Stanford University Research: The most important aspect of a student’s ideal relationship with mathematics

Source: Taken from Research by Stanford, Education: EDUC115N How to Learn Math

This word cloud was generated on August 9th based on 850 responses to the prompt “Please submit a word that, in your opinion, describes the most important aspect of a student’s ideal relationship with mathematics.”

stanford maths tuition word cloud

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. 🙂

Source: http://www2.ims.nus.edu.sg/imprints/interviews/BelaBollobas.pdf

(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.

H2 Maths 2012 A Level Solution Paper 2 Q6; H2 Maths Group Tuition

6(i)

H_0: \mu=14.0 cm

H_1: \mu\neq 14.0 cm

(ii)

\bar{x}\sim N(14,\frac{3.8^2}{20})

For the null hypothesis not to be rejected,

Z_{2.5\%}<\frac{\bar{x}-14}{3.8/\sqrt{20}}<Z_{97.5\%}

-1.95996<\frac{\bar{x}-14}{3.8/\sqrt{20}}<1.95996 (use GC invNorm function!)

12.3<\bar{x}<15.7 (3 s.f.)

(iii) Since \bar{x}=15.8 is out of the set 12.3<\bar{x}<15.7, the null hypothesis would be rejected. There is sufficient evidence that the squirrels on the island do not have the same mean tail length as the species known to her.

(technique: put in words what H_1 says!)

NUS Top in Asia according to latest QS World University Rankings by Subject

Source: http://newshub.nus.edu.sg/headlines/1305/qs_08May13.php

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

Singapore matematika kuliah

Kami penuh waktu Matematika guru, Mr Wu (Citizen Singapura), memiliki pengalaman yang luas (lebih dari 7 tahun) di les matematika. Mr Wu telah mengajar matematika sejak tahun 2006.

Mr Wu adalah pasien dengan siswa, dan akan menjelaskan konsep jelas kepada mereka. Dia mendorong untuk siswa lemah, sedangkan siswa yang lebih kuat tidak akan merasa bosan karena Mr Wu akan memberikan latihan yang cukup menantang bagi mereka untuk belajar lebih banyak. Singkatnya, setiap siswa harus mengalami perbaikan setelah kuliah.

Mr Wu lulus dengan B.Sc. (First Class Honours) dengan Mayor di Matematika (National University of Singapore).

Kami sangat percaya bahwa kepribadian dan karakter guru adalah sama pentingnya dengan kualifikasi akademik. Untuk Matematika Tutor, kesabaran ketika menjelaskan kepada siswa mutlak diperlukan.

Tutor Kualifikasi:

NUS: B.Sc. (First Class Honours) dengan Mayor di Matematika, Daftar Dean (Top 5% dari seluruh Fakultas Ilmu)

A Level: Matematika (A), Fisika (A), Kimia (A), Biologi (A), General Paper (A1)

O Tingkat: (Raffles Institution)

Bahasa Inggris (A1), Gabungan Humaniora (A1), Geografi (A1), Matematika (A1), Matematika Tambahan (A1), Fisika (A1), Kimia (A1), Biologi (A1), Bahasa Cina lebih tinggi (A2)

PSLE: (Nanyang Primer) 281, Lee Hsien Loong Excellence Award

Bahasa Inggris (A *), Bahasa Cina (A *), Matematika (A *), Sains (A *), Bahasa Cina Tinggi (Distinction), Ilmu Sosial (Distinction)

Apakah dalam Program PMP dari Pratama ke tingkat sekunder.

Terdaftar dengan MOE sebagai Guru Bantuan

(Orang tua yang ingin melihat sertifikat Mr Wu silahkan email kami. Orang tua juga dapat melihat profil StarTutor Mr Wu pada http://startutor.sg/23561, dengan sertifikat diverifikasi.)

Meskipun kualifikasi akademik Mr Wu, ia tetap seorang guru yang rendah hati dan sabar. Juga, orang tua dapat yakin bahwa Mr Wu mengajar pada tingkat yang siswa dapat sepenuhnya mengerti. Untuk A Level, kami akan mencoba untuk mengajarkannya dengan cara yang jelas dan sederhana sehingga bahkan Sec 3/4 siswa dapat mengerti. Untuk O Levels, kita akan mengajarkannya sedemikian rupa sehingga bahkan Sec 1/2 siswa dapat memahami, dan sebagainya.

