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32 votes
Accepted

Whether to tell students how difficult (you think) a problem is

In a setting where students aren't working from a book with these labels on questions, is it worthwhile for the instructor to indicate to students where the work they are asking them to do falls on a ...
Justin Skycak's user avatar
18 votes

Calculus problems arising from real research problems

Modeling Basketball Free Throws by Joerg M. Gablonsky and Andrew S. I. D. Lang, SIAM Review vol. 47, no. 4, pp. 775-798, 2005, https://doi.org/10.1137/S0036144598339555 Abstract This paper presents a ...
JRN's user avatar
  • 10.8k
13 votes

Calculus problems arising from real research problems

Consider Betz's law, which was worked out around $100$ years ago by three different scientists independently (in Germany, the UK, and Russia). It determines the maximum power that can be extracted ...
KCd's user avatar
  • 3,516
10 votes

Good Examples of Questions to have Students Ponder Over Without Paper

A possibility, requiring one definition: What is a tiling of the plane with an infinite supply of congruent copies of a single tile (technically, a monohedral tiling). This can go as deep as you'd ...
Joseph O'Rourke's user avatar
10 votes

How can you elicit the $\log x = {\log} \cdot x$ error?

Writing from a software engineer's point of view, it's a fact that mathematics uses a notation that's highly ambiguous. If you don't know that $log$ is used to denote some logarithm function, then ...
Ralf Kleberhoff's user avatar
8 votes

Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?

Here are some example questions. The graph of the function $f$ is given above. Evaluate the following limits. If the limit is infinite, write $\infty$ or $-\infty$ as appropriate. If the limit does ...
Steven Gubkin's user avatar
8 votes

How can you elicit the $\log x = {\log} \cdot x$ error?

Ask the student to critique this work: Solve for $x$: $\sqrt{x} = 3$ Easy: $x = \frac{3}{\sqrt{\phantom{x}}}$. I have tried this a small number of times, and it has worked so far. The students ...
Chris Cunningham's user avatar
8 votes

Calculus problems arising from real research problems

Consider trying to find a curve that will let you link up two straight tracks smoothly, such as two straight parts of a roller coaster track. By "smoothly" meeting, we want the derivatives ...
KCd's user avatar
  • 3,516
8 votes

Whether to tell students how difficult (you think) a problem is

A Frame Challenge I think that the premise of the question is slightly flawed: I do not think that the distinction between "exercises" and "problems" is one of difficulty—an ...
Xander Henderson's user avatar
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7 votes

Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?

I'll argue that this will be likely not feasible for a test question. There's several points in the calculus progression where there's a "hard bottleneck" of some sort, but once you get past ...
Daniel R. Collins's user avatar
7 votes

Whether to tell students how difficult (you think) a problem is

Donald Knuth certainly thought so: he graded the exercises in his monumental work The Art Of Computer Programming from 00 ‘An extremely easy exercise that can be answered immediately if the material ...
gidds's user avatar
  • 405
6 votes
Accepted

Interesting but very easy epsilon-delta problems?

I suggest using rational functions. Students are used to evaluating limits of rational functions because such examples are prevalent in most calculus courses. Moreover, I think the work required to ...
Brendan W. Sullivan's user avatar
6 votes

Are there any benefits to having an entire course's homework problems available from day one?

I teach at community college. I often publish the homework problems at the beginning of the semester, listed by section. I have never had a student work ahead (that I know of). And I have had a few ...
Sue VanHattum's user avatar
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6 votes

Good examples of functions defined as definite integrals of elementary functions?

A (cata)caustic is formed by the reflection of light, such as the cardioid in this coffee cup: G.B. Airy showed in [Airy, "On the intensity of light in the neighbourhood of a caustic," Transactions ...
user1815's user avatar
  • 5,630
6 votes

Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?

