Learn2learnmath ebook pdf by Atul Nischal

From South Asiaâ&#x20AC;&#x2122;s Best FREE Math Website and Asiaâ&#x20AC;&#x2122;s Best ICT Enabled Teacher Training Program.

As Per National Curriculum Framework 2005

Learn2Learn

Math With Free Online Learning Support From www.ganitgurooz.com

Thirty Rich Tasks help students and teachers move from AlgorithmCentric approach of learning math to Concept-Centric approach; Excellent resource for Math Labs, Math Clubs, and CCE.

Dr. Atul Nischal

Learn2Learn

Math With Free Online Learning Support From www.ganitgurooz.com

Learn2Learn

Math With Free Online Learning Support From www.ganitgurooz.com

Thirty Rich Tasks help students and teachers move from Algorithmcentric approach of learning math to Concept-centric approach; Excellent resource for Math Labs, Math Clubs, and CCE.

Dr. Atul Nischal

Elipsis Consulting Private Limited New Delhi, India.

Learn2Learn Math

First Print: August 2013 Copyright 2013 by Elipsis Consulting Private Limited All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system without permission in writing from the publisher. The author has obtained permission from owners of some third party images. From those who could not be contacted, we request permission for use of the images for education purposes. An owner of any image, who would like to deny permission to use the image for non-commercial educational purpose, may please contact the publisher so that an immediate action is taken to remove it from future prints.

Published By Elipsis Consulting Private Limited B â&#x20AC;&#x201C; 87/B Kalkaji New Delhi â&#x20AC;&#x201C; 110019 India

Printed At Union Graphics

For Private Circulation Only Not for Sale. To request a copy contact: learn2learnmath@elipsis.in

I dedicate this book to my teachers, who taught me to mathematise and to those who aspire to think mathematically.

Contents Preface ................................................................................................ i Note for Students .............................................................................. iv Note for Teachers ............................................................................. vi Part 1:Get Set ..................................................................................... 1 Learn To Learn Math......................................................................... 2 What does “learn to learn math” mean? .................................... 5 Why must you learn to learn math? ........................................... 6 How can you learn to learn math? ............................................. 8 Rich Tasks ....................................................................................... 10 Your Attitudes and Beliefs ...................................................... 11 Problem-Solving .............................................................................. 12 Record...................................................................................... 12 Prepare ..................................................................................... 13 Go For It! ................................................................................. 14 Stuck? ...................................................................................... 14 Getting Unstuck ....................................................................... 15 Introspect ................................................................................. 16 Communicate ........................................................................... 17 Part 1: Take Away ........................................................................... 18 Part 2: Kick-Off ............................................................................... 19 Staircase Math ............................................................................. 20 Playing Mathematically ............................................................... 22 Generalising Arithmetic to Algebra............................................. 24 Let‟s Play ..................................................................................... 26 Symmetry .................................................................................... 28

Choreographing a Die.................................................................. 31 Knight Riders............................................................................... 33 Experimenting with Towers ........................................................ 34 Conjectures and Proofs ................................................................ 36 Playing Mathematically - 2.......................................................... 38 Part 2: Take Away ........................................................................... 40 Part 3: Deep-Dive ............................................................................ 41 Specialising.................................................................................. 42 Always, Sometimes, Never ......................................................... 45 Probing to Learn .......................................................................... 48 Innovative Solutions .................................................................... 51 Alternative Approaches ............................................................... 55 Investigations in Mathematics ..................................................... 57 The Orange Pyramid .................................................................... 59 Existence and Uniqueness ........................................................... 63 Generalising Patterns ................................................................... 65 Generalising and Specialising ..................................................... 67 Part 3: Take Away ........................................................................... 68 Part 4: MathBuster ........................................................................... 69 Three Dimensional Tic-Tac-Toe ................................................. 70 Chains of Numbers ...................................................................... 71 Estimation .................................................................................... 73 Contextualising Equations ........................................................... 76 Searching for Alternatives ........................................................... 78 Geometric Intuition...................................................................... 79

Visualising Proofs........................................................................ 81 Abstraction: Key to Generalisation ............................................. 82 Infinite Dimensions of Mathematical Thought............................ 85 The Unexpected... ........................................................................ 87 Conclusion ....................................................................................... 89 About Ganit Gurooz ........................................................................ 90 Support Ganit Gurooz...................................................................... 91 Feedback ...................................................................................... 91 About The Author............................................................................ 92

______________________________

“Obvious” is the most dangerous word in mathematics.

_____________________ Eric Temple Bell (1883 – 1960)

Preface The decision to write this book was not easy. My team and I had several questions. Why will students, in an examination-centric environment, read a book not related to their syllabus? Why would students solve math problems that may not help them prepare for examinations? Will the teachers have time to read or share nonsyllabus related problems with their students? Can a book be relevant for students across different grade levels? Honestly, we could not find answers to most of the questions. Therefore, we decided to follow our conviction and belief to write this book. We are convinced of the need for this book. Our conviction has roots in the NCF 2005 and initiatives that CBSE, NCERT and MHRD continue to take to reform the quality of education. This book aims to be a key support for some of these initiatives. We believe that students want to develop the skills to think mathematically and teachers want to help students develop these skills. These students and teachers may not have access to resources to meet their objectives. This book will provide these students and teachers a start. The purpose of this book is to create awareness about the conceptcentric approach to learn (or teach) mathematics. This approach is radically different from the prevalent algorithm-centric approach that makes learning math ineffective, mechanical and uninteresting. All the recent initiatives to reform math education, including NCF 2005, stress on limiting the algorithm-centric approach and implementing the concept-centric approach. Writing this book was more challenging than what we imagined at the start. There were several rounds of discussions with the instructional design team to decide the content, structure, format and difficulty levels of the problems to ensure a good balance. Tasks

ii must be challenging, but if these were too challenging students would get de-motivated. Even at the authoring stage, there were several rewrites to make the book more user-friendly. Another challenge was the wide spectrum of readers that could benefit from this book. It was tough to make the content relevant, interesting, meaningful, beneficial and accessible to students of classes 9 to 12 across all boards, teachers, students of B.Ed. courses and other math enthusiasts. In the end, we made students our primary focus with the hope that others will also benefit from reading it. Though we have addressed several challenges, one challenge remains. Benefits of rich tasks depend largely on the engagement of the student in the task. This requires discussions on various aspects, peer interaction, and expert intervention. A book, by itself, is unable to deliver all of this. Thus, we decided to create discussion forums for each task on the Ganit Gurooz website where readers could interact with each other and our team. In addition, the discussion forums would provide continuous support to develop skills to learn math. This book seems to be the first book in India that comes bundled with free learning support. The experience of writing this book has been very rewarding. We conducted workshops in some schools in Delhi, where students of class 9 were engaged in some tasks from this book. Their enthusiasm and involvement was very encouraging and helpful to modify some tasks to make them more accessible. Personally, this journey has been a great learning experience. I think this is the real reward of any academic endeavour. I look forward to interact with the readers on the discussion forums at www.ganitgurooz.com. We have taken care to ensure that the book is error-free. In case we have missed something, please do let us know. Also, share with us

iii your suggestions to improve next editions. In addition, my team and I would appreciate every word of encouragement. This book evolved over the last six months. During this period, the manuscript underwent several critical reviews and revisions. I thank Simi Singh, Director, Learning Management at Elipsis Consulting Pvt. Ltd. and Dr. Shiv Kumar Sahdev, Associate Professor, Shivaji College, Delhi University for their critical reading and reviews. In the end, most humbly, I thank you for deciding to learn2learn math. Happy mathematising! Atul Nischal, PhD atul.n@elipsis.in

Note for Students I thank you for deciding to read this book. It takes a lot of courage, determination and commitment to read a math book that seems to have little correlation with your syllabus. Let me assure you that once you learn to think mathematically and begin to apply these skills to learn math concepts in your syllabus, you will achieve higher scores in mathematics tests. I recommend that you devote some regular time to study this book. Start with ONE hour every week and increase the duration and the frequency as you go along. As you engage in the tasks in this book, you will learn how to use certain skills to learn math. Initially, this may not be easy. Several distractions will come in your way. I am sure that you will not succumb to these. Let us make a promise to each other. If you promise to make a genuine attempt to study this book, I promise that what you learn will be worthwhile. My team and I are ready to support you throughout this journey of training your mind to think mathematically. We have put in a lot of effort to make this book interesting, enjoyable and fun. We have also tried to use simple language to explain the tasks in the book so that you can focus more on the mathematical aspects of the tasks. However, if there is anything that is not clear, we will be happy to discuss it on the discussion forum. There is a natural tendency to skip reading a math book and jump straight to the problems. I would strongly discourage this practice, at least for this book. The first part â&#x20AC;&#x201C; Get-Set â&#x20AC;&#x201C; explains the purpose of the book, structure of rich tasks and the basics of problem solving strategies. You must read and understand this part before you begin the tasks given in the other three parts â&#x20AC;&#x201C; Kick-Off, Deep Dive, and MathBuster. You should attempt these three parts in a sequence

v because tasks in the three parts become increasingly challenging and assume that you have done the tasks in the previous parts. Kick-Off has 10 simple tasks with some handholding. Once you have attempted at least five tasks in this part, you may move on to the next part. Deep Dive has 10 tasks that present more learning opportunities. These tasks may require more time and effort than the ones in Kick-Off. The MathBuster section has 10 tasks that require you to use the skills and knowledge that you have learnt in the previous 20 tasks. Within each part, you can follow any sequence to do the tasks. Maybe you can start with the ones that interest you. You may also begin several tasks simultaneously. In the end, I want to share the most important and useful aspect of this book. Undoubtedly, the only way to learn to think mathematically is to “do the math”. This means you must take as little help as possible and attempt the tasks on your own. However, it may sound contradictory when I say that “doing the math” is incomplete without “discussing the math”. When you discuss math you talk about your solution strategies and thought processes with peers as well as experts. Talking about mathematics is as important as doing mathematics. For this reason, Ganit Gurooz has created “Discussion Forums” for each task on their website. You can register as a student on the free website and share your strategies and thoughts related to the task. However, never put solutions on the forums. I shall post regular comments and address your doubts or queries related to the tasks. I hope you find this book interesting and recommend it to all your friends. Happy mathematisation!

