## H2 Math Tuition 2020

H2 Math can prove to be quite challenging for many students. Historically, the distinction rate is 50%, meaning that half of the student population will get an ‘A’ grade.

However, note that those 50% can be heavily represented by the top schools RI, HCI, NJC where almost all students get ‘A’. At the end of the day, the ‘A’ level students are facing heavy competition from the IP cohort (who were absent from the O levels).

Tough topics in ‘A’ level H2 Math:

Case study: I tutored a student (from China) studying in NJC for H2 Math. Although she was quite a strong student with good Math foundation (definitely above average), nevertheless the NJC H2 Math paper was set so tough that she only scored 50+ out of 100 in the prelims. Many students in her NJC class outright failed for the prelims (below 50 marks). There were some weaknesses in her H2 Math skills, for example certain topics like Integration and P&C, and also carelessness issues. After several months of tutoring, noticeable improvements were observed. In the end for the actual ‘A’ levels, she managed to score A grade for H2 Math.

Read more on student testimonials here: Testimonials

For H2 Math tuition, contact our experienced tutor Mr Wu at:

Email: mathtuition88@gmail.com

Phone:

## Online Math Tuition Singapore

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

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

Do contact our tutor Mr Wu at:

Email: mathtuition88@gmail.com

Phone (WhatsApp/SMS preferred):

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

## Homeschool Math Challenging Puzzles

The questions listed are also very suitable as PSLE Challenging Math Problem Sums.

Basically, for Grade 2-4, they are very challenging.

For Grade 6 (Primary 6, 12 year old students), they are challenging math questions.

Do give it a try and see if your child can solve it!

## Homeschool Math Challenging Questions

Mathtuition88 will be starting a series of Homeschool Math Challenging Problems, aimed at age 8 to 10 (Grade 2 to 4).

This series is targeted at kids age 8 to 10 who are strong / gifted at mathematics and wish to further stretch their potential. It is also useful for children who may not be strong in math at the moment, but have a keen interest in math nonetheless.

In particular, it is very suitable for the following purposes:

• Preparation for GEP (Gifted Education Programme) screening and selection tests
• Puzzles for kids interested in math but find school work too easy.
• PSLE challenging Math problem sums. The questions are also well within the PSLE Math Syllabus, and will be challenging to Primary 6 students as well.

This series of questions will follow the Singapore Math syllabus for Grade…

View original post 66 more words

## Homeschool Maths Puzzles with Answers: The “Average” Question

The average of 5 numbers is 73. When 2 numbers were removed, the average decreased by 3. What is the average of the 2 numbers that were removed?

This is part of a series on Homeschool Math Challenging Puzzles, suitable for Grades 2-4. (Of course, students of other grades are also welcome to try them out.) The questions are suitable for:

• Preparation for GEP (Gifted Education Programme) screening and selection tests
• Puzzles for kids interested in math but find school work too easy.

This is literally an “average” question. 😛 The word “average” appears 3 times.

The trick is to focus on the total instead.

The total of the 5 numbers is: 73×5=365.

After removing 2 numbers, the average is 73-3=70.

Hence, the total of the 3 remaining numbers is: 70×3=210.

We can then conclude that, the total of the 2 removed numbers is:

365-210=155.

Hence, the average of the 2 removed numbers is: 155/2=77.5 or 75 1/2.

## Hardest Questions in Additional Mathematics (A Math)

Additional Mathematics questions can range from standard all the way to super challenging among the secondary schools in Singapore.

Certain schools (such as IP schools), and also some schools such as Anderson, Chung Cheng High School, are well known for setting hard A Math papers.

Note that even though top schools set hard A Math papers, it is not often the case that top schools teach or prepare well their students for the tests! Often, the teachers in school teach at a basic level (due to time constraints or other factors), but still test at an advanced level. Hence, many students in top IP schools are not well prepared for their school’s tests (unless they have excellent self study skills or have a parent or tutor to guide them). It is not uncommon for a student in a top IP school to be failing his/her math tests due to the above phenomena (difficult tests which do not match what is taught in school).

Some of the more difficult types of questions in the A Math syllabus are listed below.

## Algebra

1. Conditions for ax^2 + bx + c to be always positive (or always negative).
This type of question has potential to be very tricky. Somehow, many students will assume wrongly that b^2-4ac is always positive as well (where it should be the opposite).
2. Partial Fractions with Improper Fractions.
Only top schools tend to test improper partial fractions. Many students will miss out long division or make mistakes along the way.
3. Binomial Theorem.
Many students have serious problems with this topic. Also, not many seem to know that ${n\choose 2}=\frac{n(n-1)}{2}$.

## Trigonometry

• Sketching of Tangent graphs.
90% of all sketching questions are on Sine or Cosine. Only top schools will set tangent sketching questions, and many students will be caught unaware.
• Half-angle formula sin(x/2) or Quadruple angle formula sin(4x)
Top schools like to test half-angle formula, many students who have not seen such questions will be stuck.

## Integration

• Finding area to the left of the curve, i.e. $\int x\,dy$.
Most schools kind of brush off this type of questions during teaching. But it is a hot topic for testing among top schools. Hence, students will have a hard time solving it if they lack practice for this type of questions.

## Poll Results: How many marks to get ‘A’ for H2 Math

Almost 100 people have voted on the poll on “How many marks to get ‘A’ for H2 Math“.

The results show that the majority (57.6%) of voters think it takes at least 75 marks to get ‘A’ for H2 Math. Notably, a significant percentage (27.17%) think that it takes 80 marks and above to get ‘A’ for H2 Math.

Students who have taken the H2 Math exam can actually estimate the cut-off point for ‘A’ grade quite well. Basically, the worked solutions are typically released by seniors/tutors, and students can estimate their own marks rather easily. Then, they can compare with their actual grade received.

Nevertheless, the above is just a poll, it may not be 100% accurate and it also depends on the difficulty of that year’s exam. An easier exam would naturally lead to a higher mark required for ‘A’ grade for H2 Maths.

## Is there bell curve for ‘A’ levels?

This is a tricky question. The technically correct answer is that there is no bell curve, but there is a similar thing called “grade boundaries”. It is like certain schools saying that they have no Midyear Exam, but there is a “Block Test”. Read more about whether there is bell curve for ‘O’ and ‘A’ levels.

## Key Topics for IP Additional Mathematics

The following are some of the most important topics for Integrated Programme (IP) Additional Mathematics. Also applicable for the usual ‘O’ level Additional Mathematics.

Notice that Secondary 3 topics are very important as well, for the final Promo or ‘O’ levels. This can be a major problem for students who only start to study seriously in Secondary 4 — it can be a tough job to catch up with the important Secondary 3 topics.