Mr Wu hanyalah orang biasa yang telah menguasai keterampilan dan teknik yang diperlukan untuk unggul dalam matematika di Singapura. Dia ingin mengajarkan teknik ini untuk siswa, maka memilih untuk menjadi Matematika penuh waktu guru. Mr Wu telah mengembangkan metode sendiri untuk memeriksa jawaban, mengingat rumus (dengan pemahaman), yang telah membantu banyak siswa. Banyak pertanyaan Math dapat diperiksa dengan mudah, menyebabkan siswa menjadi 100% yakin nya atau jawabannya bahkan sebelum guru menandai jawabannya, dan mengurangi tingkat kesalahan ceroboh.

Mr Wu juga kakak dari dua mahasiswa kedokteran. Adiknya sedang belajar Kedokteran di Universitas Monash, dan adiknya sedang belajar Kedokteran di Yong Loo Lin School of Medicine, NUS.

Tujuan Pengajaran:

Tujuan pengajaran adalah untuk memungkinkan siswa untuk memahami konsep-konsep dalam silabus, meningkatkan minat pada pelajaran, dan untuk menjelaskan dengan jelas metode untuk memecahkan masalah matematika. Matematika adalah subjek yang sangat kumulatif, dasar yang kuat diperlukan untuk maju ke tingkat berikutnya. Kami sangat berharap dapat membantu lebih banyak siswa membangun fondasi yang kuat di Matematika.

Untuk Matematika, kami percaya bahwa cara terbaik untuk maju adalah melalui praktek dan pemahaman. Teknik untuk memeriksa jawaban dan metode singkat untuk menjawab pertanyaan lebih cepat berguna. Ketekunan sangat penting dalam Matematika, yang penting adalah untuk tidak menyerah, dan terus mencoba!

Untuk individu Matematika kuliah, tutor dapat melakukan perjalanan ke rumah siswa.

“Didiklah anak di jalan yang patut baginya: dan ketika dia sudah tua, dia tidak akan menyimpang dari itu.”

– Amsal 22:6

คณิตศาสตร์ชั้นเรียนกลุ่มที่จะเริ่มต้นในปีหน้า 2014

คณิตศาสตร์ชั้นเรียนกลุ่มที่จะเริ่มต้นในปีหน้า 2014

ศูนย์คณิตศาสตร์เล่าเรียน

Youngest NUS graduates for 2012 – 08Jul2012

Source: http://www.youtube.com/watch?v=q-53rIy7RGg

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

Does one have to be a genius to do maths?

Source: http://terrytao.wordpress.com/career-advice/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.

Continue reading at http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/

There’s more to mathematics than grades and exams and methods

Source: http://terrytao.wordpress.com/career-advice/there%E2%80%99s-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.

Continue reading at http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-grades-and-exams-and-methods/

Maths Tuition Singapore Keywords

Just for curiosity, I went to research on the top keywords for Maths Tuition for Google search engine.

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How to Make Online Courses Massively Personal

How thousands of online students can get the effect of one-on-one tutoring

Source: http://www.scientificamerican.com/article.cfm?id=how-to-make-online-courses-massively-personal-peter-norvig&WT.mc_id=SA_SA_20130717

Educators have known for 30 years that students perform better when given one-on-one tutoring and mastery learning—working on a subject until it is mastered, not just until a test is scheduled. Success also requires motivation, whether from an inner drive or from parents, mentors or peers.

Online learning is a tool, just as the textbook is a tool. The way the teacher and the student use the tool is what really counts.

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.

stanford-maths-tuition

EDUC115N: How to Learn Math 

(Source: https://class.stanford.edu/courses/Education/EDUC115N/How_to_Learn_Math/about)

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’

Source: http://www.telegraph.co.uk/education/maths-reform/9621100/Make-Britain-Count-Stop-telling-children-maths-isnt-for-them.html

“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.

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.

The illustration for theorem about tangent and...
The illustration for theorem about tangent and secant (Photo credit: Wikipedia)

Finding equation of circle (A Maths) 8 mark Question!

