways. This helps in learning many problem-solving techniques by using arithmetic translation, scaling etc. Further general properties of the terms can also be studied.

**Outcomes**: Pattern recognition leading to arithmetic translation and scaling, recognising inherent properties in the sequence.

This is also called alphamatics. Here alphabets take the place of digits, where different digits are represented by different alphabets. Here properties of addition, carryover, multiplication are used intelligently.

**Outcomes**: Arithmetic skills development, Quick Analysis.

This can be effectively used in Geometric length inequalities, like polygonal inequalities. What is the shortest route for a cow to reach grass?

**Outcomes**: Geometric visualization, Estimating ability

## Level 2 - School Class 7 & Above

### Total number of Topics: 13 | Lecture Duration: 20 hours.

This is known as the famous box-principle or pigeon-hole principle. This is used in all Mathematical Olympiad examinations under the head combinatorics as it tests the logical and analytical skills of contestants.

**Outcomes**: Logical skill development

In any game the results are put up in a table. Here we focus on soccer and its rules. The numbers under each head, likes number of games won, lost, points etc., have some inherent properties. These are used to answer questions regarding the partially filled table with some game outcomes given. We can even answer questions regarding the game outcomes. This is similar to magic squares, sudoku, in that we fill cells with numbers based on the conditions of the game.

**Outcomes**: Logical analysis

Here we assume that a frog jumps in equal leaps on a number line. Assuming the starting point as 0 we can get numerous properties of multiples, greatest common divisor of numbers. If the starting point is a positive number, then we see other arithmetic sequences and properties thereof.

**Outcomes**: GCD, LCM and its properties

Fundamental counting principles like addition and multiplication principles are introduced. Using ten simple tips, effective counting methods are introduced. Problems using these principles in Arithmetic, Combinatorics, and other fields are discussed. Over and under counting are discussed so that students can avoid these pitfalls.

**Outcomes**: Counting skills development.

Through storytelling, several mathematical concepts and techniques can be realised. From comparing fractions in numerous and different methods to a mean value theorem, stories can be designed to create a joy and new dimensional thought about these facts.

**Outcomes**: Realising lot of maths in real life situations and its application in daily activities.

There are occasions where certain problems from elementary to higher level can be solved through a mere tabulation but with an appropriately defined header. This topic focusses on such solving process and exhibits the power of defining and tabulation.

**Outcomes**: Develops the skill of looking into alternate ways of solving other than the usual.

This is a simple two player number game where one player (say) A thinks of a number of a specified length and the other one (say B) need to make guesses. For every guess made by B, A gives hints in the form of number of cows and number of bulls with a relevant meaning for the same, as defined in the game. Player B has to think more logically as more hints are provided but should guess the complete number within limited number of trials (mostly 8 trials). This is similar to the famous Master Mind game.

**Outcomes**: Emphasises the need of strategical thinking in situations of suspense and hints.

A Polygon can be dissected in many interesting manners based on its shape and dimension. This topic in Geometry is innovative. Given a polygon with certain characteristics, how to dissect it into a given number of objects, typically, triangles, special quadrilaterals etc. The question may be to dissect it with least number of cuts or least number of objects. Many interesting problems gives an insight into dissection patterns and also gives an idea of induction-based dissection, nice designed shapes, etc…

**Outcomes**: Understanding properties of polygons, learning visualization and develops two-dimensional spatial intuition.

An innovative way of developing algebraic identities using sport events.

**Outcomes**: Imbibes the synchronisation of thought process.

An interesting way of realising algebra through posting letters in a post box.

**Outcomes**: Strong observations of binary patterns and coding in real life.

Fundamental concepts like, lines, rays, segments, angles, parallel lines, parallel postulate, triangles are introduced. Invariance properties of the sum of the angles of polygons are presented and proved. Three dimensional figures and platonic solids and their properties are introduced. An approach to geometry problem using fundamental six tips is introduced. Problems using these ideas are solved.

**Outcomes**: Basic Geometric skills, Solving simple problems in Geometry in multiple ways.

When percentages are used in profit and loss problems, we can solve these using pages of ruled note-books. We start with 0% on a line and 100% is the Cost Price line. Here again the use of algebra is avoided. Quick calculation is the result of this procedure.

