IP Math Tuition (Integrated Programme)

IP, or Integrated Programme, is well known to test more advanced mathematics than ‘O’ level.

For example:

For partial fractions, IP schools like to test improper fractions. Most of the time, ‘O’ level schools test the easier version of proper fractions.
IP schools like to test half-angle formula in trigonometry. This will absolutely stun students who have not encountered or learnt it before, since teachers rarely teach it in school.
IP schools like to test “area to the left of the curve“, or the dy version of the usual “area under the curve”. Again, this is something that is quite tricky and not usually tested in ‘O’ levels.

Even though IP students do not have ‘O’ levels, they still have to maintain a grade of around 60% (depending on school) in order to promote successfully and avoid the scenario of being transferred to the ‘O’ level track.

One problem in IP schools is that the teachers like to teach a very strange and impractical version of the syllabus. For example, they often teach stuff that are not tested, whilst things that are tested are often not taught. This leads to weaker students being totally lost in class. Also, there are many fancy extracurricular programs in IP schools that further distract the student from the actual important tests.

Hence, many students (even those who score above 250 in PSLE) cannot cope well with the IP syllabus, due to the above factors.

For IP Math Tuition (Upper secondary), contact our tutor Mr Wu at:

Email: mathtuition88@gmail.com

Phone:

Online Math Tuition Singapore

We provide online Math Tuition using Zoom, Skype or Google Hangouts for the following subjects:

E Maths (Sec 3 & Sec 4)
A Maths (Sec 3 & Sec 4)
H2 Maths
H3 Maths
University Maths (selected modules)
Olympiad Math

Do contact our tutor Mr Wu at:

Email: mathtuition88@gmail.com

Phone (WhatsApp/SMS preferred):

Face-to-face home tuition is also available at West region of Singapore (near Jurong East/ Clementi/ Bukit Batok).

Free Career Analysis (With Prizes to be Won)

Free Career Analysis (With Prizes to be Won):
Are you a Careerist, Entrepreneur, Harmoniser, Idealist, Hunter, Internationalist, or Leader?

Find out by doing this Free Career Analysis by renowned survey company Universum:
http://uledge.co/ambWWCY

Thanks a lot for your help! You will also benefit by finding out your Career Personality, which is released at the end of the survey!

1st prize: Win a week-long trip to Singapore, Stockholm, or New York.

Website: http://uledge.co/ambWWCY

The Significance of 8888: MH17+MH370+QZ8501

Although the QZ8501 plane is now confirmed to be in the Java Sea, we still hope that there may be some survivors, who may have possibly swum to a nearby island. Hopefully, the rescuers may be able to rescue some of the passengers.

This is a followup on the previous post on the Probability of adding up to 8888.

Other than the QZ 8501 incident, there are many other incidents that have relation to “8888” or “8” in general. Some of them are the “8888” Uprising in Myanmar, the May 12 Sichuan earthquake, the Blizzard of January 25, and the Tibetan Riots of March 14.

One special fact about 8, is that it is the only natural perfect power that is one less than another perfect power. (2^3=3^2-1) This is known as Catalan’s conjecture or Mihăilescu’s theorem.

Lastly, you may want to watch this video on Chinese Lucky Numbers by Numberphile.

Featured Book:

Biblical Numerology: A Basic Study of the Use of Numbers in the Bible

A sane explanation of biblical numerology. Davis explains the conventional, rhetorical, symbolic, and mystical use of numbers in this fascinating study of the structure and syntax of biblical numbers.

Numbers are present throughout the Bible, and do have some meanings. Why did God create the world in seven days? (rest on the seventh day) Why did Jesus have 12 disciples, not more or not less? Read this book to find out!

Star Wars Math

With the Star Wars Episode 7 coming up, all Star Wars fans are really excited. The trailer alone has reached 50 million views, barely a month after it was released.

$math of star wars$ Wait, can Star Wars be related to Math? Yes it can! Check out The Math of Star Wars which describes a Math related question related to Star Wars! As a Math Tutor, I try my best to relate anything and everything to Math! 😛

Christmas is ending soon, and hope everyone had a nice day, and happy new Year!

Featured Book:

Star Wars Workbook: 1st Grade Math

Does your child love Star Wars but hate math? Well, this might just work! A highly rated Workbook based on Star Wars.

NUS S/U Tips and Tricks from Provost

Source: http://blog.nus.edu.sg/provost/2014/12/23/su1101-the-science-and-art-of-the-su-option/

Check out this link by NUS Provost, Prof. Tan, who was also a Math Professor.

The Art of S/U:

Now, which grades should you keep?

If you obtained A or A+, well done and keep the grade!

If you obtained B+ or A-, I would generally encourage you to keep the grade as well. For those who may be thinking of exercising S/U on a B+ to qualify for the Dean’s List, do note that there will NOT be Dean’s Lists for the first two semesters.

If you obtained Bs and Cs, it is a little tricky. In theory, you should exercise S/U on your worst grades. However, the challenge is to do so without foresight of the grades that you will get for subsequent semesters. You should base your decisions on your academic goals and your self-assessment of your expected academic performance for the rest of your candidature. If you do not have a goal right now, your first semester CAP (before any S/U options are exercised) may be a good guide.

The S/U option is really a lifesaver for those at the borderlines, for example Borderline First Class Honours (CAP around 4.5), or Borderline Second Upper (Cap around 4.0). It makes a difference to your final grades.

The S/U option will apply to all Level 1000 modules (with or without pre-requisites) and Level 2000 modules without other NUS modules as pre-requisites, unless otherwise stipulated by the Faculties/Departments.

Taken from: http://www.nus.edu.sg/registrar/edu/UG/graduation.html#SU

This is a great improvement from the earlier system, where students could only S/U non-core modules. New NUS students have it good!

Featured Book:

The Calculus Lifesaver: All the Tools You Need to Excel at Calculus (Princeton Lifesaver Study Guide)

Proof that Square Root of Two is Irrational [Rare Constructive Proof] (YouTube Video)

This is my second ever YouTube Video, and it is about a rarely seen proof (Constructive) that Square Root of Two is irrational.

The video is based on my earlier post on: Constructive Proof that Square Root of Two is irrational

Thanks for watching! Please speed up the video according to your preference! You can speed up either 2x or 1.5x for best effect.

Featured Book:

The Irrationals: A Story of the Numbers You Can’t Count On

The ancient Greeks discovered them, but it wasn’t until the nineteenth century that irrational numbers were properly understood and rigorously defined, and even today not all their mysteries have been revealed. In The Irrationals, the first popular and comprehensive book on the subject, Julian Havil tells the story of irrational numbers and the mathematicians who have tackled their challenges, from antiquity to the twenty-first century. Along the way, he explains why irrational numbers are surprisingly difficult to define–and why so many questions still surround them. Fascinating and illuminating, this is a book for everyone who loves math and the history behind it.

Second-derivative Test For Extrema Of Functions Of Two Variables

Source: http://www.math.illinois.edu/~verahur/18.024/notesSD.pdf

Excerpt:

Proof of the second-derivative test. Our goal is to derive the second-derivative test, which determines the nature of a critical point of a function of two variables, that is, whether a critical point is a local minimum, a local maximum, or a saddle point, or none of these. In general for a function of n variables, it is determined by the algebraic sign of a certain quadratic form, which in turn is determined by eigenvalues of the Hessian matrix [Apo, Section 9.11]. This approach however relies on results on eigenvalues, and it may take several lectures to fully develop. Here we focus on the simpler setting when n = 2 and derive a test using the algebraic sign of the second derivative of the function.

The full proof can be found in the featured book below: T. Apostol, Calculus, vol. II, Second edition, Wiley, 1967

Featured book:

Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability (Volume 2)

H2 Maths Tuition: Integration by Parts

Here is a Integration by Parts helpsheet I created and uploaded on Scribd. Integration by Parts is a really useful technique, in fact it is one of the two key integration techniques in H2 Maths. The other technique is Integration by Substitution.

View this document on Scribd

Featured book:

The Humongous Book of Calculus Problems

This book is really “humongous”. If you need to source for practice problems for Calculus, this is a good place to start. 🙂

The Mystery of e^Pi-Pi (Very Mysterious Number)

Source: http://en.wikipedia.org/wiki/Mathematical_coincidence

If you have a calculator, check out the value of $e^\pi-\pi$ . It is 19.99909998…

Why is it so close to the integer 20? Is it a coincidence (few things in Math are coincidence), or is it a sign of something deeper? e and Pi are two very fundamental numbers in Math, and the very fact that $e^\pi-\pi\approx 20$ may well mean something.

