The Legendre Symbol

Math Online Tom Circle

Prove

$latex x^{2} \equiv 3411 \mod 3457 $
has no solution?

Legendre Symbol:

$latex \displaystyle
x^{2} \equiv a \mod p
\iff
\boxed{
\left( \frac {a}{p} \right)
= \begin{cases}
-1, & \text{if 0 solution} \\
0 , & \text{if 1 solution} \\
1, & \text{if 2 solutions} \\
\end{cases}
}
$

Hint: prove $latex \left( \frac{3411}{3457} \right) = -1$

Using the Law of Quadratic Reciprocity, without computations, we can prove there is no solution for this equation.

Solution:

1.
3411 = 3 x 3 x 379 = 9 x 379

$Latex \displaystyle
\boxed{
\left(\frac{a}{p}\right)
\left(\frac{b}{p} \right)=
\left(\frac{ab}{p}\right)
}
$

$latex \displaystyle
\left(\frac{3411}{3457} \right)=
\left(\frac{9}{3457} \right).\left(\frac{379}{3457} \right)=
\left(\frac{379}{3457} \right)
$
since
$latex \displaystyle\left(\frac{9}{3457} \right)=1 $
because 9 is a perfect square, 3457 is prime.

2. By Quadratic Reciprocity,
$latex \displaystyle
\boxed{
\text{If p or q or both are } \equiv 1 \mod 4 \implies
\left(\frac{p}{q} \right)=
\left(\frac{q}{p} \right)}
$

Since
$latex…

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Математика Групповые занятия класса, чтобы начать в следующем году, 2014 году.

Математика Групповые занятия класса, чтобы начать в следующем году, 2014 году.

Математика Обучение центр

The Singapore Math

Math Online Tom Circle

The famous Singapore Math for children in primary schools is based on  visual models.

The Singapore Ministry of Education has published a new 2013 Math syllabus for primary and secondary schools, which will roll out in examinations within 4 to 6 years. Todate only Primary 1 and Secondary 1 Math syllabuses are published here:

http://www.moe.gov.sg/education/syllabuses/sciences

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Algebra vs Singapore Math

Math Online Tom Circle

Who wins?

This comic video illustrates Singapore Math’s Arithmetics Polya-style problem solving process vs Algebra’s mechanical method.

The problem is as follow:
R is 3 times older than S two years ago. From now 2 years later, their total age is 32. How old is R now ?

See my previous blog (search “Monkey”) the Nobel Physicist Paul Dirac’s problem “The Monkeys and Coconuts“, 3 methods are used: 2 adanced modern math (by Sequence, eigenvector & eigenvalue), and the easiest & intuitive method (by Singapore Modelling Math). High-school Algebra method is impossible, if not cumbersome, to solve the Monkey problem !

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Khan Academy

Math Online Tom Circle

I find Khan Linear Algebra video excellent. The founder / teacher Sal Khan has the genius to explain this not-so-easy topic in modular videos steps by steps, from 2-dimensional vectors to 3-dimensional, working with you by hand to compute eigenvalues and eigenvectors, and show you what they mean in graphic views.

If you are taking Linear Algebra course in university, or revising it, just go through all the Khan’s short (5-20 mins) videos on Linear Algebra here:

In 138 lessons sequence:

http://theopenacademy.com/content/linear-algebra-khan-academy

or random revision:

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Relationship-Mapping-Inverse (RMI)

Math Online Tom Circle

Relationship-Mapping-Inverse (RMI)
(invented by Prof Xu Lizhi 徐利治 中国数学家 http://baike.baidu.com/view/6383.htm)

Find Z = a*b

By RMI Technique:
Let f Homomorphism: f(a*b) = f(a)+f(b)

Let f = log
log: R+ –> R
=> log (a*b) = log a + log b

1. Calculate log a (=X), log b (=Y)
2. X+Y = log (a*b)
3. Find Inverse log (a*b)
4. ANSWER: Z = a*b

Prove:

$latex \sqrt{2}^{\sqrt{2}^{\sqrt{2}}}= 2$

1. Take f = log for Mapping:
$latex \log\sqrt{2}^{\sqrt{2}^{\sqrt{2}}} $
$latex = \sqrt{2}\log\sqrt{2}^{\sqrt{2}}$
$latex = \sqrt{2}\sqrt{2}\log\sqrt{2} $
$latex = 2\log\sqrt{2} $
$latex = \log (\sqrt{2})^2 $
$latex = \log 2$

2. Inverse of log (bijective):
$latex \log \sqrt{2}^{\sqrt{2}^{\sqrt{2}}}= \log 2$
$latex \sqrt{2}^{\sqrt{2}^{\sqrt{2}}}= 2$

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Prof Su Buqing Problem

Math Online Tom Circle

Prof Su 苏步青, the founding pioneer Math professor of the China’s top universities (Zhejiang 浙江大学 and Fudan 复旦大学), was one of the few mathematicians who had longevity above 100 years old (the other was French Mathematician Hadammard).

http://en.m.wikipedia.org/wiki/Su_Buqing

Two men A and B are 100 km apart, walking towards each other, A at speed 6 km/hour and B at 4 km/hour.
A brings a dog which runs at 10 km/hour between them,  starting from A towards B, upon reaching B it runs back to reach A, then back to B again, and so on…

Find total distance the dog has covered when A and B finally meet ?