Give a function together with some properties of the function, but do not give a formula, and ask for the derivative. An example: Let $f: \mathbb{R} \to \mathbb{R}$ be a function which enjoys the ...
Steven Gubkin's user avatar
6 votes

Calculus problems arising from real research problems

Consider the reflective property of a parabola: rays coming into a parabola parallel to its axis of symmetric will all bounce off the parabola (angle of incidence equals angle of reflection) and meet ...
KCd's user avatar
  • 3,516
6 votes

Calculus problems arising from real research problems

There should be a lot of interesting examples in ecology. The first one that comes to mind for me is population dynamics: Wikipedia Link The simplest model is exponential growth: $$ dN/dt = rN $$ ...
anjama's user avatar
  • 221
6 votes

What to do with "wild goose chase" or "quantum leap"-types of incorrect solutions when you ask students to prove/show something?

It's quite likely to be a consequence of the belief that they have to answer the question. When they can't work it out, when they've gone around in circles and got lost, but still they have to give an ...
Nullius in Verba's user avatar
5 votes

How does Project Euler come up with such good problems so rapidly?

You can read on PE's website that questions are either Classic questions Questions that are derived from some theorem in some way Other questions are believed to be original There are now over 600 ...
orion2112's user avatar
  • 1,007
5 votes

Mathematical Task with Various Solutions

Here is one I enjoyed from middle school. This was a project: I think we had a whole week to experiment, and discuss, and come up with a solution. Consider a rectangle a 231 by 84 rectangle which ...
Steven Gubkin's user avatar
5 votes

Is there a framework to study the mathematical competence in problem-posing?

The answer to your question is yes. Check out the recent textbook: Singer, F. M., Ellerton, N. F., & Cai, J. (Eds.). (2015). Mathematical problem posing: From research to effective practice. ...
Benjamin Dickman's user avatar
5 votes

Can number theory help me create equations with nice solutions?

Begin with factored quadratic $(u-r_1)(u-r_2)=0$ so $u^2-(r_1+r_2)u+r_1r_2=0$. Choose one root positive, say $r_1>0$ whereas $r_2<0$ then we solve for $u$ using positive root, $$ u = \sqrt{(r_1+...
James S. Cook's user avatar
5 votes
Accepted

Can number theory help me create equations with nice solutions?

I think that the number theory in this case is pretty light. Let's suppose that we want some integer solution for $x$ and that $a,b,c$ are also integers. Then $x+b$ must be a perfect square, so let $N^...
Adam's user avatar
  • 5,733
5 votes

Question about the process of creation of problems and exercises in Mathematics

I will take a stab at an answer though clarifying what level of education we are talking about would help. I have never created problems for things like Qual exams (essentially masters exams) so I ...
DRF's user avatar
  • 1,018
5 votes

Good Examples of Questions to have Students Ponder Over Without Paper

Perhaps logic puzzles would work in this case. Some classic examples are: You're traveling along a road and arrive at a fork. Two guides are posted, but one always lies and the other always tells ...
Aeryk's user avatar
  • 8,021
5 votes

Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?

This might not be what you want, but sometimes a backwards problem appears more difficult than it seems and the difficulty is immediately removed when you know and apply the definition of the ...
James S. Cook's user avatar
4 votes

Good Examples of Questions to have Students Ponder Over Without Paper

Since Benjamin Dickman mentioned The Tokyo Puzzles in the comments, I'll include a couple of the questions from that book here that I thought fit the prompt nicely; neither requires a pen and paper to ...
Mike Pierce's user avatar
  • 4,845
4 votes

A question about Vector Analysis problems

You are correct. There are not nearly enough interesting problems typically given for the major theorems of vector calculus. For the most part, the problems I find are not that interesting and mostly ...
James S. Cook's user avatar
4 votes

Whether to tell students how difficult (you think) a problem is

That depends on the student level and on the objective of problem solving. For rookies just learning elementary skills, my answer would be "definitely yes". For graduate students taking a ...
fedja's user avatar
  • 3,909
3 votes

Where to find good exercises for term operations?

It costs $5/month (for educators) to use Wolfram Alpha in its practice worksheets model. It will generate a lot of problems for you, but I'm not 100% sure it gives you the granularity you want. I ...
Sciolism Apparently's user avatar

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