Monday, November 16, 2015

[Pri20151116DNGC] Quadratic Plays Second Fiddle in Product-Difference Riddle

Question

 What two numbers give a product of  21.5  and a difference of  6.1?

Introduction
This question reminded me of a question that I set myself when I was in Primary 4 (» grade 4).  I imagined a rectangle with breadth 4  and  length 2 units longer then the breadth (i.e. 6) giving a area (product) of  24.  Then I pretended that I did not know the breadth and let it be  x.  This led to a quadratic equation  x(x + 2) which I did not know how to solve (if I did not know the answer).  So I accidentally discovered quadratic equations when I was in Primary 4.  This led me to a quest to learn the method of factorisation (by “trial and error” or “guess and check”) and the quadratic formula.  I never liked trial and error.  So I continued in my quest to invent a method of factorisation that did not require “guess and check”.  I finally succeeded doing that in secondary 1 (» grade 7).  This turned out to be a Pyrrhic victory.  The method I invented was quite similar to the quadratic formula.
There is a place for “guess and check” in mathematics.  I present a simple solution to the above problem using just that.

Solution

 smaller # larger # product 2 8.1 16.2 û 3 9.1 27.3 û 2.5 8.6 21.5 ü

Solved! J

H02. Use a diagram / model    (table)
H05. Work backwards             (if the smaller number is this, what is the bigger number?)
H07. Use guess and check
H09. Restate the problem in another way      (area = product)

Suitable Levels
Primary School Mathematics
* other syllabuses that involve decimal numbers

* anyone who loves to exercise their minds