If you had been on Facebook for some time, it’s very likely that you would have come across a certain type of numbers puzzles streaming your feed. Clearly, they’re not quite the typical numerical puzzles that often appear in school math books; rather, they’re closer to the types of questions posed in aptitude or IQ tests. One such number puzzle is the following.
Unlike in textbooks that often present these logic puzzles in an uninteresting way, by seeing these colorful number puzzles on Facebook or Pinterest, and being hinted that only a small percent of the problem solvers apparently managed to get the correct answer, this entices readers to give it a try to see how well they’ll fare vis-à-vis their oft-mathematically challenged Facebook friends. Here’s another such number puzzle.
Arguably, these numbers-and-words puzzles have a certain charm to it, because they often require just simple logic to solving them, unlike similar brainteasers that may require some knowledge of elementary or middle-school math. What about the following Facebook numerical puzzle?
What is 1 + 1?
What do you make of this one?
If these on-line numbers puzzles indirectly help promote logic and number sense among math-anxious social-media addicts; and along the way, provide them with some fun and entertainment, let’s have more of them in cyberspace!
Let me leave you with half a dozen of these numerical puzzles, stolen from my Facebook feed. Note that these types of brain-unfriendly math or logic questions may have more than one mathematically or logically valid answer, depending on the rule or formula you use—they serve as numerical catalysts for promoting creative thinking in mathematics.
In my formative years, I don’t recall any elementary school teachers sharing some mathematical puns with us. I suppose that the arithmetic of yesteryear was often taught in the most uninteresting way by many who probably didn’t look forward to teaching it to a bunch of noisy kids.
Recently, while reading Stewart Francis’s Pun direction: Over 500 of his greatest gags…and four crap ones!, I came across a number of numerical puns that might even be appreciated by some nerdy seven-year-olds. Stewart Francis is considered to be the best Canadian comedian from Southern Ontario. Here are two dozen odd math-related puns I’ve stolen from his punny book. Hope you enjoy them!
The number of twins being born has doubled.
They also stole my calculator,
which doesn’t add up.
Four out of ten people are used in surveys. Six are not.
Crime in lifts is on the rise.
“QED Gravestone Small Poster” www.cafepress.com
I recently overcame my fear of calculators.
It was a twelve-step program.
Women are attracted to foreign men. I’ve heard that at least uno, dos, tres times.
All seventeen of my doctors say I have an addictive personality.
There’s a slim chance my sister’s anorexic.
Truthfully, we met at a chess match, where she made the first move.
Is my wife dissatisfied with my body? A tiny part of me says yes.
I read that ten out of two people are dyslexic.
From: wherethepunis.com
They now have a website for stutterers, it’s wwwdotwwwdotwwwdotwwwdotdotdot.
I have mixed race parents, my father prefers the 100 metres.
I’m the youngest of three, my parents are both older.
Clichés are a dime a dozen.
I’m an underachiever 24-6.
I used to recycle calendars. Those were the days.
www.cafepress.com
I’ve learned two things in life. The second, is to never cut corners.
‘Any man who lives his life in accordance to a book is a fool.’ Luke 317
I can say ‘No one likes a show off’ in forty-three languages.
I was once late because of
high-fiving a centipede.
Of the twenty-seven students in my maths class,
I was the only one who failed.
What are the odds of that, one
in a million?
I’ve met some cynical people in my twenty-eight years.
I was good at history.
Wait a minute, no, no I wasn’t.
I was terrible at school. I failed maths so many times, I can’t even count.
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?
Other than using a guess-and-guess strategy and an algebraic method, both of which offering little pedagogical or creative insight, let me repeat below one of the two intuitive methods I discussed then.
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 in order for the total number of legs to be equal.
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
The Bicycles-and-Tricycles Problem
Again, if we decided to ban any trial-and-error or algebraic method, how would you apply the intuitive method discussed above to solve a similar word problem on bicycles and tricycles?
There are 60 bicycles and tricycles altogether. The bicycles have 35 more wheels than the tricycles. How many bicycles and tricycles are there?
Go ahead and give it a try. What do you discover? Do you make any headway? In solving the bicycles-and-tricycles question, I find that there are no fewer of half a dozen methods or strategies, which could be introduced to elementary school students, three of which lend themselves easily to the model, or bar, method, excluding the Sakamoto method.
