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As a child, I made frequent use of a particular 'trick' in order to make short work of many different problems. The general form is to be presented a question which wants a definite (numerical) answer, where the question statement contains a free variable. The 'trick' was to recognize that the answer (which the teacher has told us exists by asking for it) doesn't depend on the free variable, so we now know that we can substitute any convenient value we like for the free variable and find the answer.

This FiveTrangles problem is an excellent example. The height of the non-shaded triangles is unspecified, so we know straight away that it doesn't figure in the answer to the question. Since it doesn't matter, we can set it to zero and replace this question with the trivial question of finding the area of a triangle with $base = 10$ and $height = 5$.

Does this technique have a name? Do problems which permit it have a name? Should such questions be avoided?

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  • $\begingroup$ I'm not sure I understand. Do you mean leaving the answer as an expression with a variable in it? $\endgroup$ – Karl Feb 23 '15 at 19:55
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    $\begingroup$ @Karl Another example: the question "$(x + 4)^2 - x^2 - 8x$ simplifies to a number. Find that number." can be solved with algebra, or it can be solved by plugging in x=0, since apparently the answer does not depend on x. $\endgroup$ – Chris Cunningham Feb 23 '15 at 20:35
  • $\begingroup$ That's a clearer example for me thanks. I get the triangle question too - presumably the $x$ cancels and isn't left as a variable in the expression. $\endgroup$ – Karl Feb 23 '15 at 21:13
  • $\begingroup$ More of a technique than just a 'trick', see en.wikipedia.org/wiki/Limiting_case_%28mathematics%29 $\endgroup$ – Keith Feb 24 '15 at 3:35
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    $\begingroup$ Somewhat related: for single-answer multiple choice problems, if two answers are equivalent, then they must both be wrong. $\endgroup$ – wchargin Feb 24 '15 at 5:21
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The heuristic described here is one manifestation of what Polya (1945) and others thereafter refer to as trying a special case. I do not know of a more specific term for the context that you have put forth, but this is often how one approaches a problem if you are initially unsure as to how to solve it in generality.

It is a good way to get an initial foot-hold on a problem, and especially useful in the context of some standardized tests. For example, consider the following:

Question: Player 1 flips a penny $n>0$ times, and Player 2 flips a penny $n+1$ times. What is the probability that Player 2 flips more heads than Player 1?

A. 1/2

B. 1/3

C. 1/4

D. 1/8

Since none of the choices depends on $n$, you can consider $n=1$ and see easily that the probability is $1/2$. This allows you to blaze through the question very quickly. (Indeed, this holds for all $n \geq 1$.)

Although testing well is an important life skill, your final question as to whether such problems are advisable is interesting. In my estimation, they can be good in the classroom as a way of scaffolding, as long as you press students (when appropriate) to explain why their reasoning holds for the general case.

As one small example, induction problems are done like this: You show the (hopefully easy) base case, and then complete the proof by setting up and using your inductive hypothesis.

A possible pitfall, though, is not only the absence of reasoning by students (as in the example question above) but also a possible reinforcement around misconceptions of proof by example. (I believe this relates to the recent query about counterexamples in MESE 7466.) For more about some of these misconceptions, check out work by Orit Zaslavsky (google scholar).

Lastly: The issue with guessing answers based on problem statements is pervasive in school mathematics. It ranges from only seeing examples of the form $a + b = \square$ and adopting a misunderstanding of the equal sign as an operator, to problems in which students just try to combine numbers in ways that might be "sensible" (without really engaging in sense-making). Cf. the classic:

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

Student: (thinking: 125+5, 125-5, and 125*5 are too big; it must be 125/5) The shepherd is 25.

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    $\begingroup$ I had considered trying a special case, but there's more going on than that. Something akin to trying the easiest case, and going with it, because all of the cases are the same (not that I understand why, but if it weren't true the question wouldn't have been put to me this way). That's a mouthful though. Thanks for the answer. $\endgroup$ – NiloCK Feb 24 '15 at 1:15
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    $\begingroup$ @NiloCK If you know why "all the cases are the same," then you could say that you are reducing to a special case (rather than just trying one). But there is (obviously) quite a leap between solving for one special case and understanding that the special case answers (or helps answer) a seemingly more general question. $\endgroup$ – Benjamin Dickman Feb 24 '15 at 1:25
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    $\begingroup$ The 'why' for me (as a kid) was that the problem setter put the question that way. That is, if all the cases weren't the same, then the question wouldn't (couldn't) have been asked. I knew at the time that this was 'unfair' information, but it's hard for a kid writing math contests to discard such information on the grounds that it was 'ill-gotten' or 'inadmissible'. $\endgroup$ – NiloCK Feb 24 '15 at 1:38
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    $\begingroup$ @NiloCK Benjamin addresses that "why" in the answer. Essentially, the why amounts to a guess that may or may not be true. A teacher could easily throw in questions where the real answer is "not enough information" or "none of the above" just to discourage that heuristic. In my opinion, discouraging it forces students to think through why the missing information doesn't matter, leading to a deeper understanding of the logic and the problem and the solution. For me, that sort of process is when I learn the most. $\endgroup$ – jpmc26 Feb 25 '15 at 4:28
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(continued from previous post)

In this next problem, the horizontal leg of the large right triangle and the vertical leg of the smaller right triangle are also not fixed, but setting each to zero won't lead to a solution at all: http://fivetriangles.blogspot.com/2013/04/56-obtuse-triangle-area.html (Never mind that getting in the habit of simply setting values to zero will lead to a "zero" in calculus.)

