# E Maths Formula List / A Maths Formula Sheet

Attached below are the Formula Lists for E Maths and A Maths (O Level)

Do be familiar with all the formulas for Elementary Maths and Additional Maths inside, so that you know where to find it when needed!
Wishing everyone reading this all the best for their exams. 🙂

E Maths Formula List

A Maths Formula List

## Maths Tuition

For Mathematics Tuition, contact Mr Wu at:

SMS: 98348087

Email: mathtuition88@gmail.com

## O Level E Maths and A Maths Tuition starting next year at Bishan

O Level E Maths and A Maths Tuition starting next year at Bishan
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View Mr Wu’s GEP Testimonial at

https://mathtuition88.com/group-tuition/

Despite being in the Gifted Education Programme (GEP), Mr Wu is just an ordinary Singaporean. His secret to academic success is hard work and the Maths Techniques he has discovered by himself while navigating through the education system.

He would like to teach these techniques to students, hence choosing to become a full-time Mathematics tutor. Mr Wu has developed his own methods to check the answer, remember formulas (with understanding), which has helped a lot of students. Many Math questions can be checked easily, leading to the student being 100% confident of his or her answer even before the teacher marks his answer, and reducing the rates of careless mistakes.

Mr Wu’s friendly and humble nature makes him well-liked by students. Many of his students actually request for more tuition by themselves! (not the parents)

O Level E Maths and A Maths Tuition starting next year at Bishan, the best location in Central Singapore.

Timings are Monday 7-9pm, Thursday 7-9pm. Perfect for students who have CCA in the afternoon, or students who want to keep their weekends free.

Register with us now by email (mathtuition88@gmail.com) or phone (98348087). Vacancies will be allocated on a first-come-first-serve basis.

Thanks and wishing all a nice day.

## E Maths Group Tuition Centre; Clementi Town Secondary School Prelim 2012 Solution

Q5) The speed of a boat in still water is 60 km/h.

On a particular day, the speed of the current is $x$ km/h.

(a) Find an expression for the speed of the boat

(I) against the current, [1]

Against the current, the boat would travel slower! This is related to the Chinese proverb, 逆水行舟，不进则退, which means “Like a boat sailing against the current, we must forge ahead or be swept downstream.”

Hence, the speed of the boat is $60-x$ km/h.

(ii) with the current. [1]

$60+x$ km/h

(b) Find an expression for the time required to travel a distance of 80km

(I) against the current,  [1]

Recall that $\displaystyle \text{Time}=\frac{\text{Distance}}{\text{Speed}}$

Hence, the time required is $\displaystyle \frac{80}{60-x}$ h

(ii) with the current. [1]

$\displaystyle \frac{80}{60+x}$ h

(c) If the boat takes 20 minutes longer to travel against the current than it takes to travel with the current, write down an equation in $x$ and show that it can be expressed as $x^2+480x-3600=0$   [2]

Note: We must change 20 minutes into 1/3 hours!

$\frac{80}{60-x}=\frac{1}{3}+\frac{80}{60+x}$

There are many ways to proceed from here, one way is to change the Right Hand Side into common denominator, and then cross-multiply.

$\displaystyle \frac{80}{60-x}=\frac{60+x}{3(60+x)}+\frac{240}{3(60+x)}=\frac{300+x}{3(60+x)}$

Cross-multiply,

$240(60+x)=(300+x)(60-x)$

$14400+240x=18000-300x+60x-x^2$

$x^2+480x-3600=0$ (shown)

(d) Solve this equation, giving your answers correct to 2 decimal places. [2]

$\displaystyle x=\frac{-480\pm\sqrt{480^2-4(1)(-3600)}}{2}=7.386 \text{ or } -487.386$

Answer to 2 d.p. is $x=7.39 \text{ or } -487.39$

(e) Hence, find the time taken, in hours, by the boat to complete a journey of 500 km against the current. [2]

Now we know that the speed of the current is 7.386 km/h.

Hence, the time taken is $\frac{500}{60-7.386}=9.50$ h

## Additional Maths — from Fail to Top in Class

Really glad to hear good news from one of my students.

From failing Additional Maths all the way, he is now the top in his entire class.

Really huge improvement, and I am really happy for him. 🙂

To other students who may be reading this, remember not to give up! As long as you persevere, it is always possible to improve.

# Ad: Maths Group Tuition starting in 2014

Maths can be fun too!
Build up interest in Mathematics by trying out some of these interesting Maths Riddles.

