不能秒殺的提問之在三角形內

剛下課，中五生 Woody 問：如何判別一點是否在三角形內？

即已知 A(a,b), B(c,d), C(e,f)，如何知道 D(x,y) 是否在三角形 ABC 內？

畫一個準確的圖當然可以，但這不是數學人的答案。

我只想出一個方法：

病中打字

一。釘已打

太太生日，想談更多；但這不宜，只記某刻：她去看衫，兒子上課，我去看書。

基礎數學，頗為有趣，暼見如下：

唯銀根出缺，書釘已打，匆匆離去，滿足。新近一套：《數學文化小叢書》，短小精悍，其中一本，價值 $14，題為《幾何學在文明中所扮演的角色》，著者項武義教授（姑勿論存在關於項教授的負面言論），也吸引了眼目。在起始一段：

[FW] 一高與一低

港實行聯繫匯率制度，當港匯觸及強方保證水平，金管局便會透過市場操作，沽出港元並買入美元，導致本港銀行同業拆息「HIBOR」長期低企，導致資產價格急升，亦是導致貧富差距惡化的原因。

兩個香港排名，一個跌出十大，一個排名第一，均不是好東西。

Solve DE by method of substitution

In solving ordinary differential equation

………. (*)

(where are constants)

we use the method of substitution, let

Then we have and hence

致 5E 同學

在測驗中，我隨便擬了一道極顯淺的題：

設直線 及曲線 的方程分別是 及 。若 (的圖像) 切 (的圖像) 於 ，求 的坐標。另設 上一點 ，其 坐標為 1，求 在 處的法線（normal）之方程。

奇異解

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

那麼 其實是什麼？

尋找 ，就是解微分方程的過程，上例不過用積分，得到

Find dy/dx at a point not on the curve

When distributing the marked test paper to students, one student, Carman, reminded me that there was a ‘question’ in the following question:

If , find at the point ().

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 we are obtaining? Is the number meaningless or standing for something?

Let’s consider a simple example.

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:

　（wrong！）
　（wrong！）

Here is a question in recent quiz, which involves parametric equations:

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

　正確，但不是「類似地」得到：

（錯！）

會議補充二則

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

拼出 $100

老題：利用面值 $10，$20 及 $50 紙幣若干張，要拼出 $100，問有多少組合方式？

可以拿 2 張 $50 紙幣，這是一種組合；
可以拿 3 張 $10，1 張 $20 及 1 張 $50，這是另一種組合；
……

那麼，共有多少種可能的組合？

這篇為承接昨天的發帖

閒談一些基本東西：導數符號，函數，解釋

1. 高階導數的符號

同學問，為何 D 兩次（即求二階導數）的符號是

而不是

　或　？

應用純數

有人強調數學在生活中的應用，有人指出所謂「數學在生活中的應用」是牽強的。經驗使我比較贊成後者。

無論如何，以趣味的角度向學生介紹（所謂的）數學在生活中應用的例子，或許可以達到某些教學的成效。

讓我在這裡介紹一個「偽應用」吧，起碼三本數普書籍有記載此例。

修純數的同學定會接觸介值定理：

設 f 為定義在 [a,b] 上的連續函數，已知 f(a)f(b) < 0，則在 (a,b) 內存在 c 使 f(c) = 0。

這個定理有沒有生活應用例子？嗯，或許有，見下：

今有方桌子一張，四條腿等長。若把桌子放於凹凸不平但平滑的地面上，證明一定存在某個位置，使四條腿同時著地。

橢圓規

不知有沒有授課員用過橢圓規這個教具？（實情我不知這名稱是否正確，網上找到 Ellipsograph 這個字，不知是否橢圓規的正確英文名稱。）

我「靜靜雞」用科組錢買了一個，操作見下：

點數差的期望值

擲兩顆公平骰子一次，求兩個點數差的期望值。

學生甲：

嗯，擲一顆骰子，點數的期望是 。無論擲兩顆，三顆，它們的點數期望值皆是 ，故此，兩個點數差的期望值就是 。

數數唸

一．

以下數字是「旋轉對稱」的嗎？

1961

2009 CE Additional Mathematics Paper Section B

重點題目：

以下是 2009 CE Additional Mathematics Q.18，同學，看看你可否在 16 分鐘內正確地完成它：

The following figure shows a park AED on a horizontal ground. The park is in the form of a right-angled triangle surrounded by a walking path with negligible width. Henry walks along the path at a constant speed. He starts from point A at 7:00 am. He reaches points B, C and D at 7:10 am, 7:15 am and 7:30 am respectively and returns to A via point E. The angles of elevation of H, the top of a tower outside the park, from A and D are 45 and 30 respectively. At point B, Henry is closest to the point K which is the projection of H on the ground. Let HK = h m.

