A Singapore Math Age Problem

On this Singapore's 48th National Day (or Independence Day), let's look at another question from Lorraine Walker's Model Drawing for Challenging Word Problems: Finding Solutions the Singapore Way. The author solved the following grade 4 or 5 age problem, using the bar method.

Rolff is now 4 years older than Erik. 

Five years ago Rolff was twice as old as Erik. 

How old was Rolff 5 years ago?

Her model-drawing solution is as follows:

Method 1

No doubt, the second color makes it easier to visualize what's happening, but what if students aren't given the choice to draw their model in more than one color? In fact, it wouldn't be surprising for critics of the model method to argue that there's no need to "complexify" the solution with all these bars and arrows, when the question could easily be solved without a model drawing.

Are there other valid models?

In this before-and-after word problem, is there an alternative or better model that may depict the relationship between the ages of the two persons? Let's look at two quick-and-dirty methods of solution, which a number of local students would have used in solving this question.

Method 2

In this method, we assume that we're not aware that the age difference between Rolff and Erik is a constant.

Here, we make use of the fact that the distance of "1 unit + 5" is equal to the distance of "5 + 4."

From the model drawing, 

1 unit = 4

2 units = 2 × 4 = 8

Five years ago Rolff was 8 years old. 

Method 3

Since the difference between their ages at any point of time is a constant, in the "before" diagram, the difference is 1 unit (2 units – 1 unit), and in the "after" diagram, the difference is 4. So, 1 unit must be equal to 4. 

The choice of the smaller age numbers and the relatively small number of times one person is older than the other, makes it difficult for one to appreciate the power of the model method in this before-and-after age problem. 

A slightly modified question

A modified version of the original question could probably help convince one why the bar method might serve as a useful strategy in revealing the relationship between the ages of the two persons.

Rolff is now 12 years older than Erik. 

Five years ago Rolff was three times as old as Erik. 

How old was Rolff 5 years ago?

From the model drawing, 

2 units = 12

1 unit = 12 ÷ 2 = 6 

3 units = 3 × 6 = 18

Five years ago Rolff was 18 years old. 

Conclusion 

This age problem shows that in posing word problems that lend themselves to the bar method, we'd attempt as far as possible to constructing them in such a way that the model method is seen to offer a better problem-solving strategy than other traditional (or formal) methods in solving the questions. Otherwise, there's no incentive for one to wanting to draw a model drawing than use other strategies, which might even be shorter or easier.

Reference

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

© Yan Kow Cheong, August 9, 2013.

To Count or Not to Count

Einstein said, "Not everything that counts can be counted, and not everything that can be counted counts."

One of my favorite childhood games!
Five stones or five bags containing rice.

Here are a dozen math questions that you may pose to your grades 2–3 child to see whether they've graduated from rote-learning, ready to tackle higher-order thinking questions. Don't ignore their answers off-hand, silly as some of them may appear to be. Use them to question the reasonableness of their thinking processes, or to promote sense making in problem solving.

Questions crying to be solved

1. Farmer Jones had 220 goats: 65 he-goats and 155 she-goats. 100 goats died last winter. How many he-goats did he have left?

2. Film A is only 94 minutes long. Film B is 1 hour 40 minutes. Which film is better? Explain.

\3. The largest painting in museum P is 60 centimeters square; the one in museum Q is one and a half times bigger. Which museum is worth visiting? Why?

4. There are 125 sheep and 5 dogs in a flock. How old is the shepherd?

5. If one Mona Lisa painting is worth $12 million, how much will two such paintings be worth?

6. Mr. Smoth has two children: a 9-year-old son and a 12-year-old daughter. How old is the mother?

Creative logic ©1999 The Straits Times & A. Choy 

7. A fake Rolex watch costs $2,500 at a shop in Dubai. How much would four such watches cost a businessman from a developing country, if he were to buy the original ones?

8. There are 26 sheep, 12 pigs, and 14 goats on a ship. How old is the captain?

9. Yesterday 37 boats sailed into port and 49 boats left it. Yesterday at noon there were 43 boats still in the port. How many boats were still in the port yesterday evening?

10. Tom used to have 45 Facebook friends. After some friends unfriended him for forwarding their personal e-mails without their permission, Tom is now left with 51 friends. How many friends left him?

11. On Tuesday, at 4:00 PM, Sherry was overjoyed when she had seven new Twitter followers. However, the next day, three of them unfollowed her. How many Twitter followers does she have now?

12. Ken received 8 new e-mails today. After deleting some of his e-mails, his inbox was left with 21 of them. How many e-mails were in his inbox at first?

0 friends, 3000+ Facebook friends, fiends, and foes
© 2011 Tanya Cooke

Creative problem posing 

Use these "insensible" or "unreasonable" questions to get the children to amend them, before they're deemed "solvable." Let their creative imagination run wild by posing their own versions; and be prepared for some surprises in their responses. You may be astounded how their literacy and numeracy grow by leaps and bounds.