Question:

Find the equation of the circle which passes through A(8,1) and B(7,0) and has, for its tangent at B, the line 3x-4y-21=0.

Solution:

Recall that the equation of a circle is (x-a)^2+(y-b)^2=r^2, where (a,b) is the centre of the circle, and r is the radius of the circle.

Substituting A(8,1) into the equation, we get:

\boxed{(8-a)^2+(1-b)^2=r^2} — Eqn (1)

Substituting B(7,0), we get:

\boxed{(7-a)^2+(0-b)^2=r^2} — Eqn (2)

Equating Eqn (1) and Eqn (2), we get

64-16a+a^2+1-2b+b^2=49-14a+a^2+b^2 which reduces to

\boxed{b=8-a} — Eqn (3)

after simplification.

Now, we rewrite the equation of the tangent as \displaystyle y=\frac{3}{4}x-\frac{21}{4} (make y the subject)

Hence, the gradient of the normal is \displaystyle\frac{-1}{\frac{3}{4}}=-\frac{4}{3}

Let the equation of the normal be \displaystyle y=-\frac{4}{3}x+c

Substitute in  B(7,0) we get \displaystyle 0=-\frac{4}{3}(7)+c

Hence \displaystyle c=\frac{28}{3}

Thus equation of normal is \displaystyle \boxed{y=-\frac{4}{3}x+\frac{28}{3}}

Since the normal will pass through the centre (a,b) we have

\boxed{b=-\frac{4}{3}a+\frac{28}{3}} — Eqn (4)

Finally, we equate Eqn (3) and Eqn (4),

\displaystyle 8-a=-\frac{4}{3}a+\frac{28}{3}

\displaystyle \frac{1}{3}a=\frac{4}{3}

a=4

b=8-a=4

Substituting back into Eqn (1), we get r=5

Hence the equation of the circle is:

\displaystyle\boxed{(x-4)^2+(y-4)^2=5^2}

Xinmin Secondary 2010 Prelim Paper I Q24 Solution (Challenging/Difficult Probability O Level Question)

A bag A contains 9 black balls, 6 white balls and 3 red balls. A bag B contains 6 black balls, 2 white balls and 4 green balls. Ali takes out 1 ball from each bag randomly. When Ali takes out 1 ball from one bag, he will put it into the other bag and then takes out one ball from that bag. Find the probability that

(a) the ball is black from bag A, followed by white from bag B,
(b) both the balls are white in colour,
(c) the ball is black or white from bag B, followed by red from bag A,
(d) both the balls are of different colours,
(e) both the balls are not black or white in colours.

probability maths tuition

Solution:

(a) \displaystyle\frac{9}{18}\times\frac{2}{13}=\frac{1}{13}

(b) Probability of white ball from bag A, followed by white ball from bag B=\displaystyle=\frac{1}{2}\times\frac{6}{18}\times\frac{3}{13}=\frac{1}{26}

Probability of white from B, followed by white from A=\displaystyle=\frac{1}{2}\times\frac{2}{12}\times\frac{7}{19}=\frac{7}{228}

Total prob=\displaystyle\frac{205}{2964}

(c) Prob. of ball is black or white from bag B=\displaystyle\frac{6}{12}+\frac{2}{12}=\frac{8}{12}

\displaystyle\frac{8}{12}\times\frac{3}{19}=\frac{2}{19}

(d) Prob of both red = P(red from A, followed by red from B)=\displaystyle\frac{1}{2}\times\frac{3}{18}\times\frac{1}{13}=\frac{1}{156}

P(both green)=P(green from B, followed by green from A)=\displaystyle\frac{1}{2}\times\frac{4}{12}\times\frac{1}{19}=\frac{1}{114}

P(both black)=P(black from A, followed by black from B)+P(black from B, followed by black from A)=\displaystyle\frac{1}{2}\times\frac{9}{18}\times\frac{7}{13}+\frac{1}{2}\times\frac{6}{12}\times\frac{10}{19}=\frac{263}{988}

P(both white)=\displaystyle\frac{205}{2964} (from part b)

\displaystyle 1-\frac{1}{156}-\frac{1}{114}-\frac{263}{988}-\frac{205}{2964}=\frac{1925}{2964}