## Secondary 3 topics

• Binomial Theorem
• Indices and Logarithms
• Coordinate Geometry of Circles
• Linear Law

## Secondary 4 topics

• Trigonometry: R formula and Graphs
• Differentiation and its Applications
• Integration and its applications (including area under the curve)

## Diagonalizable & Minimal Polynomial

A matrix or linear map is diagonalizable over the field $F$ if and only if its minimal polynomial is a product of distinct linear factors over $F$.

## Characteristic Polynomial

Let $A$ be an $n\times n$ matrix. The characteristic polynomial of $A$, denoted by $p_A(t)$, is the polynomial defined by $\displaystyle p_A(t)=\det(tI-A).$

## Cayley-Hamilton Theorem

Every square matrix over a commutative ring satisfies its own characteristic equation:

If $A$ is an $n\times n$ matrix, $p(A)=0$ where $p(\lambda)=\det(\lambda I_n-A)$.

## Cool Math Games

Some games are fun but not educational.

On the other hand, some games are educational but not fun!

This very popular website Cool Math Games is a combination of fun and educational!

## Benefit of Cool Math Games

• Their games are all browser-based, there is no need to download anything. It can be played on your Internet Explorer or Chrome browser.
• They have a wide range of different games, ranging from Numbers, Logic, Trivia, and Strategy.
• Their games are kid-friendly and educational.

## Cool Math Games Example

This games “Swap Sums” is suitable for lower elementary school kids (age 5-10). It starts off easy, but the later stages are actually quite challenging. For example, the stage below actually requires some thinking to get it right.

## Cool Math Games Review

Children naturally love to play games. When I was a child, my favorite activity was to play computer games. Hours can just fly by when playing games. When playing games that are educational at the same time, it can hugely benefit a child in terms of increasing his/her interest in the subject.

Hence, parents are encouraged to let their child play educational games, such as Cool Math Games, to relax as well as to learn. Most children learn better when the content is fun and engaging.

## Ex-Actress Evelyn Tan Supports Her Kids Pursuing Entertainment Careers Without A Degree

Quite interesting, and makes a bit of sense too. In this new age where almost everyone has a degree, not having a degree may in fact stand out (for unique career paths)! It carries risk though, naturally.

## Son planned his career as streamer

When her son Jairus, 12, said he wanted to be a Twitch streamer, Evelyn was confused. According to Toggle, she did not know about the online streaming platform for gamers.

She called him a “thinker” — he had told her that if he started his career now, it would be established enough to be profitable by the time he turns 21.

Her son then pointed out that a degree would be irrelevant to his career, which Evelyn said was fine.

## Evelyn Tan majored in Math in NUS

Not many people know this, but actress Evelyn Tan once majored in Math in NUS, and graduated successfully. “Upon graduating from the National University of Singapore with a degree in mathematics, Tan joined the fifth edition of Mediacorp’s Star Search in 1997.”

## Slow and Steady Snail (Homeschool Math Favorite Challenging Puzzles)

This is a favorite type of homeschool math challenging puzzle — The Snail question.

Question: An aquarium is 47 cm deep. A snail starts at the bottom of the aquarium. Each day, during the daytime the snail climbs up 8 cm, and during the nighttime the snail slides down 3 cm. How many days will it take for the snail to climb out of the aquarium?

This is part of a series on Homeschool Math Challenging Puzzles, suitable for Grades 2-4. (Of course, students of other grades are also welcome to try them out.) The questions are suitable for:

• Preparation for GEP (Gifted Education Programme) screening and selection tests
• Puzzles for kids interested in math but find school work too easy.

Solution:

A tempting answer would be 10 days. This is a trick! Those who get the answer “10 days” reason like this: each day the snail moves a net distance of 8-3=5 cm.

Hence, 47/5=9 R 2.

At the end of the 9th day, the snail moved 45 cm. Thus, rounding up will give 10 days as the answer. However, there is a tricky part to the question!

The correct answer is 9 days.

We can do a list:

Day 1 — 5 cm

Day 2 — 10 cm

Day 3 — 15 cm

Day 8 — 40 cm

Day 9 day time — 40 +8 = 48cm > 47 cm !!!

The snail is already out of the aquarium on Day 9!

## How to ensure a pair of male-female for fish & Shrimp

Given N fishes/shrimp, what is the probabillity of having at least 1 male-female pair?

Probability = 1 – (Prob. Of all Male)- (Prob. Of all Female)

If you have 4 fishes, Probability of having a breeding pair = 87.50%

If you have 5 fishes, Probability of having a breeding pair = 93.75%

If you have 6 fishes, Probability of having a breeding pair = 96.88%

If you have 7 fishes, Probability of having a breeding pair = 98.44%

If you have 8 fishes, Probability of having a breeding pair = 99.22%

## Stickers Math Question

Abby, Brian, Charles, Dennis and Eason have 50 stickers altogether.

Abby has the most stickers — she has 12 stickers.

In second place (tied) are Brian and Charles.

In third place is Dennis.

In fourth place is Eason, with 6 stickers.

How many stickers does Dennis have?

This is part of a series on Homeschool Math Challenging Puzzles, suitable for Grades 2-4. (Of course, students of other grades are also welcome to try them out.) The questions are suitable for:

• Preparation for GEP (Gifted Education Programme) screening and selection tests
• Puzzles for kids interested in math but find school work too easy.

For this type of questions, the easiest way to do is “trial and error”, also known as “guess and check”.

Firstly, lets check how many stickers B, C, and D have together:

50-12-6=32

 B C D Total (B+C+D) Comments 9 9 14 32 Wrong! Since D is more than A 10 10 12 32 Wrong! Since D is tied with A 11 11 10 32 Correct! 12 12 8 32 Wrong! Since B, C is tied with A

For guess and check, the most important thing is to be systematic, rather than guess wildly. For instance, we can systematically increase our guesses for B, C.

We can see that the only logical answer is:

B= C=11

D=10

## Marbles Math Question

Aaron and Bob had some marbles in a box.

At first, Bob had thrice as many marbles as Aaron.

Aaron sold 5 marbles and Bob bought another 35 marbles.

Then, Bob had 5 times as many marbles as Aaron.

How many marbles were there in the box at first?

This is part of a series on Homeschool Math Challenging Puzzles, suitable for Grades 2-4. (Of course, students of other grades are also welcome to try them out.) The questions are suitable for:

• Preparation for GEP (Gifted Education Programme) screening and selection tests
• Puzzles for kids interested in math but find school work too easy.

Solution:

This question can be solved using the “units” method, and “working backwards”.

At the end, Bob had 5 times as many marbles as Aaron.

We write:

Aaron –> 1u

Bob –> 5u

Next, we work one step backwards (before Aaron sold 5 marbles and Bob bought another 35 marbles.)