This was observed by a few mathematicians (Conway, Sloane, Plouffe, 1988) many years ago, but till this day there is no answer.

Do give it a thought!

Featured book:

Pi: A Biography of the World’s Most Mysterious Number

We all learned that the ratio of the circumference of a circle to its diameter is called pi and that the value of this algebraic symbol is roughly 3.14. What we weren’t told, though, is that behind this seemingly mundane fact is a world of mystery, which has fascinated mathematicians from ancient times to the present. Simply put, pi is weird. Mathematicians call it a “transcendental number” because its value cannot be calculated by any combination of addition, subtraction, multiplication, division, and square root extraction.

In this delightful layperson’s introduction to one of math’s most interesting phenomena, Drs. Posamentier and Lehmann review pi’s history from prebiblical times to the 21st century, the many amusing and mind-boggling ways of estimating pi over the centuries, quirky examples of obsessing about pi (including an attempt to legislate its exact value), and useful applications of pi in everyday life, including statistics.

This enlightening and stimulating approach to mathematics will entertain lay readers while improving their mathematical literacy.

Integration by Substitution (H2 Maths Tuition)

This is a 1 page article prepared by me for students to learn how to do Integration by Substitution, a very useful technique that can integrate many functions.

This is especially useful for students taking H2 Maths, as it is one of the two tools for integration. The other is Integration by Parts. SMU First Year Students also have to take a calculus course which includes Integration by Substitution too.

This is also my first time trying out embedding Scribd into WordPress, so that users can view the document on the website itself without downloading anything. 🙂

View this document on Scribd

Print version: Integration by Substitution

Featured book:

Calculus: An Intuitive and Physical Approach (Second Edition) (Dover Books on Mathematics)

Application-oriented introduction relates the subject as closely as possible to science. In-depth explorations of the derivative, the differentiation and integration of the powers of x, and theorems on differentiation and antidifferentiation lead to a definition of the chain rule and examinations of trigonometric functions, logarithmic and exponential functions, techniques of integration, polar coordinates, much more. Clear-cut explanations, numerous drills, illustrative examples. 1967 edition. Solution guide available upon request.

Why We Use “X” as the Unknown in Math

Why do we use “x” in algebra? Why not “a”, “b” or even “z”?

Find out the answer here: http://gizmodo.com/why-we-use-x-as-the-unknown-in-math-1657254357

Excerpt:

For hundreds of years, x has been the go-to symbol for the unknown quantity in mathematical equations. So who started this practice?

Algebra was born in the Middle East, during the Golden Age of medieval Islamic civilization (750 to 1258 AD), and its early form can be seen in the work of Muhammad Al-Khwarizmi and his 9th century book, Kitab al-jabr wal-muqabala (al-jabr later morphing into algebra in English). During this heyday, Muslim rule and culture had expanded onto the Iberian Peninsula, where the Moors encouraged scholarship in the sciences and math.

So what does this have to do with the letter “x” in math? In a recent TED talk, the director of The Radius Foundation, Terry Moore, posited that the the use of “x” in this way began with the inability of Spanish scholars to translate certain Arabic sounds, including the letter sheen (or shin). According to Moore, the word for “unknown thing” in Arabic is al-shalan, and it appeared many times in early mathematical works. (For example, you might see “three unknown things equals 15,” with the “unknown thing” then being 5.)

But since Spanish scholars had no corresponding sound for “sh,” they went with the “ck” sound, which in classical Greek is written with the chi symbol, X. Moore theorizes, as many others before him have done, that when this was later translated into Latin, the chi (X) was replaced with the more common Latin x. This is similar to how Xmas, meaning Christmas, came about from the common practice of religious scholars using the Greek letter chi (X) as a shorthand for “Christ.”

Featured book:

Algebra Survival Guide: A Conversational Guide for the Thoroughly Befuddled

Motivational Books for the Student (Educational)

Christmas is almost here, and what could be a better Christmas present than a motivational book? Books, although viewed as old-fashioned in the current world of iPhones and iPads, are still very important learning tools that can change a person’s life.

Sometimes, bright or even gifted children don’t do well in school, because of motivational issues. Many academically weak students are actually bright students, but not focused on studies due to lack of motivation. They may feel that school is boring, or see school as a chore. Teenage years are often a difficult period of time. If successfully motivated, these children can often make a wondrous turnaround to get back on track academically.

Let me share some books that are actually purchased by viewers from my website through Amazon.com:

1)
Chicken Soup for the Soul: Think Positive for Kids: 101 Stories about Good Decisions, Self-Esteem, and Positive Thinking

2)
Drive: 9 Ways to Motivate Your Kids to Achieve

3)
Empowering Underachievers: New Strategies to Guide Kids (8-18) to Personal Excellence

4)
Have a New Kid by Friday: How to Change Your Child’s Attitude, Behavior & Character in 5 Days

5)
Making Children Mind without Losing Yours

6)
Motivate Your Son: Inspire Your Boy To Be Engaged In School, Excited For College, and Energized For Success

7)
Smart Parenting for Smart Kids: Nurturing Your Child’s True Potential

8)
The Motivation Breakthrough: 6 Secrets to Turning On the Tuned-Out Child

Inspirational Video by Nick Vujicic – DVD Clip from No Arms, No Legs, No Worries – Jr. High Talk

Life Without Limits: Inspiration for a Ridiculously Good Life

Chinese maths teachers show way (Math News)

Latest Math News: Chinese maths teachers show way. Seems like Asia (especially China) is really going to be a world power in the next century, in terms of both education and economy. Students really need to improve their Chinese and Math skills in order to take advantage of this future trend!

College pays students for getting a ‘C’ in math

Interesting idea. Definitely a motivation for most students!

Source: http://www.campusreform.org/?ID=6058

Hillsborough Community College will pay up to $1,800 in cash to students who make a C or higher in three semesters of math classes.

Students also have the to win free textbooks.

HCC funds part of this experiment with additional funding coming from a George Soros organization.

Florida community college is trying to inspire students to finish their degree by doling out up to $1,800 in cash to students who make a C or higher in three semesters of math courses.

The program, called Mathematic Access Scholarship Program (MAPS), has run at Hillsborough Community College (HCC) for the past three years and is spearheaded by Manpower Demonstration Research Corporation (MDRC).

“We hoped the incentives would inspire behaviors that would lead to increased student success.”

Featured book:

Great Jobs for Math Majors, Second ed. (Great Jobs For… Series)

“What can I do with a degree in math?”

You’ve worked hard for that math degree. Now what? Sometimes, the choice of careers can seem endless. The most difficult part of a job search is starting it. This is where Great Jobs for Math Majors comes in. Designed to help you put your major to work, this handy guide covers the basics of a job search and provides detailed profiles of careers in math. From the worlds of finance and science to manufacturing and education, you’ll explore a variety of job options for math majors and determine the best fit for your personal, professional, and practical needs.

Do you want to be an actuary? Work in the banking industry? Program computers? In this updated edition, you’ll find:

Job-search basics such as crafting résumés and writing cover letters
Self-assessment exercises to help determine your professional fit
Investigative tools to help you find the perfect job
Networking tips to get your foot in the door before your résumé is even sent
True tales from practicing professionals about everyday life on the job
Current statistics on earnings, advancement, and the future of the profession
Resources for further information, including journals, professional associations, and online resources

Are you a Careerist, Entrepreneur, Harmoniser, Idealist, Hunter, Internationalist, or Leader?

Are you a Careerist, Entrepreneur, Harmoniser, Idealist, Hunter, Internationalist, or Leader?

Find out by doing this Free Career Analysis by renowned survey company Universum:
http://uledge.co/ambWWCY

Thanks a lot for your help! You will also benefit by finding out your Career Personality, which is released at the end of the survey!

http://uledge.co/ambWWCY

The Making of a Mile of Pi – Numberphile

This guy (from the Youtube channel Numberphile) actually printed a million digits of Pi! Check out how long the piece of paper actually is!