Happy problem posing and problem solving.

References

Cooke, T. (2011). Help! I'm a Facebookaholic. London: John Blake Publishing Ltd.
Schoenfeld, A. H. (2012). A modest proposal. Notices of the AMS, 59(2), 317-319.
Yan, K. C. (2012). Mathematical quickies & trickies. Singapore: MathPlus Publishing.

© Yan Kow Cheong, August 1, 2013.

How to love [Singapore] math, without liking it?

In reply to "how to motivate a contemporary teenager to love Maths? Three key words," posted in the "Singapore Math NOW" LinkedIn Group, my reply then was:

Make the most disliked subject in school inaccessible, expensive and illegal.

Here are three less-irreverent practical suggestions that have proved to be successful to a certain degree, in depicting mathematics, not as a terminal subject once it becomes optional, but as a subject worthy of study, which can help raise one's socio-economic status.

1. Reward students with money. 

A brochure to motivate citizens to
upgrade themselves mathematically.  

In Singapore, mature students (who are generally working adults who are given a second or third chance to get a formal education qualification) get a few hundred bucks from self-help groups, or from some workforce development agency, if they attain a certain level of mathematical proficiency.

In other words, Singapore tax payers reward them with cash if they do well in math, on top of their heavily subsidized part-time education. The country wants them to succeed even if they'd previously made a mess of their "mathematical life."

2. Brand the subject or topic.

In the early eighties, leveraging on some pedagogical insights from research done in mathematics education from countries, such as China, Russia, Japan, and the United States, Singapore had succeeded in extending the power and application of the strategy, "Draw a diagram" or "Draw a model," to solve a whole set of mathematics problems, which traditionally were mostly solved by analytical or algebraic means.

As a result, local math curriculum developers, headed by Dr. Kho and his team, had subliminally or subconsciously branded the "draw a diagram" strategy, by re-christening it, the "model method," or the "bar method," as it's commonly known locally.

Today, the Singapore model method is an integral part of the Singapore math curriculum, as it's being powerfully used as a visualization strategy to solve a number of challenging word problems at the elementary school level, which would normally require a formal algebraic knowledge to solve them.

3. Value math highly in society.

Numeracy is not an option.

Provide opportunities for those with math degrees to hold positions of power and influence in the country, who may act as role models. 

For instance, many key personnel in the Singapore civil service (e.g., members of Parliament, army heads, and school principals) and in the private sector (e.g., CEOs, pastors, and TV actors) have a strong mathematical background, or are quantitatively (and financially) literate. It's probably no surprise that Singapore has the world's highest percent of millionaires per capita, probably because of a significant number of its citizens and residents being blessed with an above-average financial quotient (FQ).  

On 6/9/11, I jokingly tweeted a new definition of "Singapore math":

@MathPlus: Singapore math: A term to describe the ambivalent governance of Singapore by a Prime Minister and a President, who are both math majors.

And, on 15/2/12, Republic of Math tweeted the following:

@republicofmath: Singapore president Tony Tan has applied math PhD. Prime Minister Lee Hsien Loong has math degree http://t.co/0UHDpIuE via @JohnAllenPaulos

It's probably not a coincidence to attribute good governance and economic success in Singapore to applied quantitative literacy, partly because at the top management in both government and the private sector, we've in place key personnel who can think rationally and logically—not easily swayed by race or religion politics, and who would make unpopular decisions for the benefit of the country even at the risk of not being elected in the next election.

Math as a Servant 

Just like we can love our enemies, without liking them (if we yield ourselves to the guidance of the Holy Spirit), so too can we learn to appreciate and love the most-disliked subject in school, without having quite a liking for it. Like money, we need to make math our servant, not our master—we can make math work for us, without being enslaved by it. 

© Yan Kow Cheong, July 18, 2013.

Expensive-Cheat Singapore Math Books

In recent years, a certain genre of mathematics titles had flooded the Singapore assessment (supplementary) market to meet the needs and wants of kiasu parents, who want to ensure that their children have a competitive edge over their peers.

One conspicuous series of such assessment titles in our local bookstores is published by Amazing Books. The Primary 1 - 3 titles (selling from $8.00 to $9.90) are written by a certain Newton Wong, while Primary 4 - 6 titles (selling from $11.00 to $12.50) are written by an Ernest Wong—it’s hard to confirm whether the writers are ghosts or not. These supplementary titles generally cost as much as a Ministry of Education (MOE)-approved primary textbook—however, they're value-for-money titles, if you're agnostic as to whether copyrights have been safeguarded or not.