Aaron –> 1u + 5

Bob —> 5u – 35

Now, we calculate what is 3 times of Aaron (thrice of Aaron’s marbles):

3 times of Aaron –> 3u + 15

We can conclude that:

3u+15 = 5u-35

We may draw the above model, after which we can conclude that:

2u –> 15+35=50

1u —> 25

Hence, Aaron had 1u+5 = 30 marbles at the start.

Bob had 5u-35 = 90 marbles at the start.

In total, there are 30+90= 120 marbles at first.

Ans: 120

## Homeschool Math Challenging Questions

Mathtuition88 will be starting a series of Homeschool Math Challenging Problems, aimed at age 8 to 10 (Grade 2 to 4).

This series is targeted at kids age 8 to 10 who are strong / gifted at mathematics and wish to further stretch their potential. It is also useful for children who may not be strong in math at the moment, but have a keen interest in math nonetheless.

In particular, it is very suitable for the following purposes:

• Preparation for GEP (Gifted Education Programme) screening and selection tests
• Puzzles for kids interested in math but find school work too easy.
• PSLE challenging Math problem sums. The questions are also well within the PSLE Math Syllabus, and will be challenging to Primary 6 students as well.

This series of questions will follow the Singapore Math syllabus for Grade 3 students, covering the following topics:

1. Whole Numbers
2. Fractions
3. Money
4. Measurement (Length, Mass, Volume)
5. Time
6. Area and Perimeter (rectangle/square)

Although the syllabus above is elementary, we are choosing the toughest math questions (while still remaining in the framework of the syllabus). Hence, the title of the series is “Homeschool Math Challenging Puzzles for Grades 2-4”!

As far as possible, the questions will be categorized under: https://mathtuition88.com/category/homeschool-math-challenging-puzzles/

## GEP Test Format

There is limited information on the GEP Test Format on the official MOE website:

### IDENTIFICATION OF PUPILS FOR THE GEP

The entry point into the GEP is at Primary 4.

Pupils are identified for the GEP through a two-stage exercise in Primary 3.

Stage Month Participants Papers
Screening August Primary 3 pupils enrolled in government and government-aided schools English Language

Mathematics

Selection October Only shortlisted pupils will be invited to the Selection stage English Language

Mathematics

General Ability

For the screening test, the duration for each paper is around 1.5 hours.

For the selection test, there are traditionally two papers of 2.5 hours each:

• Paper 1: English paper and a General Ability paper
• Paper 2: Maths paper and another General Ability paper

In August every year, the cohort of P3 students is invited to sit for a GEP screening test. The test is not compulsory but all children are encouraged to go for it. The test comprises an English and a Maths paper, about 1½ hours each.

Out of the cohort of approximately 50,000 kids, some 3,000 pupils (about 6%) are shortlisted for the GEP selection test.

The selection test takes place over two days in October, the first comprises an English paper and a General Ability paper, about 2½ hours in total. The second consists of a Maths paper and another General Ability paper, also about 2½ hours altogether.

## GEP Test Syllabus

Officially, the GEP Test can only test within the Primary 3 syllabus. (This is more relevant for Math than English. For English, it is more open-ended, they can test advanced GEP vocabulary like “cantankerous”.)

For example, technically they are not supposed to test “speed” questions since that is a Primary 5 topic. Similarly, they are not supposed to test area/perimeter of circle questions, or even area/perimeter of triangle questions.

Hence, the GEP Math Test Syllabus (according to the official Primary 3 syllabus) includes:

1. Whole Numbers
2. Fractions
3. Money
4. Measurement (Length, Mass, Volume)
5. Time
6. Area and Perimeter (rectangle/square)
7. Angles (basic concepts of right angle, acute angle)
8. Perpendicular & Parallel Lines (basic concepts)
9. Bar graphs

The above topics may seem deceptively easy. However, even for a simple topic like fractions, it is possible to test a question like the GEP Screening Test Question Sample: The Tap Question, which may stump many secondary school students.

Related posts:

## The Notorious Collatz conjecture

Terence Tao has uploaded his slides on the Collatz conjecture (targeted at high school students): https://terrytao.files.wordpress.com/2020/02/collatz.pdf

The best “encyclopedic” reference on the Collatz conjecture is the one listed below, published by the American Math Society. Note that the Collatz conjecture remains unsolved as of today.

## Sum of two metrics is a metric

It is quite straightforward to prove or verify that the sum of two metrics (distance functions) is still a metric.

Suppose $d_1(x,y)$ and $d_2(x,y)$ are metrics. Define $d(x,y)=d_1(x,y)+d_2(x,y)$.

## Positive-definite

$d(x,y)\geq 0+0=0$.

$d(x,y)=0 \iff d_1(x,y)=d_2(x,y)=0 \iff x=y$.

## Symmetry

$d(x,y)=d_1(y,x)+d_2(y,x)=d(y,x)$.

## Triangle Inequality

\begin{aligned} d(x,y) &\leq d_1(x,z)+d_1(z,y)+d_2(x,z)+d_2(z,y)\\ &=[d_1(x,z)+d_2(x,z)]+[d_1(z,y)+d_2(z,y)]\\ &=d(x,z)+d(z,y) \end{aligned}

Note that it follows by induction that the sum of any finite number of metrics (e.g. three, four or five metrics) is still a metric.

## IP Math Syllabus (Integrated Programme Mathematics)

Students and parents new to IP (Integrated Programme) may be confused on what is the Mathematics syllabus of IP Math. Indeed, it is very confusing as every school has its own syllabus. In general, the syllabus as a whole is not that different from ‘O’ level Mathematics, but the order in which the school teaches is unique to each school.

In general, the topics can be divided as follows, following the famous assessment book “Mathematics (Integrated Programme)” by Wong-Ng Siew Hiong who is a teacher at RI. This is one of the very few IP Math books available in local bookstores.

## Secondary 3 IP Math Syllabus

1. Geometrical Properties of Circles
3. Matrices & Simultaneous Equations
4. Quadratic Functions, Inequalities & Roots of Equations
5. Sets
6. Relations & Functions
7. Indices & Surds
8. Exponential, Logarithmic & Modulus Functions
9. Polynomials & Partial Fractions
10. Graphical Solutions & Transformations
11. Circular Measure
12. Plane Geometry
13. Coordinate Geometry & Equations of Circles
14. Linear Law
15. Trigonometry
16. Further Trigonometry

## Secondary 4 IP Math Syllabus

1. Binomial Theorem
2. Probability
3. Statistics
4. Vectors
5. Differentiation Techniques
6. Differentiation and its Applications
7. Integrated Techniques
8. Applications of Integration
9. Integration Applications — Area and Kinematics

Just received some news from Sellfy:

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We have important news about your Sellfy account and your store https://sellfy.com/mathtuition88

You have been on our transaction fee only plan (Legacy plan) since you signed up for Sellfy. We are truly thankful for being with us all this time!

