89
votes
Accepted
What's a replacement for "married couples" in combinatorics problems?
I've been using "pets" and "owners" (as in: possible pet-shelter adoptees) in recent years.
- 24.4k
84
votes
What's a replacement for "married couples" in combinatorics problems?
In the stable marriage problem, you can introduce the problem as it is. But then you ask your students how things change if you assume there are not only heterosexual but also gay and lesbian people (...
- 1,197
68
votes
What's a replacement for "married couples" in combinatorics problems?
A few possibilities off the top of my head:
Students and chairs. How many ways are there for $n$ students to sit in $k$ chairs. The game of musical chairs might be fun to play around with. One can ...
- 7,520
37
votes
What's a replacement for "married couples" in combinatorics problems?
The issue is not making problems about heterosexual married couples. The issues are:
Implicitly making the assumption that all married couples are heterosexual.
Making problems about heterosexual ...
- 857
34
votes
What's a replacement for "married couples" in combinatorics problems?
Try objects that often occur in pairs but are distinct from each other: forks and spoons (or forks and knives), left and right shoes, salt and pepper shakers, and so on (where each fork has an ...
- 10.8k
34
votes
Accepted
How can teachers warn students about common mistakes without causing the student to make the mistake?
This is a 100% subjective opinion, but it is based on teaching in various venues for close to 20 years (although none of that teaching was pure math). Also, my college calculus courses are close to ...
- 1,346
32
votes
"Real life" examples of limits of functions at finite points
First thing that comes to my mind is the limit $$\lim\limits_{x \to 0} \dfrac{\sin x}{x} = 1.$$
This limit justifies the small-angle approximation $\sin \theta \approx \theta$ (for $\theta \approx 0$) ...
- 6,262
27
votes
What's a replacement for "married couples" in combinatorics problems?
When I taught a class about the stable marriage problem last week, I replaced "men" and "women" with "medical students" and "hospitals": the classical instance in which the Gale-Shapley algorithm is ...
- 811
27
votes
Accepted
Concrete vectors spaces without an obvious basis or many "obvious" bases?
Some physical examples from physics:
Consider two spaceships that meet each other in deep space with arbitrary orientations (pitch, roll, and yaw). Even if they take the origin to be the midpoint ...
- 465
25
votes
Accepted
An introductory example for Taylor series (12th grade)
One practical reason for choosing a Taylor Series approximation of a function over the function itself is if you are able to compute using only the four arithmetic operations. For example, if you are ...
- 10.8k
22
votes
A Series of Unfortunate Examples!
Personally, I refer to this phenomenon as students "submarining" a broken understanding on a particular kind of problem.
Example #1: Our in-house elementary algebra textbook, in its first edition, ...
- 24.4k
22
votes
How would you explain what a PDE is to a very educated layman with no math background?
I would say something like this: "Often in complicated systems one needs to study multiple quantities, each of which varies at rates that depend on the other quantities and on how fast they are ...
- 17.3k
21
votes
Accepted
Big list of "interesting" abstract vector spaces
Here are some more examples:
$C[a,b]$, the set of continuous real-valued functions on an interval $[a,b]$. This abstract vector space has some very nice properties that make it very good for a first-...
- 17.3k
20
votes
"Real life" examples of limits of functions at finite points
"interesting, natural and simple"
Illustrating something dynamically might make things interesting. For example, a geometry problem involving a limit (from an old calculus book):
Consider a ...
- 9,184
19
votes
An introductory example for Taylor series (12th grade)
An excellent introductory example would be exponential function $\exp(x) = e^x$.
By definition, this is the function that is its own derivative, i.e. $\exp'(x) = \exp(x)$. That's all nice and swell ...
18
votes
Accepted
Proof by contradiction - more than one case
(1)
Here is a $3$-case proof from Larry Cusick's webpages:
Theorem. There are no rational number solutions to the equation $x^3 + x + 1 = 0$.
Proof. (Proof by Contradiction.) Assume to the ...
- 29.5k
18
votes
Are there direct practical applications of differentiating natural logarithms?
Have you thought about the fact that you’re asking this in the middle of a pandemic for which log plots are being used all over the place to visualize the growth of COVID cases?
At any rate,
$${d \...
- 5,345
17
votes
Examples of Mathematical Slang
One of the most colorful names I have heard is the Chicken Mc Nugget theorem:
for any two relatively prime positive integers $m,n$, the greatest integer that cannot be written in the form $am + bn$ ...
- 500
17
votes
Examples of arithmetic and geometric sequences and series in daily life
I tutored a student who came with a kind of problem I had never seen before and found quite refreshing. It was something like:
A child is being pushed on a swing by their father, reaching a maximum ...
- 581
16
votes
Imbuing a six year old with a sense of mathematical wonder
I remember being excited about the following at a young age.
If you add consecutive numbers you get triangle numbers.
Triangle numbers are fun.
If you put two consecutive triangle numbers together ...
- 7,999
16
votes
Simple examples that violate group axioms
Combining colored paint is an interesting example of a non-associative operation.
Define $Paint_1 * Paint_2$ to be the paint obtained by mixing the two paints in a $1:1$ ratio. It is easy to see that ...
- 1,506
15
votes
Imbuing a six year old with a sense of mathematical wonder
How about:
Numbers go the other way, too (negative)
You can cut numbers in half, forever
What if you cut a number into three pieces?
1 million is a thousand thousands (100 is ten tens)
If you don't ...
- 251
15
votes
Examples of basic non-commutative rings
The quaternion ring is a pretty simple example of a non-commutative ring (a skew-field, even).
- 3,800
15
votes
Mnemonics for some properties in mathematics
Recently, a student in my beginning algebra course offered the following to the class, regarding signed number multiplication:
Assuming positivity is like love, and negativity is like hate, then...
"...
- 9,184
15
votes
Concrete vectors spaces without an obvious basis or many "obvious" bases?
Two more examples:
The set of infinite Fibonacci-type sequences (those of the form $a_n=a_{n-1} + a_{n-2}$) (with point-wise addition and scaling) forms a 2-dimensional (real) vector space. E.g., ...
- 9,184
14
votes
Good, simple examples of induction?
Here is another one:
$\color{blue}{\text{Prove that the power of $13$ can be writen as a sum of two squares}}.
$
I will give two proofs of it. First one is more involved and includes the following ...
- 400
14
votes
What's a replacement for "married couples" in combinatorics problems?
Protons and electrons (form hydrogen atoms)
Or cations and anions (form salts), e.g. Na+ and Cl-
Pens and pen-caps
Bottles and bottle caps, etc.
Textbooks (for the course being taught) and ...
- 377
14
votes
How can teachers warn students about common mistakes without causing the student to make the mistake?
Here's another approach when there is a common pitfall that you wish the students to avoid. After teaching the correct reasoning: present the error to the class and ask a student to identify, explain, ...
- 243
14
votes
Are there direct practical applications of differentiating natural logarithms?
Whenever we measure a quantity on a log scale (such as Richter, decibels, musical pitch, or a log-plot axis), we are focusing attention on relative variation in that quantity. If $y = \ln x$, we have
$...
- 271
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