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Questions tagged [examples]

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

5
votes
1answer
108 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 ...
10
votes
3answers
338 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 ...
9
votes
2answers
657 views

Good real-life examples of transformations of function graphs

I am a gradudate student teaching college algebra at a larger state school and transformations of graphs of function, i.e.: given the graph of a function $y =f(x)$, what do the graphs $y = f(x) \pm C$,...
8
votes
6answers
314 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
137 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%...
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votes
13answers
2k 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. ...
67
votes
15answers
18k 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 (...
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7answers
2k 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) ...
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5answers
111 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
540 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 ...
13
votes
8answers
764 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 ...
10
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9answers
345 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 ...
13
votes
6answers
382 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 ...
2
votes
0answers
173 views

Benefits of knowing theory

I've got somewhat an issue: from time to time I have to teach some math to people who either avoided it or got trough by only knowing a few working algorithms. The only thing that unites all those ...
16
votes
9answers
687 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 ...
8
votes
9answers
280 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, ...
5
votes
2answers
223 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 ...
7
votes
7answers
2k 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 ...
4
votes
3answers
81 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 ...
14
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4answers
494 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 ...
2
votes
1answer
98 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 ...
6
votes
7answers
62k 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 ...
22
votes
3answers
466 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
135 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 $m$, ...
17
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3answers
524 views

A Series of Unfortunate Examples!

All of us know, when teaching, the "right" choice of examples is important. Though, making the "right" choice is one of those things that are easier said than done. Here is the story of a series of ...
7
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2answers
2k 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 ...
12
votes
3answers
445 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 ...
12
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3answers
886 views

Mathematical education slang

Amir Asghari recently asked a question about mathematical slang. He was "looking for "non-mathematical" terms or phrases that are used to refer to mathematical objects (of any kind) mainly for ...
22
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16answers
1k 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-...
9
votes
2answers
172 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 ...
35
votes
23answers
5k 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 ...
9
votes
4answers
168 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 ...
8
votes
3answers
2k 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 ...
6
votes
1answer
223 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 ...
8
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6answers
510 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 ...
6
votes
2answers
162 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-...
9
votes
3answers
190 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 ...
6
votes
2answers
197 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, ...
13
votes
3answers
222 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 ...
3
votes
2answers
184 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 ...
12
votes
7answers
560 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 ...
8
votes
3answers
656 views

Gifs of finding the volume of 3d shapes?

I'm looking for some animations (videos or gifs) of finding the volume of different 3d shapes. It would be super helpful if I could find something which stacks unit cubes into a rectangular prism to ...
9
votes
1answer
566 views

Open-ended tasks for teaching students about integration techniques

One of the best algebra-teaching games I've seen is the "Four 4's" game, where students have to take 4 fours and construct every number from 1-100 using only those fours and algebraic operations: 44/...
2
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0answers
54 views

Feedback about statistical and numerical illiteracy-examples website? [duplicate]

I have assembled a list of statistical and numerical illiteracy examples in the media, as they appeared in Chance News online (a statistics wiki) between 2005 and 2014. See "Collected Forsooths at ...
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votes
2answers
208 views

Designing a Good Question on Kinematics: Test and Develop

So I was asked in an interview to design two questions for UK Physics A Level students studying the $suvat$ equations. The first needs to 'test understanding' of the equations, while the second needs ...
13
votes
6answers
660 views

Where do you find math tasks?

What the title says. I got my degree in math last year and now I'm working on a master's independent study project through my education department finding math tasks for K-12 curriculum aligned with ...
9
votes
2answers
245 views

What are some good or neat examples of computing a function's Taylor series?

Starting from \begin{equation*} \frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots, \end{equation*} which holds when $|x| < 1$, we can conclude that \begin{equation*} \frac{1}{1 + x^2} = 1 - x^2 + x^4 - x^...
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4answers
563 views

What are your favorite instructional counterexamples on sequences?

In this article, I give counterexamples regarding real sequences. And in that one some others. In particular counterexamples answering questions like: "If for all $p \in \mathbb{N}$ $\lim\limits_{n \...
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3answers
779 views

Honors project idea for linear algebra

At my institution there is an Honors program which encourages (or requires) students to petition classes for Honors. Basically, what this amounts to is the student has to write a 10 page (not set in ...
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votes
10answers
5k 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 ...