# Algebra/trig/precalculus review questions that elicit common student errors

This semester I have decided to give students a simple question or two at the beginning of every calculus class that will trap them into making the most common errors that we all know...e.g. the freshman dream. There is a list/partial classification of such errors here. They generally involve crimes with square roots, fractions, function composition, lines (thinking they have written the equation for a tangent line that is not a straight line at all, for example), exponents, logarithms and strange other beliefs such as all functions....especially trig functions...being linear.

I always have imagined that someone would have constructed a nice list of simple "trap" problems that instructors could use to construct the kinds of short quizzes that I'm looking to give, but surprisingly there isn't a really good repository of quick, but good, problems that aim to isolate the most common student crimes.

As a specific restriction, for example, I'd really like to avoid problems that just say "simplify" or "rationalize", because these commands reinforce the cookbook thinking that got students into this mess in the first place. I would, however, like the problems to be short...i.e. take a competent student around 5 minutes to complete.

Question: Where is a good source of short problems (not of "simplify" or "rationalize" type) that are "traps" to elicit common algebra/trig/precalculus student blunders? I'm interested in online sources, or your favorite problems listed as answers.

Edit: I think it is important to point out, due to the valid concern expressed by Daniel R. Collins and echoed by others, that I am asking for these sorts of questions precisely because telling students about pitfalls seems to inevitably lead to them falling in precisely the pits you told them not to fall in. The philosophy of the question is that questions and exercises answered by the students are just that...questions and exercises answered by the students. Certainly nobody is claiming that if a student solves an exercise that elicits a particularly pernicious pitfall that the student will suffer a spectacular fall from grace and will then be unable to avoid making the mistake! The idea is that problems that show these mistakes to their would-be solver will be memorable in the right way.

This said, if indeed tricky problems like this do lead to "black hole errors" then this would explain the shenanigans I keep seeing on people's papers...

• I worry with this kind of practice: there is some reasonable claim that visually-arresting false principles are more likely to be (incorrectly) recalled after some some students have seen them. E.g., Clark, "Putting Students on the Path to Learning", American Educator, 2012: "Even after being shown the right answer, a student is likely to recall his or her discovery -- not the correction." – Daniel R. Collins Jan 23 at 4:20
• I know I've seen questions like this before.I found one similar: matheducators.stackexchange.com/questions/7249/… – Sue VanHattum Jan 23 at 5:10
• @DanielR.Collins: This is precisely why I want these to be problems and not some sort of "watch out for these pitfalls" checklist. I want the students to just see their making mistakes and try to correct them. – Jon Bannon Jan 23 at 14:49
• I think these errors come from the some deeper misunderstandings. Those we should aim at. – Peter Saveliev Jan 23 at 17:31
• absolutely, @DanielR.Collins. Spot on! – amWhy Jan 25 at 20:23

I made a handout with 10 true false questions, titled Algebra Temptations. I've put it on google drive here. I have students work in groups to decide which are true. I think it makes a good activity, though there is the risk mentioned in another answer of reinforcing the wrong idea.

A common example is that students think that $$\sqrt{ab}=\sqrt a \sqrt b$$ for all $$a,b$$; try tricking them by saying that $$1=\sqrt{1}=\sqrt{-1\cdot -1} = \sqrt{-1}\sqrt{-1} = i\cdot i = -1$$. Similarly students will not often realize that $$f(x) = \sqrt{x^2}$$ is NOT $$x$$, but rather $$|x|$$.

Extraneous solutions (i.e. solutions that pop out when solving but are actually incorrect) are also tricky; students usually do not check: https://en.wikipedia.org/wiki/Extraneous_and_missing_solutions; http://mathmistakes.info/facts/AlgebraFacts/learn/ctm/extra.html. A cool example of this is if you have some sort of equation like $$\sin(x) + \cos(x) = 1.2$$, you can solve it by making it into $$\sin(x) = 1.2-\cos(x) \implies 1-\cos^2(x) = 1.44 - 2.4\cos x + \cos^2x$$ and solving it like a quadratic equation. However, students are often so elated that this "trick" produced a solution to an equation which at first doesn't look solvable that they do not check their solutions to make sure the signs are in fact correct and that it doesn't go outside bounds of $$\sin(x)$$ and $$\cos(x)$$; i.e. they might get that $$\cos(x) = 1.1$$ and not realize that it is impossible.

When working with limits students see $$\lim_{x\to 2}\frac{(x-2)(x+1)}{(x-2)} = 3$$ but students may not realize that $$f(x) = \frac{(x-2)(x+1)}{(x-2)}$$ is NOT defined at $$2$$.

I hope these are useful.

I wouldn't necessarily call them "trap" questions, but I wouldn't be surprised if a vocabulary quiz on middle/high school math could be difficult or low scoring.

Ask students to define or to give examples for "degenerate triangle", "extraneous solution", "rational number", "principle square root", "inverse function", "quotient", "numerator", etc.

Even some true or false questions could pose difficulties, like "if positive integer $$n$$ is not prime, then it is composite" or "prime numbers are all odd"