(e)

P(neither black nor white from A, followed by neither black nor white from B)=\displaystyle\frac{1}{2}\times\frac{3}{18}\times\frac{5}{13}=\frac{5}{156}

P(neither black nor white from B, followed by neither black nor white from A)=\displaystyle\frac{1}{2}\times\frac{4}{12}\times\frac{4}{19}=\frac{2}{57}

\displaystyle\frac{5}{156}+\frac{2}{57}=\frac{199}{2964}

3D Trigonometry Maths Tuition

angle-3d-maths-tuition

Solution:

(a) Draw a line to form a small right-angled triangle next to the angle 18^\circ

Then, you will see that

\angle ACD=90^\circ-18^\circ=72^\circ (vert opp. angles)

\angle BAC=180^\circ-72^\circ=108^\circ (supplementary angles in trapezium)

By sine rule,

\displaystyle \frac{\sin\angle ABC}{30}=\frac{\sin 108^\circ}{40.9}

\sin\angle ABC=0.697596

\angle ABC=44.23^\circ

\angle ACB=180^\circ-44.23^\circ-108^\circ=27.77^\circ=27.8^\circ (1 d.p.)

(b) By Sine Rule,

\displaystyle\frac{AB}{\sin\angle ACB}=\frac{30}{\sin 44.23^\circ}

AB=\frac{30}{\sin 44.23^\circ}\times\sin 27.77^\circ=20.04=20.0 m (shown)

(c)

\angle BCD=\angle ACD-\angle ACB=72^\circ-27.77^\circ=44.23^\circ

By Cosine Rule,

BD^2=40.9^2+50^2-2(40.9)(50)\cos 44.23^\circ=1242.139

BD=35.24=35.2 m

(d)

\displaystyle\frac{\sin\angle BDC}{40.9}=\frac{\sin 44.23^\circ}{35.24}

\sin\angle BDC=0.80957

\angle BDC=54.05^\circ

angle of depression = 90^\circ-54.05^\circ=35.95^\circ=36.0^\circ (1 d.p.)

(e)

Let X be the point where the man is at the shortest distance from D. Draw a right-angle triangle XDC.

\displaystyle\cos 72^\circ=\frac{XC}{50}

XC=50\cos 72^\circ=15.5 m

Tips on attempting Geometrical Proof questions (E Maths Tuition)

Tips on attempting Geometrical Proof questions (O Levels E Maths/A Maths)

1) Draw extended lines and additional lines. (using pencil)

Drawing extended lines, especially parallel lines, will enable you to see alternate angles much easier (look for the “Z” shape). Also, some of the more challenging questions can only be solved if you draw an extra line.

2) Use pencil to draw lines, not pen

Many students draw lines with pen on the diagram. If there is any error, it will be hard to remove it.

3) Rotate the page.

Sometimes, rotating the page around will give you a fresh impression of the question. This may help you “see” the way to answer the question.

4) Do not assume angles are right angles, or lines are straight, or lines are parallel unless the question says so, or you have proved it.

For a rigorous proof, we are not allowed to assume anything unless the question explicitly says so. Often, exam setters may set a trap regarding this, making the angle look like a right angle when it is not.

5) Look at the marks of the question

If it is a 1 mark question, look for a short way to solve the problem. If the method is too long, you may be on the wrong track.

6) Be familiar with the basic theorems

The basic theorems are your tools to solve the question! Being familiar with them will help you a lot in solving the problems.

Hope it helps! And all the best for your journey in learning Geometry! Hope you have fun.