Here are some pointers I would share with my students at the start of a secondary math olympiad programme. In Singapore, most mathletes attending these enrichment classes are usually selected by the form or math teacher, who tends to choose the best three math students from each class to form a small group of 15 to 20 participants. They would then graduate to represent the school after attending a six-, eight-, or ten-session training programme, depending on the mathematical needs and wants of the school.
• Take your time! Very few contestants can solve all given problems within the time limit. For instance, in the Singapore Mathematics Olympiad (SMO), both at the junior and senior levels, [unconfirmed] feedback based on different schools’ results hints to the fact that those who can win a medal hover around five percent.
Interestingly but disturbingly, an SMO mathlete who can get six or seven out of 35 questions correct may still win a bronze medal, revealing how unmoderated olympiad math papers had been in recent years, going by the abnormally high rate of failures among the participants. This is primarily due to the fact that few, if any, faculty members who set these competition papers, are familiar with what elementary and middle school teachers are covering in local schools.
• Try the “easier” questions first. The questions aren’t set in ascending order of difficulty. It’s not uncommon to see easier questions in the second half of the paper.
• Olympiad questions don’t “crack” immediately. Be patient. Try various approaches. Experiment with simple cases. Working backwards from the desired result in some cases is helpful.
• If you’re using a contests book, and you’re stuck, glance at the “Hints” section. Sometimes a problem requires an unusual idea or technique.
• Even if you can solve a problem, read the hints and solutions. The hints may contain some ideas or insights that didn’t occur in your solution, and they may discuss intuitive, strategic, or tactical approaches that can be used elsewhere.
Remember that modeled or elegant solutions often conceal the torturous or tedious process of investigation, false starts, inspiration and attention to detail that led to them. Be aware of the behind-the-scenes hours-long dirty mathematical work! When you read the modeled solutions, try to reconstruct the thinking that went into them. Ask yourself, “What were the key ideas?” “How can I apply these ideas further?”
• Go back to the original problem later, and see if you can solve it in a different way, or in a different context. When all else fails, remember the reliable old friend, the guess-and-check strategy (or heuristic, as it’s being arguably called in Singapore)—for instance, substituting the optional answers-numbers given in an MCQ into some given equation or expression may yield the answer sooner than later.
Meaningful or creative problem solving takes practice, with insightful or elegant solutions not being the norm. Don’t get discouraged if you don’t seem to make any headway at first. The key isn’t to give up; come back to the question after a day or a week. Stickability and perseverance are two long-time buddies for full-time problem solvers.
A rectangular enclosure is 30 meters wide and 50 meters long. Calculate its area in square meters.
Solution
The My Pals Are Here Math Series
The My Pals Are Here series has been rumored to have been edited and ghostwritten by a hundred odd editors and freelancers in the last decade.
Lack of mathematical rigor was initially targeted against Dr. Fong Ho Kheong and his two co-authors by American profs in the first or/and second editions —probably by those who were “ghost advisors or consultants” for Everyday Math.
Deconstructing the Singapore Model Method
1. It’s a problem-solving strategy—a subset of the “Draw a diagram” strategy.
2. It’s a hybrid of China’s line method and Russia’s box (or US’s bar) method.
3. It’s the “Draw a diagram” strategy, which has attained a brand status in mathematics education circles.
4. It’s a problem-solving method that isn’t recommended for visually challenged or impaired learners.
5. It allows questions traditionally set at higher grades (using algebra) to be posed at lower grades (using bars).
Painless Singapore Math
Perhaps that’s how we’d promote Singapore math to an often-mathophobic audience!
A Singapore Ex-Minister’s Math Book
Dr. Yeo had the handwriting below depicted in his recreational math book, regarding his two granddaughters, Rebecca and Kathryn.
Stealing the idea from Nancy Vu’s “Just Little Things,” I tried to reflect on some mathematical things that give us delight and pleasure as we go about our hectic 24/7/365 lives.
Here are some of my personal mathematical pleasures.
It’s now your turn to reflect on some aha! mathematical moments to share with the rest of us.
References
Wu, N. (2013). Just little things: A celebration of life’s simple pleasures. New York: A Perigee Book.
In Singapore, in grades four and five, there is one type of word problems that seldom fail to appear in most local problem-solving math books and school test papers, but almost inexistent in local textbooks and workbooks. This is another proof that most Singapore math textbooks ill-prepare local students to tackle non-routine questions, which are often used to filter the nerd from the herd, or at least stream the “better students” into the A-band classes.
