[Pri20150504WNR] Emma and Francis’ Stickers

Question

Introduction

     This is another one of those Singapore primary school mathematics problems whose underlying algebraic structure is equivalent to that of a pair of simultaneous linear equations.  However, the primary school pupils are taught only simple algebra.  A very popular method is the use of bar diagrams a.k.a. “the model method”.  It is a useful tool to help pupils visualise the quantities involved.  However, some people are not comfortable with this method.  One needs to cut the bars into the correct number of sub-parts.  If the diagram is drawn wrongly, one would need to erase the diagram and redraw it.  Remember: the “model method” is just one of the many ways of drawing diagrams, which is only one of the many heuristics for solving mathematical problems.  It is worth the ship, but do not worship the “model method”.   If it works for you, go ahead.  If not, do not force it.  Try another method.  Do not cut your feet to fit the shoes (削足适履).

     Once again, I demonstrate my Distinguished Ratio Units (DRU) method as an alternative.    Let us say that a pupil who uses DRU actually drew the diagram in his/her mind, in a way.  Instead of drawing various sized multiple units, we now use numbers surrounded by different shapes to represent the different types units.  It is easy to read the passage sentence by sentence and translate them directly into the diagram without having to worry about whether the unit is drawn to the correct relative length.   To illustrate the facility of use, I shall actually present two ways to attack the problem using DRU. 

Solution 1  (Unifying units via LCM)

     I use one “circle” unit each for what Emma and Francis had at first.  Later on, after adding two “circle” units, Emma has  3  “circle” units.  After subtracting  8,  Francis ends up with, say, one “square” unit.  Emma has four times as much, so Emma has  4  “square” units, which is equal to  3  “circle” units.

     Now since the LCM of  3  and  4  is  12,  we can make Emma’s later holdings for stickers to be  12  “triangle” units, say.  We can express every of the original units in terms of this common “triangle” unit.  Just multiply the numbers in every “circle” unit by  4,  and multiply the numbers in every “square” unit by  12.  This is what we would get:-

From here, we easily see that the transition from  4  “triangle” units to  3  “triangle” units is a subtraction by  8.  Hence  1  “triangle” unit is  8  and therefore  4  “triangle” units (which represents Francis’ original number of stickers) represents  32.  So Francis had  32  stickers at first.

Solution 2  (Stepping stone)

     We model the situation in a way similar to the above solution, except that we reverse the arrow connecting  1  “circle”  unit to  1  “square” unit and replace the  “+8”  with  “–8”.  We have not changed the meaning by doing this.

Now, multiply everything in Francis’ column by  4  so as to obtain  4  square units.  This is the same as imagining what would have happened if Francis’ had  4  times his original number of stickers and he had given away  4  times (i.e. 32) the number of stickers.  Obviously, he would have  4  times the number left, as represented by  4  “square” units.  If we added  32  to the  4  “square” units,  we would get  4  “circle” units.

Notice that the  4  “square” units now serve as a stepping stone to connect the  3  “circle” units to  4  “circle” units, as highlighted in green.  Hence we can see straightway that  1  “circle” unit is the same as  32.  But this is the answer we want, because  Francis had the equivalent of  1  “circle” unit at the beginning!

Summary

     In the first method, I just use two different types of units.  Then I use the idea of Lowest Common Multiple (LCM) and then change all the quantities to a common type of unit.  In the second method, I multiply a relation by a certain number so that one type of units matches exactly and this serves as a stepping stone to find a link that solves the other unit.  I hope you like my Distinguished Ratio Units method.

Heuristics Used

H02. Use a diagram / model

H04. Look for pattern(s)

H05. Work backwards

H06. Use before-after concept

H09. Restate the problem in another way

Suitable Levels

* Primary School Mathematics

* other syllabuses that involve whole numbers and ratios