Mathematics Weblog

Student Howlers

Saturday 18 December 2004 at 8:06 pm | In Articles | 4 Comments

A couple of howlers seen in sci.math newsgroup some years ago

1. I do like the lateral (?) thinking behind this one 😀

$\displaystyle \lim_{x \to 0}\frac{\sin 7x}{5x}=\frac{\sin 70}{50}$

2. Problem

$\displaystyle \lim_{x \to 0}\left(\frac{1}{x}-\frac{1}{\sin x}\right)$

Answer

Proof

$\displaystyle \frac{1}{x}-\frac{1}{\sin x}=\frac{\sin x - x}{x\sin x}=\frac{\sin - 1}{x \sin} \text{ or } \frac{\sin - 1}{\sin x}$

$x \to 0$

Cancelling gives an example taken from Comic Sections by Desmond MacHale. Another one from the same book is:

3. Solve $\displaystyle \frac{dy}{dx}=\frac{y}{\sin x}$

Solution:

$\displaystyle \frac{dy}{dx}.\frac{\sin x}{y}=1$

$\displaystyle \frac{d\cancel{y}}{dx}.\frac{\sin x}{\cancel{y}}=1$

$\displaystyle \frac{d}{dx}(\sin x)=1$

$\cos x=1$

Do you have any favourite howlers?

I’ve found, through teaching, that a lot of kids stick with the misconception that
sin x
means
sin * x
for some mythical value ‘sin’.
It’s a nice way of making the brighter kids look like fools, I find! 😀

Comment by Ronald — Wednesday 22 December 2004 7:38 pm #
Yes I get that as well. I ask them if they use the $\times$ button on the calculator when finding $\sin 30^{\circ}$ – it doesn’t help ❗

Comment by Steve — Wednesday 22 December 2004 8:06 pm #
In my classes, it most often occurs when I teach function notation. They have such a hard time grasping the fact that f(2) means the mapping f applied to 2 and all they have to do is substitute. So many of them substitute then multiply by 2 afterwards.

Comment by Scott — Monday 27 December 2004 8:10 am #
This is what an answer I got from a student – the question is “Do the product of two rationals equal a rational? explain.”
Answer:
It can be argued that two rational numbers cannot only produce rational
numbers because two signs such as + and – can when multiplied together
produce a negative which, in some cases, dooms the product to be
irrational. Besides, where to irrational numbers come from if not from
rationals? Is it not the rules that create the unruly? Even in the
definition of irrational we find rationality.

Hence, my argument is that two rationals can equal an irrational.

Comment by Desmond — Saturday 1 January 2005 12:24 pm #