Here are two examples of these “excess-and-shortage word problems.”
Some oranges are to be shared among a group of children. If each child gets 3 oranges, there will be 2 oranges left. If each child gets 4 oranges, there will be a shortage of 2 oranges. How many children are there in the group?
A math book costs $9 and a science book costs $7. If Steve spends all his money in the science books, he still has $6 left. However, if he buys the same number of math books, he needs another $8 more. (a) How many books is Steve buying? (b) How much money does he have?
ANumerical Recipe
Depicted below is a page from a grade 3/4 olympiad math book. It seems that the author preferred to give a quick-and-easy numerical recipe to solving these types of excess-and-shortage problems—it’s probably more convenient and less time-consuming to do so than to give a didactic exposition how one could logically or intuitively solve these questions with insight.
A page from Terry Chew’s “Maths Olympiad” (2007).
Strictly speaking, it’s incorrect to categorize these questions under the main heading of “Excess-and-Shortage Problems,” because it’s not uncommon to have situations, when the conditions may involve two cases of shortage, or two instances of excess.
In other words, these incorrectly called “excess-and-shortage” questions are made up of three types:
・Both conditions lead to an excess.
・Both conditions lead to a lack or shortage.
・One condition leads to an excess, the other to a shortage.
One Problem, Three [Non-Algebraic] Methods of Solution
Let’s consider one of these excess-and-shortage word problems, looking at how it would normally be solved by elementary math students, who have no training in formal algebra.
Jerry bought some candies for his students. If he gave each student 3 candies, he would have 16 candies left. If he gave each student 5 candies, he would be short of 6 candies. (a) How many students are there? (b) How many candies did Jerry buy?
If the above question were posed as a grade 7 math problem in Singapore, most students would solve it by algebra. However, in lower grades, a model (or intuitive) method is often presented. A survey of Singapore math assessment titles and test papers reveals that there are no fewer than half a dozen problem-solving strategies currently being used by teachers, tutors, and parents. Let’s look at three common methods of solution.
Method 1
Difference in the number of candies = 5 – 3 = 2
The 16 extra candies are distributed among 16 ÷ 2 = 8 students, and the needed 6 candies among another 6 ÷ 2 = 3 students.
Total number of students = 8 + 3 = 11
(a) There are 11 students.
(b) Number of candies = 3 × 11 + 16 = 49 or 5 × 11 – 6 = 49
Jerry bought 49 candies.
Method 2
Let 1 unit represent the number of students.
Since the number of candies remains the same in both cases, we have
3 units + 16 = 5 units – 6
From the model,
2 units = 16 + 6 = 22
1 unit = 22 ÷ 2 = 11
3 units + 16 = 3 × 11 + 16 = 49
(a) There are 11 students.
(b) Jerry bought 49 candies.
The difference in the number of candies is 5 – 3 = 2.
The extra 16 candies and the needed 6 candies give a total of 16 + 6 = 22 candies, which are then distributed, so that all students each received 2 extra candies.
The number of students is 22 ÷ 2 = 11.
The number of candies is 11 × 3 + 16 = 49, or 11 × 5 – 6 = 49.
Similar, Yet Different
Feedback from teachers, tutors, and parents suggests that even above-average students are often confused and challenged by the variety of these so-called shortage-and-excess problems, not including word problems that are set at a contest level. This is one main reason why a formulaic recipe may often do more harm than good in instilling confidence in students’ mathematical problem-solving skills.
Here are two grade 4 examples with a twist:
When a carton of apples were packed into bags of 4, there would be 3 apples left over. When the same number of apples were packed into bags of 6, there would still be 3 apples left over. What could be the least number of apples in the carton? (15)
Rose had some money to buy some plastic files. If she bought 12 files, she would need $8 more. If she bought 9 files, she would be left with $5. How much money did Rose have? ($44)
Conclusion
Exposing students of mixed abilities to various types of these excess-and-shortage word problems, and to different methods of solution, will help them gain confidence in, and sharpen, their problem-solving skills. Moreover, promoting non-algebraic (or intuitive) methods also allows these non-routine questions to be set in lower grades, whereby a diagram, or a model drawing, often lends itself easily to the solution.
