# Problems which require interpreting definitions

I'm trying to find more problems suitable for early college students (students who know algebra and calculus) that involve translating words into mathematical notions. A nice example is this one:

Show that on the graph of any quadratic polynomial the chord joining the points for which $x=a$ and $x=b$ is parallel to the tangent line at the midpoint $x=(a+b)/2$

I don't mean the thing that's usually called a "word problem", where there's some "real world" situation that has to be interpreted into mathematics.

This is for a "transition to advanced math" course that's supposed to be a step between calculation based calculus and proof based analysis and algebra. What I really want them to practice is the process of solving problems that use defined notions by replacing those definitions with their meaning, since we find that's something students in the later courses struggle with.

The problem above really caught my eye, because a lot of students at that level seem to have no idea how to start with it - they see a bunch of words that don't pattern match to any problem they've solved before - but as soon as they recognize that "_ is parallel to _" means "the slope of _ is equal to the slope of _", they know how to calculate the slopes and check equality. That feature - problems which seem impenetrable when taken as a whole, but which can be reduced to something simple by breaking apart the meaning - is the main thing I'm trying to illustrate.

• The Art and Craft of Problem-Solving, by Paul Zeitz, might have good problems of this type. But if you don't have it already, good luck getting a reasonably priced copy. Oct 28 '16 at 23:26
• This is just a comment: I don't think that the problem really lies with a lack of practice and your example can be interpreted as showing that all they have been taught and therefore know how to do is, once they have been cued, to apply routines that remain in fact totally meaningless to them. Physicists are much more aware of this than mathematicians: (harvardmagazine.com/2012/03/twilight-of-the-lecture) Jul 1 '17 at 14:34
• I don't know of any pre-prepared resource which seems to be what you are looking for, but such situations are easily created in definition-problem pairs. Eg, define odd numbers then require a proof that the product of two odds is odd; define prime numbers then require a proof that, say, 7 is prime but that 27 is not. A number of competition problems define artificial terms for such purposes and this technique can be useful, too. Eg define a function to be sum_happy if f(a+b)=f(a)+f(b) and product_happy if f(ab)=f(a)f(b) for all a,b. Prove that f(x)=2x is sum_happy but not product_happy etc etc Jul 30 '17 at 12:38
• Regarding your example problem: if you wish to "cloak" the idea even further, citing theorems as well as definitions, it can be posed as follows: "For an arbitrary quadratic polynomial f defined on the interval [a,b], show that the input c guaranteed by the Mean Value Theorem is the midpoint of the interval." (I have done this problem, posed as such, with students in a Calculus 1 course.) Dec 24 '17 at 20:01

I'm trying to find more problems suitable for early college students (students who know algebra and calculus) that involve translating words into mathematical notions, [...] problems which seem impenetrable when taken as a whole, but which can be reduced to something simple by breaking apart the meaning.

I am not aware of a specific resource for problems like this, but I also wish there were one because it does seem that students (even advanced math majors) struggle with this aspect of problem-solving. I trust that we all try to model this problem-solving behavior when we work out problems for students (assessing known and unknown information, deciding what to try next, reassessing from results) but it seems like students have trouble developing these skills. It would indeed be nice to have a workbook of problems that target this skill of problem-solving. In addition, it may serve as a way to teach students to synthesize their knowledge from various courses, which can often feel to them like discrete, disconnected islands in the vast sea of mathematics.

However, I do think it would not be too challenging for you to create such problems on your own by taking some known fact and "translating/encoding" the ideas appropriately into definitions. (I do think this will be time-consuming, but not necessarily challenging.) Essentially, do the work that you expect the student to do, but in reverse!

Below, I will show a few examples of what I have in mind to not only provide example problems but also show one may create many of their own such problems. Each example is titled with the essential fact upon which it is based. (I created this list of examples by trying to conjure interesting facts and seeing what came to my mind.)

Example 1: Tangent line to a circle is perpendicular to its radius

This is a well-known fact that students likely prove in a high-school geometry course but may never revisit. The following problem has them use calculus and basic algebra to see this fact in a new light.

Consider the circle defined by the equation $x^2+y^2=r^2$, and let $P=(u,v)$ be any point on the circumference of that circle. Use implicit differentiation to find the slope of the tangent line at $P$, and then show that the tangent line is perpendicular to the line from the origin to $P$.

You could even be less explicit about expectations by removing "to find the slope of the tangent line" and expecting students to identify that "perpendicular" means "opposite reciprocal slopes".

You could even design the problem to assess their skill in dealing with arbitrariness by not providing the equation $x^2+y^2=r^2$ and instead saying: "Consider a circle of radius $r$ and any point $P$ on its circumference", and replacing "origin" later with "center of the circle". This new version relies on the student choosing an appropriate coordinate system and notation.

Example 2: The symmetry of a parabola across its vertex

Algebra and pre-calculus students are taught completing the square and "vertex form", but I find that students later in calculus do not appreciate (or think to use) the symmetry of a parabola. This problem shows them that symmetry using calculus.

Let $f$ be an arbitrary quadratic polynomial. If $x=a$ and $x=b$ are two inputs that are equidistant from the $x$-coordinate of the vertex, show that (i) $f(a)=f(b)$ and (ii) the tangent lines at the points on the graph of $f$ for which $x=a$ and $x=b$ intersect at a point whose $x$-coordinate is the same as that of the vertex.

You could change (ii) a little bit as follows: "the tangent lines to the points $(a,f(a))$ and $(b,f(b))$ and the vertical line through the vertex intersect at one common point." However, I find that recognizing that "the point on the graph of $f$ corresponding to the input $x=a$" means "the point $(a,f(a))$" is occasionally not at all obvious to a student(!).

This problem also allows for multiple approaches. Will the student take an arbitrary function $f(x)=Ax^2+Bx+C$ and work with it twice to find the tangent lines and find their intersection point? Or will they complete the square to "encode" the symmetry the problem gets at in the form of the function itself, thereby making the rest of the problem easier?

Example 3: The derivative of an inverse function at some input is the reciprocal of the derivative of the original function at its corresponding input

Okay, this idea is stated much better in theorem form (as seen in a Calculus 1 course):

If $f$ is invertible and differentiable in an interval containing $x_0$, and $x_0$ satisfies $f'(x_0)\neq 0$, then $f^{-1}$ is differentiable at $y_0=f(x_0)$ and $(f^{-1})'(y_0)=\frac{1}{f'(x_0)}$

You can construct a problem from this in many ways, by perhaps giving a specific function and having the student find its inverse as part of the problem; or, more generally, by providing a class of functions that are invertible and asking for the same steps (like $f(x)=\frac{ax+b}{cx+d}$, provided some conditions on $a,b,c,d$, that you may even ask the student to find). Here's an example of the simpler variety:

Consider $f(x)=x^3$. Show that $(2,8)$ is on the graph of $f$, and $(8,2)$ is on the graph of $f^{-1}$. Then, show that the tangent lines at those points have reciprocal slopes.

Other ideas:

1. Ask about generalizations. For example, take the problem you posed in the question and follow it up with, "Does the same property hold for cubic functions? Quartic functions?" This will hopefully encourage students to ask these kinds of questions on their own in the future.
2. Ask for examples & counterexamples. I find that creating (counter)examples to be an essential skill in my own learning and understanding of mathematics, and I find this skill is too often given short shrift in actual course-work. An example problem might be, "For each situation, give an example or explain why it is not possible: a function $f$ and a tangent line at the point $(a,f(a))$ on the graph of $f$ that intersects the graph of $f$ (i) 0 times, (ii) exactly once, (iii) exactly twice, (iv) infinitely many times."