37

In the U.S. Common Core standards, functions are supposed to be introduced in the 8th grade, i.e., around age 13-14. So arguably age 15 is a year or two behind where they ought to be. The standard for the 8th grade says: Understand that a function is a rule that assigns to each input exactly one output. So honestly that really doesn't seem like a hugely ...


31

I think you will find that almost everyone has this problem when they first starting to learn rigorous mathematics, and many students will never overcome this difficulty. The following three statements are logically equivalent to each other, but students will almost all find the first to be easier to understand than the second, and will not be mature enough ...


27

If your students have learned some statistics, then you could point out that the normal distribution's probability density function uses this double exponential. $$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right)$$


20

On the other hand, they are really struggling with injective functions. Even after spending a lot of time, they often say "a function is one-one if every element in the domain has a unique image". Have you asked your students what they mean by "unique" when they say that? The reason I'm asking is that, by some commonly used definitions ...


16

Functions are far broader and more applicable than you give them credit for. Consider the following: Country or state Capital Elevation (in meters) Bolivia Sucre 2783 Ecuador Quito 2763 Colombia Bogata 2619 Eritrea Asmara 2363 Ethiopia Addis Ababa 2362 Mexico Ciudad de Mexico 2216 New Mexico Santa Fe 2152 Wyoming Cheyenne 1856 Colorado Denver 1613 ...


12

Count all possible functions mapping $i$ input bits to $o$ output bits: For each of the $2^i$ input combinations, each output has two possible outputs ($0$ and $1$), i.e. you have $2^{2^i}$, leading to a total of $2^{o\cdot 2^i}$ binary functions. And course if you're working with $n$ different values per input/output instead of binary you end up with $n^{o\...


11

People respond well to perceived effort and investment on the part of others. When I get an email that is worded poorly from a colleague, or start reading a paper that has not been carefully revised, I am more likely to tune out and abandon the situation. It is possible that you put significant effort into creating this question, but when I read it, it feels ...


9

Some abstraction is perfect for children. Mitsumasa Anno was a Japanese writer of children's books and he had many innovative approaches to introduction functions. This is from Anno's Math Games II. You can even see the introduction of function inverse.


9

The question is probably too opinion-based to allow for a definite answer, but let me offer a few reasons why I think that we should rather not ignore domains and codomains when we teach functions to, say, highschool students (yes, this means I'm suggesting to do it differently from how it is now done in many places): (1) The OP is of course right that, from ...


8

The number of undirected graphs of order $n$ is $2^{n \choose 2} = 2^{(n^2-n)/2}$ (e.g. consider the adjacency vector as a binary-encoded number). You can describe this in terms of the number of different arrangements in which $n$ classmates can be friends. This also has the benefit of being easy to enumerate for small numbers.


8

In his Abstract Algebra book, John Fraleigh mentions that this is a common mistake beginning students make. Exercise 37 in Section 0 (7th edition) asks the reader to make a pedagogical case for using the terminology "two-to-two" instead of one-to-one. He doesn't expect the new terminology to actually take hold, of course, but just discussing it ...


7

Like most of the respondents here, I had to read this problem several times before I could understand what you are asking. The phrase "a random variable called $x$ with a function of proportional density to $x(\pi - x)$" is both nonstandard usage and potentially misleading. On my initial read of the problem I thought you were trying to describe ...


7

If 10-11 year olds can learn programming (which some have done even before Scratch was a thing), then it's hardly a leap at all to suggest that 15 year olds can learn functions since they are a common element of programming languages. Every student is going to be different and some are going to be better at math than others, but I think this notion that ...


6

You are asking a lot of them, even though each little bit of the problem isn't especially difficult and was probably covered in a class that they have taken. You need to remember a bit of trig/geometry to calculate the area from an angle, a bit of statistics/calculus to get the normalization constant and a formula for the expected value, and a bit of ...


6

Here are informal definitions of the terms that seem confusing to you: A function is a relation between two sets, usually sets of numbers. It maps elements of the first set to elements of the second set. An expression is a combination of symbols representing a calculation, ultimately a number. An equation describes that two expressions are identical (...


4

Gompertz model was created in 1825 to study human mortality curves. From the 1920's, it was used in economic fields, and from there it was also used in Biology to study cells and microorganisms, such as microbes, growth of tumours, and survival of cancer patients. The model is described by the differential equation $$\dfrac{dN}{dt} = rN \ln \left(\dfrac{K}{N}...


4

If $S$ represents the set of $n$ students of a school, then $\mathscr P(S)$ is the set of rosters for all possible clubs that that school could host (if we assume that any two clubs with exactly the same set of students merged their interests into a single club charter). In that same scenario, the yearbook that printed pictures of all of the clubs would be ...


3

It is easy to square a number. So instead of computing $\exp(z)$ you can compute $\exp(z/2^k)^{2^k}$ for some suitably large positive integer $k$. This is a very simple way of accelerating the convergence of the Taylor series for $\exp$, also known as "argument reduction". The double-exponential is important here because it allows you to get the $2^...


3

What a nice collection of answers! I am inclined to use $\lfloor A^{3^n} \rfloor$ where $A$ is Mill's constant, $$A \approx 1.3063778838630806904686144926 \;, $$ just because it is astounding that this evaluates to a prime for every $n \in \mathbb{N}$: $$ 2, 11, 1361, 2521008887, \ldots \;. $$ See A051254. But really, @JoelReyesNoche's answer is best.


3

The maximum of a large number of independent, identically distributed random variables -- e.g., the highest flood observed over a long period -- has an extreme value distribution. One common case is the Gumbel distribution, whose cdf has the double-exponential form $e^{-e^{-x}}$.


3

Most of pre 1900 mathematics can be done without the modern function concept. Hints that this was actually the case can be found in this hsm question Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function? or if you skim through Leonard Eulers books on differentiation and integration (you'll have to look very ...


2

To me, this question seems poorly worded. Not sure what "the" two sides of a triangle even means. Do you mean the two equal sides? After that, seems like we get some info about an angle and have to use that to predict info about the area. [Probably using the formula for area. Donno. Guessing.] OK. I'd say, this is medium hardish. Not end ...


2

We shouldn't need to teach functions to 15 year olds, because ideally they should have already learned programming since primary school, including mathematical and general functions and inverse functions. Programming, including demos, games and robotics, is the best motivator to learn math in my opinion.


2

Building off the circle example, you can actually work out the centripetal acceleration formula by implicitly differentiating twice. If your students aren't familiar with vectors you can just plug in x = 0 and y = 1: $$x^2+y^2=r^2$$ Differentiate with respect to t: $$2x\cdot x' + 2y \cdot y' = 0$$ Plugging in x = 0, y = r, you can solve $y' = 0$. ...


2

Inspired by your example, I like the number of matrices, tensors, binary or n-ary relations over a specified set (where the domain and range of such a relation need not match), such as asking about colorings using $k$ colors of a $n \times n$ board. Continuous or "non-combinatorial" applications feel hard to come up with, perhaps due to ...


2

I appreciate your concern and have felt similar despair when teaching about functions and the awkward game of "finding their domains" in precalculus courses. But then I lift myself up by thinking about analytic continuation. A real variable construct like $\sqrt{x}$ somehow just knows, without human intervention, what its domain should be. In this ...


Only top voted, non community-wiki answers of a minimum length are eligible