Wallet-friendly titles that may meet the
mathematical needs and wants of kiasu parents

Indeed, there is much to be gained from writing these assessment titles, as compared to, say, authoring an MOE-approved textbook which goes through the tedious reviewing exercise by the Ministry of Education, the approving body of Singapore textbooks. 

Critics (mainly authors of traditional assessment books) arguably claim that these “cheat-expensive books” or "expensive-cheat books," as they’re called in local educational publishing, are nothing but a rehash of faintly modified past exam questions from the top or popular schools in Singapore.

High Price, Low Content

Another category of expensive-cheat titles involves those whose contents don't justify their selling prices—titles that over-promise and under-deliver. Come with quality paper, some look pretty good in form but shy of substance. 

Note however that the use of the word “cheat” doesn’t in any way suggest that the writers had plagiarized the contents, although it’s not uncommon for teachers and tutors to suspect that in some cases plagiarism could have inadvertently taken place. 

At best, these expensive-cheat titles provide information for readers new to the subject matter at a high price; at worst, these titles are copycats of pirated test papers or canned contents that have been put in a form that are readers-friendly.


Piracy Goes Online

Here is a typical ad spammed probably to tens of thousands of homes every few months:

Top Schools' Test Papers (Yr 20XX) on CD 

Hi Parents Give your child a head-start at school by Practice Doing Top Schools' Test Papers This will help your kids to Score !!! *Primary 1  -  Primary 6 ( SA1 & SA2 papers ) *Data is in scan-in PDF format *English, Maths, and Science *Answers also provided *Only $5 per subject !( Min order 3 subjects @$15 only ) **3-Year Series : P5 SA2 / P6 Prelim ( Yr 20XX - 20XX )     3 yrs x 3 subjects @$15 only ( total about 50 sets testpapers ) **Also available : Sec.1 - 4 SA2/Prelim  Maths :( Yr 20XX ) @ $10/subject Call Now !! XXXX XXXX * NB *  To unlist pls click < HERE > thank you. A kiosk outside a shopping mall, selling past exam papers from allegedly top schools. Plagiarism Begets More Plagiarism Grade 4 questions that mimic those set by teachers in "top schools"—these "model answers" are quasi-edited "solutions." It’s an irony that while many fly-by-night writers had modified questions from the top schools’ exam papers with cosmetic changes; yet, often times, some teachers from these same schools are equally guilty in lifting up large chunks of content from assessment titles, by photocopying and distributing entire chapters to their students and fellow colleagues. Plagiarism begets new levels of intellectual theft. Unscrupulous vendors know too well that the MOE would find it hard to crack down on these perpetrators of piracy. After all, few parents could detect the authenticity of these questions—whether or not they’re really set by teachers from allegedly top schools. Most questions look challenging to the clueless parents, who would buy anything which would help their children do well in the subject.  Two Wrongs Don't Make a Right If half-baked or poorly edited solutions isn't much of an issue, these tips-and-tricks titles may appeal to some students who can't afford private tuition. Lest their children lose out vis-à-vis their peers, most kiasu parents wouldn’t give a second thought to buying these pirated sets of exam papers. The rationalization is that since the MOE refuses to sell them, they’ve little or no choice but to procure them from illegal vendors. The means to laying their hands on these sought-after papers justifies the end—that these non-routine questions would prepare the child confidently for his or her school test or exam. With home delivery service of these syndicated or pirated papers at no extra costs, parents can now feel less guilty, as they could avoid being seen buying these illegal papers in public. And for IT-savvy consumers, they’d now order these papers online, at a cheaper price, if they want an electronic version of them. A Copycat of Top Schools’ Exam Papers A grades 7-8 supplementary title that may be suitable for drill-and-kill specialists—it's priced at S$24.95 in retails outlets. Unknown to many of these authors, the international mathematical community of mathematics educators often poke fun at the poor quality of our local assessment or supplementary mathematics titles. Many writers have probably not read what math educators out of Singapore are saying about their dear titles, especially when they're marketed overseas.  Leaving poor design and linguistic blunders aside, these expensive-cheat local titles are often an embarrassment to the image of Singapore mathematics publishing, as the nation strives to become an educational and publishing hub in Asia. In recent years, it's an open secret that questions from so-called top schools are actually the works of syndicates, which recruit Indian and mainland Chinese nationals, and cash-strapped undergraduates, to write math questions that mimic those posed by teachers from top schools. The Works of Syndicates No one is surprised that these expensive-cheat titles are a few folds pricier than the traditional assessment titles. These often ill-written supplementary titles, which make unverifiable claims to parents and teachers, will continue to thrive if the authorities don't get down to verifying the authenticity of the questions, and as long as the consumers continue to purchase them, pretending that they’re unaware that they're MOE-copyrighted materials. While the MOE authorities should take a share of the blame pie for not making schools' test papers available for sale, very often, it’s those teachers with an entrepreneurial spirit who are probably the source of these copyrights infringements. Who can have access to these exam papers if students and outsiders are not allowed to take them out of the school compound? The non-routine questions are suitable for grades 5-6 students, but the solution methods sometimes lack rigor, and a number of them are poorly presented and edited.  Although there is no formal investigation on these assessment titles claiming to contain contents that match the questions commonly set by Singapore's top schools, it doesn’t take a DNA scientist to figure out that most of these assessment papers were probably lifted up from test questions, which have undergone some cosmetic rewriting to avoid being caught (or sued) for blatant infringement of copyrights. The sale of most past exam papers from government schools is officially illegal; yet, every local knows how and where to buy them. The public knows that schools and the MOE don’t have the resources to sue the guilty parties. It wouldn't be surprising that some education officers are themselves part of the piracy syndicate. How would syndicates have had access to those past exam papers in the first place had they not been leaked out by teachers or school personnel themselves? Indecent Photocopying Over the years, I’ve personally heard from a number of assessment authors who had complained to the MOE that some top schools’ HODs had blatantly plagiarized their contents to be used in workshops and seminars, or that some teachers had made an indecent number of copies for an army of students. Of course, the standard answer from the MOE was that they’d look into the matter, and before you know it, the whole saga has faded away.  As long as we pay lip service to the intellectual rights of math writers and authors, and the infringement of  copyrights is condoned among teachers and students, the future of math publishing in Singapore doesn't look too bright. In fact, the quality of future Singapore math titles is likely to suffer further, as more and more non-math graduates are recruited as math editors in local publishing houses.  © Yan Kow Cheong, May 29,  2013.