Due to considerable technical adjustments in our platform and supporting services, we are forced to discontinue the Legacy plan on February 1st, 2020 and switch into supporting subscription-based plans only https://sellfy.com/pricing/. We no longer offer Legacy plan for new users since March, 2016.

Most likely, we won’t be continuing with Sellfy after February 1st, 2020.

Hence, this is the last 1-2 months to purchases math notes/material from our online store: https://sellfy.com/mathtuition88 if you are interested. Thank you for the support!

## Formula for 3×3 Matrix Inverse

Let $\displaystyle A=\begin{pmatrix}a &b &c\\ d& e& f\\ g& h& i \end{pmatrix}.$

Then,
$\displaystyle A^{-1}=\frac{1}{\det(A)}\begin{pmatrix} ei-fh &ch-bi &bf-ce\\ fg-di &ai-cg &cd-af\\ dh-eg &bg-ah &ae-bd \end{pmatrix}.$

This method is about the same speed as the Gaussian-Elimination method. However, if you have already calculated det(A), using the formula may be slightly faster.

The formula provides an insight on why A is singular (not invertible) if det(A)=0. (Because, if det(A)=0, in the formula we would be dividing by 0 which is not allowed.)

## Superposition Principle

If $\mathbf{u}$ is a solution to $\bf{A}\mathbf{x}=\mathbf{b}$, and $\mathbf{v}$ is a solution to $\bf{A}\mathbf{x}=\mathbf{c}$, then $\mathbf{u}+\mathbf{v}$ is a solution to $\bf{A}\mathbf{x}=\mathbf{b}+\mathbf{c}$.

Proof:
$\mathbf{A}(\mathbf{u}+\mathbf{v})=\mathbf{A}\mathbf{u}+\bf{A}\mathbf{v}=\mathbf{b}+\mathbf{c}$

## Properties of Matrix Transpose

1) $(A^T)^T=A$
2) $(A+B)^T=A^T+B^T$
3) $(kA)^T=kA^T$
4) $(AB)^T=B^T A^T$

## RIP Sir Michael Atiyah

Rest in peace, Sir Michael Atiyah. Many scientists have called Atiyah the best mathematician in Britain since Isaac Newton.

Source: New York Times

Michael Atiyah, a British mathematician who united mathematics and physics during the 1960s in a way not seen since the days of Isaac Newton, died on Friday. He was 89.

The Royal Society in London, of which he was president in the 1990s, confirmed the death but gave no details. Dr. Atiyah, who was retired, had been an honorary professor in the School of Mathematics at the University of Edinburgh.

Dr. Atiyah, who spent many years at Oxford and Cambridge universities, revealed an unforeseen connection between mathematics and physics through a theorem he proved in collaboration with Isadore Singer, one of the most important mathematicians of the last half of the 20th century.

His work with Dr. Singer, of the Massachusetts Institute of Technology, led to the flowering of string theory and gauge theory as ways to understand the structure and dynamics of the universe, and has provided powerful tools for both mathematicians and theoretical physicists.

## Chinese High School Student “Proves” Goldbach Conjecture

According to popular Chinese Social Media, a Chinese teenager from high school has purportedly proven the difficult Goldbach Conjecture, which is that every even integer greater than two is the sum of two prime numbers.

Update: He has recently realized his mistake and deleted the post. Do check out his proof if you are interested!

## Preschool Math Tuition Books and Toys

There seems to be an increasing demand for kindergarten Maths tuition for children of ages 2-6. (I have received many such requests lately.) Many parents are worried that their child may lose at the “starting line” which is Primary 1, hence are preparing beforehand during the ages of 2-6.

Possibly, at this age the best way to learn is through play, that is why Math Toys are very useful.

Skoolzy Rainbow Counting Bears Toy is suitable for age 2, 3, 4, 5 or 6 year old preschoolers to learn to add, count, sort & stack.

Goodnight, Numbers is an award-winning bestselling bedtime story for kids, that help them familiarize with numbers.

The Pattern Blocks and Boards toy is useful for developing pattern recognition and visualization skills; all very useful skills when it comes to the GEP Logic Section at age 9. PSLE has some very tough visualization questions on nets of cubes, only children with strong visualization skills will be able to solve them. It is notoriously hard to train for such questions (though there are some tips and techniques), it is almost like either you see it or you don’t.

Times Tables would be the most challenging Math for children of the age group 3-6. Hence, early mastery of this subject will give your child a headstart to learn other more advanced math.

This 4-in-1 book bundle enables your child to learn about the alphabet, shapes, colors, math, as well as animals.

This LEGO (Duplo) set teaches numbers, basic math skills, and includes fun activity cards. Very educational and fun toy.

## The Hardest H3 Math Question (Combinatorics)

I think this may be one of the hardest H3 Math Questions in history. It is taken from RI H3 Prelim 2018. It seems that even in top schools like RI, there are less than 50 people taking H3 Maths in any given year. Part (d) is extremely hard to get the formula for general r. In fact during the exam it is probably wise to skip such questions or give partial answers (e.g. the formula for r=3) as it is not worth the time for 3 marks.

## Varignon’s Theorem (Surprising Geometry Theorem)

I first learnt it from Quora:

“Take any quadrilateral. It doesn’t have to be any special kind of a quadrilateral. Then connect the midpoints of its sides.

Surprisingly, you will always get a parallelogram!”

Quite a nice result. Some googling revealed that the name of this theorem is called Varignon’s Theorem.

An illustration is found here:

## RI vs HCI vs NUSH

Many excellent P6 students will be spoilt for choice at this stage, as they have been accepted by multiple schools under the DSA. Here are some views on RI versus HCI versus NUS High, which are the top 3 choices for boys strong in math/science:

RI emphasize “all-rounder” and “leadership” in their culture. My personal experience is that it can get a bit competitive since many people there are literally good in all forms of studies (all subjects in humanities/sciences/languages maybe except Chinese), sports, and music, etc. The culture is similar in RGS. I find that there are some cultural and personality differences between the typical RI/RGS student and the typical HCI/NYGH student.

The good points are that RI does have a lot of activities and opportunities like overseas school trips, top coaches for most CCAs, etc, that many schools don’t have. Also, many RI students are successful in securing government scholarships for top universities since their portfolio will be built up in a balanced way during their studies; there are many Community Involvement Programmes and other activities to boost the student’s portfolio.

Certainly NUSH and HCI are very good schools too. Possibly advantages of NUSH is greater focus on science/math and advantage of HCI is greater emphasis on Chinese culture and tradition, which is useful as China is becoming a world power.

Maybe you can check out this thread on RI vs HCI: https://www.kiasuparents.com/kiasu/forum/viewtopic.php?f=48&t=25209&start=370.

## Sir Michael Atiyah to reveal his awesome proof of the Riemann Hypothesis

According to the official twitter account of the Heidelberg Laureate Forum, Sir Michael Atiyah will prove the Riemann Hypothesis during the talk on Monday Sept. 25.

