Tag Archives: ratio

The Pseudo-Proportional Problem

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On May 23, 2023, @OrwellNGoode tweeted the rate question below, with the “teacher” marking the student’s answer of 20 minutes wrong, and showing why it’d be 15 minutes instead.

Who the heck takes 10 minutes to saw a board?

I’ve no idea what grade level this math question was being assigned to. Although questions on ratios and rates are formally introduced in grades 5–6 in most parts of the world, however, it’s not uncommon to spot these types of mathematical quickies in grades 1–4 Singapore math olympiad papers to trap the unwary.

Assuming that the word problem didn’t come from a bot or from ChatGPT, the teacher’s intuitive reasoning may be said to be ratio-nally sound but rationally incorrect in solving this pseudo-proportional math question.

Logically or mathematically speaking, few math teachers would disagree that the student was right and the teacher was wrong.

This arguably “badly worded” or “ill-posed” math question provides a fertile ground for a number of possible (valid or creative?) answers, probably much to the annoyance of most math teachers and editors, who often feel uncomfortable or jittery about questions with more than one possible answer.

Indeed, there is no shortage of supporters to defend a “15 minutes” answer. For example, since there is zero mention that the length of each board is of equal length (and we can’t assume it to be so), or as each sawing might take place in a different direction, the “logical” answer of 20 minutes can’t be taken as mathematical gospel truth.

That three pieces need two cuts or sawings is unanimous among problem solvers. The bone of contention is the assumed length of the second cut. Say, if the second cut was half as long as the first one, then it’d take half the time of the first cut, in which case the answer of 15 minutes would be practically plausible.

A creative solution from @A_MGregory

It looks like we’re only limited by our imagination or creativity to rationalize why the answer can’t in practice be 15 minutes or any other duration, by using a different (creative) reasoning from the flawed one provided by the “teacher.”

Like most artificial or impractical word problems in school math, this rate question debatably falls short of design thinking and is thus open to different interpretations or assumptions, which might also weaponize some “anti-woke” math educators to ban or censor these types of “confusing or tricky” math questions.

Ironically, this is why injecting a dose of realism or creativity to these oft-ill-posed or contrived math questions would help open up the minds of uncritical or unquestioning math educators.

Don’t just answer the questions, question the questions.

© Yan Kow Cheong, May 29, 2023.

A Before-and-After Singapore Math Problem

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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:

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© 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.

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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.

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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.

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.

© Yan Kow Cheong, August 4, 2013.

The Dolls Problem à la Singapour

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Following a request from a Linkedln friend to provide a solution that makes use of the Singapore model method to the question below—I couldn’t trace the origin of this word problem—here’s a quick-and-dirty sketch of a five-model-drawing solution.

Jazmine buys and sells antique dolls on the Internet. Yesterday, she focused on dolls from the Civil War period. She began the day by selling one-fourth of her dolls from that period. Then she sold six more. Just before lunch she sold one-fourth of the remaining Civil War dolls. After lunch, she bought some Civil War dolls, increasing her collection by one-sixth. Then she bought some more, doubling her collection. Just before she quit for the day, she sold two thirds of her Civil War dolls. After all that, she had fourteen of these dolls left. How many dolls did Jazmine have before she began trading yesterday?

It wouldn’t be surprising that this kind of brain-unfriendly word problem, set in a test or exam, might give some un-mathophobic grade five or six students sweaty palms, or goose pimples, if they started feeling clueless after attempting to solve it for some five odd minutes!

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A quick-and-dirty solution that makes use of the model method.

Using the “work backwards” strategy repeatedly, the model drawings show that Jazmine had 40 dolls before she began trading yesterday.

If you’re an “algebraic freak,” by all means, use algebra to check your answer—I decided to give the algebraic approach a miss this time round.

Disproportionate parts or units

Notice that I’ve loosely used “units” and “parts” alternately to represent each model drawing. And I’ve also used each unit, or part, in a rather disproportionate manner, as compared to textbooks’ modeled solutions, which generally depict the bars (or rectangles) proportionately, based on their respective numerical values—which is secondary to the reasoning or thinking processes.

The above dolls problem is similar to a question I discussed in an earlier post, except that the present one is slightly harder; otherwise, it adopts the same problem-solving strategies for  its solution.

Sakamoto and Stack Methods

My next task is to check whether the Stack method or the Sakamoto method to the above word problem is conceptually “friendlier” than the model method. Are there intuitive or elegant solutions other than the one that embraces the bar method? Meanwhile, please send us your solution(s) to the dolls problem.

© Yan Kow Cheong, March 27, 2013.

Postscript: Although math was my favorite subject in school, I don’t recall solving questions similar to the above word problem. I doubt if I would be able to solve it when I was in grade five or six. It looks like this present younger generation has been given the shorter end of the mathematical stick—worse, if math happens not to be their cup of tea! It’s no surprise that strangers, young and old, angrily tell me of their negative mathematical experiences in school—how they disliked math (and their math teachers).