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This is a detail, but getting the details right means a lot when you design your course. For the purposes of this question please assume calculators are not allowed in the course, which is true in many college calculus courses and is a debate I do not wish to have here.

I've tried things along the spectrum, from

Essentially never simplify. Simplifying doesn't matter unless it is useful. Is your answer $\frac{1}{\sqrt{256}}$? Fine, circle it. Finding the square root of 256 is not the point of this class and in the real world you won't need to know it.

to

Essentially always simplify. If you don't simplify, it betrays a lack of knowledge about what is going on. Circling $\frac{1}{\sqrt{256}}$ is embarrassing and means you haven't completed the problem.


I find it difficult to provide a well-defined policy inbetween the two extremes, but I don't like either extreme:


If simplification is not required, then students have no external incentive to simplify something like

  • $\frac{\sqrt4 - 5}{3}$

but they realize that if they simplify it incorrectly, they will lose points. The answer is -1, and the students should be able to turn that into -1. A "no-simplification-required" policy deirectly incentivizes students to circle something like that as their final answer.


On the other hand, if simplification is required, then we are spending time and effort on computational issues that really don't matter in the end, and drawing a line of what needs to be simplified is difficult. Is $\sec^2 x - 1$ simplified enough? What about $\sin(\frac{x}2)\cos(\frac{x}2)$? What about $\frac{x - 1}{\sqrt{x} - 1}$? Do I really want them worrying about any of this? I'd rather they spend more time on the content of the course.


Have you found a good simplification policy for college courses without calculators?

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  • $\begingroup$ For the numerical examples, I might ask for an answer to the nearest integer, which incentivizes simplification without requiring it. For the formulas, I'd say that $\sec^2x-1$ and $\sin(x/2)\cos(x/2)$ are good enough. However, $(x-1)/(\sqrt{x}-1)$ may not be good enough, because it's not defined at $x=1$, while $\sqrt{x}+1$ probably has the right domain of definition. $\endgroup$ – user173 Jul 2 '14 at 12:52
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    $\begingroup$ For me it depended on the situation, but one thing I always tried to do was to be very explicit as to what the final form of the answer was to be rather than say "simplify". For example, one of the expressions you gave, $\frac{\sqrt4 - 5}{3},$ would be considered simplified by many people. I would say things like "express as a quotient of two integers in reduced form" (for numerical fraction manipulations), "express using a single fraction in reduced form" (for algebraic fraction manipulations), etc. $\endgroup$ – Dave L Renfro Jul 2 '14 at 14:10
  • $\begingroup$ it is a continual frustration of mine that when students are told "do not simplify" they continue to do such like so many robots. Even in differential equations where I explicitly in bold CAPS say, do not calculate, just set-up, they still waste a bunch of time doing algebra I explicitly said to not do. I warn them ahead of time, doing that algebra does not impress me, it just shows me you can't read instructions. But, still, it happens. For simplification in calculus, I advocate a two-pronged approach, some simplify, some not. You can have both. $\endgroup$ – James S. Cook Jul 2 '14 at 15:19
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    $\begingroup$ Finding the square root might not be the point of the class, but answering the posed question is. Answering a question (how many apples are there?) with a compound expression such as $\sqrt{256}$ or $27-11$ is, in a sense, correct, but it's also a conversational non-sequitur. $\endgroup$ – NiloCK Jul 2 '14 at 15:34
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    $\begingroup$ What about the catch-all "Justify your answer" - if they have it in a certain form, and they can justify why its in that form, then that's great right? $\endgroup$ – corsiKa Jul 2 '14 at 17:13
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If you use online homework, then when it comes to symbolic answers you don't have much wiggle room. These systems generally work by generating random numbers as the values of the variables that appear in an expression, evaluating the expression, and comparing the numerical outputs. (I believe the folks developing the LON-CAPA system started out this way and then augmented it with some CAS strategies.) The software doesn't care whether the answer is entered as $x-2y$, $-2y+x$, or $x-2y+0$. It's not smart enough to know whether the answer is simplified or not. The notion of simplification is in a certain sense not definable by algorithms.

For problems with numerical answers that are based on finite-precision given data, answers should be fully evaluated with the correct number of significant figures. Example question: A square has a diagonal of length 0.76 cm. Find the length of one of its sides.