Longer Version (30 minutes):

Featured book:

Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing
This book provides a fun, hands-on approach to learning how mathematics and computing relate to the world around us and help us to better understand it. How can reposting on Twitter kill a movie’s opening weekend? How can you use mathematics to find your celebrity look-alike? What is Homer Simpson’s method for disproving Fermat’s Last Theorem? Each topic in this refreshingly inviting book illustrates a famous mathematical algorithm or result–such as Google’s PageRank and the traveling salesman problem–and the applications grow more challenging as you progress through the chapters. But don’t worry, helpful solutions are provided each step of the way.

Math Bytes shows you how to do calculus using a bag of chocolate chips, and how to prove the Euler characteristic simply by doodling. Generously illustrated in color throughout, this lively and entertaining book also explains how to create fractal landscapes with a roll of the dice, pick a competitive bracket for March Madness, decipher the math that makes it possible to resize a computer font or launch an Angry Bird–and much, much more. All of the applications are presented in an accessible and engaging way, enabling beginners and advanced readers alike to learn and explore at their own pace–a bit and a byte at a time.

Topics coming out for A Maths Paper 2

Recently, the A Maths Paper 2 just finished (today), and the A Maths Paper 2 is hot on its heels, coming tomorrow!

Usually, topics tested in Paper 1 will most likely not come out again in Paper 2, so students doing their last minute revision can use this fact to focus their revision.

Topics that came out in Paper 1:

Binomial Theorem
Trigonometry (Addition Formula, find exact value)
Rate of change
Partial Fractions
Linear Law
Prove Trigonometry
Coordinate Geometry
Integrate & Differentiate Trigonometric Functions
Discriminant (b^2-4ac)
Stationary Points
Tangent/Normal
Quadratic/Modulus

Topics likely to come out in Paper 2:

Indices/Surds
Polynomials
Exponential / Logarithmic Equations
R-formula
Sketching of Trigonometric Graphs
Circles
Proofs in plane geometry
Integration as the reverse of differentiation
Area under curve
Kinematics

All the best for those who are taking the A Maths Paper 2 exam tomorrow. 🙂

Math Olympiad Integer Sequence Question

Check out this interesting Math Olympiad Integer Sequence Question! (September 2014 Math Problem of the Month)

Main Page: http://www.fen.bilkent.edu.tr/~cvmath/Problem/problem.htm

Question: http://www.fen.bilkent.edu.tr/~cvmath/Problem/1408q.pdf

Solution: http://www.fen.bilkent.edu.tr/~cvmath/Problem/1409a.pdf

Featured book:

Fifty Lectures for American Mathematics Competitions: Volume 2

While the books in this series are primarily designed for AMC competitors, they contain the most essential and indispensable concepts used throughout middle and high school mathematics. Some featured topics include key concepts such as equations, polynomials, exponential and logarithmic functions in Algebra, various synthetic and analytic methods used in Geometry, and important facts in Number Theory.

The topics are grouped in lessons focusing on fundamental concepts. Each lesson starts with a few solved examples followed by a problem set meant to illustrate the content presented. At the end, the solutions to the problems are discussed with many containing multiple methods of approach.

I recommend these books to not only contest participants, but also to young, aspiring mathletes in middle school who wish to consolidate their mathematical knowledge. I have personally used a few of the books in this collection to prepare some of my students for the AMC contests or to form a foundation for others.

By Dr. Titu Andreescu
US IMO Team Leader (1995 – 2002)
Director, MAA American Mathematics Competitions (1998 – 2003)
Director, Mathematical Olympiad Summer Program (1995 – 2002)
Coach of the US IMO Team (1993 – 2006)
Member of the IMO Advisory Board (2002 – 2006)
Chair of the USAMO Committee (1996 – 2004)

I love this book! I love the style, the selection of topics and the choice of problems to illustrate the ideas discussed. The topics are typical contest problem topics: divisors, absolute value, radical expressions, Veita’s Theorem, squares, divisibility, lots of geometry, and some trigonometry. And the problems are delicious.

Although the book is intended for high school students aiming to do well in national and state math contests like the American Mathematics Competitions, the problems are accessible to very strong middle school students.

The book is well-suited for the teacher-coach interested in sets of problems on a given topic. Each section begins with several substantial solved examples followed by a varied list of problems ranging from easily accessible to very challenging. Solutions are provided for all the problems. In many cases, several solutions are provided.

By Professor Harold Reiter
Chair of MATHCOUNTS Question Writing Committee.
Chair of SAT II Mathematics committee of the Educational Testing Service
Chair of the AMC 12 Committee (and AMC 10) 1993 to 2000.

We are on Facebook, Twitter, and Google+ !

Thanks for reading our blog Mathtuition88.com.

We are also on:

Facebook: https://www.facebook.com/mathtuition88

Twitter: https://twitter.com/mathtuition88

Google+: Mathtuition88

If you are a subscriber, thanks for your support! We will continue our mission of blogging interesting articles on Mathematics!

Featured book:

Coding the Matrix: Linear Algebra through Applications to Computer Science

Google’s signature ranking algorithm “PageRank” is heavily based on linear algebra! Read the above book to find out more!

An engaging introduction to vectors and matrices and the algorithms that operate on them, intended for the student who knows how to program. Mathematical concepts and computational problems are motivated by applications in computer science. The reader learns by doing, writing programs to implement the mathematical concepts and using them to carry out tasks and explore the applications. Examples include: error-correcting codes, transformations in graphics, face detection, encryption and secret-sharing, integer factoring, removing perspective from an image, PageRank (Google’s ranking algorithm), and cancer detection from cell features. A companion web site,

codingthematrix.com

provides data and support code. Most of the assignments can be auto-graded online. Over two hundred illustrations, including a selection of relevant xkcd comics.

Chapters: The Function, The Field, The Vector, The Vector Space, The Matrix, The Basis,Dimension, Gaussian Elimination, The Inner Product, Special Bases, The Singular Value Decomposition, The Eigenvector, The Linear Program

Lunes (Mathematics)

What are Lunes? Check out this absolutely interesting video:

Source: http://en.wikipedia.org/wiki/Hippocrates_of_Chios

Featured book:

A History of Greek Mathematics, Volume II: From Aristarchus to Diophantus (Dover Books on Mathematics)

“As it is, the book is indispensable; it has, indeed, no serious English rival.” — Times Literary Supplement
“Sir Thomas Heath, foremost English historian of the ancient exact sciences in the twentieth century.” — Prof. W. H. Stahl
“Indeed, seeing that so much of Greek is mathematics, it is arguable that, if one would understand the Greek genius fully, it would be a good plan to begin with their geometry.”

Love Math Graphs!

How to remember graphs?

Many students have a hard time remembering how graphs look like.

Here is a humorous cartoon (suitable for Valentine’s Day) on how some graphs look like!

Remember, Maths is not just about exams, homework, or getting A1/A2. Maths, above all, is about the LOVE of learning and thinking.

$math graphs$

Featured book:

Love and Math: The Heart of Hidden Reality

A New York Times Science Bestseller

What if you had to take an art class in which you were only taught how to paint a fence? What if you were never shown the paintings of van Gogh and Picasso, weren’t even told they existed? Alas, this is how math is taught, and so for most of us it becomes the intellectual equivalent of watching paint dry.

In Love and Math, renowned mathematician Edward Frenkel reveals a side of math we’ve never seen, suffused with all the beauty and elegance of a work of art. In this heartfelt and passionate book, Frenkel shows that mathematics, far from occupying a specialist niche, goes to the heart of all matter, uniting us across cultures, time, and space.

Love and Math tells two intertwined stories: of the wonders of mathematics and of one young man’s journey learning and living it. Having braved a discriminatory educational system to become one of the twenty-first century’s leading mathematicians, Frenkel now works on one of the biggest ideas to come out of math in the last 50 years: the Langlands Program. Considered by many to be a Grand Unified Theory of mathematics, the Langlands Program enables researchers to translate findings from one field to another so that they can solve problems, such as Fermat’s last theorem, that had seemed intractable before.

At its core, Love and Math is a story about accessing a new way of thinking, which can enrich our lives and empower us to better understand the world and our place in it. It is an invitation to discover the magic hidden universe of mathematics.

Five things successful teachers do on the first day of class

$math joke$

Featured book:

Comic-Strip Math: Problem Solving: 80 Reproducible Cartoons With Dozens and Dozens of Story Problems That Motivate Students and Build Essential Math Skills

Math + Comics = Learning That’s Fun! Help students build essential math skills and meet math standards with 80 laugh-out-loud comic strips and companion mini-story problems. Each reproducible comic and problem set reinforces a key math skill: multiplication, division, fractions, decimals, measurement, geometry, and more. Great to use for small-group or independent class work and for homework! For use with Grades 3-6.

