Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now
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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 2012. There is a good chance that League of Legends is a part of many of your students' daily life, especially if you are teaching engineering calculus. It doesn'...


38

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 quantifier on the +C without staring at the notation helps some of them separate the +C from the voodoo magic. I do a similar thing in college algebra classes ...


32

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 not moving at all. Tell him that adding $x<0$ is moving $-x$ units to the left and subtracting $x<0$ is moving $-x$ units to the right.


32

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 linearization) without teaching it, because we consider it part of the standard algebra curriculum, so students who haven't seen it are at a disadvantage. Further, ...


31

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 problem you've encountered, but in none of these cases is it honestly worth the effort of solving the problem analytically. I optimize path lengths every day when I ...


29

No, it is a bad idea to avoid indefinite integrals, the reason being simply that your students will encounter them elsewhere, and therefore need to be familiar with them. Calculus is a service course. The purpose of the course is to make science and engineering majors fluent in the language of calculus as used in their fields. Rather than always using ...


28

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 no money, just debts. He has spent several years seeing $4$ greater than $3$. A debt of $4$ rupees is indeed bigger than one of $3$ rupees. But the one that ...


27

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 students. Therefore both meaning and student understanding are important. Otherwise, what's the point? So I have grounded my response there. Young students (...


26

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 is subjective. On the other hand, if you explain to students that the formula is based on $e^{i\theta}=\cos\theta+i\sin\theta$ and this essentially allows you ...


26

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 cookies does each child get? Student: Uh, two. Teacher: Yes! This is a division problem. $\frac{8}{4} = 2$. Now, if there are 8 cookies shared by only two ...


24

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 time grasping negative numbers when the number line is oriented vertically rather than horizontally. I think we as humans naturally make the 'up=greater, down=less'...


23

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 ($x-a$)". Translating between a verbal statement like this and an equation is essential. Understanding slope is essential. Point slope form of a line is essential. ...


22

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 came here by bus. Which bus was that? You say it was the 42, which is the line that goes from the main station to the university. However, was it really the 42? ...


21

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 $F(x)-G(x)=c$ hence, given an integrand which is continuous (or piecewise continuous, insert your favorite weakened set of functions here) we may write: $ \int f(x) \ dx = \{ ...


20

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 $S_n=a_1+\ldots + a_n$. I noticed that trying to stress this distinction helps a lot. To the intuition, I like R. Péter: Playing with Infinity, the chocolate bar example on ...


20

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 trig identity, blah, blah, blah..." Yet another unmotivated rule to memorize. I always present trig substitution as follows: If you see any algebraic ...


20

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 can make 3 sprogs and 2 sprakets a day. Machine B can make 1 sprog and 3 sprakets a day. We summarize this data in a table of values: $$\begin{bmatrix} 3 &...


19

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 (physics), isobars and isotherms (metereology). Probably the best-known example of this kind are topographical contour lines (lines of equal altitude, see image ...


19

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 expression in this way, BUT also when it is NOT necessary. Leading up to it by looking at fractions. Should the students write a rational number in the form $3\frac17$ ...


19

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 that I'll be able to adapt at least one of them to a similar (or maybe not so similar) theorem I'm trying to prove. Furthermore, seeing several techniques ...


18

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.


18

I have found it motivates to explain the determinant as computing a volume. One can work through and convince for $2 \times 2$ and $3 \times 3$ matrices, and perhaps only hint at the $n \times n$ generalization, when $|\det(M)|$ is the volume of the $n$-dimensional parallelepiped spanned by the column vectors of $M$.           &...


17

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 distribution at time $n$ is the $n$'th power of the transition matrix times the distribution at time $0$. You can then start talking about how to calculate powers ...


17

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 $y_1-y_2=m(x_1-x_2)$ rewritten as $f(x_1)-f(x_2)=m(x_1-x_2)$ rewritten as $f(x+\Delta x)-f(x)=m(x+\Delta x-x)$ rewritten with limits to describe the slope $m$ as the ...


17

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 is $b$. This is about not being defined. Still... why is $\frac 1 0=\infty$ not so completely wrong? Because they can see that the smaller is $a$ then the ...


16

A sketch of one idea. I think it's probably better spread over a couple of days. Day one: Start them counting, from zero, out loud to you. Write the numbers on the board as they go. Zero (0), one (1), two (2), ... , ten (10). Stop here. Prompt a discussion about what happened - how is the most recent number different than all of the previous numbers? I ...


15

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 line, where it takes 3 steps of 2 units to go from 6 to 0. Teaching the concept of division this way is just the inverse of what we have done for multiplication. ...


14

Imagine a linear mapping $f: R^2 \to R^2, e_1 \mapsto (1.5, 0.5), e_2 \mapsto (0.5, 1.5)$. (As long as $R$ contains the numbers $1.5$ and $0.5$, it could be any ring. The real numbers serve as the most convenient example, however.) Can we "see", what the mapping does? Can we "see" what $f^5$ does? Given a basis of the two eigenvectors, $(1,-1), (1,1)$ we ...


14

I think it turns out that "perfect" numbers do not interact much with other parts of number theory. Some of these very old, elementary, very ad-hoc definitions of special classes of integers have proven (and will prove) to interact interestingly with other ideas, but some seem not to. It's not easy for a beginner to guess the significance or subtlety of one ...


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