Questions tagged [examples]

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

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103 votes
35 answers
21k views

What female mathematician can I introduce to my High School students?

I enjoy talking about Pythagoras when I teach the Pythagorean theorem. I sometimes mention Descartes when introducing Cartesian coordinates. And Leibniz and Newton are mentioned in many calculus ...
48 votes
14 answers
4k 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 ...
7 votes
2 answers
172 views

Numerical / Computational problems suitable for grade 9-12 audience

I'm working with a number of students who have an interest in programming, and who represent a wide range of numeracy skills. I aim to exploit their interest in programming in order to 'trick' them ...
23 votes
8 answers
2k views

What are some good low-prerequisite examples for the heuristic advice "If you cannot prove it, prove something stronger."?

One useful trick in mathematics is to prove something stronger instead of the question asked. This works well in induction proofs (because strengthening the claim also strengthens the induction basis)...
3 votes
0 answers
74 views

Examples of Financial Institutions that Compute Interest Atypically?

Are there examples of financial institutions that compound their interest more frequently than once-a-month? Are there examples of financial institutions that consider continually compounded interest ...
13 votes
6 answers
5k views

"Real life" examples of limits of functions at finite points

This is more specific than this similar question on math.SE, since I'm not satisfied with the answers there. Question: Can you provide an interesting, natural and simple example of some physical/...
0 votes
1 answer
69 views

Seeking References on Deterministic and Stochastic Phenomena Suitable for High School Students

Can anyone recommend good and didactic references that delve into the dualism between deterministic and stochastic phenomena? Ideally, I'm seeking materials that provide a conceptual explanation along ...
9 votes
5 answers
3k 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 ...
2 votes
2 answers
196 views

Are these explanations of variance and covariance intuitive?

When tutoring, I try to simplify concepts. I came up with these examples to explain the intention behind variance and covariance. Could you please help me find conceptual, pedagogical or mathematical ...
4 votes
0 answers
174 views

What evidence is there in the literature that lessons geared towards dyslexic student help non-dyslexic students?

Dyslexic students sometimes benefit from informal analogies to things in the world which the student can see with their eyes, and/or touch with their hands. Tentatively, we can conjecture that ...
19 votes
18 answers
3k views

Concrete vectors spaces without an obvious basis or many "obvious" bases?

I am teaching a class on linear algebra to sophomore and junior science majors, and am having some trouble illustrating the difference between $\mathbb{R}^n$ and an n-dimensional vector space. The ...
3 votes
0 answers
69 views

Are there any fun toy applications of representation and character theory for finite groups to physics?

Representation theory has very deep ties with physics, leading to incredibly profound and admittedly cool results such as the classification of particles in the Standard Model via mass and spin by ...
4 votes
6 answers
413 views

Which examples should we mention when teaching the concept of derivatives?

I am teaching Calculus for non-maths major students. As far as I know, when we teach about derivatives, we should mention "the rate of change". There are some practical examples to motivate this ...
36 votes
6 answers
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 ...
43 votes
28 answers
12k views

Good, simple examples of induction?

Many examples of induction are silly, in that there are more natural methods available. Could you please post examples of induction, where it is required, and which are simple enough as examples in a ...
17 votes
4 answers
2k views

Surprising examples of Cavalieri's principle

I showed my calculus students two circular cones with the same base and height, but one of them "right" and the other slanted, and asked which had a greater volume. They all answered correctly that ...
3 votes
3 answers
247 views

Applications of Triangle Inequality for high-school students?

The Triangle Inequality ($|x+y|\leq|x|+|y|$) is useful later on in the student's math education (e.g. in proving results about limits). But for the high-school student, are there any useful and ...
4 votes
2 answers
498 views

Proof by Contradiction vs. Proof of Negation

In constructive mathematics we make a distinction between "proof of negation" and "proof by contradiction". You can read a great account of the difference in this blog post of ...
8 votes
7 answers
553 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 ...
36 votes
24 answers
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 ...
3 votes
1 answer
159 views

Favorite linear programming (not integer) examples?

I am wondering what examples you like to give when introducing linear programming, where the examples are not clearly better suited as integer linear programs. I would like a few examples where we can ...
20 votes
6 answers
12k views

Good examples of proof by contradiction?

In later courses on automata theory, many students just seem incapable of getting a proof that a language isn't regular right, be it using the pumping lemma (see also the many questions on the matter ...
9 votes
7 answers
211k 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 a lot of sequences and series every day in our regular life. I already found some examples such as the house numbers when you ...
24 votes
17 answers
4k 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-...
17 votes
8 answers
13k 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) ...
29 votes
17 answers
7k views

Examples of Innumeracy

I read Innumeracy by John Allen Paulos and would like to share more up-to-date and relevant examples of innumeracy to motivate my grade 8, 9 & 10 students. Are there any websites, blogs, books, ...
28 votes
10 answers
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 ...
9 votes
1 answer
303 views

How well can students learn abstract concepts through concrete examples?

In my own personal experience in teaching linear algebra, where many students encounter abstract ideas for the first time, I find that most students have trouble consolidating observations from ...
10 votes
2 answers
773 views

Real-world applications of taxicab metric

The taxicab metric can be used to measure distances in idealized gridded cities. However, usually this serves only as a fun exercise for students. I'm looking for engaging (as non-technical as ...
4 votes
3 answers
284 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 ...
3 votes
2 answers
189 views

Good Examples of Equations Derived from Elementary Calculus

I'm collecting additional enrichment content for my calculus students. I'm looking for examples of equations that are used in various fields, but which can be derived at least somewhat ...
6 votes
4 answers
846 views

What is the best way to introduce Laplace transforms on an Engineering Mathematics course?

Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if ...
0 votes
1 answer
221 views

Abstract math, examples and understanding or visualising

After reading some papers about special kinds of algebras and rings like Gorenstein rings, Dickson algebras, Cayley-Dickson construction, i want to ask do examples of general abstract objects in math ...
28 votes
5 answers
3k views

What are some good examples to motivate the implicit function theorem?

I always had problems teaching the implicit function theorem in advanced analysis courses. This result is motivated by later applications, but it would be great to provide easily accessible examples ...
79 votes
17 answers
20k 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 (...
7 votes
6 answers
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 ...
15 votes
7 answers
3k 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. ...
2 votes
1 answer
96 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!
7 votes
5 answers
853 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&...
26 votes
13 answers
3k views

What is a good motivation/showcase for a student for the study of eigenvalues?

Courses about linear algebra make great demands on looking for eigenvalues and transforming matrices to diagonal matrices (or, at least, to Jordan normal form). This is somehow a technical, recipe-...
4 votes
4 answers
445 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-...
25 votes
5 answers
1k 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 ...
4 votes
2 answers
187 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" ...
17 votes
8 answers
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 ...
15 votes
3 answers
5k views

What is a good physical example of Stokes' Theorem?

I find it useful to give physical examples of theorems, especially in vector calculus - for example $\nabla f$ being the direction of maximum ascent on a surface $f$. What is a good example for ...
4 votes
1 answer
646 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 ...
1 vote
0 answers
214 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 ...
11 votes
4 answers
10k views

What are easy examples from daily life of constrained optimization?

A standard example of motivating constrained optimization are examples where the setup is described in a lot of lines, e.g., when you own a company and the company is making some products out of ...
5 votes
2 answers
670 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 ...
17 votes
4 answers
726 views

What are some good ways to motivate and introduce reasoning abstractly about abstract algebra?

I've found one of the hardest topics to introduce to students early on is abstract algebra. Even if they've already written proofs, it's hard for them to work directly from axioms. They seem to have ...