Teaching first year undergraduates, I've noticed that what gives them the most trouble is simple computations like factoring, expanding, handling fractions, powers, especially when variables and other parameters come into play.

Here in France these things are supposedly taught in middle school and high school but the degree of mastery that our first year students have of these topics can be very low.

I'm currently in the process of designing a new "calculus 1" course for those students and would like to include in the first two weeks some material to have student work their computational muscles. However I'm afraid if those exercises look too much like high school the students might get bored. So I'd like to give them exercises that look like small computational challenges but rely only on these elementary precalculus topics.

One example I like is the classical 2 squares or 4 squares identities (see 1 or 2).

So the question is :

Question : What are some good resources of reasonably challenging problems of mostly computational problems which involves only the pre-calculus tools listed above ?

Disclaimer : I did browse MESE but didn't find anything, though I might just use the wrong keywords. Also I'm not too familiar with MESE and hope this question is not too broad.


I'm an assistant professor in a university in France. I need to design a calculus course for first year students in Maths, Engineering, Physics, Computer Science and Chemistry.

Up until now, calculus topics where scattered in with more formal analysis and algebra courses, which had the unfortunate side-effect that student came to know that they could focus on the computational techniques to have a decent grade rather than get hands with definitions and proofs.

French high school pupils do have a basic exposure to calculus, they know how to compute simple limits (though they have no idea of the definition) and derivatives of not too complicated (they don't know the chain rule for instance, but know how to differentiate $u^\alpha$ or $e^u$ for instance), they can study the variations of functions using the derivative and have seen that there is a thing called "integral" that you can compute using antiderivatives.

University curriculum in France is not centralized so we are not tied in the topics we can cover.

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    $\begingroup$ I've prepared many such handouts and worksheets and such for this purpose, and several used to be on the internet at Math Forum, but they seem to have taken their entire discussion groups database offline. However, since your profile leads to a real name and presumably legitimate email address, I'll send some of them to you in a few minutes. $\endgroup$ Commented Apr 14, 2020 at 18:17
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    $\begingroup$ Some old answers of mine that might contain things you could use: using the factor theorem AND applications of rationalization AND advanced level algebraic expansions AND derivative of $x^{m/n}$ by limit definition of derivative AND a challenging irrationality proof using the rational root theorem. $\endgroup$ Commented Apr 14, 2020 at 18:36
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    $\begingroup$ Have you giving the students a diagnostic test to see what they are really missing? It can help you plan? Additionally some high school problems would be in order if they are all missing the same skill. $\endgroup$
    – Amy B
    Commented Apr 23, 2020 at 18:31
  • $\begingroup$ A decent chunk of the students do formally know the rules about expansions, fractions, exponents, equations... However most of them get lost if a computation lasts more than a couple of lines or involves parameters in addition to the unknown/variable. The message they should get from those problems is that they should not be afraid to start a computation which is a bit lengthy as long as they keep track of what they want to do at each step. $\endgroup$ Commented Apr 26, 2020 at 7:27

1 Answer 1


The Art of Problem Solving has plenty of problems involving pre-calculus tools, some of them quite challenging.

However, the best way forward might not be computationally challenging exercises.

You said they lack mastery. Please correct me if I'm wrong but I interpret this as they cannot reliably perform said computations and use said pre-calculus tools in a consistent, effortless, and automatized way. This is probably because their practice was done in 'blocks', so they used those tools a lot at once and then never again, or a long time after the initial practice.

In order to improve their computational skills, my advice would be to incorporate frequent tests (spaced repetition) with questions involving a wide range of tools (interleaving). They should not know in advance what kind of tools they would be using so that you improve their performance on a steady basis.

There should be room for error, of course, so you could drop a couple of lowest scores to improve morale and give them a chance to make mistakes without overburdening their grades. Even if you repeat the types of questions they faced in high school their knowledge and automation of those tools should improve over the semester (and for everything afterwards).

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    $\begingroup$ "In order to improve their computational skills, my advice would be to incorporate frequent tests (spaced repetition) with questions involving a wide range of tools (interleaving). They should not know in advance what kind of tools they would be using so that you improve their performance on a steady basis." — could you elaborate and provide some examples? $\endgroup$
    – Rusty Core
    Commented Jun 2, 2020 at 18:20
  • $\begingroup$ @RustyCore I'll expand this answer with a few examples once I get done with the end of the online semester, which should take me around a week. $\endgroup$ Commented Jun 7, 2020 at 16:17

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