Correct answer: $0.54$ cm

Incorrect answer 1: $(0.76)/\sqrt2$ cm. This answer shows that the student has not understood the point of preserving exact expressions like $\sqrt2$ rather than evaluating them as decimal approximations. Applying that habit in this context is wrong.

Incorrect answer 2: $(0.38)\sqrt2$ cm. Another habit applied in a context where it doesn't make sense.

Incorrect answer 3: $0.53740$ cm. Ths student doesn't understand that the precision of the final result depends on the precision of the input data.

The rule of fully evaluating the numerical result in this sort of context can and should be implemented in whatever online homework software you're using.

One of my favorite types of problems to assign is of the form: (a) find a symbolic expression for $x$ in terms of $y$, $z$, ...; (b) interpret the equation's dependence on the variables. For example, the student might be asked to check whether increasing $z$ has the effect of increasing $x$ or decreasing it, and then to compare with the expected behavior based on the real-world situation being described. E.g., if $z$ is the price of a good and $x$ is the quantity sold, and our expression is $x=1/z$, then we check the mathematical behavior, which is that as $z$ gets bigger, $x$ gets smaller, and compare with the expected behavior. In tasks like these, simplification is often necessary as a preliminary step before the student can reason successfully about the behavior of the result. The requirement of simplification is sort of self-enforcing, because the student can't complete the problem without doing it.

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    $\begingroup$ I really appreciate your post here as it provides a different point of view and a neat link. My own answer to your question would probably be $(0.76)/\sqrt2$! $\endgroup$ – Chris Cunningham Jul 2 '14 at 16:44
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    $\begingroup$ I'm inclined to agree with Chris. I think the answer should be specified. "Find the length of one of its sides expressed in decimal form." They would be required to know about significant digits and why .54 is superior to .53740. I mean, if you consider building a fence, if someone says "We need a board .76/sqrt(2) meters long, you're going to get some funny looks. If you tell them .54 meters long, they'll get out the tape measure and get to work on it. But if this measurement is an input into another math function, leaving the sqrt makes sense. $\endgroup$ – corsiKa Jul 2 '14 at 17:12
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    $\begingroup$ if this measurement is an input into another math function, leaving the sqrt makes sense. Normally people retain one or two extra digits of precision at intermediate steps in order to avoid the accumulation of rounding errors in a calculation. $\endgroup$ – Ben Crowell Jul 2 '14 at 17:20
  • $\begingroup$ The diagonal example is a great one, but also not a problem I'd assign to students without access to calculators! $\endgroup$ – PurpleVermont Jul 4 '14 at 21:55
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As simplification is not definable (from Ben Cromwell's answer), there cannot be a general rule for simplification that works for all examples under all circumstances.

So, use specific rules for each task/question. Your example of $\frac{\sqrt{4}-5}{3}$ might best be changed to for example:

  • multiple choice, if just checking the given answers for correctness isn't easier or not the competence you want to check
  • comparison, under similar conditions like multiple choice
  • explicitly ask for decimal number/integers/fractions (if computations are comparably easy)
  • practical application, where uncomputed answers are useless or not customary.

And beware: algebraic simplification may also veil structure. If I teach students the base values of trigonometric functions, I give them

$$\sin(0°)=\frac{1}{2}\sqrt{0}$$ $$\sin(30°)=\frac{1}{2}\sqrt{1}$$ $$\sin(45°)=\frac{1}{2}\sqrt{2}$$ $$\sin(60°)=\frac{1}{2}\sqrt{3}$$ $$\sin(90°)=\frac{1}{2}\sqrt{4}$$

If you simplify these terms algebraically, you may lose structure.

