Monday, April 25, 2016

Visualizing Fractions

Article status:                              Draft
Time Estimate for Reading:        15 min
Learning Objectives:                   Fractions, Introductory Mathematics
Effort Required:                          low
Pedagogy Model:                        Collaborative Learning
Prior Math Tools:                        Counting numbers and basic arithmetic
Authors:                                       Rithanya, Srishti and Keerthana


Background for this topic (you may skip this section on move on the question section)
We wanted to start a group containing members from 5th standard to masters graduates and working professionals. Also a few business people. The group size is limited to 25 to begin with.

The idea is to see how different members understand and respond to a questions. It would depend on existing knowledge, own strengths, method of visualization etc

This simple topic happened to become a great experience.

The question:
How do we interpret or visualize a fraction. for example 3/5. I had problem with fractions for long..

Simple answer and visualization:
- rewrite the same as 3 * 1/5.
- reread the fraction as "divide 1 piece into 5 pieces and take 3 of them out of it".

Perspective of an artist:

Take one piece of dough, divide it into 5 balls (balls are not shown of equal size but the idea is captured) and roll 3 of them into a chappathi or poori.



Perspective of a doodler and a youngster

An youngster just getting introduced to fractions.

Got it right in dividing one piece into 5. And drawn with love to their parents. superb.

Though this representation is not a correct mathematical answer, it laid foundation for new learning.


Seeing 1/5 as dividing a circle into 5 equal parts. She just learned about degrees and circles. only children can break the monotony. She took a part of the question.. Just the 1/5 part.




Fraction Multiplication:
Actually, application of multiplication of fractions was not in the scope of this question. But once we got the cake drawing from rithu, we have got a better understanding of fractions.


Another example:
Take 3 apples, cut it into 5 equal parts and share it with 3 people.

There is nothing wrong with this answer. Mathematically correct. But, a bit difficult to visualize.

Though the topic looks trivial, visualizing fraction was a major problem in my childhood.

The world of fractions becomes interesting with irrationals (square root of 2, 3 etc) and transcendental's (pi and e). Let us get to know them in the following articles.










Idea to Commercialization 2016 - Scheme 4

Idea to Commercialization

A unique opportunity for final year students of Coimbatore district

Selected individuals and teams shall receive support for mentoring and funding through our venture building partners.
If you think your project work had one of these elements
·  Innovative (technical or business)
·  Cost effective
·  Alternate process
·  Had a research element

Who can apply?
Education:           Only current Final year students.
Stream:                Agriculture, Arts, Commerce, Diploma, Engineering, Law, Medicine, Science
Region:               Candidates representing institutions from Coimbatore District
·                                                                  Masters and PhD Candidates can also apply

Last date for registration: 4 May 2016
Shortlisted applicants will be invited for a presentation within 31st May 2016.


Make It Happen - Scheme 3

An initiative to augment the team for our science and research institute

Still open: last date for registration 30 April 2016

Water From Air - Scheme 2

An initiative launched in April 2016 to identify teams along with mentoring and funding support.

This is still open: Last date for registration is 01 May 2016.



Play with gravitational constant - scheme 1

This was a first of our outreach initiatives announced in September 2015, to identify talent and augment our team.

As a bootstrap and in early days, we got a very good reach but only one response (may be we did not ask the right question). And interestingly, the person who responded is part of our team now.

However, it ended up as an article in this blog - Field of Gravitation




Sunday, April 17, 2016

Euclid makes another point. Symbolification.

Article status:                              Draft
Time Estimate for Reading:        15 min
Learning Objectives:                   Euclid's Division Lemma, Class X, NCERT, Mathematics
Effort Required:                          low
Pedagogy Model:                        Symbolification of mathematics
Prior Math Tools:                       Secondary school level Arithmetic (long division)


It happened to me. i always thought or was told, mathematics deals with numbers. Open any mathematics book, there are less and less of numbers and more and more of symbols.  brain sees it, but mind does not accept it. a mental block. easiest way is to runaway from mathematics.

