“There should be no such thing as boring mathematics”

- Edsger Dijkstra


  Ho Soo Thong 

First Edition, Nov 2015 ;  Revised in July 2016;  April 2018 



 PSLE Math – A Bar Modelling Problem Solving Approach

Prior to 1980, some school teachers in Singapore hit upon the idea of using bars instead of line segments to represent mathematical quantities. They adopted a heuristic approach by applying basic mathematics to construct bar models  to depict and compose mathematical situations in word problems.Finally we use the Part-Whole and Comparison Concepts and apply simple Algebra  for a complete solution. The modelling approach provides a visual aid for understanding and  solving the problems.and is known as the Bar Model Method.

Bar Model Method Involves Two Major Steps:

     1. Understand Mathematics Situations  and construct Bar Model to visualise the problem. 
     2. Apply simple Arithmetic Operations and look for Algebraic Relations on bar models for unknown situations.


For Modelling “Division” Situations – Euclidean Division Algorithm 

We begin with a counting problem with mathematical linear expressions.

Example 1 A Sweet Problem

Jane, a teacher, wanted to give each of her 46 pupils a package of 6 sweets. The sweets were bought in boxes of 20. How many boxes were needed?


The bar model shows that  Jane gave away 46×6 = 276 sweets and Jane needed 13 + 1 = 14 boxes. 


Notes :  The example illustrates the repeated use of  of Euclidean Division Algorithm to depict two division situations.You may visit  a related problem (Click  The Ribbon Problem ),a variant of 2017 PSLE question where the remainder is unusable in contrast to the above problem. The two problems illustrate “Remainder” situations with usable or unusable remaining items in some word problems. For more complex problems, you may refer to the book “Bar Model Method for Job Problems


For Distributive Property Situations – Mathematical Deduction

The next problem involves a problem solving strategy for distributive problems.

Example 2 – An Apple-Orange Distributive Problem  

Kim paid $ 6.60 for 5 apples and 2 oranges. Ann paid $ 6.00 for 2 apples and 5 oranges. 

(a) Lee bought  4 apples and 3 oranges, how much did Lee pay?

(b) How much did Mei pay for 7 Apples ?


First, we construct bar models to visualise the difference between Kim’s situation and Ann’s situations:Kim has three more apples and Ann has three more orangesand deduce that an apple cost $0.20 more than an orange.



  1. The broken bar model of Ann’s situation is constructed in such a way that we can compare it visually with the bar model for Kim’s situation. 
  2. The applications of Distributive Law can be seen in Excess-Shortage Problems and variants of Chicken-Rabbit Problems.

For “Multiple” Situations – Unitary Method

 We begin the use of mathematical symbol U and simple algebra for a counting problem.

Example 3 A Changing Weight Problem

There are two bags of sugar, X and Y. If the weight of bag X is increased by 3 kg, bag X will be 2 times the weight of bag Y. If the weight of bag X is decreased by 8 kg, bag Y will be 3 times the weight of bag X

What are the original weights of each bags of sugar? 


There are three situations. 

We begin the “Before” unknown situations and then the two “after” situations. Next, we identify the common unit  of U kg  and proceed with simple Algebra.



Note : We use  the simple algebraic expression 3U for “Multiple” situations.

For “Multiple” Situations, we begin  with the concept of unit with a mathematical symbol U for algebraic expressions, simple algebraic relations and simple algebraic manipulations to solve simple algebraic equations – Linear Equations.


For Ratio Situations -  A Ratio Approach with Unitary Method

The next problem involves  the property of divisibility by two numbers.

Example 4 A Boy-Girl Ratio Problem

The ratio of the number of boys to the number of girls in a social club was initially 5 : 2. When 14 new girls joined the club, the ratio became 4 : 3. How many members were there at first?


By noting that the number of boys remained unchanged, we construct the bar models for the Before and After situations

Before situation : Number of  boys :  Number of  girl = 5 : 2. 

After situationNumber of  boys : Number of  girl = 4 : 3.



Note : The Ratio Approach is an effective method for  many PSLE questions involving ratios,  fractions and simple percentages. 


For Algebraic Variations on Bar Models – Simple Algebra

The following example illustrates a problem strategy involving varying values of two mathematics quantities values. The bar modelling approach can be applied to a  recent PSLE question(05/02/2018).


Example 5 – A Stamp Collection Problem

Ann and Tim both have stamp collections. At first, Ann had 7 more stamps than Tim. After Ann giving Tim 9 stamps, how many more stamps then Ann did Tim have?


We divide Ann’s activity of giving 9 stamps in two steps: 

Step 1 : Ann gave 7 stamps to Tim 

Step 2 : Ann gave another 2 stamps to Tim 

A visual view of a bar modelling problem solving strategy involving changing an “equal parts” and a “more than” part as shown in the following bar models.



Notes : The above illustrates a heuristic approach to an algebraic relation on the bar models. We could use bar models to visualise variables with complex algebraic relations and suggest the further use of bar models for more advanced problems involving complex algebraic relations.The above example illustrates the visual algebraic operations on the bar models.  


For Property of Relatively Prime Numbers.

Finally, we reach a higher level of mathematics to show how the use of the property of relatively prime numbers for a counting approach to a challenging problem. Pupils can use the mathematics to solve a recent PSLE question (17/02/2018).

Example 6 – A cupcake Problem 

May baked more than 100 cupcakes and she packed all of them into some small and large boxes. Each large box contains 8 cupcakes and each small box contains 5 cupcakes.

The total of cupcakes in the large boxes is 13 more than that in the small boxes.What is the least number of cupcakes she baked? 


First, we construct a bar model for the situations as shown. 

Since 5 and 8 are relatively prime numbers and the least common multiple of 5 and 8 is 5•8 = 40. Now


The least number of cupcakes is 163.


More examples can be seen in the site