Note for Teachers In 2005, NCERT published the National Curriculum Framework (NCF) as the guiding policy to shape education reforms in India. As a part of this policy, the focus group on teaching of mathematics released a position paper that strongly emphasises the objective of school math education as training studentsâ&#x20AC;&#x; mind to think mathematically. Unfortunately, even after 8 years, not much material is available to guide us on how to achieve this objective. This book aims to serve as a resource for all teachers who wish to implement the guidelines of the NCF 2005 in their classrooms. CBSE has too actively introduced several reforms in recent years. These include the mathematics laboratory, CCE, PSA (Problem Solving Assessment). Most recently, CBSE has decided to discontinue the practice of publishing sample papers. All these reforms echo that we must move away from the algorithm-centric approach of teaching mathematics that does not value (or assess) the ability of a student to think mathematically. Schools and teachers who want to support these reforms need resources that help them implement changes in the classroom. This book has 30 mathematically rich tasks. You can easily modify these tasks to use them as a project, a CCE assignment, or for designing an interesting activity for the math club or math lab in your school. The decision on how you plan to use these tasks in your school or classroom is entirely yours. To help you, we will provide worksheets for that you can download from www.ganitgurooz.com and use in your class. The most rewarding moment for teachers is when their students succeed. If success means getting a great score/grade on a particular test or examination, then we need not implement any reforms suggested by MHRD, NCERT, or CBSE. However, if success means

vii to develop an interest for mathematics, relate mathematical concepts to real life, or be able to think mathematically, then it is important that we do certain things differently. My team at Ganit Gurooz and I extend our support to move from the algorithm-centric approach of teaching math to the concept-centric approach. The 21st century has created a digital divide between the teachers and students. It is becoming increasingly difficult for teachers to communicate with students. The students are spending more time on the internet and want to learn in social environments. Teachers, on the other hand are not comfortable or committed to using technology as a tool to interact with students. This serious issue deserves immediate attention of teachers and schools who believe that communication is the basis of education. I strongly recommend that you register at www.ganitgurooz.com and participate in the discussion forums to interact with students across India on the tasks in this book. It will be even better if make this book available to all your students and encourage them to log on to the discussion forum. In addition to interacting with your students, you can also access and download thousands of free resources to use in your classrooms. This book is a part of the mission of improving the quality of math education in India. Without your support, this will be an impossible task. Therefore, I request you to support Ganit Gurooz actively by sharing it with your students. In return, I promise that Ganit Gurooz will support your initiatives to make learning math a better experience for students. I look forward to interact with you.

Learn2LearnMath

Part 1: Get Set In this part, you will: 1. Understand the process of Learning to Learn Math. 2. Learn the definition and purpose of rich tasks. 3. Learn about good practices associated with problemsolving strategies.

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1.1Learn to Learn Math Let‟s begin by understanding the answers to three important questions: 1. What does “learn to learn math‟ mean? 2. Why must you “learn to learn math‟? 3. How can you “learn to learn math”? Let us first look at how we learn math. The most common approach to learn mathematics is to solve problems using memorisation techniques or by following an algorithm (a series of steps leading to the solution). In this approach, our focus is to practice as many problems as required to memorise, retain, and recall an algorithm or a fact. This is the algorithm-centric approach to learn math. In this approach, we never connect the algorithm to the mathematical concepts or ideas that produce the algorithm. For example, to divide polynomials we learn the algorithm shown here. 3x 2  4 x  2 2 4 3 2 However, we may not x  3x  1 3x  5x  7 x  2 x  2 3x 4  9 x3  3x 2 understand the math       concepts behind some 3  4 x  10 x 2  2 x  2 of the steps of this  4 x3  12 x 2  4 x    algorithm. Why do we 2 arrange the dividend 2x  6x  2 and the divisor in 2 x2  6 x  2       decreasing order of the 0 degree of their terms

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before dividing? What would happen if we arranged them in the increasing order of the degrees of their terms? Why do we carry some terms of the dividend over to the next dividend? In the algorithm-centric approach, these questions are not important. Thus, we never get to learn the meaning behind what we are doing. Algorithm-centric approach has several serious drawbacks. To begin with, it keeps us away from learning the mathematical concepts or ideas because we never think about these. The second drawback is that we become algorithm dependent. We feel that unless we know “how” to solve a problem, we really cannot solve it. As there are so many different types of problems, we are always scared of not being able to remember or recall the „correct‟ method during the examination. This is one of the causes of math phobia. The third, and possibly the biggest drawback, is that we never learn to think mathematically. The purpose of doing mathematics is not to solve a particular type of equation or draw a particular kind of geometric figure. The real purpose is to learn to think mathematically. Even with these drawbacks, the algorithm-centric approach to learn mathematics is being followed almost universally. Why? The answer is simple. In an examination centric education system where the structure and the content of tests are extremely predictive, the algorithm-centric approach works. However, this approach fails to deliver the more important, higher objectives of learning mathematics. In fact, it never

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prepares students to learn mathematics. It only prepares students to take a specific math test. So, is there an alternate approach to learn math? Letâ&#x20AC;&#x;s find out. The other approach to learn math focuses on understanding mathematical concepts and ideas and then applying them to solve problems. In this approach, you need to seek answers to three questions â&#x20AC;&#x201C; Why, What, and How. The process of discovering answers to these questions requires you to think mathematically about a problem. This process of thinking mathematically is called mathematisation. The concept-centric approach to learn math is the problem solving approach that requires us to focus on mathematisation. This approach is an extension of the algorithm-centric approach because it focuses on the WHY and WHAT before defining HOW. Even in the concept-centric approach you need to use an algorithm but in this approach the algorithm is rooted in the understanding of mathematical concepts and a strategy to solve the problem. This is the real, and the only, way of doing mathematics that helps you achieve the higher objectives of learning mathematics. This extension provides a different meaning and stature to learning mathematics.

Learn2LearnMath

What does “learn to learn math” mean?

The transition from using the algorithm-centric approach to the concept-centric approach is neither obvious nor immediate. The concept-centric approach requires a different outlook towards solving problems. Instead of relying on your “memory skills”, you need to learn to use “problem-solving skills”. Research in cognitive behaviour has established that different regions of your brain are responsible for memorisation and problem-solving. In addition, “memorisation skills” are very different from “problem-solving skills”. Therefore, you need to train your mind for this transition.

“Learn to learn math” is the process of developing the skills required to use the concept-centric approach to learn mathematics. It is the process of learning to mathematise. Now, let us address the next question.

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Why must you learn to learn math?

If you agree with Albert Einstein, then I do not need to convince you. The concept-centric approach to learn math trains your mind to think mathematically â&#x20AC;&#x201C; the primary purpose of math education. If you do not agree with Einstein, let me share some benefits of using the concept-centric approach. The concept-centric approach addresses the major drawbacks of the algorithm-centric approach. Once you learn to use the concept-centric approach to solve problems, doing mathematics becomes an investigative journey that is full of interesting revelations, enjoyable and fun. Each problem becomes a puzzle that you can solve using various math concepts. The fulfilment of cracking a puzzle or an investigation is directly proportional to the level of challenge it involves. Similarly, as you get better in mathematising, you long for challenging math problems.

Learn2LearnMath

Another benefit of this approach is that your mind begins to form and appreciate the structure of mathematics. It learns to see how various concepts connect with each other, the limitations of each concept, and the extension of one concept into the other. The formation of this mathematical framework is essential to succeed in mathematics. The concept-centric approach is the prevention as well as the cure of math phobia, which in majority of cases, is rooted in a lack of understanding of the mathematical concepts. Every time you use mathematical concepts to crack new math problems, you develop a better understanding of these concepts. All these factors influence your achievement in mathematics tests. Thus, to achieve higher levels of success in examinations consistently and without stress, the concept-centric approach works much better than the algorithm-centric approach. I hope that you are convinced of the importance of learning to learn math. Now, let me next share how you will do this.

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How can you learn to learn math?

Your brain is very powerful. It has two lobes that use different skills to help you think. What are these skills? Letâ&#x20AC;&#x;s find out.

I am sure you have played Ludo several times. What does your mind do before every move? It makes a strategy. You can move any of your open coins. However, to decide which coin to move, you first compare the impact of moving each coin. Then you move the coin that maximises your chance of winning the game. Can you visualise a sunset or Taj Mahal? If you have seen something, you can easily visualise it. Can you also visualise things that have never happened? Try visualising that the roof of your classroom has disappeared, it is raining heavily, there is water everywhere, and you are the only person in the class

Learn2LearnMath

attending a math lecture. This has never happened and will never happen, but you can still visualise this easily. If you have to explain the directions to your house to someone, you draw a diagram as shown below.

Your mind already knows how to give convincing reasons to justify your actions. For example, if you want to go to Manali during your next summer holidays, you will try to find enough reasons to convince your family. In addition to providing justifications, in many situations, you can also find gaps in the justifications provided by others. Your mind already knows how to strategise, visualise, draw diagrams, provide reasons to justify, or find gaps in reasoning. Apart from these, you use several other skills in different situations in your daily life. You probably never thought that these skills can help you learn math. With this book, you will train your mind to use these skills to learn mathematics. How can you do this? This book consists of thirty problems called rich tasks. As you do these tasks, you will use various skills to solve mathematics

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problems. Then, you must continue to use these skills to solve problems in your textbooks. This repeated usage of these skills will train your mind to mathematise. Before you begin, let me share some important aspects about the tasks in this book.

1.2Rich Tasks Over the last 15 years, researchers have extensively studied the relationship between studentsâ&#x20AC;&#x; achievement in mathematics to the complexity of the problems they solve. Research clearly indicates that students who attempt cognitively challenging tasks score better in math tests. Therefore, to improve achievement levels in mathematics, it is important that you solve challenging tasks. However, the improvement in your achievement level is also dependent on the level at which you engage with the task. Therefore, to derive maximum benefit from a challenging task you must actively engage in the task. A rich task is a cognitively challenging task based on one or more mathematical concepts. Most rich tasks require you to use multiple skills ranging from simple to complex. The purpose of a rich task is to help you learn to mathematise. It is an activity to improve the â&#x20AC;&#x153;quality of thoughtâ&#x20AC;? by using different skills to approach the solution. To train your mind to use certain skills it is important to challenge it. The mind suggests you alternatives only when you are stuck, feel frustrated, and do not know how to proceed. Thus, every rich task has certain activities that are challenging.