“There is no royal road to Geometry.” – Euclid

Animation of a geometrical proof of Phytagoras...
Animation of a geometrical proof of Pythagoras theorem (Photo credit: Wikipedia)

Solution 2 (Eigenvalue): Monkeys & Coconuts

Math Online Tom Circle

Solution 2: Use Linear Algebra Eigenvalue equation: A.X = λ.X

A =S(x)= $Latex \frac{4}{5}(x-1)$  where x = coconuts

S(x)=λx

Since each iteration of the transformation caused the coconut status ‘unchanged’, which means λ = 1 (see remark below)

$Latex \frac{4}{5}(x-1)=x$
We get
x = – 4

Also by recursive, after the fifth monkey: $Latex S^5 (x)$ = $Latex (\frac{4}{5})^5 (x-1)- (\frac{4}{5})^4-(\frac{4}{5})^3- (\frac{4}{5})^2- \frac{4}{5}$

$Latex S^5 (x)$ = $Latex (\frac{4}{5})^5 (x) – (\frac{4}{5})^5 – (\frac{4}{5})^4 – (\frac{4}{5})^3+(\frac{4}{5})^2 – \frac{4}{5}$

 

$Latex 5^5$ divides (x)

Minimum positive x= – 4 mod ($Latex 5^{5}$ )= $Latex 5^{5} – 4$= 3,121 [QED]

 

Note: The meaning of eigenvalue  λ in linear transformation is the change  by a scalar of λ factor (lengthening or shortening by λ) after the transformation. Here

λ = 1 because “before” and “after” (transformation A)  is the SAME status (“divide coconuts by 5 and left 1”).

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Solution 1 (Sequence): Monkeys & Coconuts

Math Online Tom Circle

Monkeys & Coconuts Problem

Solution 1 : iteration problem => Use sequence
$Latex U_{j} =\frac {4}{5} U_{j- 1} -1 $

(initial coconuts)
$Latex U_0 =k$
Let
$Latex f(x)=\frac{4}{5}(x-1)=\frac{4}{5}(x+4)-4$
$Latex U_1 =f(U_0)=f(k)= \frac{4}{5}(k+4)-4$

$Latex U_2 =f(U_1)=f(\frac{4}{5}(k+4)-4)= \frac{4}{5}((\frac{4}{5}(k+4)-4+4)-4$

$Latex U_2=(\frac{4}{5})^2 (k+4)-4$

$Latex U_3=(\frac{4}{5})^3 (k+4)-4$

$Latex U_4=(\frac{4}{5})^4 (k+4)-4$

$Latex U_5=(\frac{4}{5})^5 (k+4)-4$

Since
$Latex U_5$ is integer  ,
$Latex 5^5 divides (k+4)$
k+4 ≡ 0 mod($Latex 5^5$)
k≡-4 mod($Latex 5^5$)
Minimum {k} = $Latex 5^5 -4$= 3121 [QED]

Note: The solution was given by Paul Richard Halmos (March 3, 1916 – October 2, 2006)

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Monkeys & Coconuts Problem

Math Online Tom Circle

5 monkeys found some coconuts at the beach.

1st monkey came, divided the coconuts into 5 groups, left 1 coconut which it threw to the sea, and took away 1 group of coconuts.
2nd monkey came, divided the remaining coconuts into 5 groups, left 1 coconut again thrown to the sea, and took away 1 group.
Same for 3rd , 4th and 5th monkeys.

Find: how many coconuts are there initially?

Note: This problem was created by Nobel Physicist Prof Paul Dirac (8 August 1902 – 20 October 1984). Prof Tsung-Dao Lee (李政道) (1926 ~) , Nobel Physicist, set it as a test for the young gifted students in the Chinese university of Science and Technology (中国科技大学-天才儿童班).

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Reason for Maths Tuition

My take is that Maths tuition should not be forced. The child must be willing to go for Maths tuition in the first place, in order for Maths tuition to have any benefit. Also, the tuition must not add any additional stress to the student, as school is stressful enough. Rather Maths tuition should reduce the student’s stress by clearing his/her doubts and improving his/her confidence and interest in the subject. There is a quote “One important key to success is self-confidence. An important key to self-confidence is preparation.“. Tuition is one way to help the child with preparation.

