Category Archives: Singapore Math

News and Views about the Good, the Bad, and the Ugly about Singapore Math

Algebra or Model Method

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On December 19, 2012, in her Confession from a Homeschool Mom: Singapore Math Stumped Me Today, Monise L. Seward, blogged that her 6th grader woke her up to ask her for help on a word problem in her Singapore Math book. The nonroutine question she shared was the following:

Mrs. Pappas had some apples. She sold 1/3 of the apples plus 5 more on the first day. She sold 1/3 of the remaining apples plus 5 more on the second day. She had 125 apples left in the end. How many apples did Mrs. Pappas have in the beginning?

If you use algebra to solve this problem, it’s unlikely that the working will arouse any excitement; in fact, you may find this method of solution to be somewhat uninteresting or boring. Yes, algebra does religiously solve the problem, but the solution is anything but elegant. Moreover, most average grade five or six students wouldn’t have acquired the maturity to solve it algebraically.

An algebraic check

Besides working backwards to check the answer, after solving the question, using the Singapore model method, I also checked it out by algebra.Looking at the symbolic clutter, I guessed then that even our Singapore grades 7 and 8 average students would likely be challenged to solve this problem by algebraic means.If we stick to working with only one variable, then we may end up with an unappetizing equation such as the following to solve.

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A mum’s solution

Without a good working knowledge with the Singapore model method, we can expect most teachers and parents (who still remember their school math) to solve the question in a way similar to the one sketched and tweeted by my Pinterest and Twitter friend at PragmaticMom.com, as shown below:

2/3x – 5 = 125

2/3x = 125 + 5

2/3x = 130

x = (130 x 3)/2

x = 195

You add the 5 because you have to subtract the 5 apples from the 2/3 calculation because they were added in additionally. Then you do it again …

2/3x – 5 = 195

2/3 x = 195 + 5

2/3 x = 200

x = (200 x 3)/2

x = 300

No algebra, please!

Pretend for a while that algebra is an alien language to you! And you can only use a non-algebraic explanation to communicate your solution to a ten-year-old child! How would you go about doing it? Can you think of some intuitive methods?

After coming across this grade six question via @pragmaticmom, I tweeted a quick-and-dirty solution to the above problem, using the model, or bar, method. Give it a try first, before comparing yours with mine.

Sakamoto math

I also remarked that we could also solve this question by the Sakamoto method, which I assumed most of you in the United States might not be familiar with; so I’ll skip presenting the Japanese method of solution here.

Working backwards

Had the above grade five or six question involved the fraction 1/2 instead of 1/3, then using the “work backwards” strategy, via a flow chart, would have yielded an equally decent method of solution as the model approach.

Mrs. Pappas had some apples. She sold 1/2 of the apples plus 5 more on the first day. She sold 1/2 of the remaining apples plus 5 more on the second day. She had 125 apples left in the end. How many apples did Mrs. Pappas have in the beginning?

See a sketch of a common method of solution below.

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Two bonus questions

Let me leave you with two grade 3/4 problems that lend themselves easily to the model method.1. A shop owner sold 2 more iPads than half the number of iPads in his stock. He then sold 2 fewer iPads than half of the remaining iPads. If he was left with 28 iPads, how many iPads did he have in his stock in the beginning?

2. Sarah used $8 to buy a book. She then used half of the remaining money to buy a bag. Lastly, she spent $1 more than half of what she had left on a meal. In the end, she had only $5 left. How much money had Sarah at first?
Answers: 1. 108 iPads; 2. $32

© Yan Kow Cheong, March 13, 2013.

The Chickens-and-Rabbits Problem

Problem Solving, Sakamoto method, Singapore Math, Singapore Model Method, Stack Method

In Singapore, the chickens-and-rabbits question was in vogue in the late nineties, when the Ministry of Education then wanted teachers to formally teach problem-solving strategies (or heuristics, as we commonly call them here). Two common methods of solution favored by local teachers are “guess and check” (for younger students) and “make a supposition.” And in recent years, as Sakamoto math strategies gain currency in more local and regional schools, we’ve been blessed with no fewer than three other methods of solution to solve this type of problems.

A Grade 5 Contest Problem

In math contests and competitions, it’s not uncommon to witness some variations of the chickens-and-rabbits problem, which often pose much difficulty even to students, who are fluent in the Singapore model method. Let’s look at a grade 5 chickens-and-rabbits question, with a slight twist.

There are 100 chickens and rabbits altogether. The chickens have 80 more legs than the rabbits. How many chickens and how many rabbits are there?

Give it a try first, before comparing your solution(s) with the ones I’ve exemplified below.

