56

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'...


45

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 ...


36

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 (...


36

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, ...


35

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 ...


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.


30

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? ...


29

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. ...


28

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 ...


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 ...


28

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 ...


25

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, take a geometry example first. "Is similar to" is an equivalence relation on the set of regular polygons (I'll omit a proof here). Now say I need to ...


24

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 = \{ ...


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'...


24

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 asked to find the cosine of an angle and the only computing device you have is a four-function calculator, then you can get a good approximation of the cosine ...


22

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 ...


21

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.


21

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 ...


21

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

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 ...


20

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. ...


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

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

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 ...


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 ...


19

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 the top showing If your particular Google bubble doesn't show this result, I think that there is a direct link. If you hit "Play", you can play a game of Snake, using the standard rules. ...


19

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 from a mathematical stand point, and it makes it easy to prove interesting properties of the function. But how do you actually compute it? The definition above is ...


18

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 ...


Only top voted, non community-wiki answers of a minimum length are eligible