According to him, it is a “simple” proof, based on previous work by von Neumann, Hirzebruch and Dirac. It is likely to have some relation to theoretical physics, since Dirac is a theoretical physicist. It has been rumored for quite some time that the Riemann Hypothesis is related to theoretical physics.

The Riemann Hypothesis is the most famous and most difficult problem in mathematics. Sir Michael Atiyah is a 89 year old gentleman currently. It will be really groundbreaking if the Riemann Hypothesis is found to be successfully proved.

Do check out our previous posts on the Riemann Hypothesis:

## Amazon Founder Jeff Bezos Originally Wanted to be a Theoretical Physicist

The world’s richest man is currently Jeff Bezos, founder of Amazon.

Few people know that he was an undergraduate at Princeton with the goal of becoming a theoretical physicist! What made him change his mind? Watch the video below.

Summary: Jeff Bezos was stuck on a Partial Differential Equation (PDE) question for 3 hours. Even while collaborating with his room mate, he could not find the answer. Upon consulting his Sri Lankan genius classmate, “Yosantha”, Yosantha solved the problem almost instantaneously in his mind!

Also check out our previous posts on Partial Differential Equations:

## Morse Inequalities

Let $X$ be a CW complex (with a fixed CW decomposition) with $c_d$ cells of dimension $d$. Let $\mathbb{F}$ be a field and let $b_d=\dim(H_d(X;\mathbb{F}))$.
(i) (The Weak Morse Inequalities) For each $d$,

$\displaystyle c_d\geq b_d.$
(ii)

$\chi(X)=b_0-b_1+b_2-\dots=c_0-c_1+c_2-\dots$,
where $\chi(X)$ denotes the Euler characteristic of $X$.

Proof:

The proof is by linear algebra (see Hatcher pg. 147).

By rank-nullity theorem (秩-零化度定理), $\dim C_d=\dim Z_d+\dim B_{d-1}$.

By definition of homology, $\dim Z_d=\dim B_d+\dim H_d$.

$\therefore c_d=\dim B_d+\dim B_{d-1}+b_d$.

In particular, $c_d\geq b_d$.

Taking alternating sum gives $\displaystyle \sum_d(-1)^d c_d=\sum_d(-1)^d b_d.$

Reference: A user’s guide to discrete Morse theory by R. Forman.

## Challenging P6 Math Question (Cycling)

One afternoon, 5 friends rented 3 bicycles from 5.00 p.m. to 6.30 p.m. and took turns to ride on them. At any time, 3 of them cycled while the other 2 friends rested.

If each of them had the same amount of cycling time, how many minutes did each person ride on a bicycle?

Hint: There is an “easy” way and also a “complicated” way to do this question. The “easy” way involves calculating total cycling time, while the “complicated” way involves working out a timetable to determine exactly who is cycling at which time.

(Source: Hardwarezone)

(Ans: 54)

## Basics of Partial Differential Equations Summary

PDE: Separation of Variables

1) Let $u(x,y)=X(x)Y(y)$.
2) Note that $u_x=X'Y$, $u_y=XY'$, $u_{xx}=X''Y$, $u_{yy}=XY''$, $u_{xy}=u_{yx}=X'Y'$.
3) Rearrange the equation such that LHS is a function of $x$ only, RHS is a function of $y$ only.
4) Thus, LHS=RHS=some constant $k$.
5) Solve the two separate ODEs.

Wave Equation
$\displaystyle c^2y_{xx}=y_{tt},$ where $y(t,0)=y(t,\pi)=0$, $y(0,x)=f(x)$, $y_t(0,x)=0$.

Solution of Wave Equation (with Fourier sine coefficients)
$\displaystyle y(t,x)=\sum_{n=1}^\infty b_n\sin(nx)\cos(nct)$ where $\displaystyle b_n=\frac{2}{\pi}\int_0^\pi f(x)\sin(nx)\,dx.$

d’Alembert’s solution of Wave Equation
$\displaystyle y(t,x)=\frac{1}{2}[f(x+ct)+f(x-ct)].$

Heat Equation
$\displaystyle u_t=c^2u_{xx},$
$u(0,t)=u(L,t)=0$, $u(x,0)=f(x)$.

Solution of Heat Equation
$\displaystyle u(x,t)=\sum_{n=1}^\infty b_n\sin\left(\frac{n\pi x}{L}\right)\exp\left(-\frac{\pi^2n^2c^2}{L^2}t\right),$ where $\displaystyle b_n=\frac{2}{\pi}\int_0^\pi f(x)\sin(nx)\,dx$ are Fourier sine coefficients of $f(x)$.

## Linear First Order ODE, Bernoulli Equations and Applications

Linear First Order ODE
DE of the form: $y'+P(x)y=Q(x)$.

Integrating factor: $R(x)=e^{\int P(x)\,dx}$.
\begin{aligned} R'&=RP\\ Ry'+RPy&=RQ\\ (Ry)'&=RQ\\ Ry&=\int RQ\,dx \end{aligned}
$\displaystyle \boxed{y=\frac{\int RQ\,dx}{R}}$
(Remember to have a constant $C$ when integrating the numerator $\int RQ\,dx$.)

Integration by parts
$\displaystyle \boxed{\int uv'\,dx=uv-\int u'v\,dx}$

Acronym: LIATE (Log, Inverse Trig., Algebraic, Trig., Exponential), where L is the best choice for $u$. (This is only a rough guideline.)

Bernoulli Equations
DE of the form: $y'+p(x)y=q(x)y^n$.

$y^{-n}y'+y^{1-n}p(x)=q(x)$

Set $\boxed{y^{1-n}=z}$.

Then $(1-n)y^{-n}y'=z'$. The given DE becomes
$\displaystyle \boxed{z'+(1-n)p(x)z=(1-n)q(x)}.$

Fundamental Theorem of Calculus (FTC)
Part 1: $\displaystyle \frac{d}{dx}\int_a^x f(t)\,dt=f(x)$

Part 2: $\displaystyle \int_a^b F'(t)\,dt=F(b)-F(a)$

Hyperbolic Functions
\begin{aligned} \sinh x&=\frac{e^x-e^{-x}}{2}\\ \cosh x&=\frac{e^x+e^{-x}}{2}\\ \cosh^2 x-\sinh^2 x&=1\\ \end{aligned}
\begin{aligned} \frac{d}{dx}\sinh x&=\cosh x\\ \frac{d}{dx}\cosh x&=\sinh x\\ \frac{d}{dx}\sinh^{-1}x&=\frac{1}{\sqrt{x^2+1}}\\ \frac{d}{dx}\cosh^{-1}x&=\frac{1}{\sqrt{x^2-1}} \end{aligned}
$\displaystyle \int \tanh(ax)\,dx=\frac{1}{a}\ln(\cosh(ax))+C.$

Uranium-Thorium Dating
Starting Equations:
$\displaystyle \begin{cases} \frac{dU}{dt}=-k_U U\implies U=U_0e^{-k_Ut}\\ \frac{dT}{dt}=k_UU-k_TT. \end{cases}$

$\frac{dT}{dt}+k_T T=k_U U_0e^{-k_Ut}$

$R=e^{\int k_T}=e^{k_T t}$.