Parallelogram Maths Tuition: Solution

parallelogram-maths-tuition

Solution:

(a) We have \angle APQ=\angle ARQ (opp. angles of parallelogram)

AP=RQ (opp. sides of parallelogram)

AR=PQ (opp. sides of parallelogram)

Thus, \triangle APQ\equiv\triangle QRA (SAS)

Similarly, \triangle ABC\equiv\triangle CDA (SAS)

\triangle CHQ\equiv\triangle QKC (SAS)

Thus, \begin{array}{rcl}\text{area of BPHC}&=&\triangle APQ-\triangle ABC-\triangle CHQ\\    &=&\triangle QRA-\triangle CDA-\triangle QKC\\    &=& \text{area of DCKR}    \end{array}

(proved)

(b)

\angle ACD=\angle HCQ (vert. opp. angles)

\angle ADC=\angle CHQ (alt. angles)

\angle DAC=\angle CQH (alt. angles)

Thus, \triangle ADC is similar to \triangle QHC (AAA)

Hence, \displaystyle\frac{AC}{DC}=\frac{QC}{HC}

Thus, AC\cdot HC=DC\cdot QC

(proved)

Congruent Triangles Maths Tuition: Solution

congruent-maths-tuition

Solution:

BS=BE+ES=ST+ES=ET

\angle DES=\angle ESA=90^\circ

BA=DT (given)

Thus, \triangle ASB is congruent to \triangle DET (RHS)

Hence \angle DTE=\angle SBA

Thus DT//BA (alt. angles)

(Proved)

By Pythagoras’ Theorem, we have

\begin{array}{rcl}DB&=&\sqrt{DE^2+BE^2}\\    &=&\sqrt{SA^2+ST^2}\\    &=&TA    \end{array}

Hence \triangle DEB and \triangle AST are congruent (SSS).

Hence \angle DBE=\angle STA

Thus DB//TA (alt. angles)

Therefore, ABDT is a parallelogram since it has two pairs of parallel sides.

(shown)

French Curve

Math Online Tom Circle

The French method of drawing curves is very systematic:

“Pratique de l’etude d’une fonction”

Let f be the function represented by the curve C

Steps:

1. Simplify f(x). Determine the Domain of definition (D) of f;
2. Determine the sub-domain E of D, taking into account of the periodicity (eg. cos, sin, etc) and symmetry of f;
3. Study the Continuity of f;
4. Study the derivative of fand determine f'(x);
5. Find the limits of fwithin the boundary of the intervals in E;
6. Construct the Table of Variation;
7. Study the infinite branches;
8. Study the remarkable points: point of inflection, intersection points with the X and Y axes;
9. Draw the representative curve C.

Example:

$latex \displaystyle\text{f: } x \mapsto \frac{2x^{3}+27}{2x^2}$
Step 1: Determine the Domain of Definition D
D = R* = R –…

View original post 454 more words

Cut a cake 1/5

Math Online Tom Circle

Visually cut a cake 1/5 portions of equal size:

1) divide into half:

20130513-111010.jpg

2) divide 1/5 of the right half:

20130513-133441.jpg

3) divide half, obtain 1/5 = right of (3)

$latex \frac{1}{5}= \frac{1}{2} (\frac{1}{2}(1- \frac{1}{5}))= \frac{1}{2} (\frac{1}{2} (\frac{4}{5}))=\frac{1}{2}(\frac{2}{5})$

20130513-171052.jpg

4) By symmetry another 1/5 at (2)=(4)

20130513-174541.jpg

5) divide left into 3 portions, each 1/5

$latex \frac{1}{5}= \frac{1}{3}(\frac{1}{2}+ \frac{1}{2}.\frac{1}{5}) = \frac{1}{3}.\frac{6}{10}$

20130513-174742.jpg

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Tuition That We May Have To Believe In

This insightful article makes a really good read.

Quotes from the article:

To be honest, the amount to be learnt at each level of education is constantly increasing, and tuition could just help you get that edge over others. After all, it was meant to be supplementary in nature.

The toughest part at the end of the day however, is probably this: getting the right tutor.

guanyinmiao's musings (Archived: July 2009 to July 2019)

This commentary, “Tuition That We May Have To Believe In”, is a reply to a previous article on tuition by Howard Chiu (Mr.), “Tuition We Don’t Have To Believe In” (Read).

I must say Howard’s article had me on his side for a moment. He appealed to me emotively. Nothing like a mental picture of some kid attending hours and hours of tuition immediately after school when he could well be enjoying himself thoroughly with… an iPhone or iPad (I highly doubt kids these days still indulge their time at playgrounds). But the second time I read his article, I silenced the part of my brain which still prays the best for children, so do pardon me if I sound a tad too pragmatic at times.

