# Finding the Balance in a Math Question (Teaching)

As we try to work and teach in the midst of this pandemic, some problems arise when making online math exams. My question is simple: What could be an interesting basic differentiation question such that students doing the online exam still have to work a bit to get it, rather than simply Googling it or using WolframAlpha/Mathematica to solve it for them?

Everything I can think of seems to be either too easy (meaning they can easily solve it online) or too demanding. I'm struggling to find a good balance, if there is any. This is a first-year math module I'm teaching (non-math students).

Any ideas?

• I know this isn't of the spirit of what you're asking, but if they are non-math students, I'd be perfectly happy if they used Alpha or Symbolab to compute. As long as the problem relates to an application in their field. – Aeryk Jan 11 at 20:39
• Such questions do not exist. Most of your students have had 12 or 13 years of math education where the sole goal is to become a progressively better imitation of Wolfram Alpha. It's not surprising they have zero ability to do anything Wolfram Alpha cannot. – Alexander Woo Jan 11 at 22:36
• Consider also looking at the several related questions on the site previously. – Daniel R. Collins Jan 13 at 23:58

• I think it is helpful to let students know that you are looking for their thinking while problem-solving, and not just answers. Then you can ask questions like:

Find all of the points on a circle of radius $$8$$ that have a slope of $$\frac{3}{4}$$.

so that they can explain their problem-solving. Even if they do end up using advanced calculators, they will have to manually break the problem down into steps, which demonstrates some understanding. Alternatively, you can also give them a problem that you know is too challenging, and turn the question into something like:

Do not solve, but describe the approaches you would try in order to find $$f'(x)$$ where $$f(x)=$$ [function that's too hard]

• You can also ask for their thinking directly. I'm not sure of the complexity you are looking for, but for example:

The derivative of $$\frac{x^2+4}{\sqrt{x}}$$ can be found using the quotient rule or the product rule. Show that both give the same result.

• Problems based on pictures are harder to Google. This one is based on a Khan Academy problem:

• There are also "find-the-mistake" problems, like:

Carser took the following steps when finding the derivative of $$f(x)=x^2 \cdot \sin(x)$$. Describe his mistake and explain how to correct it. $$f(x)=x^2 \cdot \sin(x)$$ $$f'(x)=2x \cdot \cos(x)$$

• This is a perfect answer! You not only suggested different questions, but also organized them into different "classes". Thank you so much for the suggestions, this will be very useful in the future. – sam wolfe Jan 12 at 21:57
• +1. I especially like the get-the-same-result question type, although I'd compare less similar techniques, e.g. quotients vs chain. – J.G. Jan 13 at 16:15
• And if the approach they would try for integrating [function that's too hard] is "See if Mathematica can figure it out"? (Which isn't an unreasonable approach, in practice. For that reason, I rather like your suggestion of giving a problem that must be broken down into Mathematicable subproblems.) – Ray Jan 14 at 5:24
• Ha @Ray well that could be the right answer! But I hope they would provide some detail. Numerical computing is the solution so often in practice, since there are infinitely many functions that we don't have "rules" for so analytical solutions are not always available. The calculus book that I teach from has a section on numerical computing. – Carser Jan 14 at 12:43

Note: It's possible that in the future, Wolfram Alpha will improve and be able to answer the questions in this answer, so it's best to actually try them in Wolfram Alpha first.

Use questions that involve conditional statements and generic expressions.

For example:

If $$\lim_{x\to\infty} f(x)=1$$, then what is $$\lim_{x\to\infty}\frac{f(x)}{x}$$ equal to?

Typing limit of (f(x)/x) as x goes to infinity if limit of f(x) as x goes to infinity is equal to 1 or limit of f(x) as x goes to infinity is equal to 1, what is limit of f(x)/x as x goes to infinity? or if limit of f(x) as x goes to infinity is equal to 1, then what is limit of f(x)/x as x goes to infinity? into Wolfram Alpha doesn't give the answer.

A more difficult one:

If $$\lim_{x\to\infty} f(x)=1$$, then what is $$\lim_{x\to\infty}\frac{x-f(x)}{x+f(x)}$$ equal to?

One involving derivatives:

If $$f'(x)=\sqrt{x}\cos(\pi x^2)$$, then what is $$f'(4)$$?

(Intentionally choose $$f(x)=\int\sqrt{x}\cos(\pi x^2)\,\mathrm{d}x$$ to be ridiculously complicated.) Typing If f'(x)=x^0.5cos(pi x^2), then what is f'(4)? into Wolfram Alpha doesn't work.

Make it clear to the students that $$f'(4)$$ (which is $$2$$), is not the same as $$\frac{\mathrm{d}}{\mathrm{d}x}\left(f(4)\right)$$ (which is $$0$$).

You can then ask (after discussing the chain rule) the more difficult question:

If $$f'(x)=\sqrt{x}\cos(\pi x^2)$$, then what is $$\frac{\mathrm{d}}{\mathrm{d}x}\left(f(x^2)\right)$$?

• Interesting approach! However, one way for the first two is to simply use a specific function $f(x)$ that satisfies the "if" half (although this does require some facility with limits), and then use WolframAlpah to compute the "then" half's limit with that specific function. – Dave L Renfro Jan 12 at 13:33
• If so, then the question is equivalent to "Find a function $f(x)$ whose limit as $x\to\infty$ is equal to $1$," which is another question that Wolfram Alpha can't answer. – Joel Reyes Noche Jan 12 at 15:07
• the question is equivalent to --- A certain constant function works or something like $\frac{x+1}{x+2}$ (does require some limit knowledge, however). Also, the student could look at calculus book (odd-numbered) answers in a library or google book, quiz, test, etc. limit problem answers and pick a function that corresponds to when the answer is $1$ (using something like "limit as $x \rightarrow \infty$ is equal to $\frac{17}{41}$" could foil this method a bit, however). – Dave L Renfro Jan 12 at 16:40
• @DaveLRenfro This can be countered by having some questions where the correct answer is: cannot be determined with the information given. For example: if f(x) goes to $1$ for $x$ goes to infinity, what is the limit of $f'(x)$ for $x$ goes to infinity? – quarague Jan 12 at 17:42
• A very curious answer. Knowing the limitations of WolframAlpha is beyond my expertise, but I find fascinating what it can do. Even though some of these questions might be "too demanding" for the module I'm teaching (in the sense that they have never seen such examples during class), they are very interesting and I will keep this answer in mind for future modules/exams. Thank you! – sam wolfe Jan 12 at 22:01

Related rate word problem question with some geometry needed as part of the problem.

But. I would caution you that if you are teaching weaker students, there is a problem in assigning tough problems in that you don't well evaluate their ability in basic methods. I would instead suggest alternatives like short duration (fixed time) remote exams. Also honor statements. None of the methods are perfect, but they help, especially the former.