Questions tagged [examples]

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

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43 votes
28 answers
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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 ...
vonbrand's user avatar
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59 votes
24 answers
72k 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 ...
Chris Cunningham's user avatar
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 ...
Mathdad's user avatar
  • 580
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, ...
David Ebert's user avatar
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13 votes
7 answers
959 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 ...
Wmol's user avatar
  • 1,061
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 ...
David Ebert's user avatar
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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 ...
vonbrand's user avatar
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16 votes
7 answers
1k 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 ...
futurebird's user avatar
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-...
Markus Klein's user avatar
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25 votes
13 answers
20k views

Ideas for high school pure maths projects

I am thinking of giving my high school students some pure maths projects to do. It is a lot easier to think of some interesting stats projects but not in pure maths. The students' maths background are ...
user71346's user avatar
  • 359
25 votes
8 answers
2k 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 ...
András Bátkai's user avatar
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-...
Amir Asghari's user avatar
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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)...
user11235's user avatar
  • 1,280
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 ...
adamblan's user avatar
  • 1,998
13 votes
8 answers
5k views

What are some fun/nonstandard examples of arithmetic/geometric series?

I am teaching those topics (arithmetic/geometric series) just now, and want some not so standard (fun) examples, which can be used essentially at high school/beginning calculus level. I'm ...
kjetil b halvorsen's user avatar
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 ...
Steven Gubkin's user avatar
37 votes
4 answers
2k views

How to convey the meaning of "mathematical maturity"?

Some university-level courses have no specific prerequisites, yet are mathematically involved to the extent that someone with little to no experience in math will probably find themselves in over ...
Adam Bjorndahl's user avatar
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 ...
user avatar
33 votes
10 answers
11k 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 ...
Steven Gubkin's user avatar
30 votes
8 answers
2k views

Good motivation for the introduction of Lebesgue integral?

When students take a course on real analysis, they have likely learned about Riemann integrals. What is a good motivation why they have to learn a new way to integrate? A student don't want to hear ...
Markus Klein's user avatar
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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 ...
Mike Shulman's user avatar
  • 6,570
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 ...
András Bátkai's user avatar
27 votes
3 answers
794 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 ...
anomaly's user avatar
  • 535
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 ...
Amir Asghari's user avatar
  • 4,428
18 votes
12 answers
22k 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 ...
Anschewski's user avatar
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18 votes
5 answers
1k views

Rigorous proofs vs. illustrative examples

No one would argue against the idea/ observation that proofs are very important in mathematics. Some people are trying to make their notations on a blackboard during a lecture as consistent as ...
user35603's user avatar
  • 281
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. ...
jng224's user avatar
  • 261
13 votes
6 answers
470 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 ...
Ovi's user avatar
  • 725
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 ...
Markus Klein's user avatar
  • 9,438
11 votes
4 answers
673 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 ...
Anschewski's user avatar
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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 ...
Michelle_B's user avatar
9 votes
1 answer
865 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/...
Brian Rushton's user avatar
8 votes
2 answers
476 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 ...
Buster Bie's user avatar
8 votes
2 answers
219 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-...
solid's user avatar
  • 183
7 votes
2 answers
823 views

Examples for reasoning by analogy going wrong

Assume you don't know the result of $(-1)\cdot 3$. You might try calculating $$ \begin{align*} 3\cdot 3=9\\ 2\cdot 3=6\\1\cdot 3=3\\0\cdot 3 = 0 \end{align*} $$ thus you may argue $$(-1)\cdot 3=-3$$ ...
Anschewski's user avatar
  • 4,811
7 votes
7 answers
8k 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 ...
Toc's user avatar
  • 171
7 votes
6 answers
849 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 ...
orion2112's user avatar
  • 1,007
5 votes
1 answer
169 views

A blog or newsfeed for innumeracy in the media?

There are a lot of good particular examples of innumeracy at this question. But in addition what I would like is to be able to consistently bring current examples in to class. Does anyone maintain a ...
Mike Shulman's user avatar
  • 6,570
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 ...
Ahmed's user avatar
  • 41
3 votes
2 answers
207 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 ...
devraj's user avatar
  • 179
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 ...
user avatar
1 vote
2 answers
361 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, that is, equations of motion with a constant acceleration. The first needs to '...
Henry Murray's user avatar