# Questions tagged [examples]

For questions about examples for some mathematical subject – usually for purposes of motivation and illustration.

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201 views

### Probability — analytical results instead of simulations

After students learn how to use probabilistic simulations, what strategies can one use to encourage them to understand analytical results anyway? For example, I'm struggling to find a compelling ...
79 views

### Looking for good examples/explanations of maximum likelihood estimators for discrete random variables

The title basically says it all. I need to prepare material for a whole classroom of elementary statistics students, so if anyone wants to help me out in the name of math education, that'd be great!
1k views

### Motivating example for sequences, sums and limits in high school

I currently work as a substitute teacher at a local high school and the topic in one of the classes is sequences, series and limits. Because I always disliked learning about a topic without having ...
2k views

### Big list of "interesting" abstract vector spaces

When introducing an abstraction it is important (in my opinion) to have a wide variety of examples of this abstraction. Since finite dimensional real vector spaces are classified up to isomorphism by ...
2k views

### An introductory example for Taylor series (12th grade)

I (a student) am doing a presentation on Taylor series in my class (12th grade, in Germany if this is relevant). I am looking for a good example where you can see when Taylor series might be useful. ...
624 views

### Example of why proof by exhaustion is inelegant

There's a nice example of why people dislike proof by exhaustion on the Wikipedia page. The problem statement is "prove that all years in which the Modern Olympics are held are divisible by 4&...
238 views

### Showing applications of calculus to intro students

So I'm wrapping up my second year teaching high school calculus, and my first as an AP course. In addition to review, I'd like to end the year by giving the students a taste of the kinds of real-...
185 views

### Pros/Cons of using a single story in multiple examples to demonstrate different points

Judea Pearl, in his book "Probabilistic reasoning in intelligent systems" uses a handful of stories over and over again, each time to demonstrate a different point. (His "Alarm" ...
205 views

### Relationships amusingly difficult to graph

Long ago, when I was the student, I encountered a bunch of stories that we were asked to illustrate with graphs. I seem to remember that it was a set of around 10 examples. One involved some two ...
200 views

### Do you avoid examples or test questions that showcase an algorithmic plug'n'chug approach?

If we accept that there's not much learning from doing the "same" questions, like find the derivative of $x^2$, and $x^3$, and $x^4$ due to the algorithmic way of how it's done, then what ...
270 views

### What are some famous problems, which are not difficult to understand, for senior high school students

I hope I am asking my question in the right forum. I am trying to introduce some mathematical problems (Better to be famous in the math community) to a group of senior high school students with a ...
2k views

### Are there direct practical applications of differentiating natural logarithms?

The textbook I am using to teach Calculus I includes in the exercises of most chapters a number of interesting real-world applications of the concepts from that chapter. However, the chapter on the ...
9k views

### How can teachers warn students about common mistakes without causing the student to make the mistake?

For example, if you're teaching integration of $\int \frac{dx}{1+x^2}$, would you mention the common wrong answer of $\ln\left(1+x^2\right)+C$? -- For myself, I very rarely mention common mistakes ...
771 views

### Proof by contradiction - more than one case

I am looking for some examples of when proof by contradiction is used in a problem with more than one case. In all the elementary examples, there are only two options (eg rational/irrational, ...
216 views

### Examples for "good" exponential growth versus linear growth

I am writing up a small text about (increasing) exponential functions versus linear functions. I want to make the point that it is not true that if an exponential is bigger than a linear function that ...
3k views

### Algorithmic thinking problems

In Norway we will have a new national mathematics curriculum for all ages including high school beginning august 2020. A fundamental change is the new focus on so called algorithmic thinking. In ...
104 views

### How to address opportunities for improvement with a teacher

My daughter’s 5th grade teacher gives some pretty impossible questions and I don’t think she understands the material she’s teaching. For example, this question has no context from previous questions ...
189 views

### Introductory exercise for the addition of large natural numbers

I'm starting a repetition with my students in 5th grade after they learned in elementary school how to sum up larger natural numbers (also 5- to 6-digits) by writing down that calculation. As ...
113 views

### The propagation of the wave equation in even versus odd dimension

I am about to teach a second year undergraduate class on applied differential equation (first time) and, while I won't have time to go into the details, I wanted to show my students the difference ...
90 views

### Examples for environmental topics in the context of terms or linear inequalities

I want to emphasize the aspect of environmental education in my math class. Now I'm reasoning whether to do that with linear inequalities or terms with two variables - these are our next topics. The ...
111 views

### Applied ODEs for Numerical Methods

I am looking for a list of ODEs to use as examples in the teaching of a numerical methods course for engineers. I am looking for first and second order examples - the more applied (to engineering) ...
526 views

### Easy examples of correspondence between global and local, as preparation for Gauss's theorem and Stokes's theorem

I'm teaching freshman electricity and magnetism this semester, and as usual in this type of course, I will need to teach my students a lot of vector calculus before they see it in a math course. The ...
399 views

### Examples of application problems of coordinate geometry in the complex plane?

I am currently writing some basic introductory texts to complex numbers for third-year high school students (Denmark). My main goal is to introduce complex numbers as a practical tool that both ...
330 views

### Examples of solving for unknowns using equivalence relations that are not equality, inequality, or boolean truth?

In a book i'm writing, i want to introduce students to equations in slightly more general terms. Solving an equation with some unknown is just one example of finding an object by fact that it is a ...
445 views