Three guests check into a hotel room. The clerk says the bill is $30, so each guest pays$10. Later the clerk realizes the bill should only be $25. To rectify this, he gives the bellhop$5 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn’t know the total of the revised bill, the bellhop decides to just give each guest $1 and keep$2 for himself. Each guest got $1 back: so now each guest only paid$9; bringing the total paid to $27. The bellhop has$2. And $27 +$2 = $29 so, if the guests originally handed over$30, what happened to the remaining $1? Try it out before looking at the answer! ## Undergraduate Study in Mathematics (NUS) # Maths Group Tuition to start in 2014! If you are interested in Mathematics, do consider to study Mathematics at NUS! Quote: # Undergraduate Study in Mathematics (NUS) # Overview The Department of Mathematics at NUS is the largest department in the Faculty of Science. We offer a wide range of modules catered to specialists contemplating careers in mathematical science research as well as to those interested in applications of advanced mathematics to science, technology and commerce. The curriculum strives to maintain a balance between mathematical rigour and applications to other disciplines. We offer a variety of major and minor programmes, covering different areas of mathematical sciences, for students pursuing full-time undergraduate studies. Those keen in multidisciplinary studies would also find learning opportunities in special combinations such as double degree, double major and interdisciplinary programmes. Honours graduates may further their studies with the Graduate Programme in Mathematics by Research leading to M.Sc. or Ph.D. degree, or with the M.Sc. Programme in Mathematics by Course Work. ## Studying at NUS Mathematics Department # Maths Group Tuition to start in 2014! Source: http://ww1.math.nus.edu.sg/ The history of the Department of Mathematics at NUS traces back to 1929, when science education began in Singapore with the opening of Raffles College with less than five students enrolled in mathematics. Today it is one of the largest departments in NUS, with about 70 faculty members and teaching staff supported by 13 administrative and IT staff. The Department offers a wide selection of courses (called modules) covering wide areas of mathematical sciences with about 6,000 students enrolling in each semester. Apart from offering B.Sc. programmes in Mathematics, Applied Mathematics and Quantitative Finance, the Department also participates actively in major interdisciplinary programs, including the double degree programme in Mathematics/Applied Mathematics and Computer Science, the double major programmes in Mathematics and Economics as well as with other subjects, and the Computational Biology programme. Another example of the Department’s student centric educational philosophy is the Special Programme in Mathematics (SPM), which is specially designed for a select group of students who have a strong passion and aptitude for mathematics. The aim is to enable these students to build a solid foundation for a future career in mathematical research or state-of-the-art applications of mathematics in industry. The Department is ranked among the best in Asia in mathematical research. It offers a diverse and vibrant program in graduate studies, in fundamental as well as applied mathematics. It promotes interdisciplinary applications of mathematics in science, engineering and commerce. Faculty members’ research covers all major areas of contemporary mathematics. For more information, please see research overview, selected publications, and research awards. ## Prime Minister Lee Hsien Loong Truly Outstanding Mathematics Student Just to share an inspirational story about studying Mathematics, and our very own Prime Minister Lee Hsien Loong. 🙂 (page 8/8) Interview of Professor Béla Bollobás, Professor and teacher of our Prime Minister Lee Hsien Loong I: Interviewer Y.K. Leong B: Professor Béla Bollobás I: I understand that you have taught our present Prime Minister Lee Hsien Loong. B: I certainly taught him more than anybody else in Cambridge. I can truthfully say that he was an exceptionally good student. I’m not sure that this is really known in Singapore. “Because he’s now the Prime Minister,” people may say, “oh, you would say he was good.” No, he was truly outstanding: he was head and shoulders above the rest of the students. He was not only the first, but the gap between him and the man who came second was huge. I: I believe he did double honors in mathematics and computer science. B: I think that he did computer science (after mathematics) mostly because his father didn’t want him to stay in pure mathematics. Loong was not only hardworking, conscientious and professional, but he was also very inventive. All the signs indicated that he would have been a world-class research mathematician. I’m sure his father never realized how exceptional Loong was. He thought Loong was very good. No, Loong was much better than that. When I tried to tell Lee Kuan Yew, “Look, your son is phenomenally good: you should encourage him to do mathematics,” then he implied that that was impossible, since as a top-flight professional mathematician Loong would leave Singapore for Princeton, Harvard or Cambridge, and that would send the wrong signal to the people in Singapore. And I have to agree that this was a very good point indeed. Now I am even more impressed by Lee Hsien Loong than I was all those years ago, and I am very proud that I taught him; he seems to be doing very well. I have come round to thinking that it was indeed good for him to go into politics; he can certainly make an awful lot of difference. ## NUS Top in Asia according to latest QS World University Rankings by Subject # Top in Asia according to latest QS World University Rankings by Subject ## 08 May 2013 NUS is the best-performing university in Asia in the 2013 QS World University Rankings by Subject. With 12 subjects ranked top 10, NUS has secured the 8th position among universities globally in this subject ranking. On the results, NUS Deputy President (Academic Affairs) and Provost Professor Tan Eng Chye said: “This is a strong international recognition of NUS’ strengths in humanities and languages, engineering and technology, sciences, medicine and social sciences.” Prof Tan noted that the rankings served as an acknowledgement of the exceptional work carried out by faculty and staff in education and research. NUS fared well, ranking among the world’s top 10 universities for 12 subjects namely Statistics, Mathematics, Material Sciences, Pharmacy & Pharmacology, Communication & Media Studies, Geography, Politics & International Studies, Modern Languages, Computer Science & Information Systems and Engineering (mechanical, aeronautical, manufacturing, electrical & electronic, chemical). Continue reading at: http://newshub.nus.edu.sg/headlines/1305/qs_08May13.php ## H2 Maths A Level 2012 Solution, Paper 2 Q5; H2 Maths Tuition 5(i)(a) $P(\text{patient has the disease and test positive})=0.001(0.995)=9.95\times 10^{-4}$ $P(\text{patient does not have the disease and he tests positive})=(1-0.001)(1-0.995)=4.995\times 10^{-3}$ $P(\text{result of the test is positive})=9.95\times 10^{-4}+4.995\times 10^{-3}=5.99\times 10^{-3}$ (b) Let A=patient has disease Let B=result of test is positive $\displaystyle\begin{array}{rcl}P(A|B)&=&\frac{P(A\cap B)}{P(B)}\\ &=&\frac{(0.001)(0.995)}{5.99\times 10^{-3}}\\ &=&0.166 \end{array}$ Note that the probability is surprisingly quite low! (This is called the False positive paradox, a statistical result where false positive tests are more probable than true positive tests, occurring when the overall population has a low incidence of a condition and the incidence rate is lower than the false positive rate. See http://en.wikipedia.org/wiki/False_positive_paradox) (ii) $\displaystyle P(A|B)=\frac{(0.001)p}{(0.001)p+(1-0.001)(1-p)}=0.75$ By GC, $p=0.999666$ (6 d.p.) ## A Level H2 Maths 2012 Paper 2 Q3 Solution; H2 Maths Tuition # A Level H2 Maths 2012 Paper 2 Q3 Solution (i) (The graph above is drawn using the Geogebra software 🙂 ) (ii) $x^3+x^2-2x-4=4$ $x^3+x^2-2x-8=0$ By GC, $x=2$ By long division, $x^3+x^2-2x-8=(x-2)(x^2+3x+4)$ The discriminant of $x^2+3x+4$ is $D=b^2-4ac=3^2-4(1)(4)=-7<0$ Hence, there are no other real solutions (proven). (iii) $x+3=2$ $x=-1$ (iv) (v) $|x^3+x^2-2x-4|=4$ $x^3+x^2-2x-4=4$ or $x^3+x^2-2x-4=-4$ $x^3+x^2-2x-8=0$ or $x^3+x^2-2x=0$ $x^3+x^2-2x-8=0 \implies x=2$ (from part ii) $x^3+x^2-2x=x(x^2+x-2)=x(x-1)(x+2)=0$ $x=0,1,-2$ In summary, the roots are $-2,0,1,2$ ## Youngest NUS graduates for 2012 – 08Jul2012 Published on Jul 9, 2012 SINGAPORE – Douglas Tan was only seven years old when he discovered a knack for solving mathematical problems, tackling sums meant for the upper primary and secondary levels. He went on to join the Gifted Programme in Rosyth Primary School and, in 2006, enrolled in the National University of Singapore High School of Math and Science (NUSHS). At 15, he was offered a place at the National University of Singapore (NUS) Faculty of Science to study mathematics. Tomorrow, the 19-year-old will be this year’s youngest graduate at NUS, receiving his Mathematics degree with a First Class Honours. This puts him almost six years ahead of those his age. Douglas, who is currently serving his National Service (NS), said the thought of going to prestigious universities overseas never occurred to him. “I was just happy doing what I was doing – solving math problems,” he said. In every class he took, Douglas was the youngest but it was neither “awkward nor tough to fit in”, he said. In fact, his age was a good conversation starter and his classmates, who were typically three to five years older, would take care of him. Seeing that he could complete his degree before he entered NS, Douglas took on three modules a semester and completed the four-year course in just two and a half years. The longest he had ever spent on a math problem was 10 hours over a few days. “I’m a perfectionist. When I do a problem, I try to do it with 100 per cent,” he noted. Douglas aspires to be a mathematician and is looking into a Masters degree but he has yet to decide if he wants to do it here or overseas. Another young outstanding graduate this year is 20-year-old Carmen Cheh, who received her degree in Computer Science last Friday with a First Class Honours and was on the dean’s list every academic year of the four-year course. Offered a place at the NUS School of Computing after three and a half years in NUSHS, Carmen was then the youngest undergraduate of the programme at 16. She was introduced to computer science and concept programming at 11 by her father, a doctor who also challenged her to solve puzzles he created. Her inability to solve them spurred her interest in the subject. Carmen, who is from Perak in Malaysia, said she decided to study for her degree in Singapore as she wanted to study in a country she felt “comfortable” in. At the same time, she was awarded an ASEAN scholarship to study in the Republic. Next month, Carmen will begin her doctoral programme in Computer Science with a research assistantship at the University of Illinois at Urbana-Champaign. The youngest ever to enrol into the NUS undergraduate programme is Abigail Sin, who entered the Yong Siew Toh Conservatory of Music at 14. She graduated in 2010 at age 18 with First Class Honours. She also received the Lee Kuan Yew gold medal. This week, NUS celebrates the graduation of 9,913 students, its largest cohort in six years. http://www.todayonline.com/Singapore/EDC120709-­0000039/Theyre-ahead-of-the-class ## Information about Mathematics Department Courses (Nanyang JC) H1 Mathematics H1 Mathematics provides a foundation in mathematics for students who intend to enrol in university courses such as business, economics and social sciences. The syllabus aims to develop mathematical thinking and problem solving skills in students. A major focus of the syllabus will be the understanding and application of basic concepts and techniques of statistics. This will equip students with the skills to analyse and interpret data, and to make informed decisions. The use of graphic calculator is expected. H2 Mathematics H2 Mathematics prepares students adequately for university courses including mathematics, physics and engineering, where more mathematics content is required. The syllabus aims to develop mathematical thinking and problem solving skills in students. Students will learn to analyse, formulate and solve different types of problems. They will also learn to work with data and perform statistical analyses. The use of graphic calculator is expected. This subject assumes the knowledge of O-Level Additional Mathematics. ## Number Theory Notes – Art of Problem Solving Excellent notes on Olympiad Number Theory! Preface: This set of notes on number theory was originally written in 1995 for students at the IMO level. It covers the basic background material that an IMO student should be familiar with. This text is meant to be a reference, and not a replacement but rather a supplement to a number theory textbook; several are given at the back. Proofs are given when appropriate, or when they illustrate some insight or important idea. The problems are culled from various sources, many from actual contests and olympiads, and in general are very difficult. The author welcomes any corrections or suggestions. ## Advice to Students # Advice to Students: Over the years, I have collected some information that I hope will help students, particularly beginning math students, to improve their study and learning habits. An important part of what you learn at college is how to learn, so that you can carry that on for the rest of your life. Find out what works for you and what doesn’t. These observations are centered around first-year calculus courses, so not everything may apply to you, but even more advanced students can benefit from some of them. As you develop your own learning habits, please think carefully about the following topics: Continue reading at http://www.math.union.edu/~dpvc/courses/advice/welcome.html ## Student Advice: Comments on Perseverance # Comments on Perseverance: One source of confusion for students when they reach college and begin to do college-level mathematics is this: in high school, it is usually pretty apparent what formula or technique needs to be applied, as much of the material in high school is computational or procedural. In college, however, mathematics becomes more conceptual, and it is much harder to know what to do when you first start a problem. As a consequence of this, many students give up on a problem too early. If you don’t immediately know how to attack a problem, this doesn’t mean you are stupid,  If you already know how to do it, it’s not really a problem. or that