Method 1

Since the chickens have 80 more legs than the rabbits, this represents 80 ÷ 2 = 40 chickens.

Among the remaining 100 – 40 = 60 chickens and rabbits, the number of chicken legs must be equal to the number of rabbit legs.

Since a rabbit has twice as many legs as a chicken, the number of chickens must be twice the number of rabbits for both their total number of legs to be equal.

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From the model drawing,

3 units = 100 − 40 = 60
1 unit = 60 ÷ 3 = 20

Number of rabbits = 1 unit = 20
Number of chickens = 2 units + 40 = 2 × 20 + 40 = 80

A check shows that the answers do satisfy the conditions of the question.

Method 2

The equations resulting from the models for Methods 1 and 2 are the same, but conceptually this method is slightly different from the previous one.

The bar representing the number of chickens must be half the length of the bar representing the number of chicken legs. The bar representing the number of rabbits must be one quarter the length of the bar representing the number of rabbit legs.

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From the model drawing,

3 units = 100 – 40 = 60
1 unit = 60 ÷ 3 = 20
2 units + 40 = 2 × 20 + 40 = 80

Therefore, the number of rabbits is 20, and the number of chickens is 80.

Let me leave you with another fertile chickens-and-legs problem, which should challenge most grade 5 or 6 students, not to say, their teachers and parents.

Mr. Yan has almost twice as many chickens as cows. The total number of legs and heads is 184. How many cows are there?

Could you use the bar method, or the Sakamoto method, to solve it?

© Yan Kow Cheong, March 3, 2013.

How do they do it?

Singapore Math
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An Annular Solar Eclipse—An “Astronomy Picture of the Day,” taken on May 19, 2012. Repined by Lesley Delux.

 

Folks in different professions often do the same thing differently; their idiosyncrasies put some of them in a different league. Let’s look at how those born with a “mathematical gene” tend to do IT.

Algebraists do IT on both sides.

Analysts do IT within limits.

Astronomers do IT in the dark.

Combinatorialists do IT discreetly and discretely.

Computer scientists do IT bit by bit.

Constructivists do IT by exploring first.

Engineers do IT exactly as the manual says.

Formalists do IT in some rigidly tedious way.

Game theorists do IT after weighing the risks and rewards.

Geometers do IT at right angles.

Group theorists do IT in fields and on rings.

IMO mathletes do IT in two days, each lasting four and a half hours.

Logicians do IT clinically and religiously.

Math writers do IT behind the scenes, showing no signs of trial and error.

Mathematicians do IT in numbers.

Number theorists do IT with no reguard to financial gain.

Numerical analysts do IT somewhat roughly or approximately.

Numerologists do IT only on their good or lucky days.

Platonists do IT in the most ideal way imaginable.

Politicians do IT the way you want it to be.

Priests and pastors mostly do IT covertly and guiltily.

Probabilists do IT using coins and dice.

Set theorists do IT in organized groups.

Singapore math teachers do IT with some models first.

Statisticians do IT with 95% confidence.

Topologists always do IT with a surprise.

What is IT? Why, solving challenging word problems, of course. What are you thinking?

Yan Kow Cheong, February 27, 2013.

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What do you see in the crop circle? © crystalinks.com

A Premature Model

Singapore Math

For math-anxious students, teachers, and parents green to the Singapore model method, figuring out how a bar or model is to be drawn, based on the information given in the word problem, often instills fear and adds stress in them.

From the model solution, how does the author know that this part of the bar, or rectangle, must be longer or shorter than the other part? How does he know that the bar representing the unknown quantity isn’t longer than the one representing the known quantity? How does she know that some part of a bar must be divided into a certain number of parts? These are some conceptually valid questions that authors of textbooks and supplementary titles seldom address, even in the teachers’ guides. The truth is that we, math educators, don’t know the answer, until we start dirtying our hands.

Below is a typical grade 3 Singapore math word problem, where the positioning of the bar model, placed right after the question, often puzzles instead of enlightens readers.

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Few students have the opportunity to witness how teachers try drawing a model, when faced with an unfamiliar word problem. They don’t see the behind-the-scenes dirty work (drawing, erasing, remodeling, erasing, …) until a sensible model is obtained, which fits the information given in the question. It’s never too early for us to help students build up their confidence, by sharing with them that mathematical success often comes after multiple failures, small as they may be.
Reference
Lee, A. (2010). Challenging word problems in primary mathematics 3. Singapore: Panpac Education Private Limited.
© Yan Kow Cheong, February 18, 2013.
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The left cover is a Singapore edition; the right cover is a US edition—from which the sample question was taken.
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This is the current US edition of the grade three 236-page supplementary math title—the bestseller can be ordered from SingaporeMath.com for about ten dollars.