### Ideas for the introduction of the derivative?

I want to introduce to my class to the derivative, but I am still searching for a good, realistic context that isn't too hard to understand, without seeming to be contrived. Do you have an ideas for ...
2k views

### Real-life exceptions to PEMDAS?

What are some real-life exceptions to the PEMDAS rule? I am looking for examples from "real" mathematical language --- conventions that are held in modern mathematics practice, such as those appearing ...
146 views

### Published papers for Intro Stat students to read

I am looking for studies and experiments in the literature that I can share with undergraduate students in an intro statistics course. I do not expect students to understand everything, and I plan to ...
972 views

### is it appropriate or beneficial to mention weird results in math?

Is it appropriate to mention weird/exciting results in math (or use as cautionary tales why one cannot apply mathematics naively) in say high school level? Examples of these results include the ...
89 views

### A Markov chain demonstration that doesn't require computers

I have a large probability class and would like to do some memorable demonstrations of Markov chains for them. Does anyone have any recommendations for a "low-tech" demo that doesn't involve ...
134 views

### A good example to show group actions and Burnside's lemma

I want to make a presentation of Burnside's lemma outside of group theory, and more as the stand-alone combinatorial tool that it can also be. My plan right now is to make it into a 15-20 minute video,...
179 views

### What is a realistic situation that illustrates precisely what a p-value is?

Students in the basic statistics courses I teach often learn a little bit of probability and then learn hypothesis testing. The core concept that ties the course together is the p-value, but most ...
127 views

### Using discrete examples in the beginning of integration

In Germany, one usual example to start teaching about integrals is to look at a simple (piecewise constant or with constant slope) functions that make up a water flow vs. time diagram and ask about ...
564 views

### Examples (for beginners) of real functions which are not given by elementary formulae

Question: What are good examples of functions $f: \Bbb R \rightarrow \Bbb R$ (or $f: D \rightarrow \Bbb R$ with $D \subseteq \Bbb R$) which are not just given by "a formula" (or finitely many formulae ...
10k views

### Good real-life examples of transformations of function graphs

I am a graduate student teaching college algebra at a larger state school, and currently I'm covering transformations of graphs of function, i.e.: Given the graph of a function $y =f(x)$, what do ...
576 views

### Simple, elegant ways to teach the idea of what functions are for the first time

The context In my country, when the concept of function needs to be introduced in math classes, most teachers will simply talk about $f(x)=c$, $f(x)=ax+b$ and $f(x)=1/x$ (constant, linear and inverse ...
291 views

### What tools are available for creating visual aids?

I would like to create some visual aids for illustrating principles in statistics, similar to the kind of graphic found here: https://en.wikipedia.org/wiki/Linear_regression#/media/File:Anscombe%...
3k views

### Mnemonics for some properties in mathematics

I am looking for various mnemonics which help students to remember some important properties or theorems. Very often students confuse signs or relations such as $\leq$ and $\geq$ in some expressions. ...
19k views

### What's a replacement for "married couples" in combinatorics problems?

Many counting problems start with the assumption that we have a certain number of men and women or a certain number of couples, with the assumption (often unstated) being that that gender is binary (...
9k views

### "Real world" examples of implicit functions

When teaching implicit differentiation in freshman calculus I lack good examples which might help students relate the theory to applications in other sciences. So I'm looking for (relatively simple) ...
127 views

### Books and worksheets on symmetry

At a local Math Circle, I loved some problems worked out through a hinged mirror to illustrate symmetry. I bought a hinged mirror from hand2mind.com, and am looking for some material, ideally books ...
2k views

### Real-world Markov chains

I will give a talk to undergrad students about Markov chains. I would like to present several concrete real-world examples. However, I am not good with coming up with them beyond drunk man taking ...
1k views

### Examples of basic non-commutative rings

I am teaching an intro to ring theory, and after grading the first quiz, I realize most of my students are under the assumption that rings must be commutative. I have given them the example of ...
518 views

### Simple examples that violate group axioms

In a course for non-math-majors at a liberal arts college, I would like to give a few lectures and activities about groups and symmetry. I think it's straightforward to explain the group axioms and ...
438 views

### Is there a simple real-world problem I can use to motivate a formula for $\displaystyle \sum_{i=1}^n i$?

I would like to know if there is a simple real-world problem which requires knowing a closed form for $\displaystyle \sum_{i=1}^n i$ and/or the sum of the first $n$ even/odd numbers. The only ...
252 views

### Benefits of knowing theory [closed]

I've got an issue: from time to time I have to teach some math to people who either avoided it, or got through by only knowing a few working algorithms. The only thing that unites all those people: ...
1k views

### Evaluating integrals geometrically, without using the fundamental theorem of calculus

I'm designing a lesson for an Introduction to Integral Calculus class, and I want to encourage students to evaluate integrals without just going straight for the antiderivative and using the ...
411 views

### Good metaphor to explain the difference between pointwise and uniform convergence

What could be a good "layman" metaphor for illustrating the difference between uniform and pointwise convergence of function series? I am teaching calculus to engineering undergrads; for many of them, ...
783 views

### Small data sets with integral sample standard deviations

I'm looking for small data sets ($N\approx10$) that have integral (or even rational) sample standard deviations. Given a list of observations $\{x_1,\ldots,x_N\}$ with mean $\bar{x}$, the sample ...