# Tag Info

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### 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.

### 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 (...

### 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 ...

### 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 ...

### 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 ...
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### 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 ...

### "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 $\theta \approx 0$) ...

### 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 ...
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### 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 ...
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### 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 ...

### 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, ...

### 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 ...
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### 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-...

### "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 ...

### 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 ...
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### 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 ...

### 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 \...

### 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 $am + bn$ ...

### 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 ...

### 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 ...

### 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 ...

### 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 ...

### Examples of basic non-commutative rings

The quaternion ring is a pretty simple example of a non-commutative ring (a skew-field, even).

### 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... "...

### 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., ...

### 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 ...

### 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 ...
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 \$...