(a) Express DK in terms of h. (1 mark)
(b) Show that AB = h m. (3 marks)
(c) Find the angle of elevation of H from C correct to the nearest degree. (3 marks)
(d) Henry returns to A at 8:10 am. It is known that the area of the park is 9450 m.
(i) Find h.
(ii) A vertical pole of length 3 cm is located such that it is equidistant from A, D and E. Find the angle of elevation of H from the top of the pole correct to the nearest degree. (5 marks)

大數值的乘階

中四同學在學期初已接觸乘階（factorial） 的運算。比如

理論上，這個正整數 ，可以「要幾大，有幾大」；但實際上，我們日常接觸的運算工具，必有其限制；用 EXCEL 2003，只可以算出的最大乘階是

這個數字有多大？

Just answer an easy additional mathematics question

A F.5 student asked me the following question some days ago, reply now.

A(-3,0) and B(-1,0) are two points and P(x,y) is a variable point such that . Let C be the locus of P.

(a) Show that the equation of C is .

(b) T(a,b) is a point on C. Find the equation of the tangent to C at T.

(c) The tangent from A to C touches C at a point S in the second quadrant. Find the coordinates of S.

(d) L is a straight line which passes through point A and makes an angle with the positive -axis, where . Q(x,y) is a point on L such that . (See the figure below)

(i) Write down the coordinates of Q in terms of r and .

(ii) L cuts C at two distinct points H and K. Let , .

(1) Show that are roots of the quadratic equation .

(2) Find the range of possible values of , giving your answers correct to three significant figures.

(HKCEE 1999)

Reply to a F.7 student

To estimate , please refer to the normal table, you may see

Just answer questions from F7 student

Q.1

sin x 同cos x 既taylor’s theorem expansion say the last term of sin x is (-1)^(n-1)*x^(2n-1)/(2n-1)! 個power of x of the remainder term 係咪2n or 2n+1 都得? similarly, the last term of cos x is (-1)^n*x^(2n)/(2n)! 個power of x of the remainder term 係咪2n+1 or 2n+2 都得?

boring discussion on limit of sequence

Here is just a typical, basic, level-zero question about the limit of sequence for formative assessment.

Let , for .

(a) Given that , show that .

(b) By squeezing principle, show that .

Minor point in differentiation

This is a simple question in differentiation.

Let for any real number , determine at (1,-1).

Some F.4 textbook questions

Students may find the following textbook questions difficult.

Question 1

Refer to the figure below, given that
and ;
;
.

Show that .

Just an old question about F.4 trigonometry

It is glad that F.4 students asked me how to evaluate

Here is a way.

Asymptotic Behavior of Solutions to Linear Equations

Apart from the interesting article in the last post, Justin also sent me the following interesting question:

Consider

– - – - – - (E)

where the constant is positive and is continuous on [).

出現 A 先於 B 的機會

常見題目：擲一顆公平骰子若干次，求得到 1 點先於得到 6 點的概率。

此例容易，因 1 點和 6 點在地位上無異，易知要求概率為

拉 curve

中二的同學問我如何「拉 curve」（即調整分數），讓我略談。

比如數學科某次考試結果，最高和最低分數分別為 79 分及 4 分，但因為及格率只有三成，太低，於是把原本得 42 分的變成及格，即 50 分，以符合某一個及格率。調整分數後，新的最高和最低分數，仍然為 79 分及 4 分。

調整分數方法有無限多種

錯在哪裡系列：零是不定義？

When giving the following basic trigonometry question in F.4 additional mathematics lesson:

Given , prove that .

It should be extremely easy,

做數雕蟲小技系列：看整體

看整體有時比看局部好。舉例

1. Factorize .

利用 Taylor’s theorem 證明 AM >= GM

設 為 個正數，命 ,

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 is divisible by 3 for any positive integer .

