Quod Erat Demonstrandum

2009/09/29

今天雜記

Filed under: Life,mathematics,NSS,School Activities — johnmayhk @ 11:02 下午
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難以轉化

科主任擬了兩班高中一 M2 的聯測卷,其中有一道題目只有三名學生懂得處理:

Given that x and y are purely imaginary numbers. If (x + y) + (2x - y)i = 12 - i, find (the values of) x and y. (more…)

2009/09/27

奇異解

Filed under: Additional / Applied Mathematics,HKALE,HKCEE — johnmayhk @ 9:11 下午
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感謝中五的 Carman 回應了上一個 post,讓我也閒聊幾句,高手見諒。

比如

\frac{dy}{dx} = f(x)

那麼 y 其實是什麼?

尋找 y,就是解微分方程的過程,上例不過用積分,得到

y = \int f(x)dx (more…)

2009/09/25

Find dy/dx at a point not on the curve

Filed under: Additional / Applied Mathematics,HKCEE — johnmayhk @ 4:50 下午
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When distributing the marked test paper to students, one student, Carman, reminded me that there was a ‘question’ in the following question:

If x^3 - 4x^2y + 3xy^2 - y^5 = 10, find \frac{dy}{dx} at the point (-2,1).

Carman said, ‘the point does NOT lie on the curve.’

Good observation! I had to say thank you to him. Although I’m not the setter, I should bear the responsibility of checking the paper.

But a natural follow-up question turns up: what is the meaning of the number \frac{dy}{dx}|_{(-2,1)} = \frac{31}{33} we are obtaining? Is the number meaningless or standing for something?

Let’s consider a simple example. (more…)

2009/09/24

Differentiation of parametric equations

It is extremely easy to set up questions about differentiation techniques (but good real life application questions are really rare, esp. at secondary school level), apart from tedious computation, when the differentiation involves parameter, students may have difficulties, like mistaking:

\frac{d^2y}{dx^2} = 1/\frac{d^2x}{dy^2} (wrong!)
\frac{d^2y}{dx^2} = \frac{d^2y}{dt^2}/\frac{d^2x}{dt^2} (wrong!)

Here is a question in recent quiz, which involves parametric equations: (more…)

2009/09/23

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Filed under: Uncategorized — johnmayhk @ 5:31 下午

(more…)

2009/09/22

Different formats of primitive functions

Filed under: HKALE,Pure Mathematics — johnmayhk @ 4:52 下午

Just take a rest from work, type something boring here…

\int \frac{dx}{\sqrt{1 - x^2}} = \sin^{-1}x + C

Students, you may regard the above as a formula or derive it by using trigonometric substitution x = \sin\theta every time.

As you may know that the expression of a primitive is not unique, we may have other forms being a primitive of \frac{1}{\sqrt{1 - x^2}}, say (more…)

2009/09/14

類似地?

小心,對一些運算法則,我們定要正本清源,不能單以一句「類似地」便隨便進行「類似」運算。

e.g. 1 循環小數

0.3 \times 0.4 = 0.12 正確,但不是「類似地」得到:

0.\dot 3 \times 0.\dot 4 = 0.\dot 1\dot 2(錯!) (more…)

2009/09/12

[FW][Sad News] 香港數理教育學會主席吳重振先生辭世

Filed under: Report — johnmayhk @ 12:32 下午
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今天偶爾上 HKASME 網,竟看到吳老師離世的消息:

香港數理教育學會主席吳重振先生辭世
http://www.hkasme.org/CCNG.pdf

已故主席吳重振先生生平簡介
http://www.hkasme.org/Ng_CC2.pdf

他不認識我,但我很敬佩他。聽他主持的講座,獲益不少,很多奧數精英也是在吳老師的栽培下成長。

願吳先生的親朋,平安。

2009/09/07

開學訓話

Filed under: Family,Life — johnmayhk @ 10:48 下午

開學頭,連續兩天(如果定義開學日期是 2009-09-01 的話,但同工知道這是假的),在晨早趕返工的港鐵車廂內,給黃毅力先生訓話:

《我對老師的訓話》

大部分學校開課了,大學在早一星期亦已開課,在這裏我希望向香港教育界的前輩分享一點點看法。
今日的教育和以前分別很大。從前的老師是教導學生,是教與育並行, 雙管齊下,為新的一代建立良好的價值觀,除了讀書學習,還要學做人,為人老師要啟發學生在心智上發展,使他們在成長過程中,學習中國傳統的價值觀,特別是德智體群美、忠信孝義及禮義廉恥。時移世易,現今的教育制度所衍生出來的現象,是大多老師只想做好一份工(more…)

2009/09/06

My first NSS math lesson

Filed under: Fun,Junior Form Mathematics,NSS — johnmayhk @ 12:01 上午

During the first NSS mathematics lesson in F.4D, I gave my students a piece of worksheet for practicing simple algebraic computation. (Well, I could collect money during the time when they were doing classwork)

Q.1 to 4 are factiorization problems: factorize the following

1. 1 - x^2(1 - 2x)^2
2. x^2 - 35x + 294
3. 2x^2 + 6x - 15z - 2xy + 5yz - 5xz
4. x^4y^4 + x^2y^2 + 1

Then I’d set the following questions in the worksheet and asked them to hand in as their homework:

5. What is your feeling about mathematics?
6. What is your expectation of mathematics teacher in this acedemic year?
7. What is your expectation on your own about the learning of mathematics? (more…)

2009/09/05

會議補充二則

Filed under: Additional / Applied Mathematics,HKALE,University Mathematics — johnmayhk @ 11:32 下午
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1.
校內的數學科會議,談到 extended reading/learning,我隨便舉例,讓同事略略看片:

但我只輕鬆帶過,大家用懷疑的眼光問:「怎會可能?」嗯,我也不知道,早前因為想找有關 tensor 的東西,翻一翻幾年前買下的數學書:

“Introduction to Topological Manifolds" by John M. Lee (more…)

2009/09/01

i 是開方負 1?

Filed under: mathematics,NSS — johnmayhk @ 11:58 下午
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新高中數學教科書出現了

i = \sqrt{-1}

這個命題。

接著是一些例題:\sqrt{-8} = \sqrt{8} \times \sqrt{-1} = \sqrt{8}i

暗暗地在 teaching note 出現了以下句子:

\sqrt{-1} = \sqrt{1} \times \sqrt{-1} = i
but
\sqrt{1} \ne \sqrt{-1} \times \sqrt{-1}

時而可以,時而不可;同學,你感到有點問題嗎?

我想說

i 不是被定義為 \sqrt{-1}

i(more…)

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