$\displaystyle \boxed{T(t)=\frac{k_U}{k_T-k_U}U_0(e^{-k_Ut}-e^{-k_Tt})}$

$\displaystyle \boxed{\frac{T}{U}=\frac{k_U}{k_T-k_U}[1-e^{(k_U-k_T)t}]}$

## Happy Pi Day!

Happy Pi Day to all readers of Mathtuition88.com!

Check out our previous posts on Pi:

Did you know, Pi day is also Einstein’s Birthday?

How to get Pi on Calculator – Without pressing the Pi Button

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

Euler’s proof of Pi^2/6 (Basel Problem)

Category Theory: How to Make Pi

Pi hiding in prime regularities

Also, check out our cooking blog for the following Pie recipes!

Beef Pie

Strawberry Cheesecake Pie

Lemon Pie

Easy Banana Pie

Fudgy Fudge Pie

## Stephen Hawking dies aged 76

RIP Stephen Hawking.

The British physicist was known for his work with black holes and relativity, and wrote several popular science books including A Brief History of Time.

“We are deeply saddened that our beloved father passed away today,” a family statement said.

At the age of 22 Stephen Hawking was given only a few years to live after being diagnosed with a rare form of motor neurone disease.

## The magic (and math) of skating on thin ice without falling in

Very interesting video. Do try to imagine the sound of the ice and check out the video to confirm your guess. Very surprising sound! It is a pity that not many countries have such ice to skate in.

The main mathematical principle is Archimedes’ Principle:

Congelation ice, while a solid form of water, does bend slightly and acts like an elastic plate buoyed by the water below. To Anje, it’s Archimedes buoyancy principle in action.

“A body partially immersed in water is buoyed by a force equal to the weight of the water displaced by the body,” he said.

Stepping onto an inch-and-a-half thick piece of lake ice — much less doing laps on it — is a no-go for most people. But for experienced Swedish skaters Henrik Trygg and Mårten Anje, few things top skating on thin ice.

In December, still photographer Trygg filmed Anje skating on 1.8 inches of fresh ice on a lake outside Stockholm. The resulting mini-documentary— filled with the eerie, laser-like sounds of bending ice — went viral in February.

One shot shows the ice, commonly called “black ice,” visibly bending under the skater’s weight. Which raised the question: Why doesn’t this thin frozen surface break?

For the answer, we turned to Anje, a 35-year veteran of nordic skating whose day job is calculating risk. He is a mathematician and actuary at a consulting firm.

## No homework, full-day school curriculum to help level playing field (Proposal, not implemented yet)

Full-day school is quite a drastic measure to combat tuition. Also, unless full-day means 7am to 7pm, it is unlikely to be different from the status quo.

Any parent with children in secondary school or JC is aware that school is already pretty much “full-day” as of today, from 7am to 5pm at the minimum on most days (including CCA). Hence, there is not much room to get more “full-day” than now. JC students are known to stay much later for CCA, probably some are already having schedules from 7am to 7pm, which more than qualifies as “full-day”.

Also, even if full-day school (say 7am to 7pm) is implemented, there seems nothing to stop students from having tuition during the weekends, or on weekdays 8pm-10pm.

Probably most students would not be too pleased at having a full-day school. If I were still a student, I would definitely be more stressed out by the full-day school. I would much rather have some homework but end school early. I would imagine teachers won’t be too happy too, full-day school for students means full-day school for teachers, since obviously some if not all teachers must stay back to supervise the students.

Most unhappy would be tutors, for obvious reasons. Probably if this is implemented, most tutors will have to change jobs. 😛

The underlying idea to level the playing field is good and makes sense though. Possibly make the full-day optional so that those who want to stay back and have the full-day can do so, those who want to leave can also do so.

Source: TodayOnline

SINGAPORE — To level the playing field for children from less advantaged socio-economic backgrounds, and break out of the country’s tuition culture, Nominated Member of Parliament Chia Yong Yong has suggested that all schools adopt a full-day curriculum.

That way, the children will complete their homework during school hours, and be able to spend more time on “push-frontier practicals” aimed at training them to become more comfortable in tackling problems and to grow an appetite for risk-taking. These qualities are essential traits for the current technological revolution, also known as Industry 4.0, she said.

In her Budget debate speech in Parliament on Wednesday (Feb 28), Ms Chia said the current academic model “runs the risk of not harnessing the potential of all our young people” who do not have access to enrichment and tuition classes. As a result, those from more advantaged socio-economic backgrounds who have access to these classes will outperform their peers.

Stressing that “every school is a good school, but not every home is equal”, the lawyer said the current system has been “abused” such that inequality continues to be perpetuated and deepened.

## Higher paying job than Doctor / Lawyer

We encourage top students to look beyond the traditional Singaporean jobs of Doctor / Lawyer as there are new emerging jobs that can equal or even surpass the pay of Doctor/Lawyer.

At the end of the day, do also consider your passion and aptitude, which may be more important than the salary. No point being stuck in a high paying job that you absolutely hate.

Do share this post with your children/relatives/classmates who may be choosing their courses after receiving their ‘A’ level results.

Source: Todayonline

SINGAPORE — A high-paying job as a doctor or lawyer has traditionally been the career path that many Singaporeans aspire to. But there is now a new kid on the block, with double degree graduates in business and computer science joining the ranks of top earners here.

According to the latest graduate employment survey released by three local universities on Monday (Feb 26), fresh graduates from Nanyang Technological University’s (NTU) business and computing science double degree programme commanded a median starting salary of S$5,000 last year, up from S$4,600 in 2016.

The median salary for the batch of 20 graduates matched that of their peers who graduated from the law and medicine faculties. They were also in demand with employers, as they recorded a 100 per cent overall employment rate.

Meanwhile, fresh computing science graduates were also among the highest paid last year. Those who graduated from this course in NTU got a median starting pay of S$3,850 last year, up from S$3,500 in 2016. Their counterparts from the National University of Singapore (NUS) received S$4,285 – S$285 more than in 2016.

However, rankings differed for 75th percentile salaries — the base salary of the top 25 per cent of the batch — as SMU-schooled lawyers emerged as top earners at S$5,840, compared to NUS doctors’ starting pays of S$5,305, and S$5,362 for NTU’s business and computer science graduates. Growth of starting salaries in law and medicine was tepid, however, as law graduates from NUS and SMU only received about S$100 and S\$150 more respectively last year, while NUS doctors banked in about the same amount as their seniors.