How to choose a toilet using Mathematics

We have seen how to cut a cake using the mathematical way, but did you know Mathematics can also be used when choosing toilets?

Here’s how:

Featured book:

Common Core Math 4 Today, Grade 4: Daily Skill Practice (Common Core 4 Today)

Build a foundation and focus on what matters most for math readiness with Common Core Math 4 Today: Daily Skill Practice for fourth grade. This 96-page comprehensive supplement contains standards-aligned reproducible activities designed to focus on critical math skills and concepts that meet the Common Core State Standards. Each page includes 16 problems to be completed during a four-day period. The exercises are arranged in a continuous spiral so that concepts are repeated weekly. An assessment for the fifth day is provided for evaluating students’ understanding of the math concepts practiced throughout the week. Also included are a Common Core State Standards alignment matrix and an answer key.

GAT DSA Past Year Paper

GAT: General Ability Test

Most schools DSA (Direct School Admissions) now requires sitting for a test called GAT.

While the actual past year papers are not to be found online, there are many similar test papers from other countries:

1) http://bettereducation.com.au/Resources/PastTestPapers.aspx

2) http://acedmy.yolasite.com/resources/gat_sample_paper.pdf

3) http://www.cpapers.com/past-papers/gat-general-test-model-questions-answers.php

4) GEP Books are an excellent source of DSA questions, since the scope of GAT testing overlaps with the Logic portion of the GEP test. Check out the myriad of GEP Books that can be used to prepare for DSA questions equally effectively.

The Logic portion of GEP test / DSA test is not taught anywhere in the MOE syllabus, and hence the most challenging to prepare for. Your child would need to solve DSA questions like the one below, which is quite obviously not taught anywhere from Primary 1 to Primary 6. However, like all skills, these kind of logic puzzles can be taught, trained, and practiced, in the Mensa book listed below (Scroll down)!

circle-traingle-puzzle-iq-test — Children can be trained to solve this type of DSA GAT questions easily

Boost your DSA GAT Scores with Mensa Book:

Match Wits With Mensa: The Complete Quiz Book

If you are looking for more DSA GAT pattern/logic questions, this is the Complete Quiz Book by Mensa. Highly rated on Amazon. These book will be helpful for those seeking for a boost in their DSA GAT scores, since GAT (General Ability Test) is just a politically correct name for IQ Test.

Furthermore, the IQ of a person is not static, it can be changed. The way to change IQ is via reading books and acquiring more knowledge.

Another good book for DSA/GAT/HAST is Ultimate IQ Tests: 1000 Practice Test Questions to Boost Your Brain Power. This book is like the “Ten Year Series” of GAT DSA tests, it will be a good and trusted book for Singaporeans who are used to studying using the practice “Ten Year Series” method, which has undoubtedly worked for generations of Singaporeans (including myself). The 1000 Practice questions (!!!) (similar to GAT) would definitely go a long way in your DSA preparation.

Subscribe to our free newsletter, for updates on Math, Education and other General Knowledge topics!
Subscribe to Blog via Email

Enter your email address to subscribe to this blog and receive notifications of new posts by email.

Email Address:

Many people think that the infamous Cheryl Birthday puzzle is very difficult. However, to a well trained Math Olympian, the Cheryl Birthday question is actually considered comparatively easy! This shows that IQ of a person can be increased by reading, learning, and practicing the relevant books.

More Books to Ramp Up your DSA GAT Score:
https://mathtuition88.com/2013/11/11/recommended-books-for-gep-selection-test/

P.S. These kind of books are rarely found in Singapore bookstores, not to mention that most decent Singapore bookstores like Borders/Page One have closed down. I have compiled the most helpful books for DSA Score-Boosting in the above link. Hope it helps!

Update (2016): Check out this Pattern Recognition (Visual Discrimination) book that is a guided tutorial for training for GEP / DSA Tests!

Motivational Books for DSA

As Singapore is a very high-tech society, there are many children who are addicted to handphones /computer games and as a result have no motivation to learn. Needless to say, this would result in rather severe consequences in exam results if not corrected early. Even for gifted children, the consequence of computer/cellphone addiction is really harmful, not to mention students who already have a weak academic foundation. Hence, motivational books like those listed here are actually of great importance. Only if a child sees the value of learning, will he be interested and self-motivated in learning. Related book: Cyber Junkie: Escape the Gaming and Internet Trap.

NUS High DSA

If you are looking for information regarding NUS High DSA, please click here.

Finally, all the best and good luck for your DSA test!

Kindle for Singaporean Students

Parents who like the idea of technology combined with education may want to check out the Kindle rather than the iPad.
Kindle Paperwhite, 6″ High-Resolution Display (212 ppi) with Built-in Light, Wi-Fi – Includes Special Offers

The problem with the iPad is that there are too many games! Children (and even adults) will find it hard to resist the games. The Kindle would be better for education, since it is primarily a reading device, and there are many educational books available at low cost or even free.

For example, this course CK-12 Algebra I – Second Edition, Volume 1 Of 2 is totally free and costs $0.00 if you have the Kindle. Hence, the Kindle is a much better alternative to iPad for students.

Buy Kindle from Qoo10 (Singapore’s Taobao/Amazon)

[S$119.90][Kindle]★ Amazon Kindle 2015! Free 8000 Ebooks.Pouch.Screen Protector.Tutorial! Best Amazon Kindle 7 Paperwhite Voyage Ebook Ereader Tablet Laptop Reader! ★

WWW.QOO10.SG

World Cup fans need math to figure out scenarios

RIO DE JANEIRO (AP) — Every four years, the World Cup forces fans to remember their math lessons.

Working out what each team needs from its final match to finish in the top two of a group and advance to the knockout rounds takes some algebra knowledge and powers of prediction.

After Brazil and Mexico played to a scoreless draw on Tuesday, the calculation became clear: Both teams just need to draw in their next matches to advance with five points in Group A. Croatia, which beat Cameroon Wednesday, would get to six points by beating Mexico. So a draw with Cameroon would still get Brazil through with five points. If Mexico beats Croatia, Brazil would advance even if it loses. But if Mexico and Croatia draw, and Brazil loses — then it gets complicated with tiebreakers.

Featured book:

The Math of Sports: Integrating Math in the Real World (Integrating Math in the Real World Series)

The Scientific (Mathematical) Way to Cut a Cake

Ever wondered if there is an alternative way to cutting cake so that it can stay fresh and softer in the refrigerator?

This is how!

Featured book:

The Calculus Lifesaver: All the Tools You Need to Excel at Calculus (Princeton Lifesaver Study Guide)

For many students, calculus can be the most mystifying and frustrating course they will ever take. The Calculus Lifesaver provides students with the essential tools they need not only to learn calculus, but to excel at it.

All of the material in this user-friendly study guide has been proven to get results. The book arose from Adrian Banner’s popular calculus review course at Princeton University, which he developed especially for students who are motivated to earn A’s but get only average grades on exams. The complete course will be available for free on the Web in a series of videotaped lectures. This study guide works as a supplement to any single-variable calculus course or textbook. Coupled with a selection of exercises, the book can also be used as a textbook in its own right. The style is informal, non-intimidating, and even entertaining, without sacrificing comprehensiveness. The author elaborates standard course material with scores of detailed examples that treat the reader to an “inner monologue”–the train of thought students should be following in order to solve the problem–providing the necessary reasoning as well as the solution. The book’s emphasis is on building problem-solving skills. Examples range from easy to difficult and illustrate the in-depth presentation of theory.

The Calculus Lifesaver combines ease of use and readability with the depth of content and mathematical rigor of the best calculus textbooks. It is an indispensable volume for any student seeking to master calculus.

Serves as a companion to any single-variable calculus textbook
Informal, entertaining, and not intimidating
Informative videos that follow the book–a full forty-eight hours of Banner’s Princeton calculus-review course–is available at Adrian Banner lectures
More than 475 examples (ranging from easy to hard) provide step-by-step reasoning
Theorems and methods justified and connections made to actual practice
Difficult topics such as improper integrals and infinite series covered in detail
Tried and tested by students taking freshman calculus

Primary 6 PSLE Angles Questions and Solutions

Having trouble with Angles questions?