I kept running away from mathematics until Isaac Asimov (The greatest science fiction author) and Prof Shankar Ramamurthy (Yale) stopped me. Isaac Asimov introduced me to the concept of formula-analysis (we have used this in our articles on science and discovery) and Prof Shankar Ramamurthy introduced me to the power of symbolification. The 'ification' may be new to you. we like it that way. just that it sounds and rhymes good. There are more of this in our Tamil language.

What better way to start than asking, "why x is the unknown?".

Invest a little of your time. 3 min and 57 sec.  watch this TED video by Terry Moore.


Having got to terms with x, let us move ahead and try to deal with more symbols. Let us begin with Euclid again. The Euclid's Division Lemma.

Euclid's division Lemma.
Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that
a = bq + r
and
0 ≤ r < |b|,
where |b| denotes the absolute value of b

Feel like running away (i used to. but not now). please resist your temptation. it is not that difficult. At a first sight, it may look incomprehensible, just give it a try. There is nothing wrong in giving it a try.

All that Euclid has done is symbolification of division.

take a few examples of division.
5/5; the quotient (q) is 1, remainder (r) is 0,5 the numerator (n) and 5 the denominator (d).
11/5; the quotient is 2, remainder is 1.
4/5;   the quotient is 0 and remainder is 4
10/5; the quotient (q) is 2, remainder (r) is 0,10 the numerator (n) and 5 the denominator (d).

We chose these examples such that we had the 3 scenarios. equal to, less than, greater than. we will be using this in most of physics.
n = d     | r =0
n > d     | r ≠ 0
n < d     | r ≠ 0

Observations:
- we find that remainder can never be greater than denominator.
- take the example 11/5.  11 (n) =  5(d) * 2 (q) + 1(r)

Now let us retain these symbols and take away the numbers

therefore, n = d*q + r; where 0 <= r <= |d|

That's Euclid's division Lemma, a = bq+r.   just replace a with n and b  with d.

The question is why do we take so much pain? why cant we leave it as such?

There are quite some advantages in symbolification.
1. It is like a rule that is followed by all integers.
2.  The Euclidean division lemma is extended into an algorithm to find out the Greatest Common Denominator. It is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 147 = 252 − 105.

The range of applications are more. please explore the reference section.

Welcome to the world of symbolification. you may want to call it algebra. it makes life easier.

Note:
We will be adding a few solved examples and exercise problems from NCERT text book later.

References:
https://en.wikipedia.org/wiki/Euclidean_division
https://en.wikipedia.org/wiki/Euclidean_algorithm
https://en.wikipedia.org/wiki/Euler%27s_totient_function

understanding the euclidean algorithm
https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/the-euclidean-algorithm

https://www.khanacademy.org/computing/computer-science/cryptography/modern-crypt/v/the-fundamental-theorem-of-arithmetic-1

Exercise
https://www.khanacademy.org/quetzalcoatl/deprecated-exercises/deprecated-bucket-5/v/the-fundamental-theorem-of-arithmetic

Method of generating primes
https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/sieve-of-eratosthenes-prime-adventure-part-4


Primes density vs natural logarithm
https://www.khanacademy.org/computing/computer-science/cryptography/comp-number-theory/v/prime-number-theorem-the-density-of-primes



                                     A critical review of textbooks by Mr Badri Seshadri











Thursday, April 14, 2016

Do exams kill Creativity?

Please post your comments for this post or mail your **responses to aatrral@gmail.com



Note:
I have to honor my words. I have decided (that's a decent word instead of  ' forced to') not write any more articles on education or entrepreneurship in my previous article drop-out and drop-in. There may be others in competition like educational institutions, parents, teachers, industry etc. You may refer to this article for some resources.

** The space below is intentionally left empty. to compile your responses. Please.