[S2_20150501AXC] 2011: Chunking and Substitution with Algebra

Question

Introduction

     This lower secondary algebra question seems complicated, doesn’t it?  Can you spot any chunk that is repeated, or almost the same?  This is one of the keys to solving the problem.  Another key that you need is the relevant algebraic identities and tricks.

Reminders

     First, let us review some of these useful formulas.                                        

This is the square-of-difference identity, illustrated here.

This swapping trick is illustrated here. 

     Looking back at the question, do you notice anything that is repeated?  Can you see any chunks that are the same or almost the same?  (n – 2011)  is almost the same as  (2012 – n)  isn’t it?  Whenever you see a repeated chunk, it is a good idea to substitute that chunk with another variable that you invent.  To name this new variable, you can use any letter that is not used before, so as not to conflict with existing letter(s).

Solution

Summary

     To solve the given problem, we have used the following:-

     (1)  square-of-difference identity

     (2)  swapping technique

     (3)  observation of repeated chunks

     (4)  using substitution with the chunks

Heuristics Used

H04. Look for pattern(s)  e.g. chunking, observation

H09. Restate the problem in another way  e.g. swapping, identities

H10. Simplify the problem e.g. substitution for chunks

H11. Solve part of the problem

H12* Think of a related problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

· Lower Secondary Mathematics

· other syllabuses that involve whole numbers and ratios