Changing Twice

Singapore Math, , ,

In his creativity book, The Forgotten Half of Change, Luc De Brabandere shares that it took about half a century to move from sailing ships to steamships. Resistance to change from sail manufacturers only led to ships with more and more masts. During that time, hybrid ships took to the waves; for example, the Sphinx was equipped with three masts but also with a funnel for a steam boiler.

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The Evolution from Sails to Steam

 

In 1833, if you were an onlooker seeing that ship sail by, how would you perceive its double source of energy? On one hand, you might see the engine as something that could be useful on a day with no wind; on the other hand, you could think of the sails as something that could come in handy should the engine break down. This means one can view the same boat from two angles; as you see more and more of these steamships, then one day, you realize that this is no longer an option, thus resulting in a break in your perception.

Or, what about an abaculator—an abacus that comes with a four-function calculator? Or, a book with an attached CD (containing an electronic version of the book)? Change is not an option—the options are what and when. As De Brabandere remarked, “You have to change twice: perception and reality.”

Change is not an option—the options are what and when. As De Brabandere remarked, “You have to change twice: perception and reality.”

© Yan Kow Cheong, Feb. 17, 2013

How to cook up numbers on an abacus

Singapore Math

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Suan Pan Zi (or abacus)—a Hakka dish served on special occasions, mostly during the Chinese New Year. © kitchentigress.blogspot.com

 

In Singapore, it’s not easy to order a plate of abacus (prepared from rice flour and ingredients like mushroom and dried shrimps) unless we go to a Hakka restaurant. A traditionally home-cooked dish, abacus is usually served on special occasions, such as the Lunar New Year and on birthdays.

There is no reason why in Singapore, best known for its ethnic cuisine and math education, we can’t marry mathematics and food to promote both items to an often-innumerate and hungry world.

A Parents’ Nite would be a good occasion to popularize the abacus, both as a calculating instrument and as a Chinese dialect dish. Parents and their children would learn to do simple arithmetical operations on an abacus, and also learn to cook and enjoy a traditional Hakka dish of abacus. At the least, it could be a calculating-and-cooking fruitful night to bond the family and school staff together.

Let’s look at some off-the-wall parallelisms, when vintage math meets traditional cuisine.

1. Learning to calculate on an abacus reminds us of our mathematical heritage—a minds-on activity that could be used to promote multicultural mathematics.

Cooking a plate of abacus introduces others to some Chinese cuisine—a hands-on activity that could help to promote and preserve a traditional Hakka dish.

 

2. Other than a few countries like China and Taiwan, and Chinatowns, the abacus may help rekindle interest in things à la Chinoise.

Other than in Hakka-speaking homes, folks may share the abacus recipe to preserve their fast-disappearing traditional dish.

3. Entrepreneurs may come up with an electrified abacus or an abaculator.

Entrepreneurs may consider canning abacus (halal and kosher) for potential overseas markets.

4. Some creative uses of the abacus involve finding the square and cube roots of a number, and differentiating and integrating functions.

Some creative and innovative cooks may dish out new abacus recipes to cater for local tastes in different locations.

5. At math conferences, the abacus dish may be served, providing healthy food for the body.

At food festivals, talks on how to use the abacus creatively, especially in mental computation, may provide intellectual food for the mind.

6. In home economics, learning how to prepare a dish of abacus will add spice to an often-boring cooking lesson.

In elementary mathematics, learning to square, cube, square root, and cube root a number on an abacus will vow students, thus leading them to pay due respect to the “primitive calculator.”

7. The abacus is used as a divination tool to welcome good luck and prosperity for superstitious businessmen or shop owners.

A plate of abacus is fearfully or reverently offered on the altar to appease the hungry ghosts during the Seventh (or Ghost) Month by Taoists and Buddhists.

Surely, math educators could come up with more interesting liaisons between the abacus as a calculating tool and the abacus as a Chinese dialect dish. Why not share with the rest of us your arithmetic and culinary thoughts?

Gastronomically and mathematically yours

K.C. Yan
www.singaporemathplus.com

© Yan Kow Cheong, Feb. 14, 2013

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These “primitive calculators” remind me of our mathematical and cultural heritage.

Math Pictures for the Young

Singapore Math

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These are some lovely “math pictures” from Richard Evan Schwartz’s “You can count on monsters.” This richly colorful children’s book makes an attempt to teach advanced concepts, such as divisibility and prime factorization, to young children.
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These pictures add a fun and creative element to introducing young children to math words like “factor” and “prime.”