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The Riemann hypothesis in various settings

What's new

[Note: the content of this post is standard number theoretic material that can be found in many textbooks (I am relying principally here on Iwaniec and Kowalski); I am not claiming any new progress on any version of the Riemann hypothesis here, but am simply arranging existing facts together.]

The Riemann hypothesis is arguably the most important and famous unsolved problem in number theory. It is usually phrased in terms of the Riemann zeta function $latex {\zeta}&fg=000000$, defined by

$latex \displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}&fg=000000$

for $latex {\hbox{Re}(s)>1}&fg=000000$ and extended meromorphically to other values of $latex {s}&fg=000000$, and asserts that the only zeroes of $latex {\zeta}&fg=000000$ in the critical strip $latex {\{ s: 0 \leq \hbox{Re}(s) \leq 1 \}}&fg=000000$ lie on the critical line $latex {\{ s: \hbox{Re}(s)=\frac{1}{2} \}}&fg=000000$.

One of the main reasons that the Riemann hypothesis is so important to number theory is that the zeroes of…

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Circle Theorems

Mathematics, Learning and Technology

A collection of excellent free resources for demonstrating the various circle theorems:
Tim Devereux has created GeoGebra applets which allow exploration of the circle theorems. You can access each theorem from the menu on the left which includes a useful summary of all the theorems.


See also these excellent demonstrations or try the following from Suffolk Maths.

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Top >10 Mathematics Websites

Mathematics, Learning and Technology

Top >10 Mathematics Websites remains a very popular post on this blog.

 

I have read various ‘Top (insert number here) Mathematics Websites’ posts and all of them have left me with the thought that so many excellent sites are missing from such lists. Any post claiming top 10 or >10 in my case is clearly the author’s top 10, notthe top 10! These are my top >10 because I really do use them – a lot – in the classroom! For my own list, I have decided to include some categories as well as individual sites which gives me the excuse to mention far more than 10! Note that every site mentioned here is free to use.

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MOOCs and TOOCs and the role of problem solving in maths education

njwildberger: tangential thoughts

A quick quiz: which of the following four words doesn’t fit with the others??

Massive/Open/Online/Courses

We are going to muse about MOOCs today, a hot and highly debated topic in higher education circles. Are these ambitious new approaches to delivering free high quality education through online videos and interactive participation over the web going to put traditional universities out of business, or are they just one in a long historical line of hyped technologies that get everyone excited, and then fail to deliver the goods? (Think of the radio, TV, correspondence courses, movies, the tape recorder, the computer; all of which held out some promise for getting us to learn more and learn better, mostly to little avail, although the jury is still out on the computer.)

It’s fun to speculate on future trends, because of the potential—indeed likelihood—0f embarrassment for false predictions. Here is the summary of my argument today:…

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NCSM-Jo Boaler-Promoting Equity Through Teaching For A Growth Mindset

Excellent article on learning maths based on a growth mindset.

Math Minds

1As you can see from the picture, it was a packed house! After waiting in line for fifteen minutes, I was so lucky (and excited) to get a seat to hear Jo Boaler speak, even if my seat was in the next to last row.

Jo opened the presentation with Dweck’s research on mindsets. “In the fixed mindset, people believe that their talents and abilities are fixed traits. They have a certain amount and that’s that; nothing can be done to change it. In the growth mindset, people believe that their talents and abilities can be developed through passion, education, and persistence.”

Jo states that the fixed mindset contributes to one of the biggest myths in mathematics: being good at math is a gift. She referenced her book, The Elephant in the Classroom (added it to my reading list) and showed the audience various television/movie clips that continue to perpetuate…

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Checking Multiplication via Digit Sums

A Narrow Margin

Last week a friend who is a fourth grade teacher came to me with a math problem.  The father of one of his students had showed him a trick for checking the result of a three-digit multiplication problem.  The father had learned the trick as a student himself, but he didn’t know why it worked.  My friend showed me the trick and asked if I had seen it before.  This post describes this check and explains why it works.

Suppose you want to multiply 231 $latex \times $ 243.  Working it out by hand, you get 56133.  Add the digits in the answer (5+6+1+3+3) to get 18.  Add the digits again to get 9.  Stop now that you have a single digit.