Try out this practice booklet (questions compiled from Prelims) with Primary 6 PSLE Angle questions!

P6 Angles Questions

Solutions

Does your child dread Math and avoid it like the plague?

Looking for a book that is fun yet suitable for young kids?

Featured book:

Bedtime Math: A Fun Excuse to Stay Up Late

Our mission: to make math a fun part of kids’ everyday lives.

We all know it’s wonderful to read bedtime stories to kids, but what about doing math? Many generations of Americans are uncomfortable with math and numbers, and too often we hear the phrase, “I’m just not good at math!” For decades, this attitude has trickled down from parents to their kids, and we now have a culture that finds math dry, intimidating, and just not cool.

Bedtime Math wants to change all that. Inside this book, families will find fun, mischief-making math problems to tackle—math that isn’t just kid-friendly, but actually kid-appealing. With over 100 math riddles on topics from jalapeños and submarines to roller coasters and flamingos, this book bursts with math that looks nothing like school. And with three different levels of challenge (wee ones, little kids, and big kids), there’s something for everyone. We can make numbers fun, and change the world, one Bedtime Math puzzle at a time.

How to be good in Additional Mathematics

Recently, I saw that many people searched the following terms on Google and landed on my website:

Why is the mid-year exams difficult and many people fail it?
How to be good in additional mathematics.

Let me try to answer the above questions:

Why is the mid-year exams difficult and many people fail it?

Usually teachers will set the mid-year exams and the prelims at a (much) higher level than the actual O Levels. This is the current trend, which may result in many people failing the mid-year exam. The idea may be to motivate students to study harder and avoid being complacent with their results. Do not be demoralized by failing the exam! On the contrary, do reevaluate your study strategies, and strive to improve your knowledge and technique in mathematics.

How to be good in additional mathematics.

The way to be good at additional mathematics is the same as the way to be good at piano, chess, and virtually any human endeavour. The key to improving is practice! Practice with understanding is the key. Would you imagine to be possible to improve in playing the piano without practicing the song? Improve in badminton without training? Definitely not! Similarly, improving in additional mathematics is not possible without practice. This is why the Ten Year Series is such a popular book: it is indeed the most useful book you can buy for studying Additional Mathematics.

Practicing with understanding helps with Application of Concepts, Increase Speed, Accuracy, which all helps in being good at additional mathematics.

In addition, during the practice sessions, try to practice checking for careless mistakes. It will help tremendously in improving your grades. Practicing with understanding means that we need to understand the method used, to the extent that if the teacher sets a slightly different question we are still able to do it. This is the secret to being good at additional maths. 🙂

Featured book:

Math Doesn’t Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail

From a well-known actress, math genius and popular contestant on “Dancing With The Stars”—a groundbreaking guide to mathematics for middle school girls, their parents, and educators

World Cup Math Prediction: Brazil 48.5% Chance of Winning!!!

Source: http://blogs.scientificamerican.com/observations/2014/06/11/world-cup-prediction-mathematics-explained/

The World Cup is back, and everyone’s got a pick for the winner. Gamblers have been predicting the outcome of sporting contests since the first foot race across the savannah, but in recent years a unique type of statistical analysis has taken over the prediction business. Everyone from Goldman Sachs to Bloomberg to Nate Silver’s FiveThirtyEight has an online World Cup predictor that uses numbers, not hunches, to generate precise probabilities for match outcomes. Goldman Sachs, for instance, gives host nation Brazil a 48.5 percent chance of winning it all; FiveThirtyEight puts the odds at 45 percent while Bloomberg Sports has concluded there’s just a 19.9 percent chance of a triumph for the Seleção.

Where do these numbers come from? All statistical analysis must start with data, and these soccer prediction engines skim results from former matches. A fair bit of judgment is necessary here. Big international soccer tournaments only come around every so often, so the analysts have to choose how to weight team performance in lesser events such as international “friendlies,” where nothing of consequence is at stake. The modelers also have to decide how far back to pull data from—does Brazil’s proud soccer history matter much when its oldest player is 34?—and how to rate the performance of individual players during their time playing for club teams such as Manchester United or Real Madrid.

Wherever the data comes from, the modeler now has to incorporate it into a model. Frequently, the modeler translates the question of “who is going to win?” into the form “how many goals will team X score against team Y?” And for this, she relies [PDF] on a statistical tool called a bivariate Poisson regression.

Read more at: http://blogs.scientificamerican.com/observations/2014/06/11/world-cup-prediction-mathematics-explained/

Statistics and mathematics is useful after all! Only time will tell if the prediction is correct.

Featured book:

Statistics in Plain English, Third Edition

This inexpensive paperback provides a brief, simple overview of statistics to help readers gain a better understanding of how statistics work and how to interpret them correctly. Each chapter describes a different statistical technique, ranging from basic concepts like central tendency and describing distributions to more advanced concepts such as t tests, regression, repeated measures ANOVA, and factor analysis. Each chapter begins with a short description of the statistic and when it should be used. This is followed by a more in-depth explanation of how the statistic works. Finally, each chapter ends with an example of the statistic in use, and a sample of how the results of analyses using the statistic might be written up for publication. A glossary of statistical terms and symbols is also included.

World Cup Math

World Cup Math: Birthday Paradox

Source: http://www.bbc.com/news/magazine-27835311

It’s puzzling but true that in any group of 23 people there is a 50% chance that two share a birthday. At the World Cup in Brazil there are 32 squads, each of 23 people… so do they demonstrate the truth of this mathematical axiom?

Imagine the scene at the Brazilian football team’s hotel. Hulk and Paulinho are relaxing after another stylish win. Talk turns from tactics to post World Cup plans.

“It’ll be one party after another,” says Hulk, confidently assuming Brazilian victory on home soil. “First the World Cup, then my birthday a couple of weeks later.”

“Your birthday’s in July?” replies Paulinho. “Me too – 25 July, when’s yours?

“No way, exactly the same day!” exclaims Hulk incredulously. “What are the chances of that?”

With 365 days in a regular year, most people’s intuitive answer would probably be: “Pretty small.”

But in this case our intuition is wrong – and the proof of that is known as the birthday paradox.

Hulk and Paulinho

Also read our earlier post on Understanding the Birthday Paradox!

Featured book:

The Math of Sports: Integrating Math in the Real World (Integrating Math in the Real World Series)

Can you read this Math Clock?

DCI Matte Black Powder Coated Metal Mathematics Blackboard Pop Quiz Clock, 11-1/2″ Diameter

If you can read this clock, you are without a doubt a geek. Each hour is marked by a simple math problem. Solve it and solve the riddle of time. Matte black powder coated metal. Requires 1 AA battery (not included). 11-1/2″ Diameter.

Who knew there was such an interesting clock? 😛

Also interesting is this mug below. If you can read it, it shows you are well versed in Mathematics!

I Ate Some Pie and It Was Delicious Mug– Perfect For Any Math Nerd!!– Funny High Quality Coffee Mug!! (11oz, Black I Ate Sum Pi)

Theorem on friends and strangers

Suppose a party has six people. Consider any two of them. They might be meeting for the first time—in which case we will call them mutual strangers; or they might have met before—in which case we will call them mutual acquaintances. The theorem says:

In any party of six people either at least three of them are (pairwise) mutual strangers or at least three of them are (pairwise) mutual acquaintances.

Featured book:

Competition Math for Middle School

Very Inspirational Math Video

Just to share a video here:

Very Inspirational Math Video

It is a video of a girl who once did a math quiz and totally blanked out for the whole quiz. However, it turned out that her teacher did not actually ask for the quiz back, and gave her as much time as she wanted to complete the quiz. Under the relaxed circumstances, she completed the quiz and got a ‘C’. (big improvement from totally blank).

Then, she went to UCLA (very good school in US), and became a mathematics major, and wrote the book that is listed below the video!

Truly inspiring. For some kids, too much pressure may result in Math anxiety and totally blank out, while for other kids a little bit of pressure is needed to ensure that they do take studies seriously. Need to find the perfect balance for each child.

**The book the girl above wrote is: Math Doesn’t Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail

Free Trial: Amazon Prime

Dear Readers,

Thanks for following our Maths Blog.

We are glad to introduce to you a Free Trial of Amazon Prime worth $99!

Random Math Fact:

Did you know:

Euler’s “lucky” numbers are positive integers n such that m² − m + n is a prime number for m = 0, …, n − 1.