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    $\begingroup$ +1 for algebraic simplification may also veil structure, and a beautiful example of what that implies. $\endgroup$ – JTP - Apologise to Monica Jul 2 '14 at 16:35
  • $\begingroup$ Do you really do this? So when you ask "compute $\int_2^4 x^2 dx$", you really either give them multiple choice or say "give your answer as a reduced fraction" in the question? This doesn't seem realistic to me. $\endgroup$ – Chris Cunningham Jul 2 '14 at 16:46
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    $\begingroup$ As simplification is not definable [...], there cannot be a general rule for simplification that works for all examples under all circumstances. I think this is going a little overboard. The linked discussion uses a specific set of assumptions, which may or may not be realistic. It's certainly possible to make a human-defined, human-understandable set of rules that works well enough for school purposes. Humans are not computer software and human rules aren't algorithms. If you apply the results at the link too literally, you prove that online homework systems are impossible. $\endgroup$ – Ben Crowell Jul 2 '14 at 17:24
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    $\begingroup$ @ChrisCunningham I'm not doing something like this at all. If a student gets picky about such an easy task like $\int_2^4x^2\mathrm{d}x$, I use it for discussion in class that should yield a mutual understanding, what I expect of the operator Compute. Or I just give the student this point: if he's able to show me, why that answer is still correct without being simplified, he deserves it. $\endgroup$ – Toscho Jul 2 '14 at 20:04
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    $\begingroup$ I have a strong philosophical disagreement with this answer. Students should understand why to do simplifications and what simplifications should be done in which cases. Making up excessively detailed rules like this on a case-by-case basis robs them of the chance to develop the mathematical maturity to figure out what is appropriate. Telling them what to do rather than why leads to the kind of silly mistakes described in my answer. $\endgroup$ – Ben Crowell Jul 2 '14 at 20:38
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First some general thoughts: I could never understand the obsession with no radicals in the denominator.Those situations where it is worth one's time to "simplify" such a thing seem remarkably rare. I feel similarly about many other "simplifications."

Further, demanding "simplification" in all cases seems very often to lead to rather irrational beliefs -- I can't tell how many students have told me that you can't divide by $\sqrt{2}$, for example. I can only surmise they feel this way because they had someone who blew a gasket every time they tried to leave a $\sqrt{2}$ in the denominator.

So that's that. For solutions, I ask (often) for an estimate at the end of the problem. That is, if I think students might try to leave their answer in the form $\frac{\sqrt{4}-3}{5}$, then I append and extra part to the problem that asks something like "is your answer closer to -2 or to 1?" Simplification then happens naturally and for the right reason -- so you can get a better idea of where your answer lies.

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Simplifying $\frac{x-1}{\sqrt x - 1}$ can be misleading insofar as while it's mostly equivalent to $\sqrt x + 1$, the unsimplified expression is undefined at $x = 1$ but not the latter. If you get the fraction expression in evaluating the problem, it's likely that the missing point on the curve is part of the solution.

But as was mentioned earlier, the whole problem of simplification is not algorithmically deterministic and the choice, e.g., of $\cos 2x$ vs $2\cos^2 x -1$ is one that must be determined based on the context.

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I'm not sure if you're asking specifically about simplifying radicals, or if you're asking a more general question, but for your radical examples, I would think that the Algebra I notion of simplifying radicals would serve here. You need to simplify radicals so that they (a) have no radical in the denominator of a rational expression, and (b) have no factors that can be pulled out of the radical.

$\frac{1}{\sqrt{256}}$ leaves a perfect square under the square root sign (and has a radical in the denominator) so it's not simplified. That should be simplified to $\frac{1}{16}$.

$\sqrt{512}$ leaves a perfect square factor under the square root sign so it's not simplified. That should be simplified to $16\sqrt{2}$.

$\frac{1}{\sqrt{512}}$ leaves a perfect square under the square root sign (and has a radical in the denominator) so it's not simplified. That should be simplified to $\frac{1}{16\sqrt{2}}$ which must be further simplified by multiplying by $\frac{\sqrt{2}}{\sqrt{2}}$ to give $\frac{\sqrt{2}}{32}$.

I would expect anyone in a college math course to be able to do this without breaking a sweat and without a calculator, as long as the numerical values you're giving them to work with are "reasonable". If you're giving them problems where they end up with things like $\sqrt{703423}$ to simplify without a calculator, that's not very nice.

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    $\begingroup$ So why is it "reasonable" to demand $\frac{1}{16\sqrt{2}}$ be transformed into $\frac{\sqrt{2}}{32}$? Other than the fact that I had someone tell me the latter was simpler than the former at some point, I actually don't see why it is. Nor do I understand what good is achieved by demanding this. $\endgroup$ – Shay Jul 7 '14 at 16:34
  • $\begingroup$ @Shay I think the point is to have some agreement on a single "simplest" form of an expression in cases where it's practical to do so. Much discussion here: math.stackexchange.com/questions/26080/… $\endgroup$ – PurpleVermont Jul 7 '14 at 19:18

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