# Quod Erat Demonstrandum

## 2008/11/21

### Exist or does not exist

Is giving hints a good way to help students in solving mathematics problems? Urm, sometimes it may not.

Here is a common m.i. question in recent F.4 additional mathematics regular test:

Show that $n^3 - n + 3^n$ is divisible by 3 for any positive integer $n$.

(Note, it is quite a stupid question, because, $n^3 - n = (n - 1)n(n + 1)$, i.e. $n^3 - n$ is a product of 3 consecutive integers, and of course, it is divisible by 3. Also $3^n$ is divisible by 3 trivially. Hence $n^3 - n + 3^n$ is divisible by 3, Q.E.D.)

To HELP students to get some marks, I’d posted the question like

(a) Show that $(k + 1)^3 - (k + 1) = k^3 - k + 3(k^2 + k)$ for any value of $k$.
(b) By using m.i., show that $n^3 - n + 3^n$ is divisible by 3 for any positive integer $n$.

Do you think that part (a) is a helpful part? At the beginning, I think it is helpful in terms of getting marks. BUT, the real case is, many students (esp. those were not in my class) did the part (a) wrongly! Why F.4 students did not know how to cope with this F.2-standard question? Because, they tried to do it BY M.I.

Oh, my! M.I. is invalid in solving (a).

I come across similar things (not helping, but hurting) sometimes. May be, it is not a fault to set up ‘hint-giving’ steps, but the problem is, the so-called hints were inappropriate and irrelevant.

I think I need to learn more in setting up good paper.

Another thing could be interesting about the recent additional mathematics regular test. Some arguments in “exist or does not exist".

Here is a part of the question I set:

Write down the coefficient of $x^9$ in the expansion of $(1 - 4x^2 + 4x^4)^{10}$.

It is easy to answer because $(1 - 4x^2 + 4x^4)^{10} \equiv (1 - 2x^2)^{20}$,

and there is no term in $(1 - 2x^2)^{20}$ involving odd power of $x$, hence the coefficient is ZERO.

BUT, some students gave the answer : “does not exist"

Quite an interesting reply! Do you agree with the students?

## 13 則迴響 »

1. Regular test results (2008-11-21) of F.4E

Full mark = 25

Mark no. of students
25　　2
24　　2
23　　2
22　　2
21　　4

Mean = 17.6
SD = 4.22
no. of passing = 38 (out of 43)
no. of abs. = 1

迴響 由 johnmayhk — 2008/11/21 @ 4:40 下午 | 回應

2. John Sir,
我想問, 如何在 wordpress 用 LaTex 打 maths symbols?

迴響 由 joesir — 2008/11/22 @ 5:07 下午 | 回應

3. The x^9 term does not exist, but the coefficient does exist. So the student is wrong.

迴響 由 杜輝 — 2008/11/22 @ 6:54 下午 | 回應

4. @joesir

Try to read the following post

http://faq.wordpress.com/2007/02/18/can-i-put-math-or-equations-in-my-posts/

迴響 由 johnmayhk — 2008/11/22 @ 11:25 下午 | 回應

5. John Sir,
感激我遇見!

迴響 由 joesir — 2008/11/23 @ 8:30 上午 | 回應

6. joesir 係伊健 fans??

迴響 由 johnmayhk — 2008/11/24 @ 4:18 下午 | 回應

7. john sir 你是草蜢fans嗎?

迴響 由 joesir — 2008/11/24 @ 9:24 下午 | 回應

8. 可以咁講，但始終鍾意明哥多Ｄ。

迴響 由 johnmayhk — 2008/11/24 @ 10:12 下午 | 回應

9. 不如考慮下 23/12 扮明哥唱歌, 樂善好施.

迴響 由 joesir — 2008/11/25 @ 2:41 下午 | 回應

10. 籌款表演當然要睇新秀(即係 Joesir 等等)啦，我已是 OC (old cake)，沒有市場價值了！

迴響 由 johnmayhk — 2008/11/25 @ 5:53 下午 | 回應

11. 我們很想看 John Sir 跳"三分鐘計數"呢

迴響 由 joesir — 2008/11/28 @ 11:06 下午 | 回應

12. 迴響 由 johnmayhk — 2008/11/29 @ 8:02 下午 | 回應