Learn2LearnMath

You may get stuck while doing these activities. Putting effort to figure out a way to get unstuck will train your mind to think mathematically. Your Attitudes and Beliefs The word „problem‟ shares its root with the word „probe‟, which means, “to enquire”. If you are curious or do not have the zeal to enquire, then doing rich tasks will not be as beneficial as they can be. In addition, you also need a positive attitude and belief that the concept-centric approach to learn mathematics is important and that you can learn how to use it.

You will possibly begin the tasks in this book with the attitude, “I‟ll try to do it”. To motivate you, I have provided ample handholding for the tasks in Part – 2. However, if you are

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stuck, pose these tasks to your friends or your teachers and see how they do it. You may want to post them on facebook too! The tasks in Part 3 may require patience and more effort than the ones in Part 2. These tasks are a little more complex, longer and provide more opportunities to practice various skills to learn mathematics. You must spend enough time on tasks in Part 2 and Part 3 before attempting the tasks in Part 4. These are complex than other tasks. However, do not get discouraged when you are stuck because we are available on the discussion forums to guide you through these activities. To help you do these tasks successfully let me share some key steps of problem-solving strategy.

1.3Problem-Solving Solving rich tasks is often not a single step, or a linear process. You may be required to create a different strategy for each one of them. Although each strategy is unique, we can define some successful practices associated with problem-solving techniques. I will describe these practices briefly and invite you to learn more about these on www.ganitgurooz.com. Record

As you work through these tasks, you must record your progress in a notebook â&#x20AC;&#x201C; you can call it The L2LM Journal. This has two benefits. First, you may not be able to complete a

Learn2LearnMath

rich task at one go. So, if you want to return to that task after a few days, the journal will help you to refresh your thoughts. Second, writing mathematics is as important as doing mathematics. As you learn to record your progress regularly, you will find better and efficient ways to write mathematics. Prepare

If you have played chess, or seen someone play chess, you will know the importance of analysing the chessboard before making a move. You need to think of all the possible moves, pre-empt your opponentâ&#x20AC;&#x;s move for each of these options. Then decide the best move. Before you start a rich task, you need to prepare. It is in this stage that you answer the WHYâ&#x20AC;&#x;s and HOWâ&#x20AC;&#x;s related to the problem. Read the task several times to make sure that you understand it. Try rephrasing the problem in your own words or create a visual representation. For some tasks, you may need to discuss parts of problems that you do not understand. This is where the discussion forum can help you. Then, try to think of the math concepts that you may use to solve the problem. Think of some strategies that you could

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follow. Remember, rich tasks may have different approaches to a solution. Evaluate each strategy and decide which one you want to try first. Go For It!

Once you have understood the WHY and devised some strategies to answer the HOW, it is time to focus on the WHAT. Start with the simplest strategy. Rich tasks usually have multiple steps and you need to move carefully from one step to the next. As you cover each step, write it down in your L2LM journal. If you are happy with what you have achieved write “WAH” on the journal. If you are not satisfied, write “AAH”. When you make some progress, try to share your work with your friends or teachers. Stuck?

Being stuck during a rich task is inevitable. Do not let this dampen your enthusiasm or discourage you. On the contrary, whenever you are stuck, think of it as an opportunity to enhance your knowledge. There could be several reasons why you might be stuck. Let‟s look at some possible reasons. In some cases, you may not have understood the problem itself while in other cases the mathematical concept that you thought of using may not be applicable to the task. It is also possible that your strategy

Learn2LearnMath

needs some modification or you need to check your calculations. All of these are learning opportunities. It is important to accept that you are stuck. You can do this by writing “NOW WHAT” or “STUCK” in your journal. Getting Unstuck

To get unstuck, you can either seek help or attempt the task again. Getting help is easier while redoing the task is more enriching. The choice is yours. It is a pleasure to discover a way of getting unstuck on your own. And, you should not deprive yourself of this wonderful feeling. Try the problem yourself with alternate approaches or try to find what went wrong with your approach. Here is a checklist that may be useful to identify where you made a mistake or missed something. 1. Read the problem again. a. Did you miss out any information? b. Did you interpret any information incorrectly? 2. Check each step of the solution in your journal. a. Did you make any wrong assumptions? b. Did you apply the facts correctly? c. Did you make a calculation error? 3. Examine your strategy. a. Can you justify each step? b. Are your reasons valid? We want you to get unstuck on your own but there could be times when you need help. The team of experts at Ganit Gurooz will be happy to help you. We will provide a hint,

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suggest an alternate strategy, help identify errors in your reasoning, discuss the problem or share an example. Introspect

One of the most important aspects of learning to learn math is to look back and learn from everything that you did right and from every error that you made. Remember, I asked you to put a “WAH” and an “AAH” at certain places. You must go back to these regularly and ask questions that can help you introspect. Here are some questions you can ask: a) What kind of errors did you make? What lead to your making these errors? What helped you identify the errors? Did you learn anything new while fixing the errors? b) What was the most exciting part of the task? Why did you find it exciting? c) What was the most challenging part of the task? What helped you overcome the challenge? d) What skills did you use to do the task? Why were these skills critical or important for the task? In what other kinds of problems, would you be able to use these skills? e) How did other readers approach this task? What was different in their approach?

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Communicate

The last step is to be able to communicate the final solution to yourself and others. Write the final solution in your journal in a systematic manner. Write everything as if you were explaining it to someone who has not interacted with the problem. Explain and justify every step of the solution. For example, if you decided to make a diagram, write down why you thought the diagram would be helpful. As I said earlier, learning to write math is as important as learning to do math. This is your first step to learn logical reasoning. This skill is extremely important to do math and develops only through practice.

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Part 1: Take Away Congrats! You are ready to attempt the first set of rich tasks. Letâ&#x20AC;&#x;s look at some major points that we discussed in this part. 1. Learning to learn math means learning to use the concept-centric approach to solve problems. 2. You can learn to learn math because you are already using most of the skills required to do this. 3. You can only learn to learn math by doing rich tasks that vary in degree of complexity. 4. While attempting rich tasks, the quality of your thought and your approach to the solution is important. 5. You must follow the steps of the problem solving strategy to do rich tasks. After a few days, read part 1 again to re-enforce the above points. Now, letâ&#x20AC;&#x;s kick-off this journey. Good Luck!

Learn2LearnMath

Part 2: Kick-Off This part has the following tasks: 1. Staircase Math 2. Playing Mathematically 3. Generalising Arithmetic to Algebra 4. Letâ&#x20AC;&#x;s Play 5. Symmetry 6. Choreographing a Die 7. Knight Riders 8. Experimenting with Towers 9. Conjectures and Proofs 10. Playing Mathematically - 2

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2.1Staircase Math Staircases are present in every building that has different levels. Most modern buildings that have more than 3 floors now have elevators or escalators. While on an escalator, most people stand on a step. Sometimes to save time you climb or descend the steps as the escalator moves. How can you measure the time that you save? What does this time signify? Do you think the time saved depends upon whether you are going up or down the escalator?

While using the stairs, you can save time by climbing or descending 2 steps at a time? And sometimes even take a long jump for the last 4 or 5 steps. Is there any math behind climbing stairs? Letâ&#x20AC;&#x;s find out. A staircase has 12 steps. You can descend 1 or 2 steps at a time and jump a maximum of 4 steps in the end. Here are three possible ways to descend the staircase: [1, 1, 2, 2, 1, 1, 1, 3]; [1, 1, 2, 2, 1, 1, 4]; [2, 1, 1, 2, 1, 1, 4] These sequences are called partitions of 12 because the sum of the numbers in each sequence is 12. Observe the last two partitions. How are they different? Two partitions that differ only in the order of the summands are considered the same

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partitions. For example, the second and third sequence above are the same partitions. In the activities below, you will explore partitions of numbers.

How many partitions of 12 can you create using the numbers 1, 2 and 4 such that the number 4 appears only in the end? How many partitions of 12 can you write if you only use the numbers 1 and 2? Why has the number of partitions changed?

Suppose P(n) denotes the number of partitions of the positive integer â&#x20AC;&#x17E;nâ&#x20AC;&#x; . Explain the meaning of P(5) and find its value. Find P(n) for n = 1, 2, 3, ...10 Do you spot a pattern?

A prime partition of a positive integer involves prime numbers only. Write the prime partitions of the first 10 positive integers. Can you find all numbers that do not have a prime partition? How many such numbers exist? Why? Find a rule that allows you to determine if a number will have a prime partition or not?

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2.2Playing Mathematically In the movie MunnaBhai MBBS, one of the characters plays a fabulous carrom shot in the hospital. You can see this shot on YouTube at http://www.youtube.com/watch?v=KXxQOlsfkcs Do you think such shots are possible? The character in the movie visualises the shot before making it. All of us have the ability to analyse a situation visually. And this is an essential skill to learn mathematics.

The objective of Carrom is to pocket “coins” by striking them with a “striker”. The coin, striker and the pocket are all circular having different radii.

Compare the diameters of the coin, striker, and the pocket using ratios and percents.

In the figure you can see two carrom shots. If the striker „B‟ hits the coin „A‟, it goes into the pocket. But, if the striker „D‟ hits the coin „C‟, it bounces off the side and does not go into the pocket. In this task, we will use angles, lines, tangents to a circle and laws of reflection to visualise different carrom shots.

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Draw an illustration of a carrom board. What mathematical concepts did you use to draw an accurate illustration?

In the illustration that you have drawn, draw a coin in the center of the board. Join the center of the top right pocket to the center of the coin and extend it to meet the boundary of the coin at the point P. What does point P signify? For different positions of the striker, draw the line of movement of the striker such that it touches the coin at the point P. With a dotted line, extend the line of movement of the striker to meet one of the sides of the board. Which position of the striker would you prefer for a shot? Why?

How can you predict the movement of the coin if you know the position of the striker and the line of strike?

Rebound shots involve high precision and skills. There are three types of rebounds: only the coin rebounds; only the striker rebounds; and both the striker and the coin rebounds.

You want to pocket a coin placed at the center of the board in the pocket to your right using a rebound shot. Describe the shot that you will play. Why did you choose this shot?

Mathematically describe the carrom shot in the movie MunnaBhai MBBS.