**Outcomes**: Profit and Loss problems solutions; Skills required for competitive exams.

The two important characteristics of any 2D shaped closed figure are area and perimeter. This topic deals with variety of interesting problems on these two characteristics.

**Outcomes**: Problem solving skill enhancement.

## Level 3 - School Class 9 & Above

### Total number of Topics: 20 | Lecture Duration: 30 hours.

Several interesting non-routine problems are dealt in this familiar topic.

**Outcomes**: Exploration of learning the unknown challenges in a familiar topic.

This is a game of building numbers from given conditions, with limited usage of arithmetic symbols, but unlimited usage of arithmetic operators and functional characters like factorial, square root, floor, ceiling, concatenation.

**Outcomes**: Constructing functions and realising its usage of different kinds.

Geometric configurations of specified characteristic are generated based on the given input. Configuration could be arithmetic also.

Outcomes: Logical reasoning skill

This is famously known as Invariance Principle in the pure math circuit. Interesting problems using these principles in several fields are discussed.

**Outcomes**: Recognising famous scientific laws, mathematical theorems, properties, as invariance facts in a varying system.

Through several types of grids, mathematical concepts can be realised. The topic emphasises this idea and also tests through variety of problems.

**Outcomes**: Builds up the skill of developing theories by means of grid visualisation.

Very interesting facts about the geometric shapes are discussed in this topic. It poses questions related to this idea and thereby provides opportunity to discover shape properties of geometric figures while answering.

**Outcomes**: Gives a broader perception about geometric shapes.

This topic poses variety of problems on weighing masses where answering involves logical conclusions at each step.

**Outcomes**: Improves the logical thinking skill.

This topic in Geometry is of new kind where the focus is on estimation of length, area, angle measure, volume, etc… not necessarily the exact value.

**Outcomes**: Develops estimation skill set.

There is content symmetry in algebraic expressions that makes it special. There is also geometric symmetry where the object is invariant and not affected due to transformation such as reflection, rotation, etc…

**Outcomes**: Gives an insight into symmetry.

Several algebraic problems that require usually unknowns such as x,y,z, etc… can be solved without the use of even a single variable. This creative technique is illustrated in this topic.

**Outcomes**: Builds manipulative techniques and develops non-routine thinking.

Two or more players play number games following the rules with a specific winning target. Problems are mostly on the winning strategy of a player in the game.

**Outcomes**: Emphasise the importance of mathematics theory in developing strategy.

There are numerous special equations that cannot be solved by ordinary ways. It requires special techniques to solve such equations. These are illustrated in this topic.

**Outcomes**: Develops very good observation and higher order thinking.

Unfamiliar but very useful applications of ratios and proportions in solving problems are discussed in this topic.

**Outcomes**: Problem solving skill enhancement.

Congruency and similarity - two important characteristics of certain geometric shapes, not necessarily triangles, are discussed in detail both on concepts and its applications.

**Outcomes**: Deeper understanding of the concepts and their applications.

Using ten simple tips, effective counting methods were introduced earlier in basic counting. The same tips are used effectively and innovatively in solving problems here.

**Outcomes**: Assimilating several techniques in counting.

Introduction of coordinate geometry with coordinates as an identity of a point in a plane or space. We discuss elegant solving methods, more from a geometric sense, to problems in coordinate geometry.

**Outcomes**: Develops elegant problem-solving skills.

Basic Trigonometric ratios and properties are introduced. Interesting problems are solved.

**Outcomes**: Effective usage of trigonometric formulae.

Introduction of the necessity of measures of central tendency and measures of dispersion. Illustrates how the concept itself helps in developing varied techniques in the solution process.

**Outcomes**: Establishes the fundamental basis of statistics.

Topic explains about modulo theory (an important branch of number theory) and its applications. Very interesting problems are dealt with.

**Outcomes**: Learning the power of cyclic remainders in number theory.

This is a very powerful creative tool in factorisation of algebraic expressions and the theory of polynomials.

**Outcomes**: Speed of solving is enhanced.

## Level 4 - School Class 11 & Above

### Total number of Topics: 20 | Lecture Duration: 30 hours.

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 Baye’s theorem are 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 Baye’s 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 are 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 coloured using two or three colours and questions regarding certain geometrical objects are raised. Many questions in combinatorics can be solved using colouring. Covering planar objects, problems using polyominoes can be solved using colouring.

**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 realised. 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 are 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.