The Chickens-and-Cows Problem

At the end of an earlier post entitled "The Chickens-and-Rabbits Problem," I tickled readers whether the following question could be solved using the Singapore model method, or the Sakamoto method.

Mr. Yan has almost twice as many chickens as cows. 

The total number of legs and heads is 184. How many cows are there?

On his "Johnny and Mary Do Maths" blog, Chris Patterson followed up with an algebraic solution to the word problem in a post entitled Visualisation versus Algebra?. He asked me whether I have a visual proof to the question, particularly one using the Sakamoto method.

I'm not sure whether I've succeeded in answering Chris's question, but here are three quick-and-dirty non-algebraic attempts to the chickens-and-cows problem.


Method 1 (Singapore model method)

From the model,

11 units = 184 + 3 ▌= 16 × 11 + (8 + 3 ▌)

If ▌= 1, then 11 units = 16 × 11 + 11

          1 unit = 17

2 units – 1 = 2 × 17 – 1 = 33

A quick check shows that for other integral values of ▌other than 1, 11 units cannot be an integer.

Hence, Mr. Yan has 33 chickens and 17 cows.


As a student of the Singapore model method, still trying to learn how to make wise use of this visualization strategy, especially when it lends itself well to a particular word problem, I would like to hear from members of the Singapore mathematical brethren about their alternative models in solving this chickens-and-cows problem, assuming that we're discussing this question with a group of grade 5 or 6 students—with no knowledge of Diophantine equations or advanced algebraic techniques.


Method 2 (Sakamoto method)

1. Grasp the relation

Let ① represent the number of cows, and △ be a variable quantity that is much less than ①.

                                           Chickens            Cows

                                              ② – △                ①

Number of legs                 (② – △) × 2         ① × 4

                                            = ④ – 2△            = ④

                                   _________________________

                                                           184

2. Diagram

3. Number sentences

② – △ + ④ – 2△ + ① + ④ = 184
⑪ – 3△ = 184
⑪ = 184 + 3△ = 16 × 11 + (8 + 3△)

If △ = 1, then ⑪ = 16 × 11 + 11 = 17 × 11
                       ① = 17
② – △ = 2 × 17 – 1 = 33

So, there are 17 cows and 33 chickens.


Unless there's a more elegant Sakamoto solution, I find that it doesn't differ much conceptually from its algebraic cousin.


Method 3 (Using the "Make a supposition" strategy)

Suppose there were exactly twice as many chickens as cows.

Then each group of 2 chickens and 1 cow would have a total of [(2 + 2 × 2) + (1 + 1 × 4)] = 11 legs and heads.

Now, 184 = 11 × 16 + 8

16 groups of 2 chickens and 1 cow would have a total of 16 × 11 = 176 legs and heads.

How many chickens and cows have 8 legs and heads altogether?

Clearly, 1 chicken and 1 cow have 8 legs and heads altogether.

So, Mr. Yan has (16 × 1 + 1) = 17 cows and (16 × 2 + 1) = 33 chickens.


Conclusion

With Chris's algebraic solution, we're sharing four methods of solution to the chickens-and-cows problem. Which method do you prefer? Which one would you use with your students? Share yours with the rest of us.


© Yan Kow Cheong, April 24, 2013.