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2.3Generalising Arithmetic to Algebra Moina has just moved to Delhi from her village. She is amazed that Delhi has so many buses to help people commute within the city. She asks Nitish how she can identify the route for each bus? Nitish tells her, “All buses have a sign that shows the route number”. Do you think Nitish has checkedALL DTC buses to be sure that his statement is true? Why is Nitish confident about his response? Nitish generalised the pattern that he noticed on some DTC to all DTC buses. Many students who love arithmetic in primary classes get overwhelmed with algebra in middle school. But, in reality algebra is just a generalisation of arithmetic. Let‟s see how. You know that addition of numbers is commutative. Thus, for two numbers, say 2 and 3, you know that 2 + 3 = 3 + 2. This statement tells us, “The sum of 2 and 3 is the same as the sum of 3 and 2”.

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If we assign different letters to the numbers 2 and 3, how will the equation change? Let‟s see. 2 x x x

3 y x+1 x2 – 1

2+3=3+2 x+y=y+x x + (x +1) = (x +1) + x x + (x2 – 1) = (x2 – 1) + x

The first and second algebraic equations obtained above tells us that: a) The sum of a number xwith another number y remains same irrespective of the order in which they are added. b) The sum of a number and its successor is the same as the sum of the successor of a number and the number.

For each equation/expression/statement given below: a) Identify the rule, concept, or property that is being illustrated; b) Write an algebriac form using a different variable for each number; c) Write an algebraic form using only one variable. a) 2 ( 3 – 8) = 2 x 3 – 2 x 8 c) – (- 7) = 7 e) 5% of 120 = 6

b) 1/(1/4) = 4 d) 42 + 0 = 42

g) 4 + 5 is odd.

f) 23 x 1 = 23 h) 9 = 3 1 1 1 j) 2 × 3 = 6

1 2

+3 =6

Which of the above parts were more difficult? Why? How did you overcome the difficulty?

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2.4Let’s Play Games involve luck, skill, and strategy. Snakes and Ladders is a game of luck. Tic-tac-toe is a game of strategy. Chess requires both strategy and skill. Scrabble requires all three. What about cricket? Games that involve chance or strategy are studied in two different branches of mathematics. A game of chance is studied using Statistics and Probability while a game of strategy is studied in Game Theory.

To play Tic-Tac-Toe, two players take turns to mark noughts and crosses in a 3 by 3 grid.A “winning line” is drawn when three marks of the same kind occur in a row – either vertically, horizontally or diagonally. The player who draws a winning line first wins the game. Let‟s try to look at the mathematics in Tic-Tac-Toe.

How many winning lines are there? Devise a symbolic notation to identify the different winning lines. Write each winning line using this notation.

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How many winning lines will remain for your opponent if you start by putting a cross in: a) The middle cell, b) Any corner cell, c) Any other cell. Did you work out your answers by drawing visuals or by using the mathematical notations developed in the first activity? Why did you prefer the technique that you used?

A winning strategy for a game defines your moves that are most likely to result in a win for you. But, to do this, we must analyse the game thoroughly by observing each possible move. Letâ&#x20AC;&#x;s do this for tic-tac-toe.

Play tic-tac-toe with a friend and after every move: a) Count the number of winning lines remaining for you and your friend. b) Draw out all the possible moves that your friend can make.Count the number of winning lines for each of these possibilities. c) Which of these moves did your friend make? Why?

Create winning strategies for Tic-Tac-Toe if: ď&#x201A;ˇ You make the first move. ď&#x201A;ˇ You make the second move. How are the two strategies different? Do you think that the player starting the game has an advantage over the second player? What is the advantage?

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2.5Symmetry Rahul, Riti and their uncle went out for a dinner. While waiting for food, they decided to play a game based on observation. Rahul decided to start the game. He asked Riti and his uncle to look away from the table while he made some change to the objects on the table. Then he asked them to identify the change he made. After some time, Riti spotted the spoon that had been flipped by Rahul. Now, it was Ritiâ&#x20AC;&#x;s turn. She also moved an object, but Rahul and his uncle could not figure out the object that she moved. Which object do you think Riti moved and how? Symmetry has always inspired mathematicians because it symbolises beauty and order. Symmetry is a gift of nature, a gift of God. It is said that symmetry is a common characteristic of everything beautiful, including human faces.

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Observe and explain the symmetry in the structure of:Honeycombs, Sunflower, Nautilus Shell, Spider Webs, Snowflakes, Animals (Lion, Dragonfly, Butterfly, Snakes), Humans, Fruits, Nuts, Vegetables, Plants, Leaves, Flowers.

Have you guessed what Riti could have moved? She rotated the plate. Rahul and her uncle could not notice the change because the plate was rotationally symmetric. If I can rotate an object to a position such that it still appears the same as it did orginally, the object is said to possess rotational symmetry. The number of positions to which I can rotate the object is called its order of rotational symmetry. Thus, the order of rotational symmetry of a circle is infinity, while that of a square is four. Symmetry is the property of an object to appear unchanged even when it undergoes a change. Another word for â&#x20AC;&#x201C; unchanged â&#x20AC;&#x201C; is invariant. So, if an object remains invariant under a change or a series of change, it is said to be symmetric. Architects create symmetric designs to improve the visual appeal of structures they build. You must be aware of the symmetries of The Taj Mahal. The picture on the right shows the ceiling of Lotfollah mosque in Iran. It has rotational symmetry of order eight and eight lines of reflection.

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Make a list of 10 famous structures of the world. What kind of symmetries do they exhibit?

You do not need to be an architect or a designer to create symmetrical designs. If you study the images below, you will see how symmetrical designs can be created using letters of the English Alphabet.

How many lines of symmetry are there in each of the design shown above? Which designs possess rotational symmetry? Explain the rotational symmetry of each design.

Create symmetric designs using alphabets and numerals of the English or Hindi language. Your designs may contain one of the elements or a combination of elements. Describe the symmetry of your designs.

Create symmetric designs using any one or a combination of circles, squares, triangles and pentagons. Describe the symmetry of your designs.

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2.6Choreographing a Die Choreographers divide the stage into 9 regions as shown in the image below. Then they decide how the dancers would move from one region to another during the performance. So, it seems that apart from creativity, some mathematics is part of every performance. In this task, you are going to discover the mathematics involved in choreographing a die over a grid of squares. A die is a cube with each of its faces having 1, 2, 3, 4, 5, or 6 dots.

Do you think a particular pattern is followed to place the dots on the faces of a die? Describe this pattern. Why do you think this pattern is followed for all dice?

Place a die (as shown in the image), with four dots toward you and 6 dots toward your left. If you roll over this die to the right, the top face will have 6 dots. What number will the top face have if you roll this die toward you?

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Make a 4 by 4 grid as shown here. Place a die on the top left hand cell of the grid so that the top face has 1 dot. You can move the die onto a cell that is either to its right or below it, by “rolling it over” the edge toward the cell.

You have to “roll” the die to the right most cell in the bottom row. How many different paths can you create? Draw these different paths.

Draw different paths so that the face up has a 6 dots when it reaches the last cell. How many paths did you find? Repeat this by replacing 6 with 1, 2, 3, 4 and 5. Do the number of paths for different number of dots change? Why?

Repeat activity „B‟ on a 3 by 3 grid and a 5 by 5 grid. Do you see any pattern that relates the dimension of the grid to the number of paths? How many paths will be there on a n by n grid?

Explain how visualisation helped you understand the problem and explore the activities A, B, C and D.

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2.7Knight Riders A knight is a chess piece. It can be moved to a square that is two squares horizontally and one square vertically, or two squares vertically and one square horizontally from its original position. The complete move therefore looks like the letter L. At the start of the game, the white knight is placed in the cells 1b and 1g as shown in the image.

Starting from 1b, in which cells can we move the knight in one turn? If all cells where the knight can be moved are empty, what are the minimum number of turns in which the knight can be moved to the cell 1c?

How many cells are there on the chess board to which the knight can never be moved legally?

Find the least number of moves required for the knight at 1b to be moved to each of the remaining cells on the chess board. Explain the strategy you used to find these numbers.

How was this task similar or different from the task â&#x20AC;&#x153;Choreographing a dieâ&#x20AC;??

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2.8Experimenting with Towers Raju hired an auto rikshaw to go to the railway station. Here is the conversation that happeed between him and the autorikshaw driver. Raju: How much time will it take to reach the station? AD: About 1 hour 15 minutes. Raju: Why would it take so much time?. AD: Because it‟s a week day and we will driving through Old Delhi during peak hours. Raju was amazed that in exactly 1 hour and 15 minutes he reached the station. How could the driver predict the duration with such accuracy?Possibly, he had timed himself on this route under similar situations many times. On several occasions we make our decisions on data that we collect through repetitive experiments. Let‟s use this skill to learn mathematics.

Experiments have been a tool to discover mathematical results since the Babylonian civilisation. Most of the early results in arithmetic and algebra have been motivated by experiments. Experiments are used for several reasons – including studying relations between variables. In this task, you will use an experiment to discover mathematics. You will require an inclined plane, about 15 sugar cubes, a ruler and a protractor. [You can replace sugar cubes with match boxes or dice.]

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In this experiement, we will study the stability of a tower made using sugar cubes. Here is the procedure to conduct the experiment. 1. Mark a point on the inclined plane. Measure its distance (d) from its fixed edge. 2. On this point, build a tower by placing n sugar cubes one on top of the other. 3. Lift the inclined plane very slowly. 4. Measure the angle, , at which the tower topples. 5. Record your observation in a table: d n



Why does the tower topple? What are the factors that make the tower topple? How does each of these factors effect the tower as the inclination of the plane is increased?

Describe the relation between the variables 1. n and  2. d and 

Would there be any value of n for which the tower does not topple? How do you explain this?

How can you predict the value of  for a given value of d and n? Justify your prediction?

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2.9Conjectures and Proofs Raman goes to a restuarant with his friends. His friend says, â&#x20AC;&#x153;This restuarant serves excellent ice-tea. Everyone who visits this restuarant orders ice-tea with their meals â&#x20AC;?. Do you think that the statement made by Ramanâ&#x20AC;&#x;s friend is true or false? How would you prove this statement to be true or false? Conjectures are statements that need to be proved or disproved. In this task, you will make a conjecture and prove it. You will also learn how to disprove conjectures.

