57 votes

Optimization problems that today's students might actually encounter?

Here's the example I had which inspired me to post the question in the first place: The game League of Legends was the most-played PC game, in number of hours played, in North America and Europe in ...
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  • 19.1k
48 votes

Should we avoid indefinite integrals?

On quizzes, homeworks, and tests, I repeatedly ask questions like this: Find three different functions that have derivative equal to $x^2 + x$. Forcing them to do antiderivatives and deal with the ...
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  • 19.1k
37 votes
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Dividing by zero

You asked: "How do/would you explain why division by zero does not produce a result." Any such explanation that is not rooted in student understanding would be talking to ourselves, not to ...
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  • 7,375
36 votes
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Optimization problems that today's students might actually encounter?

Bad Optimization Problems I thought that Jack M made an interesting comment about this question: There aren't any. There may be situations where it's possible to apply optimization to solve a ...
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  • 7,930
36 votes

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

As someone who teaches calculus to college students, I expect my students to have seen point-slope form. We just start using it (because it's the right way to talk about tangent lines and ...
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32 votes
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How to teach someone that $-3>-4$?

Draw a number line and label all the integers. Tell him that adding $x>0$ is moving $x$ units to the right and subtracting $x>0$ is moving $x$ units to the left. Tell him that adding $0$ is ...
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30 votes
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How to motivate equivalence classes

I think that the following story is quite illuminating for introduction. Naturally, one can/should adapt/change it to better fit the audience, I just wanted to sketch the general idea. Suppose you ...
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  • 8,729
29 votes

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

Point slope form emphasizes the actual meaning of slope. Literally, $$ y - b = m(x -a) $$ Says "The change in the outputs ($y-b$) is equal to the slope ($m$) times the change in the inputs ($x-a$)"...
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28 votes
Accepted

Is $e^{i\pi}+1=0$ a good motivation for introducing $e$ or $i$? Why (not)?

I have never quite understood why it was impressive and/or beautiful, and it always frustrates me when people claim that it is. Therefore, I would say "no, it is not good motivation", because beauty ...
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  • 1,513
28 votes

How to teach someone that $-3>-4$?

Again and again he finds $-4$ greater than $-3$. Ask him who is richer, he who has a smaller debt $($like $3$ rupees$)$, or he who has a bigger debt $($like $4$ rupees$)$, assuming both persons have ...
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  • 397
28 votes

Dividing by zero

Have the students tell you that division by zero is a non-sequitur. This is possible at any age where division is understood at all. Teacher: If there are eight cookies and four children, how many ...
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  • 4,850
27 votes
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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, ...
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25 votes
Accepted

What are some good examples to motivate the implicit function theorem?

Isolines and isosurfaces Isolines and isosurfaces (i.e., lines and areas of equal whatever) correspond to the graphs of implicit functions and are relevant in many sciences, e.g., isopotentials (...
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  • 2,366
24 votes

Should we avoid indefinite integrals?

I go a step further than Thomas (see Henry Towsner's answer). In my view, $$ \int f(x) \ dx = \{ F(x) \ | \ F'(x)=f(x) \} $$ On a connected domain, it is true that $F'(x)=G'(x)$ implies $F(x)-G(x)=c$ ...
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24 votes

How to teach someone that $-3>-4$?

Apologies, this should be a comment on the answer provided by @Jasper Loy but I don't have enough rep on this site. I just wanted to add that in my experience, struggling students have an easier ...
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  • 341
24 votes
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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 ...
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22 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 ...
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21 votes
Accepted

What is a good motivation/showcase for a student for the study of eigenvalues?

Obtaining formulas for the $n$-th term in a linear recurrence, such as Fibonacci numbers, is one application that certainly does not overtly mention linear algebra in the set-up.
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  • 13.3k
21 votes
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For calculus students, what should be the intuition or motivation behind series?

In my experience, one of the problems with series is that usually you have two sequences if you investigate the series $\sum(a_n)$: the sequence $(a_n)$, and the sequence of partial sums $S_n=a_1+\...
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21 votes

Direct applications and motivation of trig substitution for beginning calculus students

In my opinion, trig substitution is presented in a terrible fashion in every calculus book I have ever seen. "If you see $\sqrt{a^2 - x^2}$, substitute $x = a \sin \theta$, and then use such-and-such ...
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  • 2,045
21 votes
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Motivating the study of matrices

Even without explicitly introducing the language of "linear maps", "vectors", and so on, you can still develop matrices as a shorthand for such maps, thought of as exchange rates. Example: Machine A ...
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20 votes

Dividing by zero

In third grade we taught division using repeated subtraction. To divide 6 by 2, subtract 2 until you get to 0. 6-2=4, 4-2=2, 2-2=0. It took 3 steps so 6÷2=3. This can also be shown on a number ...
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  • 6,735
19 votes
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How to justify teaching students to rationalize denominators?

My thinking is that it is just so damn useful for students to be aware of these tricks. The examples/exercise should allow them to develop a sense of when and how it is helpful to simplify an ...
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19 votes

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

As someone else teaching calculus and higher math to college students, I use point slope form repeatedly: In the/a definition of derivative, I use point-slope form. I have students think about $y_1-...
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  • 191
19 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 ...
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  • 359
19 votes
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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 ...
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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 ...
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18 votes

What is a good motivation/showcase for a student for the study of eigenvalues?

I like Markov chains and Google PageRank (which is essentially a special kind of Markov chain). It doesn't take very long to explain and motivate Markov chains and to argue that the probability ...
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