Questions tagged [examples]

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

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6
votes
2answers
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 ...
7
votes
6answers
973 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 ...
4
votes
2answers
90 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 ...
4
votes
1answer
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 ...
3
votes
2answers
135 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,...
31
votes
10answers
9k views

Simple "real world" l'Hôpital's rule problem?

I am on a team which is writing a set of lecture notes for differential calculus. I am using a format of "Break ground" which poses a problem, "Dig in" which develops the tools to solve the ...
8
votes
7answers
188k views

Examples of arithmetic and geometric sequences and series in daily life

In this part of the course I am just trying to show that we actually see alot of sequences and series everyday in our daily life. I already found some examples such as the housenumbers when you drive ...
4
votes
1answer
128 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 ...
13
votes
3answers
569 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 ...
7
votes
6answers
586 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 ...
4
votes
3answers
293 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%...
54
votes
24answers
61k views

Optimization problems that today's students might actually encounter?

Our students are not fencing in farm fields, cutting wires and folding them, or designing windows, so they are often uninspired by the optimization problems we give them. They seem like something that ...
12
votes
3answers
661 views

Example of function with *all* the features of differential calculus at first-year level

I'm teaching a first-year calculus course, that is mid-way between a first intro to university-level calculus, and intro to real analysis (I'm based in Australia, for reference). We assume the ...
13
votes
7answers
834 views

Where can I find realistic data for college-level elementary statistics problems?

I'm creating a large number of practice problems for my statistics students. These problems are for an elementary stats course where students: measure central tendency measure dispersion use linear ...
15
votes
13answers
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. ...
8
votes
2answers
201 views

Second Order Differential Equation Example Request

I am looking for some non-complicated second order differential equations to illustrate certain techniques for control engineering. It doesn’t matter if the differential equations are linear or non-...
17
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12answers
19k views

Real-World Applications of Logic

When introducing logic in a first semester university course, the examples I use are often quite artificial. One example: One of three kids (Annie, Bob, Chris) has broken a window. Annies says "it was ...
4
votes
5answers
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 ...
12
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4answers
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 ...
9
votes
2answers
323 views

Good examples of non-convex optimization

I am looking for a function of on an interval with several local optima that appears in some mathematical model and which you can at least imagine that you want to optimize. I am teaching a calculus ...
13
votes
7answers
704 views

Non-Mathematical Examples of Orders

Different properties of different types of orders including partial, total, scattered and well-orders are a part of any graduate/undergraduate set theory course. I am looking for interesting "non-...
13
votes
6answers
439 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 ...
23
votes
8answers
1k views

Counterintuitive consequences of standard definitions

Let me motivate my question with the following situation. While teaching the concept of continuity, I usually start with motivating the concept. Then, when we see that there is an important and ...
11
votes
9answers
526 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 ...
16
votes
9answers
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 ...
15
votes
4answers
602 views

How would you explain what a PDE is to a very educated layman with no math background?

Is every mathematical concept, even the complex ones, explainable? As someone who will be needing to explain my line of work for a position to a committee who is very, very, educated, just not in ...
6
votes
1answer
253 views

What is a less anglo-centric collection of persons than Andy, Beth, Carl, Debby and Earl?

These five imagined persons have accompanied me for some time. We've had a bunch of laughs and a few tears. I love them dearly. That said, I'd like to retire them in favor of a more culturally diverse ...
26
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8answers
2k views

Unusual applications of integration

I am trying to teach my calculus students to apply integration by thinking about what they are integrating rather than just applying formulas. Calculus books are full of formulas like "to find the ...
6
votes
2answers
827 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 ...
27
votes
11answers
3k views

Impressive examples where a "proof by picture" goes wrong

There are many proofs where the whole idea can be expressed by a picture and often naturally translated into a correct formal proof. Often one has to argue with students that a picture is not a proof ...
23
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16answers
3k views

Examples of Mathematical Slang

Unless you have taught highschool algebra in Iran, you could not make sense of the phrase: Elephant and Teacup Identity! This is what teachers use to refer to the following identities: $ (a+b)(a^2-...
14
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4answers
421 views