Mathematics is structured progressively. This means that the math you learn in middle school is also required in senior school. And, similarly the math you learn in senior school, will be required in math courses during higher education. The first concept of mathematics that you learn is numbers. By the time you are in senior school, you have studied various types of numbers and their properties, rules of divisibilities, even and odd numbers, prime and composite numbers. You have also understood how to find squares, cubes, square-roots and cube-roots of numbers. In this task, you will make a conjecture about numbers and then use some basic concepts of numbers to prove the conjecture.

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Evaluate: a) 53 − 5, b) 63 − 6, c) 73 − 7. Do you observe anything that is true for all your answers? If not, evaluate more expressions of these kinds. Make a list of statements that you think are true for all your answers.

What kind of conjectures can you make based on your observations in the activity above? Why do you think these conjectures should be true? Can you also write some conjectures that may not be true?

C D

Prove or disprove the conjecture you have made.

Prove or disprove the conjecture: If a number leaves 1 as the remainder when divided by 3, so will its cube. On the discussion forum we will discuss how to prove or disprove conjectures by taking several interesting examples.

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2.10Playing Mathematically - 2 Sahil and Monish went to an amusement park where they saw an AIR HOCKEY table for the first time. It was 4 feet wide and 8 feet long and had tiny holes on its surface. They asked the table

attendant about the purpose of those holes. The attendant explained that during the game these holes release air to make the surface of the table frictionless.

To play air hockey, each player uses a striker to hit a â&#x20AC;&#x153;puckâ&#x20AC;? in the opponents goal. A puck is a disc with a diameter of 3.25 inches. And, each goal is 18 inches wide. Sahil and Monish found the game interesting and decided to play.

To play Air Hockey, you can hit the puck at any time as long as it is in your half of the table. Here are a few ways by which you can modify your shots: a) You can strike the puck at any distance from your goal post;

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b) You can strike the puck at different angles so that it hits either one or both side rails before targetting the opponentâ&#x20AC;&#x;s goal.

Observe the position of the puck in the image shown above. Suppose a player wants to hit the puck on the side rail in such a way that after rebounding , the puck goes straight inside the opponents goal. a)

At what point on the side rail should the puck be hit?

How many such points are there on the side rail?

For three different positions of the puck on the table, find the points on the side-rail from where the puck can rebound into the opponentâ&#x20AC;&#x;s goal.

Repeat (c) if it is desired that the puck should rebound on both side rails before going into the opponentâ&#x20AC;&#x;s goal.

Mathematically compare Air hockey and Carrom to determine their similarities and differences.

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Part 2: Take Away Here is a list of what you may have learnt while doing these tasks. Can you recall the tasks in which you used these concepts or skills? 1. Mathematical Concepts Covered a. Partitions of Numbers; b. Permutation and Combinations; c. Angles d. Laws of reflections e. Ratios and proportions f. Circles and Tangents g. Game Theory h. Number Theory i. Symmetry and Invariance j. Graph Theory 2. Learning Skills Practiced a. Observation of patterns of numbers; b. Generalisation; c. Visualisation; d. Analysis of mathematical concepts and situations e. Mathematical Reasoning f. Creating and using mathematical notations g. Using diagrams to understand solutions h. Describing diagrams. i. Making mathematical comparisons j. Observing mathematical concepts in real life k. Experimenting with mathematics l. Making and proving conjectures m. Establishing relations between variables

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Part 3: Deep-Dive This part has the following tasks: 1. 2. 3. 4. 5. 6. 7. 8. 9.

Specialising Always, Sometimes, Never Probing to Learn Innovative Solutions Alternative Approaches Investigations in Mathematics The Orange Pyramid Existence and Uniqueness Generalising Patterns 10. Generalising and Specialising

How would you describe this shape?

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3.1Specialising Every class in your school follows a different time table. The time table defines a general rule according to which you study different subjects during the day. A special case of this general rule is the subject that you studied in the 3rd period on 12th August. Thus, a special case gives specific information about a general rule. Let‟s see how you can use your ability to specialise in mathematics. Mathematical facts are usually stated and proved in their generalised form. Here is an example of one such fact. The sum of the first n odd numbers is n2. This fact is true for any natural number n. Three special cases (for n = 2, n = 3, and n = 4) of this fact are: 1 + 3 = 22 1 + 3 + 5 = 32 1 + 3 + 5 + 7 = 42 Try to write a few more special cases.

Write three special cases for the fact “The reciprocal of the reciprocal of a number is the number itself”.

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The process of identifying special cases of a general rule is called specialising. Specialising is a great tool to discover a solution to a general problem. Let me show you how. Let us try to devise a strategy to prove the general result: The sum of the first n odd numbers in n2. First, we consider the special case for n = 2, which is 1 + 3 = 22 Now, let us create a geometrical model of this equation. Take ONE square of a particular colour and THREE squares of a different colour. Arrange these squares as shown in the figure to get two rows of two squares each. Using this geometric model, let us try to write an argument for the equation given above. First, we observe that the left side of the equation represents the squares of two different colours; while the right hand side represents the squares in different rows. Thus, we split the number of squares of each colour into the number of squares that lie in each row, to get 1 + 3 = (1 + 1) + (2) [Two parenthesis ~ Two rows] =2+2 =2x2 = 22 Now, you try the same thing with a few more special cases.

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a) Create a geometric model of the statement 1 + 3 + 5 = 32 b) Write an argument to show why this is true without computing the values of each side of the equation.

Prove that: 1 + 3 + 5 + 7 + ...+ (2n – 1) = n2

Now, let‟s try these tasks to practice specialising.

Find a positive integer that can be written as a difference of two squares. How many such integers can you find? What do you think is common between these integers? How did you find these integers? Can you generalise this method to find all positive integers that can be written as a difference of two squares?

Find all positive integers that can be written as the difference of two cubes.

How many positive integers of the form 4n + 1, 4n + 2, or 4n + 3 can be written as the difference of two squares?

At this point, I must highlight the beauty of using the conceptcentric approach to problem solving. Students of class 11 may use the Principle of Mathematical Induction to do activity C. In this case, they would be using the algorithm-centric approach to problem solving because they will not connect the steps of their solution to a mathematical concept. In fact, they will not even see how the “inductive” process works and miss the geometry behind this fact.

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3.2Always, Sometimes, Never An „action‟ can either occur always, sometimes or never. For example: a) Kanav eats cereal for breakfast sometimes. b) Ravi never watches television after 10 PM. c) Ashwani always writes with a fountain pen. Very often we are interested in determining the truth of the statements we hear. These statements can be “always true”, “sometimes true”, or “never true”. For example, a) Mango has two seeds – is never true. b) I carry an umbrella when it rains – is sometimes true. c) There are many students in my school – is always true. Let‟s classify math statements as “Always True”, “Sometimes True” or “Never True”. In doing this, you will develop a better understanding of some math concepts. Whenever you read a math statement, you must probe it to determine if it is always, sometimes, or never true. Testing the truth of a statement is one of the most important things. Always True

Sometimes True

Never True

True for all possible cases;

True for some cases; not true for others

Not true for all possible cases;

Proof Required

Examples Required

Proof Required

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Here is one example of each type of statement. Always True

The successor of a natural number lies to its right on the number line.

Proof: The successor of any natural number is 1 more than the natural number. Since, it is greater than the given number, it lies to its right on the number line. Sometimes The square root of the product of two numbers is True equal to the product of the square roots of the numbers.

Never True

• True for 4 and 5 • False for – 4 and – 5 For any integer n, the number 2n + 1 is divisible by 2. Proof: For every integer n, 2n is an even number. Therefore 2n + 1 is an odd number. Thus it cannot be divisible by 2.

Always? Sometimes? Never?

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Determine if each statement is always, sometimes or never true? a) If p is a prime number then p + 1 is also a prime number. b) The centroid of a triangle lies inside the triangle. c) The difference between the nth powers of two numbers is a multiple of the difference between the two numbers. d) Three chords of a circle divide the circle into 4 regions. e) Zeros of polynomials with real coefficients are real numbers. f) The graph of a linear equation is a straight line. g) Mean is a better measure of central tendency of a data than its median. h) If the volume of two spheres is equal, then the two spheres are congruent. i) If the volume of two cones is equal, then the two cones are congruent. j) The perpendicular bisector of the longest side of a triangle divides the triangle into two regions having equal areas. k) A geometric shape with four vertices is a quadrilateral.

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3.3Probing to Learn Namita is member of the music club at her school. If she likes a particular compositions, she wants to know everything about it â&#x20AC;&#x201C; names of the singer,lyricist, music composer, the year in which it was recorded and released. Sometimes she gets this information from her music teacher, while at other times, she searches the internet or ask friends. A mind that learns to ask questions can never cease to learn. Your mind is trained to ask questions, to probe into various situations that interest you. In this task, you will learn to probe the mathematical aspects of different siutations. Mathematics is everywhere, but very few of us can actually see it. Everything from the structure of the Milky Way to the formation of the smallest piece of diamond involves mathematics. So, why canâ&#x20AC;&#x;t we notice it? To observe the math in daily life situations you need to be curious. This curiosity will train your mind to ask questions that help you explore the situations mathematically. Everyday lakhs of passengers go to a railway

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station to board a train. Only a mathematically curious mind would ask, â&#x20AC;&#x153;Why does the number of coaches in different trains varyâ&#x20AC;?? The number of coaches in a train depends on several factors. One of the factors is the number of people that travel on it. This, in turns depends upon several factors, one of which is, the distance between the station of origin and the destination. Therefore, it is clear that to determine the number of coaches required for a train, we have to know the distance between the station of origin and the destination. Asking questions helps you appreciate mathematics in real life. It is also the beginning of your learning to think mathematically. Letâ&#x20AC;&#x;s try to create questions for the three contexts given below. Balancing Skywalkers balance themselves on beams or a rope at a very high altitude. In villages, people often balance loads on their heads to carry them from one place to another. A basketball player balances a rotating ball on his finger.

Write 20 questions to explore the mathematics behind balancing objects.

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Racquet Sports Table Tennis, Badminton, Squash, and Lawn Tennis are sports in which a player hits a ball or a shuttle with a racquet.

Write 20 questions to mathematically compare racquet sports.

Viral Videos Did you know that the video â&#x20AC;&#x153;Why this Kolaveri Diâ&#x20AC;? received more than 30 million views within 40 days of its release on Youtube? A video becomes viral when people start sharing it by emails or through social networking sites.

Write 20 questions to explore the mathematics behind viral videos.

Look at the questions you made for each context. What kind of difficulty did you face in making questions? Was some context/situation easier to probe than others? Why?