Alternatively, do this digit adding beforehand.  Adding the digits of 231 together, we get 6.  Adding the digits of 243 together, we get 9.  Multiply 6 $latex \times$ 9…

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How should logarithms be taught?

Gowers's Weblog

Having a blog gives me a chance to defend myself against a number of people who took issue with a passage in Mathematics, A Very Short Introduction, where I made the tentative suggestion that an abstract approach to mathematics could sometimes be better, pedagogically speaking, than a concrete one — even at school level. This was part of a general discussion about why many people come to hate mathematics.

The example I chose was logarithms and exponentials. The traditional method of teaching them, I would suggest, is to explain what they mean and then derive their properties from this basic meaning. So, for example, to justify the rule that xa+b=xaxb one would say something like that if you have a xs followed by b xs and you multiply them all together then you are multiplying a+b xs all together.

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Use of mathematics II

Gowers's Weblog

Today I had an experience that I have had many times before, and so, I imagine, has almost everybody (at least if they are old enough to be the kind of person who might conceivably read this blog post). I was in a queue in a chemist (=pharmacy=drugstore), and I knew that my particular item would be quick and easy to deal with. But I had to wait a while because in front of me was someone who had an item that was much more complicated and time-consuming. In this instance the complexity of the items was not due to their sizes, but a more common occurrence of the phenomenon is something that often happens to me in a local grocery: I want to buy just a pint of milk, say, and I find myself behind somebody who has a big basket of things, several of which have to be…

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On multiple choice questions in mathematics

What's new

Now that the project to upgrade my old multiple choice applet to a more modern and collaborative format is underway (see this server-side demo and this javascript/wiki demo, as well as the discussion here), I thought it would be a good time to collect my own personal opinions and thoughts regarding how multiple choice quizzes are currently used in teaching mathematics, and on the potential ways they could be used in the future.  The short version of my opinions is that multiple choice quizzes have significant limitations when used in the traditional classroom setting, but have a lot of interesting and underexplored potential when used as a self-assessment tool.

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Why aren’t all functions well-defined?

Gowers's Weblog

I’m in the happy state of just having finished marking exams for this year. There is very little of interest to say about the week that was removed from my life: it would be fun to talk about particularly bizarre mistakes, but I can’t really do that, especially as the results are not yet known (or even fully decided). However, one general theme emerged that made no difference to anybody’s marks. There seems to be a common misconception amongst many Cambridge undergraduates that I’d like to discuss here in the hope that I can clear things up for a few people. (It is an issue that I have discussed already on my web page, but rather than turning that into a blog post I’m starting again.)

The question where the misconception made itself felt was one about functions, injections, surjections, etc. I noticed that a lot of people wrote things…

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Article in the New York Times, and maths education

What's new

I’ve received quite a lot of inquiries regarding a recent article in the New York Times, so I am borrowing some space on this blog to respond to some of the more common of these, and also to initiate a discussion on maths education, which was briefly touched upon in the article.

Firstly, some links:

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Displaying maths online, II

What's new

As the previous discussion on displaying mathematics on the web has become quite lengthy, I am opening a fresh post to continue the topic.  I’m leaving the previous thread open for those who wish to respond directly to some specific comments in that thread, but otherwise it would be preferable to start afresh on this thread to make it easier to follow the discussion.

It’s not easy to summarise the discussion so far, but the comments have identified several existing formats for displaying (and marking up) mathematics on the web (mathMLjsMath, MathJaxOpenMath), as well as a surprisingly large number of tools for converting mathematics into web friendly formats (e.g.  LaTeX2HTMLLaTeXMathML, LaTeX2WPWindows 7 Math Inputitex2MMLRitexGellmumathTeXWP-LaTeXTeX4htblahtexplastexTtHWebEQtechexplorer

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Gamifying algebra?

What's new

High school algebra marks a key transition point in one’s early mathematical education, and is a common point at which students feel that mathematics becomes really difficult. One of the reasons for this is that the problem solving process for a high school algebra question is significantly more free-form than the mechanical algorithms one is taught for elementary arithmetic, and a certain amount of planning and strategy now comes into play. For instance, if one wants to, say, write $latex {\frac{1,572,342}{4,124}}&fg=000000$ as a mixed fraction, there is a clear (albeit lengthy) algorithm to do this: one simply sets up the long division problem, extracts the quotient and remainder, and organises these numbers into the desired mixed fraction. After a suitable amount of drill, this is a task that can be accomplished by a large fraction of students at the middle school level. But if, for instance, one has to solve…

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How should mathematics be taught to non-mathematicians?