you don’t understand what’s going on; that’s just how real problems work. After all, if you already know how to do it, it’s not really a problem, is it? You should expect to be confused at first. There’s no way you can know ahead of time how to solve every problem that you will face in life. You’re only hope, and therefore your goal as a student, is to get experience with working through hard problems on your own. That way, you will continue to be able to do so once you leave college. One of the first steps in this is to realize that not knowing how, and the frustration that accompanies that, is part of the process. Then you have to start to figure out the questions that you can ask to help you to break down the problem, so that you can figure out how it really works. What’s really important in it? What is the central concept? What roles do the definitions play? How is this related to other things I know? Continue reading at http://www.math.union.edu/~dpvc/courses/advice/perseverance.html ## Does one have to be a genius to do maths? Better beware of notions like genius and inspiration; they are a sort of magic wand and should be used sparingly by anybody who wants to see things clearly. (José Ortega y Gasset, “Notes on the novel”) Does one have to be a genius to do mathematics? The answer is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does not need some sort of magic “genius gene” that spontaneously generates ex nihilo deep insights, unexpected solutions to problems, or other supernatural abilities. ## There’s more to mathematics than grades and exams and methods When you have mastered numbers, you will in fact no longer be reading numbers, any more than you read words when reading books. You will be reading meanings. (W. E. B. Du Bois) When learning mathematics as an undergraduate student, there is often a heavy emphasis on grade averages, and on exams which often emphasize memorisation of techniques and theory than on actual conceptual understanding, or on either intellectual or intuitive thought. There are good reasons for this; there is a certain amount of theory and technique that must be practiced before one can really get anywhere in mathematics (much as there is a certain amount of drill required before one can play a musical instrument well). It doesn’t matter how much innate mathematical talent and intuition you have; if you are unable to, say, compute a multidimensional integral, manipulate matrix equations, understand abstract definitions, or correctly set up a proof by induction, then it is unlikely that you will be able to work effectively with higher mathematics. However, as you transition to graduate school you will see that there is a higher level of learning (and more importantly, doing) mathematics, which requires more of your intellectual faculties than merely the ability to memorise and study, or to copy an existing argument or worked example. This often necessitates that one discards (or at least revises) many undergraduate study habits; there is a much greater need for self-motivated study and experimentation to advance your own understanding, than to simply focus on artificial benchmarks such as examinations. ## Maths Tuition Singapore Keywords Just for curiosity, I went to research on the top keywords for Maths Tuition for Google search engine. Keywords: mathematics tuition singapore a maths tuition singapore maths tuition singapore singapore maths tuition a math tuition singapore a level maths tuition a maths tuition a level maths tuition singapore o level maths tuition singapore maths tutors singapore maths tuition in singapore maths tutor singapore maths tuition a level maths tutoring tuition maths o level maths tuition o level maths tutor mathematics tuition e math tuition a level tuition singapore maths a level tuition tutor in maths online mathematics tuition maths tuition teacher singapore a level maths tutor singapore tutor home maths tuition best maths tuition singapore online math tuition a level mathematics tutors ip maths tuition singapore tuition for maths a level tuition maths tutor a level online maths tuition best math tuition singapore ip math tuition h2 maths tuition singapore maths tutor a level tutor a level math tutor math tuition online o level tuition singapore as maths tutor mathematics tuition online e maths tuition tutor maths maths tutor for a level secondary maths tuition singapore h2 math tuition singapore tuition singapore maths a level tutor maths singapore o level tutor a maths tutor tuition a level primary maths tuition singapore tutor singapore maths tuition online h2 maths tuition teach tuition singapore looking for tuition teacher singapore ip maths tuition singapore tuition best maths tutor singapore be a tutor singapore maths in singapore home maths tutor primary maths tuition o level a maths maths for a level maths tuition teacher maths at a level a levels maths online tuition singapore maths home tuition singapore maths online tuition maths tuition fees tutor in singapore secondary math tuition tutor maths online ib maths tuition ib maths tuition singapore singapore online tuition home tutor for maths singapore math tuition home tuition in singapore tutor tuition one to one maths tuition o level math that math tutor advanced level maths best maths tutor secondary maths tuition experienced maths tutor chemistry tuition singapore o level a math maths tutor online o level math questions ## The Key To Career Success “The essence of mathematics is not to make simple things complicated, but to make complicated things simple.” ~ Stan Gudder, Mathematician Math, at its core, is about solving problems — about breaking a challenge into its basic elements to be investigated, tested, manipulated and understood. Math can give you the tools to find a winning formula. And, it can create the path to your career. Math is the key to unlocking possibilities. It frees you up to think creatively about solutions and to focus your attention on what truly matters at the end of the day. Finally, math empowers you to be a better leader and to remain open to new ideas. It sparks creativity and learning. It gives you confidence and conviction to say “YES!” when you’re asked to take on a new challenge. It helps you attract and energize the people you hire to help you. In a marketplace that’s moving so fast, it’s important to constantly listen, learn, analyze and formulate new ways to serve customers. Math provides the foundation for doing just that. Want to succeed? It’s simple … math. ## EDUC115N: How to Learn Math (Stanford Online Maths Education Course ) I will be attending this exciting online course by Stanford on Math Education. Do feel free to join it too, it is suitable for teachers and other helpers of math learners, such as parents. EDUC115N: How to Learn Math ## About This Course In July 2013 a new course will be available on Stanford’s free on-line platform. The course is a short intervention designed to change students’ relationships with math. I have taught this intervention successfully in the past (in classrooms); it caused students to re-engage successfully with math, taking a new approach to the subject and their learning. ## Concepts 1. Knocking down the myths about math. Math is not about speed, memorization or learning lots of rules. There is no such thing as “math people” and non-math people. Girls are equally capable of the highest achievement. This session will include interviews with students. 