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3.4Innovative Solutions To play cricket you need a bat, a ball and three wickets. Sometimes the three wickets are replaced by three sticks, a drawing of three wickets on a wall, a temporary structure using bricks, or just two stones.

When was the last time you used an alternate method, outof-the-box approach, or an innovative method to tackle a situation. There are exceptions to every rule. Although, you can use more than one technique to solve most math problems, some problems require a particular technique. Let‟s look at one such „historic‟ problem. This diagram illustrates how to bisect an angle using a compass and a ruler. We can repeat this process to divide an angle into 4, 8, 16, or 32 equal parts.

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Can we trisect an angle using a compass and ruler? This problem remained unsolved for thousands of years. Finally, it was established that there is no standard way to construct a third of an arbitrary angle. Does that mean we can never trisect an angle? Well, I did not say that. We can trisect angles by using a different method. And, this method is â&#x20AC;&#x201C; origami or paper folding. Here are the steps to trisect an angle ď ą: Take a square piece of paper. Draw the angle PBC of measure ď ą, so that the point B is in the corner of the square as shown in the first image. Then make a horizontal fold anywhere across the square, to mark the line EF, as show in the second image.

Now, fold the line BC up to the line EF and unfold, creating line GH as shown in this image.

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Next, fold the bottom left corner up so that the point E touches the line BP and the point B touches the line GH.

With the corner still up, fold both layers to continue the crease that ends at point G all the way to J, then unfold. Now, unfold the corner B.

As shown in the first image below, fold along the crease that runs to the point J, extending it to the point B. Fold the bottom edge BC up to line BJ and unfold. The two creases BJ and BK divide the original angle PBC into thirds.

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Verify that the above method works for angles with measure – 81, 60, 45? Do you think the above method can be used to trisect an angle of measure 135? What kind of modifications would be required?

Explain why this paper folding activity works to trisect an angle.

Using a paper folding activity, find a method to bisect a 90 angle. How would you use this method to make an angle of 11.25 ?

Try making this paper pyramid using origami.

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3.5Alternative Approaches Your family is organising a party. To arrange the food, there are three options: a) Cook at home, b) Order from a resturant, or c) Hire someone to cook.

Each option has its own advantages and disadvantages. Similarly, in mathematics, there are more than one methods to solve a problem. You must find all these methods and then choose the one that suits you the most. To solve math problems you usually try to find â&#x20AC;&#x153;the right strategyâ&#x20AC;?. But, in reality there can be multiple correct strategies. You may approach the problem algebraically, visually or even experimetally. A strategy is flawed only if it defies the rules of logic and mathematical concepts involved.

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Consider this problem: Given a triangle, find the radius of the circle that can be inscribed in it. You can solve this problem by using: 1. 2. 3. 4.

Geometry. An experiment. Coordinate geometry. Calculus.

If you know different approaches to solve a problem, you can choose an approach that works best for you. In this task, letâ&#x20AC;&#x;s try to find different approaches to a problem.

What are the different methods to find the solutions of the equation: đ?&#x2018;Ľ 2 + 5đ?&#x2018;Ľ â&#x2C6;&#x2019; 27 = 0 a) Which of these methods can be extended to find solutions of higher degree equations? b) What do you like (or not like) about each method? c) Which method do you like the most? Why?

Log on to discuss alternate methods to solve the problems that you are doing in class.

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3.6Investigations in Mathematics Jasmeetâ&#x20AC;&#x;s father got transferred to Chandigarh from Dharamshala. They found Chandigarh very different from Dharamshala. They had so many questions about Chandigarh. Which is the best place to buy groceries? How to find a reliable maid to help them with the house-work? How to commute within the city? Which is the best school in the city for Jasmeet? It was not surprising that the family spent a lot of time investigating everything about the city. What do you do to seek answers to the questions that come to your mind in different situations? You investigate. In this task, we are going to use our investigation skills to learn some math.

Very simple situations sometimes present opportunities to learn a lot of math. Let us understand this by conducting an experiment that was possibly one of the first scientific investigation in history. For this task you need a table, a piece of cardboard or a plank of wood, tape, some books, steel balls of different sizes (weights) and a small bowl (katori). Create an inclined plane using the cardboard or the wood plank. This can be done by putting some books at one end of the board. Position the inclined plane at some distance from the edge of the table in such a way that when a ball is rolled on the

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inclined plane, it travels some distance on the table before falling on the ground.

To predict where a ball of given size (weight) will fall for a particular inclination, we will need to do some math. Letâ&#x20AC;&#x;s begin our investigation.

What are the variables in this experiment? Can you predict how these variables are related to each other? What is the basis of your prediction?

Fix the position of the inclined plane on the table as well as the angle of inclination. How does the weight of the ball affect the position where it falls on the floor?

For a specific ball, how does the inclination of the plane effect the position where it falls on the floor?

For a specific ball and the inclination of the plane, how does the distance the ball travels on the table effect the position where it falls on the floor?

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3.7The Orange Pyramid At a sweet shop, Gulab Jamuns are often stored in a bowl that has sugar syrup. What happens if you add more GJs to the bowl? Can you imagine that this seemingly trivial question has tremendous mathematics hidden in it? The image shows some oranges stacked on a table. Assume that the oranges are congruent spheres.

What strategy would you use to count these oranges? How many oranges are there?

The number of oranges on the table is a square pyramidal number. A square pyramidal number is the sum of squares of natural numbers.

Write the 6 smallest square pyramidal numbers. How did you obtain these numbers? How do these numbers connect with the image of oranges shown above?

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How many oranges will be there on the table if the pyramid had n layers? For the next activity, you need a chessboard.

What strategy would you use to count the number of squares on the chessboard? How many squares does a chessboard have? What is the relation between the number of squares on a chessboard to the numbers you found in activity „A‟ and „B‟?

Let‟s come back to spheres at a sweet shop.

a) Rani and Joy order 4 GJs at Mukherjee Sweets. Surprisingly, their waiters serve them GJs in two different ways as shown below.

Compare the placements of GJ‟s chosen by the two waiters. b) The owner of Mukherjee Sweets wants you to design a box to pack four GJs. What kind of box would you design and why?

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Rashmi goes to a sweet shop to buy some GJ‟s. She observes 28 GJ‟s in a vessel that has a round top. She wonders how many more GJ‟s can be added to the vessel such that all the GJs are floating on the surface of the sugar syprup. How can Rashmi find this number?

The densest spherical packing of equal spheres is the most natural way of packing spheres. To create a dense packing of spheres, we:   

Create a layer L of spheres such that three spheres touch each other. Create another layer M by placing a sphere on top of each set of three spheres. Build a third layer N, using the same rule. You can create the layer N in two ways shown in the images below. Can you spot the difference between the two images?

Now, you may be able to answer what happens when more GJ‟s are added to a bowl of sugar syrup at a sweet shop? You should also be able to tell when the second layer of GJ‟s will form in the bowl? For now, let‟s go back to the oranges on the table.

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a) If the number of oranges (spheres) on the table were

packed in the densest manner, would you still be able to form a pyramid? Why? b) How many layers of oranges will you get if the

bottom layer in the densest packing has the same number of oranges that were there in the pyramid structure? How many oranges would be there in the top layer?

Spherical Packing of Spheres of Different Sizes

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3.8Existence and Uniqueness Suppose, you are travelling from Dehradun to Delhi. There is a massive traffic jam on the highway because of a major accident. You find out that the jam may not clear for the next 6 hours. However, if you get delayed you will miss your cousinâ&#x20AC;&#x;s wedding. What would you do? First you must ponder if there exists another route to Delhi from your current position. If there is no such route, you canâ&#x20AC;&#x;t do much. But, if there is a route, then you try to figure out if there are more than one routes. If there is only one route, you dont have a choice. But, if there are more than one routes, then you try to figure out which one suits you the most. Solving equations is very common in mathematics. To find a solution means to find values of the variables for which the equation is true. There are three questions that you must ask before you try to solve an equation: a) Do the solutions exist? b) Is the solution unique? c) In case the solution is not unique, how many solutions exist for that equation? The equation sin2 đ?&#x2018;Ľ + cos 2 đ?&#x2018;Ľ = 1 is satisfied by all real numbers. It has infinite solutions. Since every value of x satisfies this equation we call it an identity. This means that the LHS and the RHS of the equation are identical.

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The equation đ?&#x2018;Ľ 2 + đ?&#x2018;Ś = 1 is not satisfied by all real numbers. So, it is not an identity. But it still has infinite solutions.

You are given some equations. For each equation answer the following: a) Does it have a solution? Why are you sure of your response? b) Does it have a unique solution? Justify your response. c) Does it have a finite number of solutions? Why are you sure of your response? How can you determine the exact number of solutions an equation can have? d) Is the equation an identity? How can you prove it? e) Solve each equation using any method? Equations: 3đ?&#x2018;Ľ + 5(đ?&#x2018;Ľ â&#x2C6;&#x2019; 1) = 8 1 + sin â&#x2C6;&#x2026; + 3 = 3 â&#x2C6;&#x2019;(4đ?&#x2018;Ľ â&#x2C6;&#x2019; 7) = đ?&#x2018;Ľ 2 đ?&#x2018;Ľ 3 +đ?&#x2018;Ľ =1 3 sin2 â&#x2C6;&#x2026; + cos2 â&#x2C6;&#x2026; = 2 (sin3 đ?&#x2018;Ľ â&#x2C6;&#x2019; tan3 đ?&#x2018;Ľ) = (sin đ?&#x2018;Ľ â&#x2C6;&#x2019; tan đ?&#x2018;Ľ)(sin2 đ?&#x2018;Ľ + sinđ?&#x2018;Ľtanđ?&#x2018;Ľ+tan2đ?&#x2018;Ľ) G. đ?&#x2018;Ľđ?&#x2018;Ś â&#x2C6;&#x2019; đ?&#x2018;Śđ?&#x2018;Ľ = 8 A. B. C. D. E. F.