References
Chew, T. (2008). Maths olympiad: Unleash the maths olympian in you — Intermediate (Pr 4 & 5, 10 – 12 years old). Singapore: Singapore Asian Publications.
Chew, T. (2007). Maths olympiad: Unleash the maths olympian in you — Beginner (Pr 3 & 4, 9 – 10 years old). Singapore: Singapore Asian Publications.
“The Parrot’s Theorem” (translated from “Le Théorème du Perroquet”) was an instant bestseller in France when it was published in 1998.
Sylvia Nasar’s A Beautiful Mind and G. H. Hardy’s A Mathematician Apology are two nonfiction mathematical classics for both mathematicians and mathematics educators. Lesser known are the mathematical novels which often feature characters whose speciality is number theory, also known as higher arithmetic, and elevated as math’s purest abstract branch.
If you’re looking for math, women, sex, and back-stabbing, The Wild Numbers is a math melodrama unlikely to disappoint.
Winner of the New South Wales Premier’s Prize
Fictional math
Who are these mathematical fiction books targeted? Math and science teachers? Educated laypersons? Pure mathematicians may like to read them, yet at the same time they may complain that the mathematics discussed in these books is anything but rigorous.
These books seldom fail to convey the following subtle messages:
• The thin line between mathematical genius and madness.
• The search for mathematical truth at all costs, and the heavy price of finding it.
• The arrogance and pride of pure mathematicians who look down on their peers, most of whom work as applied mathematicians and research scientists.
• The relatively high divorce rate among first-rate mathematicians as compared to their peers in other disciplines.
• Mathematics is apparently a young’s man game; one has past one’s prime if one hasn’t written one’s best paper by the age of 40.
• Mathematicians are from Mars; math educators are from Venus.
• Pure mathematicians (or number theorists) are first-rate mathematicians; applied mathematicians are second- or third-rate mathematicians. To the left of the “mathematical intelligence” bell curve are math educators from schools of education, and high-school math teachers.
“Reality Conditions” is collection of 16 short stories, which is ideal for leisure reading—it’s suitable for promoting quantitative literacy, or it’d serve as the basis for a creative course on “Mathematics in Fiction.”
The joy of reading mathematics
Let’s rekindle the joy of appreciating mathematics for mathematics’s sake. Let’s welcome poetry, design thinking, and creativity, whatever ingredient that may help to draw the community into recognizing and appreciating the language of science and of technology. These “pure-math-for-poets” titles have a place in our mathematics curriculum, as they could help promote the humanistic element of mathematics.
Here are ten titles you may wish to introduce to your students, as part of a mathematics appreciation or enrichment course.
The New York Times Book of Mathematics
The Best Writing on Mathematics 2010
Clifton Fadiman’s Fantasia Mathematica
Clifton Fadiman’s The Mathematical Magpie
Don DeLillo’s Ratner’s Star
Edwin A. Abbott’s Flatland
Hiroshi Yuki’s Math Girls
John Green’s An Abundance of Katherines
Philip J. Davis’s The Thread: A Mathematical Yarn
Thomas Pynchon’s Gravity’s Rainbow
Juvenile fiction—A child prodigy and his friend tried to create a mathematical formula to explain his love relationships.
References
Green, J. (2006). An abundance of Katherines. New York: Dutton Books.
Guedj, D. (2000). The parrot’s theorem. London: Orion Books Ltd.
Kolata, G. & Hoffman, P. (eds.) (2013). The New York Times book of mathematics: More than 100 years of writing by the numbers. New York: Sterling.
Hiroshi, Y. (2011). Math girls. Austin, Texas: Bento Books.
Pitici, M. (ed.) (2011). The best writing on mathematics 2010. Princeton, New Jersey: Princeton University Press.
Wallace, D. F. (2012). Both flesh and not: Essays. New York: Little, Brown and Company.
Woolfe, S. (1996). Leaning towards infinity: A novel. NSW, Australia: Random House Australia Pty Ltd.
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 SeventhMonth 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”?
Reference Yan, K. C. (2006). Aha! math! Singapore: SNP Panpac Education.
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 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.
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.
Practice
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?
Answers
1. $300 2. $86 3. $196
Reference
Walker, L. (2010). Model drawing for challenging word problems: Finding solutions the Singapore way. Peterborough, NH: Crystal Springs Books.
