[S2_20151029QFCS] Chunking and Piggy-back

Question 

Introduction

     We have seen here, here and here that recognising chunks is useful in mathematics.  It is an example of a having what Carolyn Kieran calls a structural view of algebra, which is a type of pattern recognition.  In this article, I show how, using chunking and substitution, we can piggy-back or ride on solutions of a simpler equation to obtain solutions of a more complicated equation.

     The solution for part (a) of is easy enough.  Just factorise (or AmE. “factor”)

                            (y + 2)(y – 7) = 0

Hence                 y + 2 = 0   or   y – 7 = 0

                                 y = -2   or   y = 7

This is Standard Operating Procedure.  But part (b) seems like a monster of an equation.  Oh dear!  What shall we do?

Observation

     Can you observe anything appearing more than once?

     Now do you notice any chunks that are repeated?  Once you can see the connection, you can make a substitution  y = x³ – 1  and then make use of the answer in part (a).  We  copy and paste the chunk (shown in green) into the previous solution and proceed from there.

Solution to part (b)

                                 x³ – 1 = -2     or     x³ – 1 = 7

                                       x³ = -1     or           x³ = 8

                                        x = -1     or            x = 2

Solved!

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* Lower Secondary Mathematics (Sec 2 ~ Grade 8-9)

* any syllabus that includes algebraic factorisation (factoring) and substitution

* anyone who is interested in and ready for algebra