Explaining subjects whose justification requires demanding technical content

This is my first question and I hope it's appropriate. Often in the process of teaching a subject I start with examples of a phenomenon, exhibiting similar properties between the examples and ...
36
votes
23answers
6k views

Imbuing a six year old with a sense of mathematical wonder

My six year old started school a few months back and he's loving it. This first year is more about social skills than anything academic and I like that approach. But we're spending some time at home ...
8
votes
9answers
416 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, ...
4
votes
3answers
106 views

Example illustrating use of additivity of (probability) measures before introducing independence and conditioning

I would like to be able to illustrate why additivity is a natural property to assume for probability measures. While it is relatively simple to give a short hand waving intuitive analogy to volume or ...
7
votes
7answers
7k views

Monte Carlo real life examples

I want to introduce Monte Carlo methods for a group of 16-18-years-old high school students. Besides classic examples (coin flips and count of heads/tails, rolls of a pair of dice) which other ...
2
votes
1answer
126 views

Proving convergence or divergence of series: tips and recommendations

This is a follow up of my question on MSE. Which tips and recommendations would you give students who want to investigate series about convergence or divergence? So far we have collected: It is ...
15
votes
3answers
81k views

What are good survey questions for statistics students to ask each other in class?

I think this is a nontrivial question, because to avoid embarrassment or singling-out, I cannot ask students about: their age anything about their body (height, weight) anything involving money ...
25
votes
3answers
656 views

Breaking students from the habit of relying on examples

One of the most frustrating things about my experiences teaching math (at the university level, if that matters) is that students seem very reluctant to actually learn the material. Instead, they seem ...
7
votes
0answers
264 views

Examples of multiple induction

It is easy to find/construct cases that can be proven by nested induction, i.e., some variation of the theme to prove the statement $P(m, n)$ you prove $P(1, n)$ by induction as a base case for $m$, ...
8
votes
2answers
5k views

Counterexamples to the Greedy Algorithm

In the graph theory section of my Discrete Math course I'll be covering Prim's Algorithm for finding the minimal spanning tree. I'd like to impress upon the students just how special it is that the ...
9
votes
4answers
188 views

Specific examples (like elementary proofs,or simple problems) which appear rich in abstractions when observed through the lens of abstraction

I am looking for pedagogically motivated examples (like elementary proofs,or simple problems) of "abstraction in action" ? I am looking for good specific examples (pre-university level or first year ...
11
votes
4answers
613 views

Example for a theorem where the (more) formal proof is easier than other argumentation (e.g. imagination)

When students ask me for the use of the formal and abstract theory, I often would like to give answers they wouldn't understand. For instance, one application of abstract vector spaces and the banach ...
9
votes
3answers
3k views

Questions for Oral Examination in Mathematics

I am working on some research on implementation of oral assessment in mathematics classroom, and I was wondering are there any questions/problems/concepts in mathematics that can be only assessed ...
8
votes
6answers
637 views

Convincing a high schooler that $i$ is a number

I would like to convince a high school student that $i$ is a number, broadly put. I'm not going to define what I mean by "number" unless he asks, but I just want to convince him that it's somehow ...
9
votes
3answers
223 views

Teaching and motivating the use of Eigenvectors

I would like to know how to better demonstrate Eigenvectors. The texts that I have display the properties and methods to calculate them. There are plenty of great elementary examples to follow through ...
13
votes
3answers
257 views

Counterexamples to "stable digit" theory of error estimates

When covering issues related to error estimates in a calculus course, students find the technique of making estimates (definition of limit, Newton's method, numerical integration, remainder formula ...
5
votes
2answers
228 views

Examples where roots are necessary for the solution

I currently write an article where I want to introduce roots. Thus I need to motivate them. Here I said, they can be used to find solutions of equations like $x^n=a$. Now I want to make some examples, ...
3
votes
2answers
198 views

How to teach mathematically about Fourier analysis and synthesis? [duplicate]

I have recently started teaching. It gets totally blank in front of the big crowd. Now, I am quite confused about how to start teaching the Fourier transform and Fourier series chapter. I want ...
6
votes
5answers
449 views

Hypothesis Tests in Students' Lives

Consider, in a basic statistics class, the difference between the following two examples: A six-sided die is rolled twenty times. It comes up a "two" eight times, which seems unfair. The owner of ...