Leonhard Euler published the polynomial x² − x + 41 which produces prime numbers for all integer values of x from 0 to 40. Obviously, when x is equal to 41, the value cannot be prime anymore since it is divisible by 41. Only 6 numbers have this property, namely 2, 3, 5, 11, 17 and 41.

Source: http://en.wikipedia.org/wiki/Lucky_numbers_of_Euler

Why is e irrational?

Anyone who has taken high school math is familiar with the constant $\boxed{e=2.718281828\cdots}$ .

e-irrational

Today we are going to prove that e is in fact irrational! We will go through Joseph Fourier‘s famous proof by contradiction. The maths background we need is to know the power series expansion: $\displaystyle \boxed{e=\sum_{n=0}^{\infty}\frac{1}{n!}}$ . The proof is slightly tricky so stay focussed!

(Reference: http://en.wikipedia.org/wiki/Proof_that_e_is_irrational)

Suppose to the contrary that e is a rational number, so $\displaystyle e=\frac{a}{b}$ .

Using the power series formula mentioned above, we have $\displaystyle\sum_{n=0}^\infty \frac{1}{n!}=\frac{a}{b}$

Multiply both sides by $b!$ , $\displaystyle \sum_{n=0}^{\infty}\frac{b!}{n!}=\frac{ab!}{b}=a(b-1)!$

Now, we split the sum into two parts:

$\displaystyle \sum_{n=0}^b \frac{b!}{n!}+\sum_{n=b+1}^\infty \frac{b!}{n!}=a(b-1)!$

Rearranging,

$\displaystyle \sum_{n=b+1}^\infty \frac{b!}{n!}=a(b-1)!-\sum_{n=0}^b \frac{b!}{n!}$

Now, denote $\displaystyle x=\sum_{n=b+1}^\infty \frac{b!}{n!}>0$ . $x$ is an integer since both $\displaystyle a(b-1)!$ and $\displaystyle\sum_{n=0}^b \frac{b!}{n!}$ are integers and their difference (which is x) will be an integer.

We now prove that $x<1$ . For all terms with $n\geq b+1$ we have the upper estimate

$\displaystyle\begin{array}{rcl} \frac{b!}{n!}&=&\frac{1\times 2\times \cdots \times b}{1\times 2\times \cdots \times b \times (b+1) \times \cdots \times n}\\ &=&\frac{1}{(b+1)(b+2)\cdots (b+(n-b))}\\ &\leq& \frac{1}{(b+1)^{n-b}} \end{array}$

This inequality is strict for every $n\geq b+2$ . Changing the index of summation to $k=n-b$ and using the formula for the infinite geometric progression $S_\infty = \frac{a}{1-r}$ , we obtain:

$\displaystyle x=\sum_{n=b+1}^\infty \frac{b!}{n!} < \sum_{n=b+1}^\infty \frac{1}{(b+1)^{n-b}}=\sum_{k=1}^\infty \frac{1}{(b+1)^k}=\frac{\frac{1}{b+1}}{1-\frac{1}{b+1}}=\frac{1}{b}\leq 1$

We have that $x$ is an integer but $0<x<1$ . This is a contradiction (since there is no integer strictly between 0 and 1), and so $e$ must be irrational. (QED)

Interesting? 🙂

Featured book:

Euler: The Master of Us All (Dolciani Mathematical Expositions, No 22)

Did you know the constant e is sometimes called Euler’s number?

Learn more about Euler in this wonderful book. Rated 4.9/5 stars, it is one of the highest rated books on the whole of Amazon.

Leonhard Euler was one of the most prolific mathematicians that have ever lived. This book examines the huge scope of mathematical areas explored and developed by Euler, which includes number theory, combinatorics, geometry, complex variables and many more. The information known to Euler over 300 years ago is discussed, and many of his advances are reconstructed. Readers will be left in no doubt about the brilliance and pervasive influence of Euler’s work.

Watch this video for another proof that e is irrational!

Books for Gifted Children

Featured book:

Match Wits With Mensa: The Complete Quiz Book

This is the #1 Top-Selling book recommended on my website! It includes Mathematical Logic Puzzles from Mensa. Highly recommended for gifted children. Parents, if your child is gifted and you want to stretch his or her learning potential, you may want to buy this book as it is the most complete quiz book on the market. It doesn’t matter whether you are in the Gifted Education Programme, as long as you have an interest in logic puzzles this book is for you.

Maths and Science is essentially about logical thinking, so logic puzzles will directly benefit studies in maths and science. Above all, logic puzzles are meant to be fun and a good and healthy pastime.

Puzzle fans have bought more than 650,000 copies of the Mensa Genius Quiz series—the only books that let readers “match wits with Mensa,” comparing how well they do against members of the famous high-IQ society. Here, in a giant omnibus edition, are four best-selling titles: The Mensa Genius Quiz Books 1 & 2, The Mensa Genius Quiz-A-Day Book, and The Mensa Genius ABC Book. Here are more than 800 fun mindbenders to exercise every part of your brain—word games, trivia, logic riddles, number challenges, visual puzzles—plus tips on how to improve your thinking skills. All the puzzles have been tested by members of American Mensa, Ltd., and include the percentage of Mensa testers who could solve each one, so that you can score yourself against some of the nation’s fittest mental athletes.

Dyscalculia — A Parent’s Guide

Dyscalculia specialist Ronit Bird talks about the difficulties some children have in developing number sense and learning basic arithmetic. She explains some of the common symptoms and indicators for dyscalculia and offers suggstions for how parents can help their children at home. For more information on Dyscalculia please visit http://www.ronitbird.com/

Featured book:

The Dyscalculia Toolkit: Supporting Learning Difficulties in Maths

‘The new dyscalculia toolkit has a great introduction that is broken down into manageable chunks, brilliant explanations and interesting reading. The new tables explain what each game entails at the start of the book, making planning and using the toolkit much easier and effective especially if short on time! Very enjoyable to read, and highly recommended’
-Karen Jones, Chartered Educational Psychologist, The Educational Guidance Service

With over 200 activities and 40 games this book is designed to support learners aged 6 to 14 years, who have difficulty with maths and numbers. Ronit Bird provides a clear explanation of dyscalculia, and presents the resources in a straightforward fashion.

The Monty Hall Problem

This is the clearest and most interesting explanation of the Monty Hall Problem I have ever seen:

What is the Monty Hall Problem? It is basically a game show with 3 doors. Behind one of the doors is a car, while behind the other two doors are two goats. Most people will want to get the car of course.

The player gets a chance to choose one of the doors. Then, the host will open a door which contains a goat. Now, the player is allowed two choices: either stick to his original choice, or switch to the other unopened door. Which choice is better?

Watch the video to find out!

Featured book:

The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser

Mathematicians call it the Monty Hall Problem, and it is one of the most interesting mathematical brain teasers of recent times. Imagine that you face three doors, behind one of which is a prize. You choose one but do not open it. The host–call him Monty Hall–opens a different door, always choosing one he knows to be empty. Left with two doors, will you do better by sticking with your first choice, or by switching to the other remaining door? In this light-hearted yet ultimately serious book, Jason Rosenhouse explores the history of this fascinating puzzle. Using a minimum of mathematics (and none at all for much of the book), he shows how the problem has fascinated philosophers, psychologists, and many others, and examines the many variations that have appeared over the years. As Rosenhouse demonstrates, the Monty Hall Problem illuminates fundamental mathematical issues and has abiding philosophical implications. Perhaps most important, he writes, the problem opens a window on our cognitive difficulties in reasoning about uncertainty.

Singapore Education News

PSLE tweaks will come but as part of broader changes to education system: Heng
Straits Times
SINGAPORE – Changes being made to the Primary School Leaving … be done in the light of the broader changes to Singapore’s education system, …

SMU to broaden learning for freshmen
Straits Times
Freshmen entering the Singapore Management University (SMU) in August next year will go through a revamped syllabus, in the university’s bid to …

MOE to focus on tertiary, secondary education before turning to PSLE
Channel News Asia
SINGAPORE: With the Character and Citizenship Education syllabus being rolled out in all schools, the Ministry of Education (MOE) will tilt its focus …

AM-GM inequality

A very useful inequality in Mathematics is the AM-GM Inequality.

The arithmetic mean of numbers $x_1, x_2, \cdots, x_n$ is $\displaystyle \boxed{\frac{x_1+ x_2+\cdots+x_n}{n}}$ .