Create an m.i. question

It is not difficult to create questions like:

Prove by mathematical induction that

正割餘割

中四開新課，同學認識一下新「朋友」吧：

正弦 = sine ()
餘弦 = cosine ()
正切 = tangent ()
餘割 = cosecant () 定義：
正割 = secant () 定義：
餘切 = cotangent () 定義：

我今天才知「正割」「餘割」這兩個譯名，慚愧。

Initial guess

數值法解方程的題目，通常會提供 initial guess。但有時就算沒有提供，我們也可以約略估計，比如解

同學可以估計其中一個實數解，大概是什麼嗎？（先問：為何知道實數解一定存在？）

Integrate polynomial of degree less than 4

To find the definite integral of a polynomial of degree less than 4, we can use the following formula.

e.g.

Nothing special, it is just something about Simpson’s rule.

不能秒殺的提問

在堂上，一般情況下，學生提問，在下多半手起刀落，秒殺解之。今天 4E 同學問了兩個問題，在下不能秒殺：

1. 誰發明 這個符號？
2. 畢氏數組中是否必然存在 3 或 5 的倍數？

Assuming step in mathematical induction

Just share a minor point in the presentation of M.I.

To prove that a proposition P(n) is true for all positive integers n by using M.I.

We need ‘4′ steps, namely

Say something about series in Applied Mathematics (II)

Here is just a typical question in AL Applied Mathematics (II).

For natural numbers ().

Let , evaluate .

Just a question of applied math. from a F.7 boy

Just discuss an easy AL Applied Mathematics (II) Question with students.

Let be twice differentiable functions such that .

(a) Let , where . Show that .

(b) Using (a), or otherwise, find Taylor’s expansion of the function about , up to the term in .

利用插值法找餘式

近日，中六純數課談的是餘式定理，中七的應數課談的是插值法，恰巧，兩者有一點點關聯。

Prerequisite of F.4 algebra

As usual, we start with basic algebraic computation drilling at the beginning of F.4 mathematics lessons. But, I just did some useless mathematical chatting and gave something about infinity (e.g. the discussion of 1 – 1 + 1 – 1 + … ) and they showed their excitment with hands clapping. Well, of course, I switched to serious matter very soon. As expected, most of the students did not know the fact that “zero is an even number” (even, they did not know that zero is an integer). I went through some ‘prerequisite’ questions with them, one is:

(a) 　[True or false?]
(b) If , then .　[True or false?]
(c) The square roots of 9 are 3 and -3.　[True or false?]
(d) If , then and .　[True or false?]

F.4 students, here are the answers.

利用對稱性解概率問題

想收手，無奈自己按奈不住，趁開學還未忙死，偷偷又講數：

對稱性有時可以幫助解題，舉兩例。

例一（頗常見的技巧）
兩顆公平骰子，一顆是紅色，一顆是藍色。現獨立地各擲兩次，求紅色骰子的總點數大於藍色骰子的總點數之概率。

Seminar on teaching statistics

講座題為
Hong Kong Statistical Society Seminar on “teaching statistics for A-level and AS-level Examinations”

我無緣參加那個講座，只是多年前，從同事的舊物的中接過一個資料夾，當中盛載著的，就是那個在 1996 年 10 月 12 日舉行的講座講義。轉眼已是十多年。講座的主講人包括 Mr. Fung Hing Wang, Dr. Cheung Siu Hang, Dr. Shen Shir Ming, Dr. Chiu Wing Kin

先貼一貼趙博士寫的有關 Hypothesis Testing 的詩。

閒談相關係數 Correlation Coefficient

在校內模擬會考放榜活動中，同事以 EXCEL 計算出所謂 Coefficient of Correlation（相關係數），從而告訴學生，學校估計的會考成績比同學自己估計的準確，因為校方的相關係數較同學的接近 1。

慚愧地，身為數學授課員，也不能深入認識何謂相關係數（或許是很久以前曾經認識過，現在已歸還教授們），現在只能以我這個統計學行外人，泛泛而談一些廢話。

首先，相關係數（或應說樣本相關係數）愈接近 1，是否代表所謂「愈準確」？

正多邊形作圖法

我在工業學校渡過中學生活。已往在工業學校，繪圖是必修科（現在好像稱為『圖象傳意科』）。從中一開始，我們便手拿大大塊的繪圖木版，插著 T 尺上課，每星期也要寫 lettering（功效類似我年代幼稚園練字用的 copy book，不知現在還有嗎？）以打好寫字的基本功。

中華民族會滅亡嗎？

中華民族會滅亡嗎？放心，這裡不談政治，不存在政治正確與否的問題。我只希望利用數學處理這個問題。想證明，如果「一孩政策」長此下去，並落實在全球華人群族中，則問題的答案是肯定的。嗯，這似乎是頗「常識」的，但我始終想借其他東西包裝一下數學。