2. Math and Mindset. Participants will be encouraged to develop a growth mindset, they will see evidence of how mindset changes students’ learning trajectories, and learn how it can be developed. 3. Mistakes, Challenges & Persistence. What is math persistence? Why are mistakes so important? How is math linked to creativity? This session will focus on the importance of mistakes, struggles and persistence. 4. Teaching Math for a Growth Mindset. This session will give strategies to teachers and parents for helping students develop a growth mindset and will include an interview with Carol Dweck. 5. Conceptual Learning. Part I. Number Sense. Math is a conceptual subject– we will see evidence of the importance of conceptual thinking and participants will be given number problems that can be solved in many ways and represented visually. 6. Conceptual Learning. Part II. Connections, Representations, Questions. In this session we will look at and solve math problems at many different grade levels and see the difference in approaching them procedurally and conceptually. Interviews with successful users of math in different, interesting jobs (film maker, inventor of self-driving cars etc) will show the importance of conceptual math. 7. Appreciating Algebra. Participants will learn some key research findings in the teaching and learning of algebra and learn about a case of algebra teaching. 8. Going From This Course to a New Mathematical Future. This session will review the ideas of the course and think about the way towards a new mathematical future. ## Make Britain Count: ‘Stop telling children maths isn’t for them’ “The title comes from the central argument of the book,” says Birmingham-raised Boaler, “namely the idea that maths is a gift that some have and some don’t. That’s the elephant in the classroom. And I want to banish it. I believe passionately that everybody can be good at maths. But you don’t have to take my word for it. Studies of the brain show that all kids can do well at maths, unless they have some specific learning difficulty.” But what about those booming Asian economies, with their ready flow of mathematically able graduates? “There are a lot of misconceptions about the methods that are used in China, Japan and Korea,” replies Boaler. “Their way of teaching maths is much more conceptual than it is in England. If you look at the textbooks they use, they are tiny.” Professor Boaler’s tips on how parents can help Make Britain Count. 1 Encourage children to play maths puzzles and games at home. Anything with a dice will help them enjoy maths and develop numeracy and logic skills. 2 Never tell children they are wrong when they are working on maths problems. There is always some logic to what they are doing. So if your child multiplies three by four and gets seven, try: “Oh I see what you are thinking, you are using what you know about addition to add three and four. When we multiply we have four groups of three…” 3 Maths is not about speed. In younger years, forcing kids to work fast on maths is the best way to start maths anxiety, especially among girls. 4 Don’t tell your children you were bad at maths at school. Or that you disliked it. This is especially important if you are a mother. 5 Encourage number sense. What separates high and low achievers in primary school is number sense. 6 Encourage a “growth mindset” – the idea that ability changes as you work more and learn more. ## Mathematics is an art The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such. Everyone understands that poets, painters, and musicians create works of art, and are expressing themselves in word, image, and sound. In fact, our society is rather generous when it comes to creative expression; architects, chefs, and even television directors are considered to be working artists. So why not mathematicians? Part of the problem is that nobody has the faintest idea what it is that mathematicians do. The common perception seems to be that mathematicians are somehow connected with science— perhaps they help the scientists with their formulas, or feed big numbers into computers for some reason or other. There is no question that if the world had to be divided into the “poetic dreamers” and the “rational thinkers” most people would place mathematicians in the latter category. Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood. So let me try to explain what mathematics is, and what mathematicians do. I can hardly do better than to begin with G.H. Hardy’s excellent description: A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. . ## Tangent Secant Theorem (A Maths Tuition) Nice Proof of Tangent Secant Theorem: http://www.proofwiki.org/wiki/Tangent_Secant_Theorem Note: The term “Square of Sum less Square” means $a^2-b^2=(a+b)(a-b)$ The proof of the Tangent Secant Theorem, though not tested, is very interesting. In particular, the proof of the first case (DA passes through center) should be accessible to stronger students. ## How to use tables in CASIO FX-9860G Slim (H2 Maths Tuition) Tables in CASIO FX-9860G Slim The most popular Graphical Calculator for H2 Maths is currently the TI-84 PLUS series, but some students do use Casio Graphical Calculators. The manual for CASIO FX-9860G Slim is can be found here: http://edu.casio.com/products/fx9860g2/data/fx-9860GII_Soft_E.pdf The information about Tables and how to generate a table is on page 121. Generating tables is useful to solve some questions in sequences and series, and also probability. It makes guess and check questions much faster to solve. ## H2 JC Maths Tuition Foot of Perpendicular 2007 Paper 1 Q8 One of my students asked me how to solve 2007 Paper 1 Q8 (iii) using Foot of Perpendicular method. The answer given in the TYS uses a sine method, which is actually shorter in this case, since we have found the angle in part (ii). Nevertheless, here is how we solve the question using Foot of Perpendicular method. (Due to copyright issues, I cannot post the whole question here, so please refer to your Ten Year Series.) Firstly, let F be the foot of the perpendicular. Then, $\vec{AF}=k\begin{pmatrix}3\\-1\\2\end{pmatrix}$ ——– Eqn (1) $\vec{OF}\cdot\begin{pmatrix}3\\-1\\2\end{pmatrix}=17$ ——– Eqn (2) From Eqn (1), $\vec{OF}-\vec{OA}=\begin{pmatrix}3k\\-k\\2k\end{pmatrix}$ $\begin{array}{rcl}\vec{OF}&=&\vec{OA}+\begin{pmatrix}3k\\-k\\2k\end{pmatrix}\\ &=&\begin{pmatrix}1\\2\\4\end{pmatrix}+\begin{pmatrix}3k\\-k\\2k\end{pmatrix}\\ &=&\begin{pmatrix}1+3k\\2-k\\4+2k\end{pmatrix}\end{array}$ Substituting into Eqn (2), $\begin{pmatrix}1+3k\\2-k\\4+2k\end{pmatrix}\cdot\begin{pmatrix}3\\-1\\2\end{pmatrix}=17$ $14k+9=17$ $k=4/7$ Substituting back into Eqn (1), $\displaystyle\vec{AF}=\frac{4}{7}\begin{pmatrix}3\\-1\\2\end{pmatrix}$ $\displaystyle|\vec{AF}|=\frac{4}{7}\sqrt{14}$ ## JC Junior College H2 Maths Tuition If you or a friend are looking for Maths tuitionO level, A level H2 JC (Junior College) Maths Tuition, IB, IP, Olympiad, GEP and any other form of mathematics you can think of. Experienced, qualified (Raffles GEP, NUS Maths 1st Class Honours, NUS Deans List) and most importantly patient even with the most mathematically challenged. So if you are in need of the solution to your mathematical woes, drop me a message! Tutor: Mr Wu Email: mathtuition88@gmail.com Phone: 98348087 ## O Level E Maths Tuition: Statistics Question Solution: From the graph, Median = 50th percentile =$22,000 (approximately)