## The results of the 2017 Singapore-Cambridge General Certificate of Education Advanced Level (GCE A-Level) examination will be released on Friday, 23 February 2018.

Good luck to all collecting their A Level results today!

Check out our post on BMAT Book Recommendations for NTU Medicine, and also Alternate Admission Route to NUS Computing.

1. The results of the 2017 Singapore-Cambridge General Certificate of Education Advanced Level (GCE A-Level) examination will be released on Friday, 23 February 2018. School candidates may collect their results from their respective schools from 2.30pm that day.

2. Private candidates will be notified of their results by post. The result slips will be mailed on 23 February 2018 to the postal address provided by the candidates during the registration period. Private candidates who have SingPass1accounts can also use their SingPass to obtain their results online via the internet Examination Results Release System (iERRS) on the Singapore Examinations and Assessment Board’s website (www.seab.gov.sg) from 2.30pm on 23 February 2018.

## Heroic math teacher saved her students from Florida shooting

Heroic math teacher saved her students from Florida shooting by covering classroom door’s window, ordering kids to the floor and refusing to let anyone in… even the SWAT team. Read more: http://www.dailymail.co.uk/news/article-5399143/Heroic-teacher-saved-students-Florida-shooting.html#ixzz57i0Hcvi9

# Heroic math teacher saved her students from Florida shooting by covering classroom door’s window, ordering kids to the floor and refusing to let anyone in… even the SWAT team

• Mrs Viswanathan realised something was wrong after two fire alarms sounded
• Instead of letting pupils out her math class, she told them to duck in the corner
• Mrs V refused to let SWAT teams in, so they had to enter through the window
• A mother of a pupil in the class said Mrs V’s actions helped to save student’s lives

## Linear System of Differential Equations, Solutions, Phase Portrait Sketching

Solutions of Homogeneous Linear System of DE
$\displaystyle \mathbf{y}'=\mathbf{A}\mathbf{y}$
$\displaystyle \mathbf{y}(t)=\mathbf{v}e^{rt}$
where $r$ and $\mathbf{v}$ are eigenvalue and eigenvector for $\mathbf{A}$ respectively.

Superposition Principle
If $\mathbf{x_1}(t)$ and $\mathbf{x_2}(t)$ are two solutions to a homogeneous SDE $\mathbf{y'}=\mathbf{Ay}$, then $\displaystyle \mathbf{y}=c_1\mathbf{x_1}(t)+c_2\mathbf{x_2}(t)$ is also a solution for any scalars $c_1$, $c_2$.

Euler’s formula
$\displaystyle e^{i\theta}=\cos\theta+i\sin\theta$

General Solutions (Complex Eigenvalues)

1) Let $r_1=a+bi$ be an eigenvalue corresponding to eigenvector $\mathbf{v_1}$. (The eigenvectors are complex conjugates: $\mathbf{v_1,v_2}=\mathbf{p}\pm \mathbf{q} i$.)
2) Construct
$\displaystyle \mathbf{x}_\text{Re}(t)=e^{at}(\mathbf{p}\cos bt-\mathbf{q}\sin bt)$
$\displaystyle \mathbf{x}_\text{Im}(t)=e^{at}(\mathbf{p}\sin bt+\mathbf{q}\cos bt)$
3) The general solution is $\displaystyle \mathbf{y}=c_1\mathbf{x}_\text{Re}(t)+c_2\mathbf{x}_\text{Im}(t).$

## How to Sketch Phase Portrait

Probably the best video on how to sketch Phase Portrait:

## Characteristic Polynomial, Eigenvalues, Eigenvectors

Characteristic Polynomial, $\det(\lambda I-A)$
\begin{aligned} \lambda\ \text{is an eigenvalue of }A&\iff\det(\lambda I-A)=0\\ &\iff \lambda\ \text{is a root of the characteristic polynomial}. \end{aligned}

Eigenspace
The solution space of $(\lambda I-A)\mathbf{x}=0$ is called the eigenspace of $A$ associated with the eigenvalue $\lambda$. The eigenspace is denoted by $E_\lambda$.

Sum/Product of Eigenvalues
– The sum of all eigenvalues of $A$ (including repeated eigenvalues) is the same as $Tr(A)$ (trace of $A$, i.e. the sum of diagonal elements of $A$)
– The product of all eigenvalues of $A$ (including repeated eigenvalues) is the same as $\det(A)$.

## Finding Least Squares Solution Review and Others

Rotation Matrix

The rotation matrix
$\displaystyle R=\begin{pmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{pmatrix}$
rotates points in the $xy$-plane counterclockwise through an angle $\theta$ about the origin.

For example rotating the vector $(1,0)$ 45 degrees counterclockwise gives us:
$\displaystyle \begin{pmatrix} \cos 45^\circ & -\sin 45^\circ\\ \sin 45^\circ & \cos 45^\circ \end{pmatrix} \begin{pmatrix} 1\\ 0 \end{pmatrix} = \begin{pmatrix} \frac{\sqrt{2}}{2}\\ \frac{\sqrt{2}}{2} \end{pmatrix}.$

Finding Least Squares Solution

Given $Ax=b$ (inconsistent system), solve
$\displaystyle A^TAx=A^Tb$ instead to get a least squares solution of the original equation.

Projection

If we know a least squares solution $\mathbf{u}$ of $A\mathbf{x}=\mathbf{b}$, we can find the projection $\mathbf{p}$ of $\mathbf{b}$ onto the column space of $A$ by $\displaystyle \mathbf{p}=A\mathbf{u}.$

Dimension Theorem for Matrices (Also known as Rank-Nullity Theorem)

If $A$ is a matrix with $n$ columns, then $\displaystyle rank(A)+nullity(A)=n.$

($rank(A)$=number of pivot columns,

$nullity(A)$=number of non-pivot columns.)

Linear Independence and the Wronskian
A set of vector functions $\vec{f_1}(x), \dots, \vec{f_n}(x)$ from $\mathbb{R}$ to $\mathbb{R}^n$ is linearly independent in the interval $(\alpha,\beta)$ if $\displaystyle W[\vec{f_1}(x),\dots,\vec{f_n}(x)]\neq 0$ for at least one value of $x$ in the interval $(\alpha,\beta)$.

# 5 Ways To Make Math More Fun And Meaningful For Kids

Fun and meaningful – these are two words that children rarely use to describe math. There are several reasons why many kids dislike math, but according to kids and learning experts, the top reasons always include:

• They always have to memorize mathematical formulas and concepts
• They often have to make numerous complex and lengthy calculations (such as finding the surface area of cuboid or cylinder)
• They always feel pressure to get perfect quiz or test scores
• They have a hard time finding practical applications for the advanced mathematical formulas and concepts they’re learning

Because of these reasons (and more), parents always struggle to get kids to like math and excel in this subject.

