[Pri20150510OAC] An Oranges-to-Apples Comparison

Question

Introduction

     This primary school mathematics question is slightly tricky as the pieces of fruit* are packaged.  I present two solutions: the first uses the “assumption” or “supposition” method together with the concept of unit costs.  The second solution illustrates my Distinguished Ratio Units (DRU) method.

Solution 1 (using an Assumption / Supposition)

 Solution 2 (using my Distinguished Ratio Units method)

I use  3  “circle” units to denote the number of oranges (since I know it can be divided by 3) and

I use  2  “triangle” units to denote the number of apples (since the number is divisible by 2).  Then the cost of the oranges would is two “circle” units and the cost of the apples is  3  “triangles” units.  Let us equalise the “circle” units by multiplying the first row by  2  and the second row by  3.  With both “circle” units equal to  6,  the  6  “circle” units can be used as a stepping stone to connect  4  “triangle” units with  9  “triangle” units (highlighted in yellow).  We now know that  5  “triangle” units corresponds to  20.  It is easy now to work one  1  “triangle” unit and then  2  “triangle” units, which corresponds to  8.

Ans:  Abigail bought  8  Apples

Commentary

     Note that my DRU solution does not use any fractions!  The astute reader will note that my DRU method is actually the equivalent to the  u and  p (“units vs parts”) method used by many Singapore teachers/tutors.  Both are actually algebraic methods in disguise, just as the bar modelling (“the model method”) is.  In case you are wondering what the difference between units and parts is: units and parts actually have the same meaning, except that they refer to differently-sized unknowns and we need different names for different units.  Primary school mathematics in Singapore is actually rather challenging because many of these problems are equivalent to solving a pair of simultaneous equations in two unknowns.  It is possible to use one unknown unit, but you would kill a few brain cells in the process of formulating the problem in terms of only one unknown.

*A Note on Singlish and standard English

     Most Singaporeans would call, for example, 2 apples and 3 oranges as 5 “fruits”.  Actually, in standard English, “fruit” (as an uncountable noun) refers to general fleshy food that comes from flowering plants.  So if you eat “2 apples and 3 oranges”, you are eating fruit.  Yes, fruit is food.  The word “fruit” can also be used as a countable noun.  This refers to fruit coming from different botanical species.  So in the aforementioned example, “2 apples and 3 oranges” would be considered as 2 fruits (2 types of fruit) and  5  pieces of fruit.

Heuristics Used

H02. Use a diagram / model

H04. Look for pattern(s)

H05. Work backwards

H08. Make suppositions (assume, “what if”, imagine if ...)

H09. Restate the problem in another way

Suitable Levels

* Primary School Mathematics

* other syllabuses that involve ratios, fractions and money