Thursday, October 29, 2015

[S2_20151029QFCS] Chunking and Piggy-back


     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?

     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

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

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