The mean is lower than $22000 because from the graph, there is a large number of people with income less than$22000, and fewer with income more than \$22000. (From the wording of the question, calculation does not seem necessary)

Hence, the median is higher.

The mean is a better measure of central tendency, as it is a better representative of the gross annual income of the people. This is because more people have an income closer to the mean, rather than the median.

## Ten Year Series: How many questions or papers to practice for Maths O Levels / A Levels?

This is a question to ponder about, how many questions or papers to practice for Maths O Levels / A Levels for the Ten Year Series?

If you searched Google, you will find that there is no definitive answer of how many questions to practice for Maths O Levels/ A Levels anywhere on the web.

For O Level / A Level, practicing the Ten Year Series is really helpful, as it helps students to gain confidence in solving exam-type questions.

Here are some tips about how to practice the Ten Year Series (TYS):

1) Do a variety of questions from each topic. This will help you to gain familiarity with all the topics tested, and also revise the older topics.

2) Fully understand each question. If necessary, practice the same question again until you get it right. There is a sense of satisfaction when you finally master a tough question.

3) Quality is more important than quantity. It is better to do and understand 1 question completely than do many questions but not understanding them.

Back to the original query of how many questions or papers to practice for Maths O Levels / A Levels for the Ten Year Series, I will attempt to give a rough estimate here, based on personal experience.

5 Questions done (full questions worth more than 5 marks) will result in an improvement of roughly 1 mark in the final exam.

(The 5 Questions must be fully understood. )

So, if a student wants to improve from 40 marks to 70 marks, he/she should try to do 30×5=150 questions (around 7 years worth of past year papers). Repeated questions are counted too, so doing 75 questions (around 3 years worth of past year papers) twice will also count as doing 150 questions. In fact, that is better for students with weak foundation, as the repetition reinforces their understanding of the techniques used to solve the question.

If the student starts revision early, this may work out to just 1 question per day for 5 months. Of course, the 150 questions must be varied, and from different subject topics.

 Marks improved by Long Questions to be done Approx. Number of years of TYS OR (even better) 10 50 2 1 year TYS practice twice 20 100 4 2 year TYS practice twice 30 150 6 3 year TYS practice twice 40 200 8 4 year TYS practice twice 50 250 10 5 year TYS practice twice

This estimate only works up to a certain limit (obviously we can’t exceed 100 marks). To get the highest grade (A1 or A), mastery of the subject is needed, and the ability to solve creative questions and think out of the box.

When a student practices TYS questions, it is essential that he/she fully understands the question. This is where a tutor is helpful, to go through the doubts that the student has. Doing a question without understanding it is essentially of little use, as it does not help the student to solve similar questions should they come out in the exam.

## 积少成多: How can doing at least one Maths question per day help you improve! (Maths Tuition Revision Strategy)

We all know the saying “an apple a day keeps the doctor away“. Many essential activities, like eating, exercising, sleeping, needs to be done on a daily basis.

Mathematics is no different!

Here is a surprising fact of how much students can achieve if they do at least one Maths question per day. (the question must be substantial and worth at least 5 marks)

This study plan is based on the concept of 积少成多, or “Many little things add up“. Also, this method prevents students from getting rusty in older topics, or totally forgetting the earlier topics. Also, this method makes use of the fact that the human brain learns during sleep, so if you do mathematics everyday, you are letting your brain learn during sleep everyday.

Let’s take the example of Additional Mathematics.

Exam is on 24/25 October 2013.

Let’s say the student starts the “One Question per day” Strategy on 20 May 2013

Days till exam: 157 days  (22 weeks or 5 months, 4 days)

So, 157 days = 157 questions (or more!)

Each paper in Ten Year Series has around 25 questions (Paper 1 & Paper 2), so 157 questions translates to more than 6 years worth of practice papers! And all that is achieved by just doing at least one Maths question per day!

A sample daily revision plan can look like this. (I create a customized revision plan for each of my students, based on their weaknesses).

 Topic Monday Algebra Tuesday Geometry and Trigonometry Wednesday Calculus Thursday Algebra Friday Geometry and Trigonometry Saturday Calculus Sunday Geometry and Trigonometry

(Calculus means anything that involves differentiation, integration)

(Geometry and Trigonometry means anything that involves diagrams, sin, cos, tan, etc. )

(Algebra is everything else, eg. Polynomials, Indices, Partial Fractions)

By following this method, using a TYS, the student can cover all topics, up to 6 years worth of papers!

Usually, students may accumulate a lot of questions if they are stuck. This is where a tutor comes in. The tutor can go through all the questions during the tuition time. This method makes full use of the tuition time, and is highly efficient.

Personally, I used this method of studying and found it very effective. This method is suitable for disciplined students who are aiming to improve, whether from fail to pass or from B/C to A. The earlier you start the better, for this strategy. For students really aiming for A, you can modify this strategy to do at least 2 to 3 Maths questions per day. From experience, my best students practice Maths everyday. Practicing Ten Year Series (TYS) is the best, as everyone knows that school prelims/exams often copy TYS questions exactly, or just modify them a bit.

The role of the parent is to remind the child to practice maths everyday. From experience, my best students usually have proactive parents who pay close attention to their child’s revision, and play an active role in their child’s education.

This study strategy is very flexible, you can modify it based on your own situation. But the most important thing is, practice Maths everyday! (For Maths, practicing is twice as important as studying notes.) And fully understand each question you practice, not just memorizing the answer. Also, doing a TYS question twice (or more) is perfectly acceptable, it helps to reinforce your technique for answering that question.

If you truly follow this strategy, and practice Maths everyday, you will definitely improve!