Gowers's Weblog

Michael Gove, the UK’s Secretary of State for Education, has expressed a wish to see almost all school pupils studying mathematics in one form or another up to the age of 18. An obvious question follows. At the moment, there are large numbers of people who give up mathematics after GCSE (the exam that is usually taken at the age of 16) with great relief and go through the rest of their lives saying, without any obvious regret, how bad they were at it. What should such people study if mathematics becomes virtually compulsory for two more years?

A couple of years ago there was an attempt to create a new mathematics A-level called Use of Mathematics. I criticized it heavily in a blog post, and stand by those criticisms, though interestingly it isn’t so much the syllabus that bothers me as the awful exam questions. One might…

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Quiz

Math Online Tom Circle

image

In the diagram, the circumference of the external large circle is
1) longer, or
2) shorter, or
3) equal to,
the sum of the circumferences of all inner circles centered on the common diameter, tangent to each other.

Answer: 3) equal

circumference = π. diameter

Let d be the diameter of the external large circle C
Let dj be the diameter of the inner circle Cj

$latex \displaystyle d = \sum_{j} d_j$
$latex \displaystyle \pi. d = \pi. \sum_{j} d_j= \sum_{j}\pi.d_j$

Circumference of the external circle
= sum of circumferences of all inner circles

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What maths A-level doesn’t necessarily give you

Gowers's Weblog

I had a mathematical conversation yesterday with a 17-year-old boy who is in his second year of doing maths A-level. Although a sample of size 1 should be treated with caution, I’m pretty sure that the boy in question, who is very intelligent and is expected to get at least an A grade, has been taught as well as the vast majority of A-level mathematicians. If this is right, then what I discovered from talking to him was quite worrying.

The purpose of the conversation was to help him catch up with some work that he had missed through illness. The particular topics he wanted me to cover were integrating $latex \log x$, or $latex \ln x$ as he called it, and integration by parts. (Actually, after I had explained integration by parts to him, he told me that that hadn’t been what he had meant, but I don’t think…

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Stratified sampling vs. quota sampling

SOC 382: the blog

Shawn asked a good question in class yesterday about the differences between stratified sampling and quota sampling. In terms of sampling mechanism (i.e. the actual process by which cases are chosen from the population), it is clear that these two samples are different. Unclear, however, is why they would lead to different results.

Recall that stratified sampling is conducted by dividing a population into two or more strata by virtue of some characteristic, and taking random samples from each strata. This is done when a simple random sample of an entire population will likely not generate enough analyzable cases for a given group of particular interest.

Let’s say we want to study the income differences between blacks and whites in the United States. Unfortunately, we only have enough funding to distribute 500 questionnaires. Given that 10% of the population is black (made up, but reasonably approximate), a simple random…

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Landau’s Beautiful Proofs

Math Online Tom Circle

Landau’s beautiful proofs:
1= cos 0 = cos (x-x)

Opening cos (x-x):
1 = cos x.cos (-x) – sin x.sin (-x)
=> 1= cos² x + sin² x
[QED]

Let cos x= b/c, sin x = a/c
1= (b/c)² + (a/c)²
c² = b² + a²
=> Pythagoras Theorem
[QED]

Landau (1877-1938) was the successor of Minkowski at the Gottingen University (Math) before WW II.

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Open Cubic Root

Math Online Tom Circle

Mental Trick

It was discovered by the Martial Art writer Liang Yusheng 武侠小说家 梁羽生 (《白发魔女传》作者), who met Hua Luogeng (华罗庚) @1979 in England:
2³= [8]
8³= 51[2]
3³= 2[7]
7³= 34[3]

The last digit pairs :
[2 <->8] , [3 <-> 7]
Others unchanged.

Example:

$latex \sqrt[3]{658503} = N$
Last three digits 503 <-> …[7]
First three digits 658:
 (8³ =512)< 658 < (729 = 9³)
=>  8
Answer : $latex \sqrt[3]{658503} = N$= 87
Note: Similar trick for opening $latex \sqrt[23] {200 digits}$ by an indian lady Ms Shakuntala (83) dubbed “Human computer”.

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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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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 –…

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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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Tip: Using Google “filetype” filter to search for Free Exam Papers

This is a handy tip to search for Free Exam Papers on Google.

(Note: Google is very powerful, but it can only search for exam papers that are already online in the first place)

Since most exam papers are in PDF format, we can restrict our search to PDF files by adding the phrase “filetype:pdf” to the Google search.

For example, searching “hwa chong maths sec 2” in Google does not yield many exam papers.

Searching “hwa chong maths sec 2 filetype:pdf” returns a much better result, including some worksheets and test papers.

Screenshot:

google-hwa-chong-maths-tuition

Sec 3 Hwa Chong Institution Maths Test Papers and Resources

Site: http://anglc.wiki.hci.edu.sg/space/content

Description: Includes Sec 3 Hwa Chong Institution Maths Test Papers and Resources

 

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