The geometric mean of numbers $x_1, x_2, \cdots, x_n$ is $\boxed{\sqrt[n]{x_1\cdot x_2 \cdots x_n}}$ .

The AM-GM Inequality states that:

For any nonnegative numbers $x_1, x_2, \cdots, x_n$ ,

$\displaystyle\boxed{\frac{x_1+x_2+\cdots+ x_n}{n}\geq\sqrt[n]{x_1\cdot x_2 \cdots x_n}}$ , and equality holds if and only if $x_1=x_2=\cdots=x_n$ .

am-gm-inequality

How to Apply?

Let say we have three (nonnegative) numbers a, b, c that add up to 30, i.e. $a+b+c=30$ . Can we know what is the largest possible product $abc$ ?

Yes! Using the AM-GM inequality we have just learnt above, we know $\displaystyle \frac{a+b+c}{3}\geq \sqrt[3]{abc}$ .

$\displaystyle 10\geq \sqrt[3]{abc}$

Cubing both sides, we have, $\displaystyle abc\leq 10^3=1000$ .

Also, the AM-GM inequality tells us that there is equality only when $a=b=c$ , i.e. $a=b=c=10$ . Hence, the largest possible product $abc$ is 1000.

Featured book from Amazon:

Competition Math for Middle School

Written for the gifted math student, the new math coach, the teacher in search of problems and materials to challenge exceptional students, or anyone else interested in advanced mathematical problems. Competition Math contains over 700 examples and problems in the areas of Algebra, Counting, Probability, Number Theory, and Geometry. Examples and full solutions present clear concepts and provide helpful tips and tricks. “I wish I had a book like this when I started my competition career.” Four-Time National Champion MATHCOUNTS coach Jeff Boyd “This book is full of juicy questions and ideas that will enable the reader to excel in MATHCOUNTS and AMC competitions. I recommend it to any students who aspire to be great problem solvers.” Former AHSME Committee Chairman Harold Reiter

Math Handheld Computer Game

Featured Item:

Educational Insights Math Whiz

Is your child disinterested in Math? Looking for some fun and educational Math games?

Math Whiz plays like a video game and teaches like electronic flash cards. This portable ELA quizzes kids on addition, subtraction, multiplication and division, AND works as a full-function calculator at the press of a button. Problems are displayed on the LCD screen. Features eight skill levels, as well as lights and sounds for instant feedback. Two AAA batteries required (not included).

The Boy With The Incredible Brain – Autism Math Documentary

This is the breathtaking story of Daniel Tammet. A twenty-something with extraordinary mental abilities, Daniel is one of the world’s few savants. He can do calculations to 100 decimal places in his head, and learn a language in a week.

He also meets the world’s most famous savant, the man who inspired Dustin Hoffman’s character in the Oscar winning film ‘Rain Man’.

This documentary follows Daniel as he travels to America to meet the scientists who are convinced he may hold the key to unlocking similar abilities in everyone.

Featured books by Daniel Tammet:

Born On A Blue Day: Inside the Extraordinary Mind of an Autistic Savant

Bestselling author Daniel Tammet (Thinking in Numbers) is virtually unique among people who have severe autistic disorders in that he is capable of living a fully independent life and able to explain what is happening inside his head.

He sees numbers as shapes, colors, and textures, and he can perform extraordinary calculations in his head. He can learn to speak new languages fluently, from scratch, in a week. In 2004, he memorized and recited more than 22,000 digits of pi, setting a record. He has savant syndrome, an extremely rare condition that gives him the most unimaginable mental powers, much like those portrayed by Dustin Hoffman in the film Rain Man.

Thinking In Numbers: On Life, Love, Meaning, and Math

The irresistibly engaging book that “enlarges one’s wonder at Tammet’s mind and his all-embracing vision of the world as grounded in numbers.” –Oliver Sacks, MD
THINKING IN NUMBERS is the book that Daniel Tammet, mathematical savant and bestselling author, was born to write. In Tammet’s world, numbers are beautiful and mathematics illuminates our lives and minds. Using anecdotes, everyday examples, and ruminations on history, literature, and more, Tammet allows us to share his unique insights and delight in the way numbers, fractions, and equations underpin all our lives.

Notes on Coordinate Geometry by Hwa Chong Institution

Coordinate Geometry Notes

Source: http://mathace2012.wiki.hci.edu.sg/

Check out these Formulas for:

Distance between 2 points
Midpoint between 2 points
Gradient Formulas
and more

at http://mathace2012.wiki.hci.edu.sg/

Featured book: The Ultimate book of Formulas (2000+ Formulas!)

Schaum’s Outline of Mathematical Handbook of Formulas and Tables, 4th Edition: 2,400 Formulas + Tables (Schaum’s Outline Series)

This Schaum’s Outline gives you

More than 2,400 formulas and tables
Covers elementary to advanced math topics
Arranged by topics for easy reference

Chinese Lucky Numbers – Numberphile

8 and 6 are lucky but 4 is unlucky… if you’re Chinese!

Featuring Xiaohui Yuan from the University of Nottingham.

Website: http://www.numberphile.com/
Numberphile on Facebook: http://www.facebook.com/numberphile
Numberphile tweets: https://twitter.com/numberphile

Videos by Brady Haran

Brady John Haran is an Australian independent film-maker and video journalist who is known for his educational videos and documentary films produced for BBC News and for his YouTube channels. (http://en.wikipedia.org/wiki/Brady_Haran)

Highly recommended to subscribe to Numberphile on Youtube for fun and interesting Math videos!

Featured book:

Number: The Language of Science

Number is an eloquent, accessible tour de force that reveals how the concept of number evolved from prehistoric times through the twentieth century. Tobias Dantzig shows that the development of math—from the invention of counting to the discovery of infinity—is a profoundly human story that progressed by “trying and erring, by groping and stumbling.” He shows how commerce, war, and religion led to advances in math, and he recounts the stories of individuals whose breakthroughs expanded the concept of number and created the mathematics that we know today.

– Rated 4.5/5 on Amazon

News: Math, reading performance is stagnant among US 12th-graders, assessment finds

Math, reading performance is stagnant among US 12th-graders, assessment finds
Washington Post
The nation’s high school seniors have shown no improvement in math and reading performance since 2009, and large racial achievement gaps …

Math, reading performance stagnates among US 12th-graders, assessment finds
Business Mirror
WASHINGTON—The United States’s high-school seniors have shown no improvement in math and reading performance since 2009, and large racial …

Social Math: Why Learning Math Involves More Than Writing Numbers
Huffington Post
My lifelong passion for creating better ways to learn math got its start in a school not unlike the Israeli one I visited — and at the very same age.

Math games aim to keep kids sharp over summer
Scranton Times-Tribune
JAKE DANNA STEVENS / STAFF PHOTOGRAPHER Kaylynn Howe, 8, left, and her sister, Destine Howey, play a math game at Bancroft Elementary …

Common Core math gains are worth the pain
New York Daily News
As more than a million New York students in grades 3-8 took the state math exams last week, a small but vocal cadre of parents railed at the new …

Lindblom Math and Science principal among Golden Apple winners
Chicago Tribune
The organization quickly realized that Mather, principal of Lindblom Math and Science Academy In Chicago’s West Englewood neighborhood was the …

Math Learning – A Universal Language?
Huffington Post
Fifth-grade students at Woodward Elementary School had an interesting math assignment this fall: watching college football games. Though …

US students’ reading, math show no progress
TheChronicleHerald.ca
WASHINGTON — In an abysmal showing, only about one-quarter of U.S. high school seniors performed solidly in math in a major assessment known …

Using math in the fight against cancer
WNYT
But a local college professor says many people don’t like math because they don’t see a connection to it. In this Friday’s STEM 13 report, learn how the …

Math Day at Molly Stark honors memory of Gail Harwood
Bennington Banner
BENNINGTON — The Molly Stark School honored a longtime teacher on Monday with “Gail Harwood Math Day.” Harwood passed away in January …

Monster Group

Check out this Youtube video on the Monster Group (related to Group Theory, a branch in Mathematics)

In the mathematical field of group theory, the monster group M or F₁ (also known as the Fischer–Griess monster, or the Friendly Giant) is a group of finite order. (See Wikipedia: http://en.wikipedia.org/wiki/Monster_group)

Featured book:

The Symmetries of Things

This book is written by John Conway, one of the mathematicians who worked on the Monster Group. Rated highly on Amazon.