Hardwork $\times$ 100% = Success! (^_^)

There is no substitute for hard work.” – Thomas Edison

## Exam Time Management and Speed in Maths (Primary, O Level, A Level)

Time management is a common problem for Maths, along with careless mistakes.

For Exam Time management, here are some useful tips:
1) If stuck at a question for some time, it is better to skip it and go back to it later, rather than spend too much time on it. I recall for PSLE one year, there was a question about adding 1+2+…+100 early in the paper, and some children unfortunately spent a lot of time adding it manually.
2) Use a exam half-time strategy. At the half-time mark of the exam, one should finish at least half of the paper. If no, then need to speed up and skip hard questions if necessary.

To improve speed:
1) Practice. It is really important to practice as practice increases speed and accuracy.
2) Learn the faster methods for each type of question. For example, guess and check is considered a slower method, as most questions are designed to make guess and check difficult.

Sincerely hope it helps.
For dealing with careless mistakes (more for O and A levels), you may read my post on How to avoid Careless Mistakes for O-Level / A-Level Maths?

## Challenging Geometry E Maths Question — St Andrew’s Sec 3 Maths Tuition Question

Question:

ABCD is a rectangle. M and N are points on AB and DC respectively. MC and BN meet at X. M is the midpoint of AB.

(a) Prove that $\Delta CXN$ and $\Delta MXB$ are similar.

(b) Given that area of $\triangle CXN$: area of $\triangle MXB$=9:4, find the ratio of,

(i) DN: NC

(ii) area of rectangle ABCD: area of $\triangle XBC$. (Challenging)

(ii) 20:3

Suggested Solutions:

(a)
$\angle MXB=\angle NXC$ (vert. opp. angles)

$\angle MBX = \angle XNC$ (alt. angles)

$\angle BMX = \angle XCN$ (alt. angles)

Therefore, $\Delta CXN$ and $\Delta MXB$ are similar (AAA).

(b) (i) $\displaystyle\frac{NC}{BM}=\sqrt{\frac{9}{4}}=\frac{3}{2}$

Let $BM=2u$ and $NC=3u$

Then $DC=2\times 2u=4u$

So $DN=4u-3u=u$

Thus, $DN:NC=1u:3u=1:3$

(ii)

We now have a shorter solution, thanks to a visitor to our site! (see comments below)

From part (a), since $\Delta CXN$ and $\Delta MXB$ are similar, we have $MX:XC=2:3$

This means  that $MC:XC=5:3$

Thus $\triangle MBC:\triangle XBC=5:3$ (the two triangles share a common height)

Now, note that $\displaystyle\frac{\text{area of }ABCD}{\triangle MBC}=\frac{BC\times AB}{0.5 \times BC \times MB}=\frac{AB}{0.5MB}=\frac{2MB}{0.5MB}=4$

Hence area of $ABCD=4\times\triangle MBC$

We conclude that area of rectangle ABCD: area of $\triangle XBC=4(5):3=20:3$

Here is a longer solution, for those who are interested:

Let area of $\triangle XBC =S$

Let area of $\triangle MXB=4u$

Let area of $\triangle CXN=9u$

We have $\displaystyle\frac{S+9u}{S+4u}=\frac{3}{2}$ since $\triangle NCB$ and $\triangle CMB$ have the same base BC and their heights have ratio 3:2.

Cross-multiplying, we get $2S+18u=3S+12u$

So $\boxed{S=6u}$

$\displaystyle\frac{\triangle BCN}{\triangle BDC}=\frac{3}{4}$ since $\triangle BCN$ and $\triangle BDC$ have the same base BC and their heights have ratio 3:4.

Hence,

$\begin{array}{rcl} \triangle BDC &=& \frac{4}{3} \triangle BCN\\ &=& \frac{4}{3} (9u+6u)\\ &=& 20u \end{array}$

Thus, area of $ABCD=2 \triangle BDC=40u$

area of rectangle ABCD: area of $\triangle XBC$=40:6=20:3

## H2 Maths Tuition: Foot of Perpendicular (from point to plane) (Part II)

This is a continuation from H2 Maths Tuition: Foot of Perpendicular (from point to line) (Part I).

## Foot of Perpendicular (from point to plane)

From point (B) to Plane ( $p$)

## Equation (I):

Where does F lie?

F lies on the plane  $p$.

$\overrightarrow{\mathit{OF}}\cdot \mathbf{n}=d$

## Equation (II):

Perpendicular

$\overrightarrow{\mathit{BF}}=k\mathbf{n}$

$\overrightarrow{\mathit{OF}}-\overrightarrow{\mathit{OB}}=k\mathbf{n}$

$\overrightarrow{\mathit{OF}}=k\mathbf{n}+\overrightarrow{OB}$

## Final Step

Substitute Equation (II) into Equation (I) and solve for k.

## Example

[VJC 2010 P1Q8i]

The planes $\Pi _{1}$ and $\Pi _{2}$ have equations $\mathbf{r\cdot(i+j-k)}=6$ and $\mathbf{r\cdot(2i-4j+k)}=-12$ respectively. The point $A$  has position vector  $\mathbf{{9i-7j+5k}}$ .

(i) Find the position vector of the foot of perpendicular from  $A$ to $\Pi _{2}$ .

## Solution

Let the foot of perpendicular be F.

### Equation (I)

$\overrightarrow{\mathit{OF}}\cdot \left(\begin{matrix}2\\-4\\1\end{matrix}\right)=-12$

### Equation (II)

$\overrightarrow{\mathit{OF}}=k\left(\begin{matrix}2\\-4\\1\end{matrix}\right)+\left(\begin{matrix}9\\-7\\5\end{matrix}\right)=\left(\begin{matrix}2k+9\\-4k-7\\k+5\end{matrix}\right)$

Subst. (II) into (I)

$2(2k+9)-4(-4k-7)+(k+5)=-12$

Solve for k,  $k=-3$ .

