[S2_20151226EFQF] Factorisation without Trial and Error?

Problem

 

Introduction

     This problem was posed by a student going on to Secondary 1 (~ grade 7) next year.  This sort of problem is usually done at Secondary 2 or 3 (about grade 8 or 9).  This reminds me of my personal story.

     I accidentally discovered quadratic equations when I was in Primary 4.  I imagined a rectangle whose length is  2 cm  longer than the breadth.  If the breadth is  4 cm, the length is  6 cm and the area is obviously  24 cm².  But if I pretended that I knew the area but did not know the dimensions, I did not know how to solve it with the knowledge that I had at that time.  This started me on a quest to find out the answer.  I read secondary school guidebooks, asked my friend’s brothers and sisters, and even asked my Chinese teacher (who, after exams, offered to answer any question we had)!  Basically, I was offered two choices: (1) trial and error factorisation  and  (2) the quadratic formula.  I did not like guess and check (or hit and run?), and the quadratic formula looked formidable to me.

     So I started a quest to find a method of factorisation that did not require trial-and-error.  By secondary 1, after fiddling around with algebra, I managed to do it.  I reconstruct my derivation below.  And then I use my method to solve the above factorisation problem.

Derivation

Solution

Remark

     This looks like a Pyrrhic victory.  But like they say, it’s the journey and not the destination that matters.  Doing my own explorations prepared me for future learning and made me understand better.

     Nowadays, the new models of calculators give solutions to the associated equations and you can work backwards to get the factorisation.  Unfortunately, many students just blindly use this and forget to work backwards, giving the wrong factorisation.  If calculator gives 9 and ^-248/₂₉, and you write your factorisation as (x – 9)(x + ²⁴⁸/₂₉), your answer is wrong. Moral of the story: you still need to use your brain.

[OlymLSec_20150507BDP] A Partition of Unity via Factorisation

Question

Introduction

     A premier school in Singapore is said to have set a question similar to this question in a regular school test.  I suspect it was meant to select pupils to represent the school for mathematics Olympiad.
     I present two solutions: one using the idea of prime factor, the other one using the possible factorisation of some number into two factors.

Solution 1

Solution 2

Remark

     The Factorisation Grid helps us to perform the factorisation.  It is presented in the form that I learned it, in which the cross terms are written in the middle.  I prefer it this way.  Nowadays, the textbooks and the schools put these terms on a column on the right instead of in the middle.

     The word “unity” in mathematics is just a fancy word for the number  1.  It is a “partition” because the number is being split into two separate parts (fractions).  The “partition of unity” concept can be extended, and is useful in advanced mathematics including signal processing, weighted averages, spline functions and topology.

Suitable Levels

* Lower Secondary Mathematics Olympiad

* any student who loves a challenge in algebra or number theory