Start with a single shape. Repeat it in some way—translation, reflection over a line, rotation around a point—and you have created symmetry.

Symmetry is a fundamental phenomenon in art, science, and nature that has been captured, described, and analyzed using mathematical concepts for a long time. Inspired by the geometric intuition of Bill Thurston and empowered by his own analytical skills, John Conway, with his coauthors, has developed a comprehensive mathematical theory of symmetry that allows the description and classification of symmetries in numerous geometric environments.

This richly and compellingly illustrated book addresses the phenomenological, analytical, and mathematical aspects of symmetry on three levels that build on one another and will speak to interested lay people, artists, working mathematicians, and researchers.

How to prove square root of 2 is irrational?

A rational number is a number that can be expressed in a fraction with integers as numerators and denominators.

Some examples of rational numbers are 1/3, 0, -1/2, etc. Now, we know that $\sqrt{2}\approx 1.41421\cdots$ .

Is the square root of 2 rational? Or is it irrational (the opposite of rational)? How do we prove it? It turns out we can prove that the square root of two is irrational using a technique called proof by contradiction. (One of the earlier posts on this blog also used proof by contradiction to show that there are infinitely many prime numbers.)

First, we suppose that $\displaystyle\sqrt{2}=\frac{p}{q}$ , where $\displaystyle\frac{p}{q}$ is a fraction in its lowest terms.

Next, we square both sides to get $\displaystyle 2=\frac{p^2}{q^2}$ .

Hence, $2q^2=p^2$ . We can conclude that $p^2$ is even since it is a multiple of 2. Thus, $p$ itself is also even. (the square of an odd number is odd).

Thus, we can write $p=2k$ for some integer k. Substituting this back into $2q^2=p^2$ , we get $2q^2=4k^2$ , which can be simplified to $q^2=2k^2$ .

Hence, $q^2$ is also even, and hence $q$ is also even!

But if both $p$ and $q$ are even, then $\displaystyle\frac{p}{q}$ is not in the lowest terms! (we could divide them by two). This contradicts our initial hypothesis!

Thus, the only possible conclusion is that the square root of two is not a rational number to begin with!

irrational

Featured book:

Math Jokes 4 Mathy Folks

Who says math can’t be funny? In Math Jokes 4 Mathy Folks, Patrick Vennebush dispels the myth of the humorless mathematician. His quick wit comes through in this incredible compilation of jokes and stories. Intended for all math types, Math Jokes 4 Mathy Folks provides a comprehensive collection of math humor, containing over 400 jokes.

– Highly rated on Amazon.com

What to do if fail Mid Year Exams?

Source: http://news.asiaone.com/News/Education/Story/A1Story20080601-68205.html

When the Mid Year Exams are over, students will receive their results nervously. What to do if one fails the Mid Year Exams?

In many schools, it is common to have a significant portion of the school actually failing the Mid Term exam. “40 per cent of his school cohort failed Social Studies and 30 per cent English.” in the school mentioned in the above article.

“Such significant failure rates have become common in schools here when mid-year or preliminary exams roll around, especially for those with a big national exam – PSLE, O or A levels – at the year-end.”

Here are 5 tips on what are the best actions to take for one who fails the Mid Year Exams, especially for Mathematics:

Do not be discouraged! Try to maintain a positive attitude on Math. There is still time before the final exams. With proper time management, you will be able to set aside time for revision, which will definitely help.
Analyse what went wrong. Are you studying Math the correct way? (i.e. practising with understanding) Are you studying Math just by reading the textbook? (not effective for studying Math as Math needs practice.) Is time management an issue? Or is the main issue careless mistakes?
Work out a new study strategy and stick to it religiously. For better results, you need to change your study habits for the better. This may include better time management, or seeking help from Math tutors.
O Level Exams are not about intelligence, it is more about good study techniques. The content for O Levels can definitely be mastered by any student given the right amount of time and effort. The key is to put in time and effort to the studies. Even an average student is capable of scoring an A1 in O Levels if he or she works hard. Whereas, a very intelligent but lazy student may not do well for the exams.
It is possible to improve tremendously for Maths if you study enough and using the right method. This is a truth that many people can attest to. I have seen students going from fail to A1. Improving one or two grades is also very common.

There are usually two types of students, the ones who are more playful and laidback, and the very perfectionistic student but is prone to stress. For the more playful students, the tough Mid Year exams are actually meant as a wake up call to start studying before it is too late. “‘Papers must be a bit challenging so that they can shake one out of complacency and make one study harder,’ said Mr Lak Pati Singh, 56, principal of St Patrick’s School.”

For students who are too stressed up and already trying their best, the way to improve may be to study more efficiently using the right methods (especially for Maths, the right way to study is practice with understanding). A healthy lifestyle balance may also be very helpful. Again, seeking help from Math tutors may be a choice to be considered, which can alleviate stress from not understanding the subject material.

Studies and Studying: How do top students study?

Source: http://www.quora.com/Studies-and-Studying/How-do-top-students-study/answer/Qiaochu-Yuan-1

Check out this post by MIT almost perfect-scorer, on how to study. His secret is to study the material in advance, before the lessons even start! This is really a useful strategy, if implemented correctly. Imagine being in Primary 3 and already knowing the Primary 4 syllabus! Primary 3 Math will be a breeze then. This is one of the reasons why China students are so good at Math – they have already studied it back in China, where the Math syllabus is more advanced!

Do try out this strategy if you are really motivated to improve in your studies. The prime time to do this is during the June and December holidays – take some time to read ahead what is going to be learnt during the next semester.

This is an excerpt of the thread:

I graduated from MIT with a GPA of 4.8 (out of 5.0) in mathematics. I had two non-As, both of which were non-math classes.

That doesn’t imply that I have good study methods, but anyway, here’s how I studied at MIT. My main study method as an undergraduate, for math classes, was knowing a sizable chunk of the material in advance.

This isn’t a method that will work for everybody. I did a lot of mathematics outside of the classroom both in high school and at MIT, and I often saw a substantial portion of the material in a given class before I took it. I can’t emphasize enough how much easier this makes a class, and not just for the reasons you might expect: one of the most valuable things you get out of knowing a lot of the material already is just not being intimidated by it. (And you can get this benefit even if you’ve only seen some of the material before and possibly forgotten some of it too.) You’re much more relaxed, and that makes it easier to process the part of the material that you don’t know.

What that translates to in terms of practical advice is this:

cultivate a sense of curiosity,

don’t restrict your learning to the classroom,

only take classes that actually seem really interesting to you, and

try to learn something related to those classes the semester before.

None of this is advice for studying for a class you’re taking now, but it’s advice for reducing the extent to which you will need to study for classes you’ll take in the future.

– Qiaochu Yuan

Free Career Analysis (With Prizes to be Won): Are you a Careerist, Entrepreneur, Harmoniser, Idealist, Hunter, Internationalist, or Leader?

The Art of S/U:

GAT: General Ability Test

Boost your DSA GAT Scores with Mensa Book:

Subscribe to our free newsletter, for updates on Math, Education and other General Knowledge topics! Subscribe to Blog via Email Enter your email address to subscribe to this blog and receive notifications of new posts by email. Email Address: Subscribe

Subscribe to Blog via Email

More Books to Ramp Up your DSA GAT Score: https://mathtuition88.com/2013/11/11/recommended-books-for-gep-selection-test/

NUS High DSA

Kindle for Singaporean Students

Buy Kindle from Qoo10 (Singapore’s Taobao/Amazon)

Why is the mid-year exams difficult and many people fail it?

How to be good in additional mathematics.

Why is the mid-year exams difficult and many people fail it?

How to be good in additional mathematics.

World Cup Math: Birthday Paradox

AM-GM inequality

How to Apply?

Coordinate Geometry Notes

Free Career Analysis (With Prizes to be Won):
Are you a Careerist, Entrepreneur, Harmoniser, Idealist, Hunter, Internationalist, or Leader?

Subscribe to our free newsletter, for updates on Math, Education and other General Knowledge topics!
Subscribe to Blog via Email

Enter your email address to subscribe to this blog and receive notifications of new posts by email.

Email Address:

More Books to Ramp Up your DSA GAT Score:
https://mathtuition88.com/2013/11/11/recommended-books-for-gep-selection-test/