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3.9Generalising Patterns What is your house number? What are the house numbers of your immediate neighbours? Do you think that house numbers are allotted using a pattern?What would have happened if the house numbers were allotted randomly? What is the registration number of your car? What are the registration numbers of the cars owned by your immediate neighbours? Do you think that these numbers are issued using some pattern? What would have happened if these numbers were allotted randomly? Have you ever thought about the purpose of a calendar? What are the benefits of a calendar? Do you see other patterns and structures around you? What is the purpose and benefit of these patterns and structures? In this task, we shall study some mathematical structures and patterns. Spotting mathematical patterns can be a lot of fun and challenge . Sometimes it takes time and effort to spot a pattern. But, the joy of discovering a pattern â&#x20AC;&#x201C; that others cannot notice â&#x20AC;&#x201C; makes up for all the hardwork.

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Let‟s see if you can spot a pattern in the following tasks.

Write down the multiplication tables of numbers from 2 to 11. What kind of patterns do you see in each table?

What kind of patterns do you see in the three lists shown below:  List A: 1, ½, 1/3, ¼, ...  List B: 1, ½, 2, 3, 1/3, ¼, 2/3, 3/2, 4, 5, 1/5, 1/6, ... a) Write a positive rational number that is neither in List A or List B. b) Write an expression that represents the general form of numbers in both lists.

Observe the three statements below: 2+2 =2 ×2 1 1 3 + 12 = 3 × 12 1

a) b) c) d)

4 + 13 = 4 × 13 Are all the statements true? Write the next few statements for this patttern. Are they also true? Write a generalised statement that represents this pattern? Prove the generalised result.

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3.10Generalising and Specialising I am sure you cut a cake on your last birthday. Before you cut the cake did you count the number of people you wanted to serve it to? If not, looks like you missed a chance of doing some math on your birthday. The most traditional way of cutting a cake is to cut it into slices that meet in the center. But there are other ways to cut cakes. The picture shows an alternate way to cut a cake. Let‟s explore the math involved in cutting cakes. The activities in this task require you to specialise and generalise. It is possible that you may have to use these two skills simultaneoulsy as you progress through the activities.

What is the least number of cuts required to get „n‟ pieces of a cake?

On the boundary of the cake mark n points. What are the maximum number of pieces that you can cut so that each cut joins two of these points?

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Part 3: Take Away Here is a list of what you may have learnt while doing these tasks. Can you figure out the tasks in which these concepts and skills were involved? 1. Mathematical Concepts Covered a. Sequences of Numbers; b. Trisecting Angles c. Quadratic Equations d. Spherical Packing e. Pyramidal Numbers f. Permutation and Combinations; g. Number Theory h. Number Patterns 2. Learning Skills Practiced a. Specialising; b. Generalising; c. Geometric Model of an Algebra problem d. Analysis of Mathematical Statements; e. Asking Questions f. Origami/Paper Folding as a tool to understand math g. Multiple approaches to solve problems h. Visualisation; i. Experimenting with mathematics j. Establishing relations between variables k. Mathematically analysing situations l. Mathematical Reasoning

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Part 4: MathBuster Congratulations! You have reached the final ten tasks in this book. I am sure you enjoyed this journey and have developed a better understanding of the processes involved in learning mathematics. Hopefully, you appreciate mathematics more than you did earlier. These tasks may appear to be more challenging. However, if you use the skills that you have developed while doing the tasks in Part 2 and 3, you will not struggle much. If you are stuck, please visit the discussion forums. The tasks in this section are: 1. Three Dimensional Tic-Tac-Toe 2. Chains of Numbers 3. Estimation 4. Contextualising Equations 5. Searching for Alternatives 6. Geometric Intuition 7. Visualising Proofs 8. Abstraction: Key to Generalisation 9. Infinite Dimensions of Mathematical Thought 10. The Unexpected...

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4.1Three Dimensional Tic-Tac-Toe You have earlier made a winning strategy for the2-dimensional tic-tac-toe. Are you ready to tackle its 3-dimensional version? A 3D tic-tac-toe can be thought of as three 2D tictac-toes stacked in three layers as shown in the image. Instead of crosses and circles, two balls of different colours are placed in circular slots. A player can keep the ball in any available slot and at any level. A winning line is drawn when three balls of the same colour are placed in a line in any direction either in one plane or across planes. So, you could have 1 ball in the top layer, 1 ball in the middle layer, and 1 ball in the bottom layer as a winning line. The player who makes more winning lines is declared the winner.

How many winning lines are there in the 3D Tic-TacToe? Through which point do the maximum number of winning lines pass? How would you devise a winning strategy for this game?

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4.2Chains of Numbers Numerologists use the digits 1 to 9 to predict the future, luck or personal traits of individuals. To calculate this digit, addition is performed repeatedly on a series of numbers starting from the original number. For example, if your birth date is 29.11.1989, then we add the digits 2, 9, 1, 1, 1, 9, 8 and 9 to get 40. Then we add the digits 4 and 0 to get 4. The number 4 represents your profile. So, you get a chain of numbers: 29111989 ď&#x192; 40 ď&#x192; 4 Do you know that if your original number is a multiple of nine, then at the end of the chain you will always get the digit 9? The difference between a numerologist and a number theorist (student of number theory - a branch of mathematics) is that the former is interested only in the answer 9, whereas a number theorist will immediately ask â&#x20AC;&#x201C; Why does this happen? Maybe you can figure it out.

Prove that on successively adding the digits of a multiple of 9, the final sum is always 9.

For any given number, how can you predict the final sum without actually creating the chain?

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Adding digits is not the only way to get new numbers from existing numbers. For example, for a two digit number, with digits a and b, we can define the next number as a + 2b.

Starting from any two digit number create the chain of numbers obtained by adding the units digit to twice of the digit in the tenâ&#x20AC;&#x;s place. What is the digit at the end of the chain? Does this digit change if the original number changes?

Consider the number 56. Define a rule to combine the digits 5 and 6 such that its repeated application creates an infinite (but recurring) chain of numbers. How many such rules can you define? Will this rule work for other numbers too?

Consider different numbers and find a rule for successively combining digits to get infinite chains. Is there a pattern that you can find? Can this pattern help you generalise this procedure?

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4.3Estimation Estimation is an integral part of our daily lives because in most scenarios, we are more concerned with an estimate rather than the exact value of a measure. For example, if we have to go from destination A to destination B, we estimate the duration of the journey rather than calculate the exact time of completion of the journey. Estimation improves through practice. For example a fruit juice vendor, through experience, can provide a better estimate of the number of oranges required for five glasses of juice.

Jameel is making a cot for which he needs a certain amount of rope. The dimensions of the cot are 3 ft wide and 6 ft long. He uses two different patterns for the cot. For an area of 3 ft by 1.5 ft toward the feet, he weaves the rope parallel to the longer side of the cot. For the remaining portion, he uses a diagonal weave. How would you estimate the length of rope required to weave this cot? What assumptions did you make to arrive at this estimate?

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A least upper bound (LUB) of an expression is the smallest number which is not smaller than any possible value of the expression. Similarly, the greatest lower bound (GLB) is the largest number that is not larger than any possible value of the expression. The value of sin x cannot be larger than 1 or smaller than â&#x20AC;&#x201C; 1. Thus, the LUB of sin x is 1 and its GLB is â&#x20AC;&#x201C; 1.

Using the facts that â&#x2C6;&#x2019;1 â&#x2030;¤ sin đ?&#x153;&#x192; â&#x2030;¤ 1 and â&#x2C6;&#x2019;1 â&#x2030;¤ cos đ?&#x153;&#x192; â&#x2030;¤ 1, Estimate the Least Integral Upper Bound (M) and Greatest Integral Lower Bound (m) of each of the following expressions. a) sin ď ą ď&#x20AC;Ť cos ď ą b) 2sin 2 ď ą ď&#x20AC; 1 c) 8sin ď ą ď&#x20AC; 4cos ď ą d)

3 4 sin ď ą ď&#x20AC; cos ď ą 5 5

sin ď ą ď&#x20AC;Ť cosď ą

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The map of Rajasthan shows all its districts. The area of Barmer district is 28,387 square kilometre. Find a method to estimate the area of each district using the map below. When you are done with estimating areas, look at the exact areas from a reliable internet source and compare these with the estimates you found.

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4.4Contextualising Equations Word problems are always one of the most difficult aspects of the syllabus. And, the best way to master word problems is to learn how to create them. In this task, you will learn how to create context based word problems. Letâ&#x20AC;&#x;s consider the equation 4x + 3 = 32. This equation models the situations in the following two word problems. 1. Rohit had Rs. 32. After buying four candies, he was left with Rs. 3. Find the price of each candy. 2. Vaani took an online quiz that had four questions. She spent 3 minutes reading the instructions and then spent an equal amount of time to answer each question. If she spent 32 minutes on the quiz, how much time did she spend answering each question? Now, itâ&#x20AC;&#x;s your turn to try similar problems.

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From the following table, create an equation by selecting a collection of expressions with the following rules: a) Two or more expressions chosen from the same row are always added and must lie on the same side of the „=‟ sign; b) Expressions from different rows should be on different sides of the „=‟ sign.

10.50

3/5

2.5x

5x/6

5x – 8

34x – 12

5.5x + 1.7

(2/3)x – 4/5

2x2

x2+ 1

x2– 2

c) Frame a word problem that describes the equation.

What did you find difficult or easy about writing a word problem? Where were you stuck? How realistic is the context that you chose? How can you make it more realistic?

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4.5Searching for Alternatives A real line (1 dimensional space) has infinite points. Each point can be uniquely described by the distance of that point from a fixed point (marked as 0).

Rene Descartes extended the above logic to provide a way to uniquely represent a point on the real plane (2 dimensional space) by an ordered pair of numbers (a, b). Here, the numbers a and b were distances of the point from two mutually perpendicular fixed lines (called axis) that intersected at the point (0, 0).

In what other ways can we uniquely determine the position of a point on a plane?What are the advantanges and disadvantages of the various ways you have found?

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4.6Geometric Intuition Just like there is a mathematical intuition that allows a person to “visualize” a mathematical result, geometric intution allows a person to “visualize” a geometric result. The first step toward developing geometric intuition is to be able to visualise geometric diagrams. And, then analyse the diagramsmentally. For people with good creative sense, this skill is easy. Others, may need to practice and spend more time to master this skill.

In this task, you are required to visualise a diagram that is explained to you. Please do not draw the diagram on paper, till you have completly visualised it. a) Draw three circles C1, C2, C3. b) Select any two circles, say C1 and C2. c) Draw the two common external tangents T12 and t12 to these circles. d) Let P12 be the point of intersection of T12 and t12. e) Now, select another pair of circles, say C2 and C3. f) Draw the two common external tangents T23 and t23 to these circles. g) Let P23 be the point of intersection of T23 and t23. h) Now, select the last pair of circles, C1 and C3.

