Sunday, November 6, 2016

[AM_20161105ITFF] False Friends in Integration (Calculus)

Question

Introduction
          False friends are words in two languages that look/sound alike, but differ significantly in meaning.  Do you know that there are also false friends in mathematics?  Can you distinguish and explain the difference between the two integrals?

Solution

          The integrand on the left has the variable  x  as the base and the constant  e  as the index.  So we integrate it using the Power Law.
          By contrast, for the integrand on the right, the base  e  is a constant whereas the index is the variable  x.  Integrating  e  to the power of  x  is the eeeeeeeeeeeeeeeasiest.  You just get back the same thing, plus the arbitrary constant of course.

Remark
          Many students make the mistake of trying to apply the Power Law for the exponential.  As a learner of mathematics, one needs to cultivate the habit of being observant and paying attention to detail.  This is part of developing one’s identity and character which is important in life.

Suitable Levels
GCE ‘O’ Level Additional Mathematics
GCE ‘A’ Levels (revision)
* revision for IB Mathematics HL & SL (revision)
* Advanced Placement (AP) Calculus AB & BC
* University / College Calculus
* other syllabuses that involve integral calculus

* whoever is interested

Thursday, November 3, 2016

[Enrich20161103NRP] The Napkin Ring Problem

Problem

     Two rings are made by drilling a cylindrical hole through a small sphere and a hole through the large sphere, such that the resulting rings have the same height (2h).
     Which ring has the larger volume of remaining material?

Solution
     The answer is: both rings have the same volume.  How can we know?
     There is a way to show this using integration.  But calculus is not necessary.

     Let  r  be the radius of any chosen sphere and let  a  be the radius of the cylindrical hole.  By Pythagoras’ Theorem,  h² = r² – a².  Consider a cross-section of the ring sliced a distance  x  from the centre of the sphere, perpendicular to the axis of the cylindrical hole.  The outer radius of this cross section is the square root of  r² – x².  Hence the area of the material in the cross-section is
               p [(r² – x²) – a²]  =  p (r² – a² – x²)  =  p (h² – x²)
Note that  r  does not appear in the formula.  That means the cross-section does not depend on  rA bigger (or smaller) sphere would have the same cross-sectional area for each distance  x  away from the centre.  By Cavalieri'sPrinciple, the other sphere will have the same volume!
     By the way what is this volume?  It is the same as that of a sphere without hole  i.e.  where  a = 0  and  r = h.  This works out to be  4/3 p h³,  where  h  is half the height of the ring.

H02. Use a diagram / model
H09. Restate the problem in another way
H11. Solve part of the problem
H13* Use Equation / write a Mathematical Sentence

Suitable Levels
GCE ‘A’ Levels H2 Mathematics (Number patterns, with algebra)
* revision for IB Mathematics HL & SL
* Advanced Placement (AP) Calculus AB & BC
* University / College Calculus
* other syllabuses that involve volumes and Pythagoras’ Theorem
* any learner who is interested






Saturday, October 22, 2016

[Pri1_20161021DIV] Labelling as a strategy for Division

Introduction
          The Singapore mathematics syllabuses are very well designed, especially the primary school syllabus.  Fundamental concepts and skills are introduced before going on to complex calculations and problem solving.  At primary 1, pupils learn the idea of multiplication and division of small numbers by grouping (or partitioning).  They are not made to recite the times tables meaninglessly.
          Division is easy if the number of things in each group is known.  You just keep on circling the known number of objects until everything is circled.  However, if the number of groups is required but the number of things in each group is not given, and if the objects are not arranged in a convenient way, the task can be a bit more challenging.  Remember: they have not memorised the multiplication tables yet.

Problem / Question


Solution (Suggested)
          One way to solve this problem is to label the fish 1, 2, 3, 1, 2, 3, ... in a cyclic fashion, assigning fish to each of the three friends one at a time, thereby ensuring that each person gets the same number of fish.  Start with “1” somewhere on the left, “3” on the right and “2” somewhere in the middle.  Assign the next “1” close to the previous “1”, the next “2” close to the previous “2” and the next “3” close to the previous “3”.  So all the 1s are close together, the 2s are close together and the 3s are close together.  After all the fish have been labelled, the partitioning (or grouping) becomes obvious.

