28 votes
Accepted

What's the point of learning equivalence relations?

Perhaps emphasize to students the spirit of equivalence relations. They partitions sets into equivalence classes--cutting down the amount of cases necessary to prove something. To illustrate this, ...
Andrew Sansom's user avatar
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 ...
JRN's user avatar
  • 10.8k
21 votes

teach that $\frac10$ not defined properly

What is $\frac 1 a$? It is the unique (real) number such that $a\cdot \frac 1 a=1$. Does there exist a real number that multiplied by $0$ gives $1$? No. Why is this? Because if $0\cdot b=0$ which ever ...
Nicola Ciccoli's user avatar
20 votes

Why do we care about multiple proofs of the same theorem?

I'd say different proofs usually employ different techniques, which in turn might be applicable to different sets of other theorems. So the more proofs I know for one theorem, the higher the chances ...
MvG's user avatar
  • 369
19 votes
Accepted

Replacement for the Pac-Man grid analogy

A possibility is to show your students Google's implementation of the game Snake. If you enter the term snake into Google's search engine, the there is a box at ...
Xander Henderson's user avatar
  • 8,185
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 ...
cmaster - reinstate monica's user avatar
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 \...
user1815's user avatar
  • 5,545
17 votes

How can we motivate that Newton's method is useful?

I'm going to respond from an applied math (or maybe CS) perspective: Part of the problem is that the functions that you look at in high school and the standard calculus sequence are unrealistically ...
Adam's user avatar
  • 5,703
16 votes

What's the point of learning equivalence relations?

In general, all applications are going to be more of the "here is what we have to check to make sure that our algorithm/theory/definitions work". We don't usually encounter practical ...
Misha Lavrov's user avatar
15 votes

How can we best motivate the study of polynomials to high-school students?

Using puzzles to attract attention: "Think of a number, subtract 7, multiply 3, add 30, divide by 3. Then subtract the original number. The result will always be 3. Why does this magic work?"...
Spai's user avatar
  • 299
14 votes
Accepted

Good way to explain fundamental theorem of arithmetic?

The way I have explained the fundamental theorem of arithmetic in the past is by establishing what it means to be prime (has exactly two positive divisors) and then having students construct factor ...
Benjamin Dickman's user avatar
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 $...
nanoman's user avatar
  • 271
14 votes

How can we motivate that Newton's method is useful?

In video game animation, each second thousands of equations need to be solved with code. You can't tell a computer "look at a graph"! Your students need to think beyond the setting of ...
KCd's user avatar
  • 3,446
13 votes

How to teach the Pythagorean theorem in a satisfying way to high school students?

I really like the idea of a "discovery fiction" -- it gives a name to something I often try to use when teaching. Here is one suggestion. I will try to come back and write a more elaborated ...
mweiss's user avatar
  • 17.4k
13 votes

Does induction really avoid proving an infinite number of claims?

The "avoidance of proving an infinite number of claims" explanation for the need for induction has not yet resonated with me because there are obviously many universally quantified ...
Steve's user avatar
  • 1,514
12 votes

An introductory example for Taylor series (12th grade)

I feel it a bit strange that no one mentioned it, but a famous example is how Newton quickly estimated $\pi$ by the Taylor series. Here is a quick sketch: First note that the area of a quarter circle ...
tryst with freedom's user avatar
12 votes

Any meaning/interpretation for $\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\dots (= \mathrm e)$ (sum of reciprocals of factorials)?

You could ask this series of questions; I don't know how "real world" it is but it does indicate the specific contributions of each individual term of the series. Suppose we have a car, and ...
Chris Cunningham's user avatar
11 votes

How to teach binary numbers to 5th graders?

When teaching binary to any age group I always start by taking a set of kitchen weights into the classroom. 2lb (32oz) 1lb (16oz) 8oz 4oz 2oz 1oz. Then make a table asking what weights yo1u would use ...
C Wren's user avatar
  • 119
11 votes
Accepted

How can I motivate the formal definition of continuity?

Have a look at the paper written by Nunez et all: EMBODIED COGNITION AS GROUNDING FOR SITUATEDNESS AND CONTEXT IN MATHEMATICS EDUCATION. In essence, they argue that it is better to be causious if ...
11 votes
Accepted

Why should we study continuity?

Most functions that are studied by physicists and other scientists are continuous. However, more and more discontinuous functions are appearing in the various sciences. This is due to: Computers and ...
Rory Daulton's user avatar
  • 2,582
11 votes

Good way to explain fundamental theorem of arithmetic?

To really understand why the integers $\mathbb{Z}$ have unique prime factorization, it helps to understand how unique prime factorization can fail in other settings. For example, $$(2 + \sqrt{10}) \...
Daniel Hast's user avatar
  • 4,893
11 votes
Accepted

Why do we care about multiple proofs of the same theorem?

Worth noting here is that there is an additional level of certainty in the validity of a statement when you have multiple proofs of said statement. By having multiple ways to solve the same problem, ...
Brevan Ellefsen's user avatar
11 votes

What's the point of learning equivalence relations?

Equality vs. Identity Alexei touched on this topic by mentioning hash tables, but I would like to spell it out more explicitly, because this is a critical and fundamental topic in software engineering,...
Lawnmower Man's user avatar
11 votes

Does induction really avoid proving an infinite number of claims?

My personal take on this, is that all the talk about "infinite this, and infinite that" is only mudding the waters. The emphasis should not be on wanting to prove $P(n)$ for all all $n$, but ...
Martin Argerami's user avatar
10 votes

Should I be teaching point-slope formula to high school algebra students?

Point slope form teaches a very important concept in a very simple way - that of translation. When you get into quadratics or any other equation, you can move the equation around by taking a standard ...
johnnyb's user avatar
  • 1,249
10 votes
Accepted

How can one motivate the adjugate matrix?

Here is one way to put the rabbit in the hat before pulling it out: Derive the general formula for the inverse of an invertible matrix. It ends up having the form $A^{-1} = \frac{1}{\textrm{Det}(A)} ...
Steven Gubkin's user avatar
10 votes

Why do we study ordinary differential equations?

ODEs are used in many models to determine how the state of this model is changing (regarding time or another variable). […] Am I missing another application […]? This may be somewhat pedantic, but I ...
Wrzlprmft's user avatar
  • 2,548
10 votes

Why do we study ordinary differential equations?

Of course I agree that one motivation for studying ODEs is that they have applications. But it might be useful to also point out another fact that students do not always think about: ODEs are often ...
Gustav's user avatar
  • 201
10 votes

Against introducing precise definitions first

At the secondary level, students have not yet mastered formal mathematics and most will need to continue learning concepts before definitions in many cases. The van Hieles (the Dutch educators who ...
Scott Eberle's user avatar

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