[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

[S2 Expository] Negation and Swapping a difference

Introduction

     In this article, I introduce a neat and very useful trick for algebra and arithmetic.  Let us say you want to buy something that costs  $5,  but you only have  $2.  You need to borrow from your friend to help pay for it.  How much do you owe?  $3  Obviously.  Using negative numbers to denote debts, we can write out the story as  2 – 5 = -(5 – 2) = -3.  It is a ‘-’ because you know you are owing something.  To calculate the amount you owe, you just swap the  2  and the  5.

     In other words, you are instinctively using the following identity without realising it.                                             

This algebraic identity (something that is always true, not just sometimes true) actually works for all types numbers that the student will encounter, including positive and negative numbers.  It basically says that swapping a difference is the same as negating it. 

     How do we know it is always true?  This is easily shown:-

                      RHS = -(b – a) = -b + a = a + (-b) = a – b = LHS

The equality relationship is symmetrical.  Since  RHS = LHS,  we have  LHS = RHS.  Equations work both ways.

How to use it?

     (1) You can use it for calculations involving positive and negative numbers, as above.

     (2) Whenever you see difference e.g.  pet – cat,  you can swap the quantities, put brackets around it and attach a negative sign in front to get  -(cat – pet).  Remember that equations work both ways.  You can use it in the other direction.  Hence

     (3) Whenever you see a negated difference e.g.  -(train – van),  you can imagine that the negative sign causes the two quanties to swap.  After swapping, the ‘-’  is used up and you end up with  van – train.
     May the swapping power be with you!

Suitable Levels

* Lower Secondary Mathematics

* other syllabuses that involve algebra and negative numbers