## How to Help Kids Change Their Attitude towards Math

According to a study published on the website of Stanford University’s Graduate School of Education, kids who outgrow their dislike and fear of of this subject will find it easier to do better on this subject.

If you are a parent or teacher, you can help children change their attitude towards math by making it more fun and meaningful for them. You can do this through the following ways:

### 1.    Enable your kids to realize the importance of math

When children understand that math is not all about theories and principles, they will start viewing the study of math as a valuable learning opportunity and thus become more interested in it. As such, you need to constantly show them how useful math is in real life.

For instance:

• Train younger kids to sort coins and bills and how to use them when buying individual or small amounts of items
• Allow your older children to help find the best prices for the items on your shopping list. Ask them to tally simple sums while grocery shopping.

Other activities that will help kids understand the relevance of math in real life include using measurements and basic operations when cooking and baking, telling time, checking temperatures, etc.

Although these activities seem simple, they are still effective ways of teaching kids the importance of knowing the right concepts and applications of certain mathematical operations.

### 2.    Take math outdoors

If you’re an educator, when you take math learning outside the classroom, you provide kids excellent ways of realizing that math can be found and used everywhere. This will also allow you to transfer lessons outside the classroom, and vice versa.

Below are examples of fun activities that will enable you to take math outdoors:

• Treasure or scavenger hunt
• Multiplication hopscotch
• Leaf logic
• Counting maze (for preschoolers)

### 3.    Enroll your kids in an after-school tutoring program

Sometimes, children need outside help to discover that math is interesting and meaningful. If you and your kids decide to get help from a tutor, find a tutoring center that specializes in teaching kids math.

The right math tutoring center will follow a unitary method that will help their students make sense of all the theories and concepts they are learning. They will assess the needs of the students and design a personalized learning program that will address their specific requirements.

Most tutoring centers today do not simply provide additional explanations and activities for kids to learn a particular concept. Tutors tailor their teaching techniques to ensure the students learn by heart and apply their knowledge.

As such, they also employ fun and creative methods to teach their students. They also check progress along the way to make sure kids truly understand, apply, and retain the concepts they learned.

### 4.    Incorporate math in games

Bring out your board games, a pack of cards, a puzzle, or even or old blocks and turn the game into a family competition. Activities and games that incorporate or focus on math are great in reinforcing the right mathematical skills and concepts.

Regardless of the activity, you can reward even small accomplishments and help your kids know that they just completed a fun math-related task. Children will love the recognition and prize, especially if they can compete with their siblings. They will also realize that knowing mathematical operations can be fun and applying them can be rewarding.

### 5.    Be supportive

Lastly, although you may want to empathize with your kids, saying things like “I was also never good at math” won’t do anything good for them. It is best to encourage your children to embrace challenges and see the fun in learning even if they are having a hard time with some mathematical concepts.

Be as involved as you can be in your children’s schoolwork and show enthusiasm. When you help your kids learn to associate math with fun, pleasure, parental love and attention, they will be excited about the subject throughout their learning years.

As a parent or educator, your support and willingness to think outside the box will go a long way in helping your kids think differently about math and eventually excel in the subject.

AUTHOR BIO

Maloy Burman is the Chief Executive Officer and Managing Director of Premier Genie FZ LLC. He is responsible for driving Premier Genie into a leadership position in STEM (Science, Technology, Engineering and Mathematics) Education space in Asia, Middle East and Africa and building a solid brand value. Premier Genie is currently running 5 centers in Dubai and 5 centers in India with a goal to multiply that over the next 5 years.

## Dot Product and Span Summary

Dot Product
$\mathbf{u}\cdot\mathbf{v}=\|u\|\|v\|\cos\theta$
$\cos\theta=\frac{\mathbf{u}\cdot\mathbf{v}}{\|u\|\|v\|}$

Span
$\text{span}\{\mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k}\}=\{c_1\mathbf{u_1}+c_2\mathbf{u_2}+\dots+c_k\mathbf{u_k}\mid c_1,c_2,\dots,c_k\in\mathbb{R}\}=\text{set of all linear combinations of }$ $\{\mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k}\}.$

Subspaces
$V\subseteq\mathbb{R}^n$ is a subspace of $\mathbb{R}^n$ if
1) $V=\text{span}\{\mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k}\}$ for some vectors $\mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k}$.
2) $V$ satisfies the closure properties:

(i) for all $\mathbf{u},\mathbf{v}\in V$, we must have $\mathbf{u}+\mathbf{v}\in V$.

(ii) for all $\mathbf{u}\in V$ and $c\in\mathbb{R}$, we must have $c\mathbf{u}\in V$.

3) $V$ is the solution set of a homogeneous system.

(Sufficient to check either one of Condition 1, 2, 3.)

Remark:
For $V$ to be a subspace, zero vector $\mathbf{0}$ must be in $V$. (Since for $\mathbf{u}\in V$, $0\in\mathbb{R}$, we have $0\mathbf{u}\in V$.)

Linear Independence and Dependence
$\mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k}$ are linearly independent if the system $\displaystyle c_1\mathbf{u_1}+c_2\mathbf{u_2}+\dots+c_k\mathbf{u_k}=0$ has only the trivial solution, i.e. $c_1=c_2=\dots=c_k=0$.

If the system has non-trivial solutions, i.e. at least one $c_i$ not zero, then $\mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k}$ are linearly dependent.

## Gaussian Elimination Summary

Row echelon form (REF)
For each non-zero row, the leading entry is to the right of the leading entry of the row above.

E.g. $\begin{pmatrix} 0 & \mathbf{1} & 7 & 2\\ 0 & 0 & \mathbf{9} & 3\\ 0 & 0 & 0 & 0 \end{pmatrix}$

Note that the leading entry 9 of the second row is to the right of the leading entry 1 of the first row.

Reduced row echelon form (RREF)
A row echelon form is said to be reduced, if in each of its pivot columns, the leading entry is 1 and all other entries are 0.

E.g. $\begin{pmatrix} 1 & 0 & 0 & 2\\ 0 & 1 & 0 & 3\\ 0 & 0 & 1 & 4 \end{pmatrix}$

Elementary Row Operations
1) $cR_i$ — multiply the $i$th row by the constant $c$
2) $R_i \leftrightarrow R_j$ — swap the $i$th and the $j$th row
3) $R_i+cR_j$ — add $c$ times of the $j$th row to the $i$th row.

Gaussian Elimination Summary
Gaussian Elimination is essentially using the elementary row operations (in any order) to make the matrix to row echelon form.

Gauss-Jordan Elimination
After reaching row echelon form, continue to use elementary row operations to make the matrix to reduced row echelon form.