$\overrightarrow{\mathit{OF}}=\left(\begin{matrix}3\\5\\2\end{matrix}\right)$

## H2 Maths Tuition

If you are looking for Maths Tuition, contact Mr Wu at:

SMS: 98348087

Email: mathtuition88@gmail.com

## Foot of Perpendicular is a hot topic for H2 Prelims and A Levels. It comes out almost every year.

There are two versions of Foot of Perpendicular, from point to line, and from point to plane. However, the two are highly similar, and the following article will teach how to understand and remember them.

## H2: Vectors (Foot of perpendicular)

From point (B) to Line ( $l$)

(Picture)

### Equation (I):

Where does F lie? F lies on the line  $l$.

$\overrightarrow{\mathit{OF}}=\mathbf{a}+\lambda \mathbf{m}$

### Equation (II):

Perpendicular:

$\overrightarrow{\mathit{BF}}\cdot \mathbf{m}=0$

$(\overrightarrow{\mathit{OF}}-\overrightarrow{\mathit{OB}})\cdot \mathbf{m}=0$

### Final Step

Substitute Equation (I) into Equation (II) and solve for  $\lambda$.

## Example:

[CJC 2010 P1Q7iii]

Relative to the origin $O$ , the points $A$ , $B$ and $C$  have position vectors  $\left(\begin{matrix}1\\2\\1\end{matrix}\right)$ , $\left(\begin{matrix}2\\1\\3\end{matrix}\right)$ and $\left(\begin{matrix}-1\\2\\3\end{matrix}\right)$ Find the shortest distance from  $C$ to $\mathit{AB}$ . Hence or otherwise, find the area of triangle $\mathit{ABC}$ .

[Note: There is a 2nd method to this question. (cross product method)]

## Solution:

Let the foot of perpendicular from C to AB be F.

Equation (I):

$\overrightarrow{\mathit{OF}}=\overrightarrow{\mathit{OA}}+\lambda \overrightarrow{\mathit{AB}}=\left(\begin{matrix}1+\lambda \\2-\lambda \\1+2\lambda \end{matrix}\right)$

Equation (II):

$(\overrightarrow{\mathit{OF}}-\overrightarrow{\mathit{OC}})\cdot \overrightarrow{\mathit{AB}}=0$

$\left(\begin{matrix}2+\lambda \\-\lambda \\-2+2\lambda \end{matrix}\right)\cdot \left(\begin{matrix}1\\-1\\2\end{matrix}\right)=0$

$\lambda =\frac{1}{3}$

$\overrightarrow{\mathit{CF}}=\overrightarrow{\mathit{OF}}-\overrightarrow{\mathit{OC}}=\left(\begin{matrix}2\frac{1}{3}\\-{\frac{1}{3}}\\-1\frac{1}{3}\end{matrix}\right)$

$\left|{\overrightarrow{{\mathit{CF}}}}\right|=\sqrt{\frac{22}{3}}$

Area of  $\Delta \mathit{ABC}=\frac{1}{2}\left|{\overrightarrow{\mathit{AB}}}\right|\left|{\overrightarrow{\mathit{CF}}}\right|=\sqrt{11}$

For the next part, please read our article on Foot of Perpendicular (from point to plane).

## H2 Maths Tuition

If you are looking for Maths Tuition, contact Mr Wu at:

SMS: 98348087

Email: mathtuition88@gmail.com

## How to avoid Careless Mistakes for Maths?

Many parents have feedback to me that their child often makes careless mistakes in Maths, at all levels, from Primary, Secondary, to JC Level. I truly empathize with them, as it often leads to marks being lost unnecessarily. Not to mention, it is discouraging for the child.

Also, making careless mistakes is most common in the subject of mathematics, it is rare to hear of students making careless mistakes in say, History or English.
Fortunately, it is possible to prevent careless mistakes for mathematics, or at least reduce the rates of careless mistakes.

From experience, the ways to prevent careless mistakes for mathematics can be classified into 3 categories, Common Sense, Psychological, and Math Tips.

Common Sense

1. Firstly, write as neatly as possible. Often, students write their “5” like “6”. Mathematics in Singapore is highly computational in nature, such errors may lead to loss of marks. Also, for Algebra, it is recommended that students write l (for length) in a cursive manner, like $\ell$ to prevent confusion with 1. Also, in Complex Numbers in H2 Math, write z with a line in the middle, like Ƶ, to avoid confusion with 2.
2. Leave some time for checking. This is easier said than done, as speed requires practice. But leaving some time, at least 5-10 minutes to check the entire paper is a good strategy. It can spot obvious errors, like leaving out an entire question.

Psychological

1. Look at the number of marks. If the question is 5 marks, and your solution is very short, something may be wrong. Also if the question is just 1 mark, and it took a long time to solve it, that may ring a bell.
2. See if the final answer is a “nice number“. For questions that are about whole numbers, like number of apples, the answer should clearly be a whole number. At higher levels, especially with questions that require answers in 3 significant figures, the number may not be so nice though. However, from experience, some questions even in A Levels, like vectors where one is suppose to solve for a constant $\lambda$, it turns out that the constant is a “nice number”.

Mathematical Tips

Mathematical Tips are harder to apply, unlike the above which are straightforward. Usually students will have to be taught and guided by a teacher or tutor.

1. Substitute back the final answer into the equations. For example, when solving simultaneous equations like x+y=3, x+2y=4, after getting the solution x=2, y=1, you should substitute back into the original two equations to check it.
2. Substitute in certain values. For example, after finding the partial fraction $\displaystyle\frac{1}{x^2-1} = \frac{1}{2 (x-1)}-\frac{1}{2 (x+1)}$, you should substitute back a certain value for x, like x=2. Then check if both the left-hand-side and right-hand-side gives the same answer. (LHS=1/3, RHS=1/2-1/6=1/3) This usually gives a very high chance that you are correct.

Thanks for reading this long article! Hope it helps! 🙂

I will add more tips in the future.

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

This book is a New York Times Bestseller by actress Danica McKellar, who is also an internationally recognized mathematician and advocate for math education. It should be available in the library. Hope it can inspire all to like Maths!