Tag Archives: model drawing

Hungry ghosts don’t do Singapore math

In Singapore, every year around this time, folks who believe in hungry ghosts celebrate the one-month-long “Hungry Ghost Festival” (also known as the “Seventh Month”). The Seventh Month is like an Asian equivalent of Halloween, extended to one month—just spookier.

If you think that these spiritual vagabonds encircling the island are mere fictions or imaginations of some superstitious or irrational local folks who have put their blind faith in them, you’re in for a shock. These evil spirits can drive the hell out of ghosts agnostics, including those who deny the existence of such spiritual beings.


Hell money superstitious [or innumerate] folks can buy for a few bucks to pacify the “hungry ghosts.”

During the fearful Seventh Month, devotees would put on hold major life decisions, be it about getting married, purchasing a house, or signing a business deal. If you belong to the rational type, there’s no better time in Singapore to tie the knot (albeit there’s no guarantee that all your guests would show up on your D-Day); in fact, you can get the best deal of the year if your wedding day also happens to fall on a Friday 13—an “unlucky date” in an “unlucky month.”

Problem solving in the Seventh Month

I have no statistical data of the number of math teachers, who are hardcore Seventh Month disciples, who would play it safe, by going on some “mathematical fast” or diet during this fearful “inaupicious month.” As for the rest of us, let’s not allow fear, irrationality, or superstition to paralyze us from indulging into some creative mathematical problem solving.

Let’s see how the following “ghost” word problem may be solved using the Stack Method, a commonly used problem-solving strategy, slowing gaining popularity among math educators outside Singapore (which has often proved to be as good as, if not better than, the bar method in a number of problem-situations).

During the annual one-month-long Hungry Ghost Festival, a devotee used 1/4 and $45 of the amount in his PayHell account to buy an e-book entitled That Place Called Hades. He then donated 1/3 and $3 of the remaining amount to an on-line mortuary, whose members help to intercede for long-lost wicked souls. In the end, his PayHell account showed that he only had $55 left. How much money did he have at first?

Try solving this, using the Singapore model, or bar, method, before peeking at the quick-and-dirty stack-method solutions below.


From the stack drawing,
2 units = 55 + 13 + 15 + 15 = 98
4 units = 2 × 98 = 196

He had $196 in his PayHell account at first.

Alternatively, we may represent the stack drawing as follows:


From the model drawing,
2 units = 15 + 15 + 13 + 55 = 98
4 units = 2 × 98 = 196

The devotee had $196 in his account at first.

Another way of solving the “ghost question” is depicted below.


From the stack drawing,
6u = 55 + 13 + 15 + 15 = 98
12u = 2 × 98 = 196

He had $196 in his PayHell account at first.

A prayerful exercise for the lost souls

Let me end with a “wicked problem” I initially included in Aha! Math, a recreational math title I wrote for elementary math students. My challenge to you is to solve this rate question, using the Singapore bar method; better still, what about using the stack method? Happy problem solving!


How would you use the model, or bar, method to solve this “wicked problem”?
Yan, K. C. (2006). Aha! math! Singapore: SNP Panpac Education. 
© Yan Kow Cheong, August 28, 2013.


A businessman won this “lucky” urn with a $488,888 bid at a recent Hungry Ghost Festival auction.

A Before-and-After Singapore Math Problem


A Singapore math primer for grades 4–6 students, teachers, and parents

In Model Drawing for Challenging Word Problems, one of the better Singapore math primers to have been written by a non-Singaporean author for an American audience in recent years, under “Whole Numbers,” Lorraine Walker exemplified the following before-and-after problem, as we commonly call it in Singapore.

Mary had served $117, but her sister Suzanne had saved only $36. After they both earned the same amount of money washing dishes one weekend, Mary noticed she had twice as much money as Suzanne. What was the combined total they earned by doing dishes?

The solution offered is as follows:


© 2010 Crystal Springs Books

The author shared that she did two things to make the model look much clearer:

• To add color in the “After” model;
• To slide the unit bars to the right.

This is fine if students have easy access to colored pens, and know which parts to shift, but in practice this may not always be too convenient or easy, especially if the question gets somewhat more complicated.Let me share a quick-and-dirty solution how most [elementary math] teachers and tutors in Singapore would most likely approach this before-and-after problem if they were in charge of a group of average or above-average grades 4–5 students.


From the model drawing,

1 unit = $117 – $36 = $81
1 unit – $36 = $81 – $36 = $45

2 × $45 = $90

They earned a total of $90 by doing dishes.

Analysis of the model method

Notice that the placement of the bars matters—whether a bar representing an unknown quantity is placed before or after another bar representing a known quantity.

In our model, had we placed the [shaded] bar representing the unknown unit on the right, it would have been harder to deduce the relationship straightaway; besides, no sliding or shifting is necessary. So, placing the bar correctly helps us to figure out the relationship between the unknown unit and the known quantities easier and faster.

In general, shading and dotting the bars are preferable to coloring and sliding them, especially when the problem gets harder, with more than two conditions being involved.

The Stack Method

This word problem also lends itself very well to the Stack Method. In fact, one can argue that it may even be a better method of solution than the bar model, especially among visually inclined below-average students.

Take a look at a quick-and-dirty stack solution below, which may look similar to the bar method, but conceptually they involve different thinking processes. To a novice, it may appear that the stack method is just the bar method being depicted vertically, but it’s not. Perhaps in this question, the contrast isn’t too obvious, but for harder problems, the stack method can be seen to be more advantageous, offering a more elegant solution than the traditional bar method.


From the stack model diagram, note that the difference $81(= $117 – $36) must stand for the extra unit belonging to Mary.

1 unit = $81
$36 + ▅ = $81
▅ = $81 – $36 = $45
2 ▅ = 2 × $45 = $90

So, they had a total of $90.

The Sakamoto Method

This before-and-after problem also lends itself pretty well to the Sakamoto method, if the students have already learned the topic on Ratio. Try it out!

Let me leave you with three practice questions I lifted up from a set of before-and-after grades 4–6 problems I plan to publish in a new title I’m currently working on, all of which encourage readers to apply both the bar and the stack methods (and the Sakamoto method, if they’re familiar with it) to solving them.


Use the model and the stack methods to solve these questions.

1. At first, Joseph had $900 and Ruth had $500. After buying the same watch, Joseph has now three times as much money as Ruth. How much did the watch cost?

2. Moses and Aaron went shopping with a total of $170. After Moses spent 3/7 of his money and Aaron spent $38, they had the same amount of money left. How much money had Aaron at first?

3. Paul and Ryan went on a holiday trip with a total of $280. After Paul had spent 4/7 of his money and Ryan had spent $52, the amount Paul had left was 1/4 of what Ryan had left. How much money did Ryan have at first?

1. $300 2. $86 3. $196

Walker, L. (2010). Model drawing for challenging word problems: Finding solutions the Singapore way. Peterborough, NH: Crystal Springs Books.

© Yan Kow Cheong, August 4, 2013.