# Tag Info

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, ...
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 ...
• 10.9k

### 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 ...
• 1,715

### 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 ...
• 369
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 ...
• 8,299

### 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 ...

• 4,893
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, ...

### 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,...

### 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 ...
• 1,078

### 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 ...
• 1,249

### 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 ...
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### 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 ...
• 201

### 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 ...
• 421
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### Is there a toy example of an axiomatically defined system/ structure?

This may go beyond what you are asking for, but there is a wonderful book called Introduction to the Foundations of Mathematics by Raymond L. Wilder. I provided its axioms and an example of how they ...
• 18.7k

### Is there a simple real-world problem I can use to motivate a formula for $\displaystyle \sum_{i=1}^n i$?

Students sometimes find applications to computer programming interesting. The sum in question often pops up in determining the complexity of various algorithms. For a simple example, selection sort ...
• 1,536
As a simple illustration of the most basic case, $\mathbb{Z}/2\mathbb{Z}$, I sometimes push the light switch of the classroom about a dozen times in a row, too fast for them to count. Then I ask: &...