Finally, the following proof uses the identical triangle area concept as in the original "concave quadrilateral" problem, but the "trick" doesn't work at all: http://fivetriangles.blogspot.com/2013/10/104-equal-area-proof.html

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    $\begingroup$ You can combine both answers into one. There is no character limit (within reason) in answers. $\endgroup$ – Chris C Feb 23 '15 at 21:54
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    $\begingroup$ Stack exchange proscribed more than 2 links in one post if 0 ≤ rating < 10. $\endgroup$ – Five Triangles Feb 23 '15 at 22:11
  • $\begingroup$ You can't control the order in which answers appear, so calling another answer the "previous post" is not good. For instance, if this answer gets more points it will even appear above the other one. Call your other answer "my other post" rather than "previous" one. $\endgroup$ – KCd Feb 25 '15 at 11:28
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In my SAT tutoring, I refer to such problems as underspecified. For instance, consider this problem:

If $\frac{a}{2} + \frac{b}{3} = 0$, then find $\frac{a}{b}$.

I teach the student that since two variables are presented but only one equation, this makes the problem underspecified. In other words, a and b are not uniquely determined. This gives the student license to make up a number for one out of $a$ and $b$, then use the given equation to solve for the other variable, and finally substitute both $a$ and $b$ into the expression $\frac{a}{b}$ to obtain the answer.

There are also many examples of underspecified SAT geometry problems. Often the problem will ask the student to find the value of $a+b$ or some other expression involving two variables. Example: The Official SAT Study Guide (second edition) by CollegeBoard, page 799, problem 13.

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That some solutions are independent of a variable is not uncommon, but it does not necessarily trivialise a problem or negate its teaching value. The following problem presents similar possibilities, but we'd argue that solving it with such a "trick" is actually more circuitous than finding the answer directly (and may be more enlightening, too): http://fivetriangles.blogspot.com/2012/04/area-problem.html Here is a nice animation someone posted of the situation:

The shortcoming in such "tricks", we agree, is that they may lead to tunnel vision, but worse, their utilisation is at the user's risk.

(continued in next post)

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    $\begingroup$ I assume that you're the owner of the Five Triangles blog? Great to have you on the site; I love yours. Many beautiful questions. $\endgroup$ – NiloCK Feb 24 '15 at 0:56
  • $\begingroup$ Now that you have the necessary reputation, perhaps you could merge your two answers into one? Your other post (at this time) has 3 up-votes, whereas this one has only 1; so it would make better sense to merge into the higher voted one. (FWIW: I have not up-voted the other one, and will happily do so if I see them merged; in this way, deletion will not cause you any net-loss in reputation!) $\endgroup$ – Benjamin Dickman Mar 26 '15 at 5:50
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I don't know the answer to the questions about the potential names for such things.

As to whether such questions should be avoided: I think some types of these questions can maybe help students to think about quantifiers, or to really think about the concepts involved in the question.

I like Chris Cunningham's example: if you tell the student that $(x+4)^2 - x^2 - 8x$ simplifies to a number, and ask them to find the number, they may just do the algebra of simplifying. But if you show them that you can get the answer by plugging in $0$, then you can teach them something about what it really means to say that "$(x+4)^2 - x^2 - 8x$ simplifies to a number". Now you can talk about what it means for a statement to hold "for all $x$". Things like this are useful to be familiar with. Many college students struggle with this when I ask them, for example, to show that a function is not linear!

It's also nice to have this kind of logic on hand for when you need to remember things. For example, say you were taking a linear algebra exam and you knew the determinant was either the sum or the product of the eigenvalues. Well, you could just consider the $2\times2$ identity matrix and figure it out pretty quickly! This is because you knew the statement didn't depend on the matrix or even the size of the matrix!

I (and this is just opinion) don't like questions like the triangle one that you linked to. Because there, the "trick" from the solver's perspective is just to realize that the question asker left out some words! I don't know how useful it is to have someone apply the logic "If they asked me this, it has an answer, and if they didn't say it depended on quantity $A$ then it must not depend on quantity $A$.'' It is not clear to me what the mathematical benefit is to that.

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  • $\begingroup$ RE: last paragraph. One benefit to the problem solver/student is to consider the importance of the givens; alternatively, a benefit to the problem poser/instructor is to consider more carefully how problems are phrased. $\endgroup$ – Benjamin Dickman Feb 25 '15 at 2:51
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I would be surprised if the solution strategy or problem types have a name. While it can be useful to find the correct answer, what does it accomplish from a learning perspective? Take, for instance, Chris Cunningham's problem of "$(x+4)^2 - x^2 -8x$ simplifies to a number. Find that number." As an instructor, I would be looking to make sure the students accomplished the appropriate learning outcomes for the problem. For this problem, I would more than likely be looking for (1) can they multiply out $(x+4)^2$ and (2) can they simplify the addition of terms. Such a technique would not accomplish (1) or (2), so I would avoid writing problems that can (knowingly) make use of it nor would I ever teach such meta-strategies.

This reminds me of a multiple choice exam question on trig identities: "Which of the following is an equivalent identity to $\cot(x) + \tan(x)$?" with $\csc(x)\sec(x)$ being in the list of answers. Instead of using trig identities, many students plugged in a value like $x=10$ and compared. While it works for this case, it dodges the actual learning and testing of the concept.

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Discovering the irrelevance of $x$ in the expression could be achieved by getting the students (perhaps in different groups) to substitute different values in for $x$ and compare results. Was this what they expected? Which number would they choose to substitute knowing the answer will always be the same? Having aroused their curiosity the expanding brackets and simplifying can be motivated and discussed. This could be a rich activity if the emphasis is on discovery and not just sub in $x=0$

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