A Vintage Age Problem

While surveying some age problems in Joseph Degrazia's Maths Is Fun, I came across the following 1949 word problem.

Bill is twice as old as Sam was when Bill was as old as Sam is. 

When Sam is as old as Bill is now, both of them together will be 90. 

How old are they?

Since no solution was given in the book, I assume that the author then would most likely expect the problem solver to use an algebraic approach to solving it.


Language and Logic

I needed to read the question a few times, before I felt confident that I was in control of the problem situation. That made me believe that one major obstacle for students attempting to solve this vintage problem would be to really understand what the question is all about—how the ages of Bill and Sam are related at different times.

However, regardless of the method of solution used, this word problem might also prove to be quite a challenge to those who are learning English as a second or third language.


The Model Method as an Ideal Choice

Among competing non-algebraic approaches, especially if this question is assigned to grade 5 or 6 students, my methodological bias in favor of the Singapore model method convinces me that a visual approach would offer an elegant, if not better, solution to the problem, as compared to an algebraic method of solution; in fact, the word problem does lend itself quite well to the model, or bar, method. Try solving the question, using the model method, on your own first, before peeping at a suggested model-drawing solution below—you may even come up with a better model than mine.

Here's a quick-and-dirty Singapore model solution to the above vintage age problem.

Observe that the age difference between Bill and Sam is always one unit.

From the model,

9 units = 90

1 unit = 90 ÷ 9 = 10

3 units = 3 × 10 = 30 ➝ Sam

4 units = 4 × 10 = 40 ➝ Bill

Bill is 40 years old.

Sam is 30 years old.

When Bill was 30, Sam was 20.

When Sam is 40, Bill will be 50 and both will be 90.

An inexpensive supplementary title to learn
the Singapore model method confidently. 
Visit singaporemath.com

Conclusion

Even for students who like math, the trouble with this age problem isn't just about exhibiting clear thinking and reasoning, but also about language proficiency. Language and logic, particularly in this case, each play an important part in the thinking process in producing the correct product. Moreover, a model-drawing solution to this vintage problem also shows that this question, which would normally be solved algebraically at a higher level, could be posed to students at a lower grade.


Reference
Degrazia, J. (1949). Maths is fun. London: George Allen and Unwin Ltd.

© Yan Kow Cheong, April 12, 2013

Immoral Algebra

Immoral ALGEBRA

Here is an online algebra question, forwarded to me almost every year, which has been circulating the globe many times:

Maya is 21 years older than her son Raju.

In 6 years from now Maya will be 5 times as old as Raju.

Question: Where's Maya's husband?

My reply to the sender was: My limited algebraic knowledge leads me to infer that the "normal-child-to-be" was some one year away to see the world (Answer: x = –1). Challenge: Try solving this problem using the Singapore model method.

(

So I can only conclude that the "conservative couple" was in some "missionary position" then.

My sender's forwarded answer was: 

Current age of Raju is -0.75 ie -9 months ie he is 

being conceived therefore husband is on top of wife

And my reply following the e-mailer's answer was:

 

I bet is that if we've a few of these CRE8TIVE word problems in our traditionally sterile MOE-approved textbooks, more students would be excited to reading algebra until they're old enough to have kids of their own. Students would be begging teachers and tutors for more algebra and more word problems – two bêtes-noires of school mathematics. Who knows? This may turn out to be an inexpensive cure to addressing our declining birthrate! When most students' favorite four-letter word would then be MATH MATH MATH ... and MORE MORE MORE!

Stripping Off Your Maths

The above online algebra word problem reminds me of a math workshop organized by the Singapore Science Centre in late eighties, when the speaker-and-author, Mr Ang Tok Woon, shared with fellow teachers that a good metaphor could help weak or uninterested students recall mathematical concepts or formulas rather effectively.

He used the idea of ‘stripping off’ some unknowns when teaching the topic on "Subject of Formula" to a group of academically challenged [ITE or SAF mature] students. For instance, given E = mc², ‘making c the subject of formula’ means expressing c in terms of E and m. And he was surprised that years later when some of his ex-students met him, they reminded him that they had forgotten everything else he taught them except for that part on stripping off.

Sexing Up the “Sixth Factor”

Only a lack of imagination and creativity would deprive hundreds of thousands of pubescent teenagers from experiencing the joys (and pains) of algebra.

As we explore how best to incorporate the sixth factor — the ‘context knowledge’ — as suggested by Assoc. Prof. Wong Khoon Yoong from the National Institute of Education, Singapore, to extend the present two-decades-old Singapore Mathematics Curriculum Framework comprising of the five factors (Skills, Attitudes, Concepts, Processes and Metacognition), diagrammatically placed along the sides of a pentagon, I think posing ‘real-world problems’ with a recreational or cultural flavor would go a long way to arousing the algebraic interests of our students who might otherwise give algebra a miss, once it becomes an optional topic. 

Indeed, we need a fertile dosage of algebra to debunk the myth that the common language of mathematics is difficult and boring. Algebra, or any topic in mathematics, for that matter, is amoral and neutral. Mathematics can only be boring as long as we’ve boring teachers and writers (and editors, math consultants and math specialists) — or all five — churning out unexciting textbooks. However, matters would have been a notch above boring if we had had fewer boring reviewers and curriculum specialists.

Some Nudity May Promote Numeracy 

Another episode proves that infusing some perceived elements of decent nudity – art for mathematics’ sake – to enhance numeracy (quantitative literacy, as it is commonly known in the US) may not be a far-fetched idea. Years ago I was attending a two-day teachers’ course entitled Andragogical Principles, an in-service program for those teaching working adults or mature students, when the lecturer confided to us that by letting his students write about their [wild?] fantasies, they actually improved their language proficiencies. In fact, I wouldn’t be surprised that some of them are now working as freelance writers or editors for some adult publications!

Indeed, numeracy spiced with some exciting part of human biology may act as a catalyst (‘mathematical aphrodisiac’) to attracting even mathematically dull students to sign up for some extra-mathematical activities to enhance their quantitative literacy. As mathematics educators, we critically need creative methods to promote our mathematical trade; otherwise, we’ll end up losing potential candidates with a mind for the abstract to major in other disciplines that often promise more tangible rewards.

MATH$ I$ MONEY

Like guerrilla marketers, we need to go all out to promote and sell the idea that MATHEMATIC$ I$ MONEY! Some born with the ‘mathematical gene’ definitely stand a high probability of becoming millionaires, by cracking some unsolved mathematical nuts – for example, a few million-dollar prizes are on offer by the Clay Mathematics Institute to whoever can find the solution to some fiendishly difficult mathematical puzzles. For the majority of us, we can at least hope to secure some decent jobs thanks to the logical, analytical and deductive skills acquired through years of toying around with notions and notations.

If the oldest profession on earth continues to thrive because of the money factor, there is no reason why it cannot share a common denominator with the most disliked subject in school. With some financial carrots, more people would be motivated to speak the language of science and technology. Like sex, mathematics is money — and it can be enjoyable, too!