H02. Use a diagram / model
H04. Look for pattern(s)
H09. Restate the problem in another way

Suitable Levels
Primary School / Elementary School Mathematics
* any precocious or independent learner who is interested




Wednesday, June 8, 2016

The passing of a Great Giant - Jerome Bruner

Jerome Bruner, a great psychologist, has died recently.  We thank him for his theories that guided the development of Singapore Mathematics education.

One of his greatest theories that is useful for children's learning is the Concrete-Pictorial-Abstract (Enactive-Iconic-Symbolic) approach.

It is impossible to use only words to explain simple mathematical concepts to a young child, since those concepts cannot be further explained using words.  You will run out of words to explain!  What definitely does not work is to start from use just words and they don't "get it", then scold them for being stupid.

They have to learn buy touching and playing with things, then from looking at pictures and then using abstract words or reasoning.

Saturday, February 27, 2016

[S1_20160227FZCK] Factorisation by Chunking

Problem / Question
 

Solution

Commentary
     Here I illustrate the usefulness of chunking to factorise (AmE: factor) an algebraic expression.  Observe that  3a – 2b  is a repeated part of the expression.  I call it a “chunk”.  To make it clear, I rewrite  (3a – 2b)²  as   (3a – 2b)(3a – 2b)  so that you can see it as two copies of the same chunk.  I highlight in yellow one copy of  (3a – 2b)  from each of 
(3a – 2b) (3a – 2b)   and  -3(3a – 2b).  The remaining stuff are highlighted in blue and green.  Take out the yellow chunk as common factor by writing it out on the left in the third line, shown in yellow.  You can pull out the common factor by writing it out to the right if you want, but here I chose to put it on the left.  The result would be equivalent anyway.  Once you have written out the common factor,  you write out the other stuff (shown highlighted in blue and green) into another other bracket.
     Once you understand how it works, you can actually do the second line mentally and write down the answer straightaway.  Chunking is a very useful technique in mathematics.  Here are some more examples of the technique of chunking: (1), (2), (3).

H04. Look for pattern(s)
H10. Simplify the problem
H11. Solve part of the problem

Suitable Levels
Lower Secondary Mathematics (Sec 1 ~ grade 7)
GCE ‘O’ Level “Elementary” Mathematics
* other syllabuses that involve algebra and factorisation (factoring)
* any learner who is interested





Tuesday, February 23, 2016

[Pri20160223FPDM] Pernicious Portion Problem? Shift Happens!

Problem / Question

Strategy
     This seems to be a confounding question on decimals.  What shall we do with the triangles?  Is there a short cut?

     Yes!  What you can do is to imagine putting the two triangles together to form a rectangle.  And then the solution becomes easy!  This is because the area is unchanged and hence the proportion of the shaded area is unchanged, is the same as before.  We can make use of fractions and convert it to a decimal.


Solution


H02. Use a diagram / model
H04. Look for pattern(s)
H09. Restate the problem in another way
H10. Simplify the problem

Suitable Levels
Primary School Mathematics
* other syllabuses that involve fractions and decimals
* any learner who is interested


Monday, February 22, 2016

[Maths Education] Mathematical Journalling

     Nowadays, I see some schools / textbooks asking students to search the internet and to write on a certain problem on their “mathematics journal”.  It's so very guided.  It's so artificial.  The questions should come from the learners themselves, out of their own curiosity.  The learn then seeks to answer their own questions.  The journal can serve as to document and summarise their process of learning.

     The best maths journals are self-initiated.  Great mathematician Karl Friedrich Gauss and renowned scientist Richard Feynman kept math journals on their own accord, not because some teacher told them to do it. 

     When I was a student, I borrowed books from the National Library on things out of the normal curriculum.  I kept notes of things I learned.  I also did my own investigations.  I accidently discovered quadratic equations when I was in Primary 4.  I read guidebooks, asked my friend's elder brothers and sisters, my Chinese teacher (!) and other people to find out more.  I did not like factorisation by trial-and-error.  Neither did I like completing the square nor using the quadratic formula.  So I did my own research to find a sure-fire way to factorise without trial-and-error.  I finally managed to find a way, but my method had an uncanny similarity to the quadratic formula.  It was a Pyrrhic victory, but it was fun!  I thoroughly enjoyed it.

     If students need to be told or goaded to write mathematics journals, then we as educators need to ask ourselves:  Why?  What is their conception of mathematics and education?  What experiences have they gone through that lead them to these beliefs?

     Some food for thought, eh?