送你一條鏈，馬可夫鏈

約廿年前，在 hku 某 O-camp 中，組長帶了一些消磨時間的遊戲。其中一個，嗯，都是叫阿仔同大家玩，準備好未？

以大數定律證明維爾斯特拉斯定理

伯努利（Jacob Bernoulli）的大數定律，是概率論中的著名定律，略表如下：

錯在哪裡之數學歸納法

對於任意自然數 ，證明

婚姻定理

看港台節目《改革開放三十年系列 – 中國新面貌》之《入贅女婿》才知近年國內『入贅女婿』這個『社會現象』頗為普遍。據《說文解字》，「贅，以物質錢，從敖貝，敖者獲放貝，當復取之也。」即是贅，有抵押、交換的意思。男子入贅成婚，就是以身作『抵押、交換』。愈看節目，心中愈有不可名狀的納悶。抽水完畢，還是說數學好了。

組合小談

某班有 30 名學生進行互訪，每個學生在一個晚上可以進行多次出訪。但在客人來訪的晚上，他必須留在家中。
證明：為了使每個學生都訪問了他的每一位同學，七個晚上已經足夠。

[FW] 如何造假資料？

http://chanlikhangnick.googlepages.com/writing20080703.htm

簡單有趣的介紹，推薦大家一看！

Just answer an applied mathematics question

Just answer an applied mathematics question from a F.6 student.

Book 4A Ex 2(a) No.19.

An urn consists of n tickets labelled 1, 2, 3, …, n. Tickets are drawn one by one without replacement. What is the probability that in these n draws, there is at least one for which the number on the ticket corresponds to the number of draw.

Marker’s report on SFXC F.6 Applied Mathematics (II) Final Exam 2007-2008

Marker’s report on SFXC F.6 Applied Mathematics (II) Final Exam 2007-2008

Passing rate : 53.33% (with regular test score considered)
Max : 64
Min : 12
Mean : 40.3
SD : 14.7

附加數題之最大四邊形

這是一道中四附加數學題。（相信這樣的好題，在將來的公開試會漸漸式微。）它的根本要問的是：一個固定四邊長度的四邊形，在什麼情況下面積最大？題目在教科書可找，詳表如下：

習，數飛也 – 四階拉丁方及四階數獨

習，數飛也（參考《說文解字》全文檢索測試版）

香港數學課改『總舵手』李柏良先生（教育局總課程發展主任（數學）個人網頁）在 29/4 的數學教育研討會上，把『數飛』，聯想到數算麻將的『飛數』，即係『呢副牌可以叫幾多飛』。台下的聽眾雀躍，大概是因為看到『生活的數學』吧。我慘，我是火星人，不懂麻將（其實初中時，家父教過在下，但自己沒有『操練』，所以現在已不懂了），如果將來的數學考卷問什麼『大四喜』，『十三么』之類，我的情況同看『會考潮文』一樣，死得。

Wrong method, but correct answer

Obviously, the methods throughout the following sets are completely wrong in general; however, correct answers will be obtained. Try to verify them and explain why it happens. You may read the explanation at the end for confirmation.

(Set 1) Erase the indices

(Set 2) Erase the logarithm

(Set 3) Product to sum

(Set 4) Out of radical sign

(Set 5) Exchange indices

(Set 6) Sum to product

Explanation
Try to prove the following identities to see the reasons why we have equations from set 1 to set 6.

Gini coefficient

今天考完 Applied Mathematics. 其中 2008 Applied Mathematics (II) Q.7 這道 5 marks 的短題是有關 Gini coefficient（堅尼系數），頗『生活化』吧，但論難度，它應是一道 Additional Mathematics 的題。詳述如下：

In a city of households, let be the sum of the first lowest household incomes for . Let be the area of the polygon with vertices . As a measure of disparity in household income, the Gini coefficient () is defined as .

(a) Show that , where is the mean of .
(b) Suppose all household incomes are the same. Show that .
(c) Will be changed if all household incomes are double?

解說 Applied Math 小題

Nothing special, just answer a question from some students in my class.

Ex. 5(a) Q.35

Rods are made of nomial length of 8 cm but in fact they form a normal distribution with mean 8.02 cm and standard derivation 0.03 cm. Each rod costs $9 to make and may be used immediately if its length lies between 7.96 cm and 8.04 cm. If its length is less than 7.96 cm, the rod is useless but has but has a scrap value of $1. If its length exceeds 8.04 cm it may be shortened and used at a further cost $2. Find the average cost per usable rod.