New answers tagged

6 votes

How can I improve my Retake Policy?

As a college math professor, I have a retake policy. I've been trying to encourage other professors to develop retake policies, and it seems to be becoming more common with resources emerging like The ...
user avatar
  • 1,919
-2 votes

How can I improve my Retake Policy?

Many high school classes don't have a "retake policy", so you could just devolve to that. I.e. no retakes. It's sort of nice what you are doing now, but definitely not required, in my ...
user avatar
  • 1
4 votes

How can I improve my Retake Policy?

I'm not strictly familiar with how the US school year works, so take my suggestions as needing adaptation. I allow much restricted retakes in my classes, based on a system I met in undergrad: For ...
user avatar
2 votes
Accepted

Importance of exploring skills in mathematics

"How can you improve exploring skills [skills involved in discovering new mathematics] in high school mathematics education...?" I'm still feeling fuzzy on the question. It seems related to ...
user avatar
  • 17k
1 vote

Examples of relations that are not functions

Let $p(x,y)$ be a two-variable polynomial (or really any function of two variables). Set $a R b$ if $p(a,b) = 0$. This is a binary relation that typically is not a function. Example: $p(x,y) = x^2 - ...
user avatar
  • 1,968
2 votes

Importance of exploring skills in mathematics

I think by being upbeat and motivated and a good example, as a teacher. A lot of people are inspired by HS teachers who were particularly magnetic. I probably should/could have been a physicist, but ...
user avatar
-1 votes

Examples of relations that are not functions

I just think of graphs, rather than input-output machines. A circle is a relation. xsq + ysq = rsq. Gives you more than one y, for a given x. [provided |x| is <r.] Lot's of good graphs to draw ...
user avatar
0 votes

Explaining why volume of cone is a third of cylinder

A possibility may be to show how a cube consists of 6 equivalent pyramids. The picture below shows three of them, and has empty slots in between for the missing three (I felt this would make the ...
user avatar
2 votes

Examples of relations that are not functions

I found this example in "Finite Mathematics" by Rolf: When Sarah buys six apples ($x=6$), the checker determines the weight in order to know the cost. The six apples of the customer behind ...
user avatar
  • 6,350
1 vote

Examples of relations that are not functions

Maybe your students don't yet know this (but it might be instructive to revisit this when it comes up): The indefinite integral (antiderivative) can be considered a map from functions to functions, ...
user avatar
  • 111
5 votes

Examples of relations that are not functions

A classic example, and perhaps the reason that people started thinking about "multivalued functions" in the first place: Try to define $\theta(x,y) = \displaystyle\int_{(1,0)}^{(x,y)} \frac{-...
user avatar
3 votes

Examples of relations that are not functions

If you are also teaching sets, or a student brings this up, consider the mapping from sets to their elements: $f(\{1, 2\}) = 1\\f(\{1, 2\}) = 2$ So this is not a function. On the other hand, there is ...
user avatar
  • 131
2 votes

Examples of relations that are not functions

Map each country to their official languages. This might look like a (not injective) function depending on which countries you start out with but there are definitely countries with more than one ...
user avatar
  • 129
16 votes

Examples of relations that are not functions

An example that should be natural to the students is the square root over the positive real numbers. If one simply says "square root of $4$" then there are two equally nice roots, $2$ and $-...
user avatar
2 votes

Examples of relations that are not functions

As you are teaching to students in a classroom, and presumably they are not your only students nor you their only teacher, a variety of mappings between the teachers, students, classes, and rooms ...
user avatar
  • 227
1 vote

Examples of relations that are not functions

A confidence, tolerance, or predictive interval maps data to a range. Note that there are additional lessons here because there are an infinite number of confidence intervals. Any function that ...
user avatar
12 votes

Examples of relations that are not functions

Given points $P = \{x_1,x_2,\ldots \}$ on a line $L$, let $nn(x)$ for $x \in L$ be the nearest neighbor to $x$: the point in $P$ closest to $x$. For most $x$, there is a unique $nn(x)$, but if $x$ is ...
user avatar
9 votes

Examples of relations that are not functions

I’ll give a purely mathematical example that I often used with students beyond the elementary calculus level. (See last paragraph below regarding college algebra and precalculus level students.) At ...
user avatar
2 votes

Examples of relations that are not functions

Another "real life" example is "result of throw of a die" (namely, from one to six pips).
user avatar
  • 13.4k
23 votes

Examples of relations that are not functions

The example I use: $f(x) = x$'s sister Looks fine, has a formula. I can write "$f($Chris$) = $Jessica" since I have a sister. I can talk about the domain of $f$ by asking for someone to ...
user avatar
  • 19.1k
1 vote

Equation of a straight line on two dimensional Cartesian plane

TL;DR I try not to get overly hung up on notation, and prefer to teach some kind of principle. In this setting, the principle is one of geometric transformation using basic techniques. Many ...
user avatar
5 votes

Explaining why volume of cone is a third of cylinder

Without Calculus, you can still be sort of rigorous. With Euclidean geometry we can find the centroid of a right triangle by finding the point where all the medians intersect. A cardboard triangle ...
user avatar
10 votes

Explaining why volume of cone is a third of cylinder

One framework of understanding and building towards proof that we teach to mathematics teachers at primary school level is (and translated to English on fly): Naive empiricism: The pupil tries an ...
user avatar
  • 4,544
27 votes
Accepted

Explaining why volume of cone is a third of cylinder

This is an experiment which can lead you to guess that the volume of a cone is approximately $\frac{1}{3}$ the volume of a cylinder with the same base and height. It is not a proof in any sense of ...
user avatar
7 votes
Accepted

Equation of a straight line on two dimensional Cartesian plane

Assuming we are talking about an algebra class because of the secondary-education tag, the reason it is okay to use "point-slope form," i.e. $y - y_1 = m(x - x_1)$ as the default way to ...
user avatar
  • 19.1k
4 votes

Equation of a straight line on two dimensional Cartesian plane

@TomKern mentioned the "standard" form above. I believe what he has in mind is ax+by=c, which allows for both horizontal and vertical lines. Another form, rarely used, but more like the ...
user avatar
  • 17k

Top 50 recent answers are included