The overarching assertion that Howard projects his points from is that there is “huge over consumption of this good”. Firstly, private tutoring…

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Generalized Analytic Geometry

Math Online Tom Circle

Generalized Analytic Geometry

Find the equation of the circle which cuts the tangent 2x-y=0 at M(1,4), passing thru point A(4,-1).

Solution:

1st generalization:
Let the point circle be:
(x-1)² + (y-4)² =0

2nd generalization:
It cuts the tangent 2x-y=0
(x-1)² + (y-4)² +k(2x-y) =0 …(C)

Pass thru A(4,-1)
x=4, y= -1
=> k= -2
(C): (x-3)² + (y-1)² =…
[QED]

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Math Chants

Math Online Tom Circle

Math Chants make learning Math formulas or Math properties fun and easy for memory . Some of them we learned in secondary school stay in the brain for whole life, even after leaving schools for decades.

Math chant is particularly easy in Chinese language because of its single syllable sound with 4 musical tones (like do-rei-mi-fa) – which may explain why Chinese students are good in Math, as shown in the International Math Olympiad championships frequently won by China and Singapore school students.

1. A crude example is the quadratic formula which people may remember as a little chant:
ex equals minus bee plus or minus the square root of bee squared minus four ay see all over two ay.”

$latex \boxed{
x = \frac{-b \pm \sqrt{b^{2}-4ac}}
{2a}
}$

2. $latex \mathbb{NZQRC}$
Nine Zulu Queens Rule China

3. $latex \boxed {\cos 3A = 4\cos^{3}…

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What is “sin A”

Math Online Tom Circle

What is “sin A” concretely ?

1. Draw a circle (diameter 1)
2. Connect any 3 points on the circle to form a triangle of angles A, B, C.
3. The length of sides opposite A, B, C are sin A, sin B, sin C, respectively.

Proof:
By Sine Rule:

$latex \frac{a}{sin A} = \frac{b}{sin B} =\frac{c}{sin C} = 2R = 1$
where sides a,b,c opposite angles A, B, C respectively.
a = sin A
b = sin B
c = sin C

20130421-193110.jpg

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成语数学

An alternative answer to Q1) 20 除 3 is “陆续不断”.

20除以3,因为它的答案接近于6.6666,所以这道题的答案是陆续不断,或者是六六大顺都行,百分之一就是百里挑一,9寸加1寸等于一尺即是得寸进尺,12345609,七零八落,1、3、5、7、9无双数所以叫做举世无双,或者你把它答出天下无双都行,如此小升初的难题您答对了吗?

Source: http://politics.people.com.cn/n/2013/0528/c70731-21638931.html

Math Online Tom Circle

中国的小学离校考试 (PSLE) 「神题」: 猜成语
1) 20 除 3
2)1 除100
3)9寸+1寸=1尺
4)12345609
5)1,3,5,7,9

答案::
1) 20/3= 6.666 六六大顺
2)百中挑一
3)得寸進尺
4)七零八落
5)举世无双

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O Level E Maths Tuition: Statistics Question

statistics-olevel-tuition-graph

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.

Secondary Maths Tuition: Kinematics Question

kinematics-question-o-levels

Solution:

acceleration of car =\frac{12}{6}=2m/s^2

\frac{v}{15-5}=2

v=2\times 10=20

Let T be the time (in seconds) when the car overtakes the truck.

Total distance travelled by car at T seconds = area under graph = \frac{1}{2}(T-5)(2(T-5))

(2(T-5) is the velocity of car at T seconds, it is obtained in the same way as we calculated v.)

Total distance travelled by truck at T seconds = \frac{1}{2}(6)(12)+\frac{1}{2}(15-6)(12+16)+16(T-15)

Equating the two distances will lead to a quadratic equation T^2-26T+103=0

Solving that gives T=21.12s or T=4.876s (rejected as car only starts at t=5)

21.12-5=16.1s (3 s.f.)

Maths Resources available for sale!

Do check out our Maths Resources available for sale! All Maths notes and worksheets are priced affordably, starting from just $0.99!
All the resources are personally written by our principal tutor Mr Wu.

https://mathtuition88.com/maths-notes-worksheets-sale/

 

 

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