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i) Draw the two common external tangents T13 and t13 to these circles. j) Let P13 be the point of intersection of T13 and t13. Draw the figure that you visualised? Do you think that others would have visualised this figure in any other way? What could be the differences visualisations of this problem?

How would you describe this bridge?

various

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4.7Visualising Proofs Proofs do not have to be written down, they can be drawn too. A typical example of this procedure is the gemeotric proof of Pythagoras theorem. In the last 20 years, the acceptance of a diagram or a series of diagrams as a proof has attracted a lot of debate. In this task, I will help you construct a visual proof. For this we will refer to task 4.6 â&#x20AC;&#x201C; Geometric Intuition

Draw 3-4 diagrams for the task 4.6 such that you keep the size of two circles fixed and vary the size of the third circle. Also, the centers of the three circles must not lie on the same line.

In each figure, join the points P12, P23, and P31. What do you observe about these points in each of the diagrams that you drew?

There is always a unique line that joins any two distinct points. How can you use this fact to provide a visual proof of what you have observed in the activity above?

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4.8Abstraction: Key to Generalisation Evaluate: a) 5 + 4 b) ½ of 21 c) 6 x 3 I am sure the problems above did not require much thinking. You probably got 9, 11.5 and 18 as answers. Did you stop and question what the numbers in the problems represented? Now try to answer these problems: a) How much is 5 oranges and 4 apples? b) If 21 elephants are arranged in two rows, how many elephants are there in each row? c) What will be the product of a line segment that is 6 cm long and a line segment that is 3 cm long? You may have found it difficult to answer the second set of questions. And your answer would definitely not be 9, 11.5 and 18. But, did you realise that both the sets of questions required the same computations? When you were introduced to numbers, each number represented a collection. For example, 1 nose, 2 eyes, 3 apples, 4 coins and 5 fingers. Very soon you realised that „5 fingers‟, „5 apples‟, and „5 continents‟ had something in common. So, you dropped the objects and the number „5‟ became the focus. Now, if someone said “Show me 5”, you could show them any set of 5 objects.

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Unknowingly, you entered the “Abstract Zone” of mathematics.Abstraction is a celebrated aspect of mathematics. And, knowingly or unknowingly, you are already familiar with it. Abstract is the opposite of concrete. Look around you and spot a cylindrical object. You may see a tubelight, portion of a pipe, a portion of the gas lighter used in the kitchen, a glass, the lid of a bottle, etc. These are all concrete forms of a cylinder. You can touch, feel, or measure these objects. Now, let‟s practice some geometric visualisation. 1. Think of two parrallel lines. You can‟t draw complete lines – because a line has an infinite length. So, you can only create a mental picture of the two parallel lines. 2. Now, imagine that one of these lines is fixed and the other one is revolving around the fixed line such that the distance between the two lines never changes. 3. What kind of shape would you get? The infinite cylinder obtained by revolving a straight line around another is an „abstract‟ form.

Describe the „abstract‟ form of the following concrete objects: a) A marble, basketball, gulabjamun, the Earth. b) An ice-cream cone, a conical party hat.

Look around for different shapes. Can you describe these shapes in an „abstract‟ manner? What is common and different between various descriptions?

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The ability to think at abstract levels is the key to generalisation. Thinking at the abstract level allows you to figure out the difference between the „crucial‟ and the „immaterial‟ details of a problem. As you retain the crucial details and keep playing with the other information you get various specialised cases. These specialisations lead you to discover beautiful generalisations.

Draw a right angled triangle. On each of its sides draw an equilateral triangle. What relationship can you establish between the areas of the equilateral triangles?

Draw a right angled triangle. On each of its sides draw a regular pentagon. What relationship can you establish between the areas of the regular pentagons?

Draw a right angled triangle. On each of its sides draw a semicircle. What relationship can you establish between the areas of the semicircles?

Can you generalise the conclusions of the above three tasks? How many more specialised cases exist? Is there a famous specialisation of this result that you can recall?

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4.9Infinite Dimensions of Mathematical Thought Mathematics exists in all dimensions – 1 to infinite. A lot of results that are true for a particular dimension may also be true for other or all dimensions. However, there are certain results that are true only for a particular dimension. Let‟s see how the shape of the sphere changes as we move from one dimension to the next. 1 – dimension 2 points

2 – dimensions Circle

3 – dimensions Ball

What would a 4 dimensional sphere look like? Does it really exist? If yes, how can we describe its shape? Do spheres exist in higher dimensions? Is there something called an infinite dimensional sphere? These are questions that are relevant to study mathematics. Let‟s see how the definition and form of a unit sphere centered at origin changes as we move from one dimension to another.

Write the equations that describe the points at a distance of 1 unit from the origin a) on the number line. b) on the cartesian plane. c) on cartesian space.

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Write the equations of a unit sphere centered at origin in the: d) 4-dimensional space e) 5-dimensional space f) n-dimensional space g) Infinite dimensional space.

A linear equation in two variables given by ax + by = c is represented by a line in the 2 窶電imensional plane. Write the standard form of a linear equations in a: a) 3-dimensional space b) 4-dimensional space c) n-dimensional space d) Infinite dimensional space.

The real line has infinite points. A circle also has infinite points. Does this mean that the number of points on the real line and the circle are equal? If they have unequal points, which one has more? What is the difference between the number of points on a line and the number of points on the circle? How can you prove your hypothesis?

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4.10The Unexpected... When you are reading a math book or attending a lecture, it seems that mathematics is a very systematic body of knowledge. Concepts, theorems, formulae seem to be welldefined. However, that is not how all new results in mathematics are discovered. The reality is that a vast amount of mathematics has been discovered by accidents. While trying to discover the solution to a problem, mathematicians â&#x20AC;&#x17E;accidentallyâ&#x20AC;&#x; discover new theorems some of which are truly remarkable. Similarly, as you try tasks in this book, be very vigilant and observe everything minutely. You may discover completely new things about math, things that were not even intended to be discovered. In 1899, Frank Morley discovered a very interesting fact associated with trisecting angles of a triangle. Let us see if you can figure out what he discovered.

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Make any triangle. Trisect its angles. You will get a diagram as shown below.

Draw the triangle ACE. What kind of triangle is it? Do you think this happened by accident? Do you think this will always happen? How can you prove your observation?

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Conclusion I thank you for attempting the tasks in this book. It does not matter, if you left out a few tasks or could not complete some of them. Just the fact that you attempted some of them is great and you must congratulate yourself. Within the last 10 years, research in Cognitive Load Theory (CLT) has clearly indicated that the cognitive gain from a task is directly proportional to the level of challenge involved in the task. Thus, I am sure that if you have attempted majority of these tasks sincerely, you have learnt to learn math. Now, you must use these skills in curriculum topics too. To do this, begin with the topic that you are currently studying in class. Think of a skill that you have used to attempt the tasks in this book that can help to better understand the concepts or the problems in that topic. If you cannot figure out what skills can be used to study any topic, send a mail to learn2learnmath@elipsis.in and one of the experts at Ganit Gurooz will guide you. In the end, there are two things I want you to remember for life. First, you can develop and master new skills only if you keep using them regularly. Second, always aim for perfection in everything you do. This way, you will definitely reach excellence. See you at Ganit Gurooz!

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About Ganit Gurooz Ganit Gurooz is an innovative CSR initiative of Elipsis Consulting Pvt. Ltd for improving the quality of math education and empowerment of school mathematics teachers. This initiative was launched in 2009, with the objective of providing free learning resources for mathematics teachers. www.ganitgurooz.com is an effort of a team of mathematicians and math education experts to improve student achievement in mathematics by improving the quality of teaching. The gaining popularity of our online portal has encouraged us to extend the services to students. Ganit Gurooz provides teachers and students an opportunity to interact with each other and mathematics experts across the nation. Thousands of links to content-tutorials, multimedia modules, interactivities, videos, practice assignment and assessment tools have been tagged to chapters of the NCERT textbooks from class 1 to 12. These learning resources have been approved by our experts so that teachers and students can access useful and relevant information easily. Within the first three years, Ganit Gurooz bagged the Manthan Award SouthAsia 2010 for the Best Math Website for Teachers and the eASIA 2011 Award for the Best ICT Enabled Teacher Training Program. However, the real encouragement for our team is the increasing number of visitors to our site. More than 1500 teachers across all cities and towns of India use Ganit Gurooz to improve the math education of lakhs of students.

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Support Ganit Gurooz The mission of Ganit Gurooz cannot be accomplished without the support of every school, principal, teacher, student and parent. In addition, organisations working in education (including NGOâ&#x20AC;&#x;s, government bodies, and corporate entities) who share a common vision are welcome to contact us. You can support us by spreading the word around so that your friends, colleagues and students register and use ganitgurooz.com to learn and teach mathematics. To begin with, we request you to spend some time in sending us a feedback about this book by answering the following questions. You may email your feedback at learn2learnmath@elipsis.in or send a request to access an online feedback form.

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About The Author Dr. Atul Nischal completed his PhD in Mathematics at Tulane University, New Orleans (USA). While pursuing his research in Differential Topology, he taught undergraduate courses for several years. His desire to contribute towards improving quality of school mathematics instruction brought him back to India in 1997. Since his return, Dr. Nischal has been actively engaged in various aspects of school math education. He supports the use of technology to teach mathematics and has been part of several eLearning initiatives in the country. In 2006, he presented a paper “Delivering Lectures Digitally” at the National Seminar on Perspectives in Educational Technology organised by NCERT. In 2007, Dr. Ashok Ganguly, then Chairman, CBSE and President, COBSE invited him to author the theme paper for COBSE‟s conference on Reforms in Mathematics and Science Education. In this paper, Dr. Nischal emphasised the importance of addressing the issues in math education at four levels – Infrastructure, People, Processes and Technology. Dr. Nischal regularly conducts workshops for teachers and students on various aspects of mathematics education and assessment. He regularly visits leading universities to teach undergraduate courses or interact with students at several occasions. He is currently collaborating with the British Council on the prestigious TESS-I project of the Open University (UK) and UK-Aid. This initiative aims to provide open source content to teachers and teacher educators across seven states. Dr. Nischal is authoring Teacher Development Units for secondary mathematics.

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