Quod Erat Demonstrandum

2017/05/28

Transformation of graphs

Filed under: NSS,Teaching — johnmayhk @ 8:48 下午
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Share some points about the topic ‘transformation of graphs’. Nothing new.

Transformation of graphs in the HKDSE syllabus refers to

– translation
– reflection (with respect to the x-axis or the y-axis)
– enlargement or contraction (along the x-axis or the y-axis)

and all happen in the xy-plane.

(more…)

2016/07/22

有理函數和矩陣

Filed under: NSS,Pure Mathematics,Teaching,University Mathematics — johnmayhk @ 12:06 下午
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Core mathematics 介紹過 rational function(有理函數),即形如 \frac{P(x)}{Q(x)} 者(其中 P(x)Q(x) 皆為多項式)。

P(x)Q(x) 皆為線性(linear),即形如 \frac{ax+b}{cx+d} 者,稱之曰 fractional linear function(FLF)。

中四教 function(函數)時,偶談下例,設 f(x)=\frac{ax+b}{cx+d},求 f(f(x))

解之曰

f(f(x))

=\frac{af(x)+b}{cf(x)+d}

=\frac{a\times \frac{ax+b}{cx+d}+b}{c\times \frac{ax+b}{cx+d}+d}

=\frac{(a^2+bc)x+b(a+d)}{c(a+d)x+bc+d^2}

仍舊是 FLF。

現看看 2×2 矩陣 (more…)

2016/07/10

論商餘(三)

Filed under: mathematics,NSS,Teaching — johnmayhk @ 9:45 下午
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一. 帶餘除法

(a) 多項式與數字

聞說以下是某校的試題:

一看,個心離一離。 (more…)

2016/06/13

變易圖式

Filed under: Teaching — johnmayhk @ 2:44 下午
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在教大上了一兩課節「變易理論」(variation theory),當中 V3 是關於教學設計的主導原則「變易圖式」(pattern of variation),希望透過四種辨識體驗:對比(contrast)﹑區分(separation)﹑類比(generalization)和融合(fusion)以刺激學生對新知識或技巧之覺知。

比如要學「紅色」這個概念,可呈現紅色及非紅色的東西,此之謂「對比」:

johnmayhk-v3-1-contrast (more…)

2014/11/03

存在非平凡解的齊次線性方程組

Filed under: NSS,Teaching — johnmayhk @ 4:08 下午
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以下是一道普通的 M2 題目:

已知以下線性聯立方程有非平凡解(non-trivial solution)

\left \{ \begin{array}{ll} 2x+(2+k)y+2z=0\\(4+k)x+2y+5z=0\\7x+3y+(6+k)z=0\end{array}\right.

k 值。

因方程組是齊次的(homogeneous),要有非平凡解,只要設 \Delta=0,即

\left|\begin{array}{ccc}2 & 2+k & 2\\4+k & 2 & 5\\7 & 3 & 6+k\end{array}\right|=0

便可,從而解出

k=1,-1,-12

但有同學利用 Gaussian elimination,得

\left(\begin{array}{cccc}2 & 2+k & 2 & 0\\4+k & 2 & 5 & 0\\7 & 3 & 6+k & 0\end{array}\right)

~\left(\begin{array}{cccc}1 & \frac{k}{2}+1 & 1 & 0\\7 & 3 & 6+k & 0\\4+k & 2+k & 5 & 0\end{array}\right)

~\left(\begin{array}{cccc}1 & \frac{k}{2}+1 & 1 & 0\\ 0 & -\frac{7k}{2}-4 & k-1 & 0\\ 0 & -\frac{k^2}{2}-3k-2 & 1-k & 0\end{array}\right)

~\left(\begin{array}{cccc}1 & \frac{k}{2}+1 & 1 & 0\\ 0 & -\frac{7k}{2}-4 & k-1 & 0\\ 0 & -\frac{k^2}{2}-\frac{13k}{2}-6 & 0 & 0\end{array}\right)

因方程組有非平凡解,觀察上述第三式,即

-\frac{k^2}{2}-\frac{13k}{2}-6=0

解出

k=-1,-12

咦,奇怪了,一早知 k=1,-1,-12,為何用 Gaussian elimination,得不到 k=1 這個可能值?

::: 停一停,想一想 ::: (more…)

2013/11/14

上課偶拾之 sd

Filed under: mathematics,NSS,Teaching — johnmayhk @ 3:46 下午

今天開始教 standard deviation(標準差),課堂設計都是一般時序:

1. Range 和 Inter-quartile range 作為量度離散程度的工具之不足。
2. 創作可以涉及全體數據的量度工具,由最簡單考慮

\displaystyle{\frac{\sum_{i=1}^n(x_i-\overline{x})}{n}}

到 mean deviation (more…)

2012/10/05

Rationalization

Filed under: NSS,Teaching — johnmayhk @ 3:20 下午

何謂 inequality?

對,不平等。例如社經地位的不平等,性別上的不平等。但相信中學生和數學授課員多數想到:不等式。

何謂 rationalization? (more…)

2012/04/21

圓錐截線切線

Filed under: NSS,Teaching — johnmayhk @ 2:40 下午

Core Mathematics 習題:

Let C:x^2+y^2-6x+2y-15=0. Show that P(6,3) lies on C and find the equation of the tangent to C at P.

解法一:

Let L: y-3=m(x-6) \Rightarrow y=mx+(3-6m) (more…)

2012/02/16

Core Math 某題:數算 ABC

Filed under: NSS,Teaching — johnmayhk @ 8:58 上午

那天我代體育堂。一心期待上體育堂的 F.5C 學生,知我代堂,心情極度低落,課室之混亂可想而知,最重要是那時是午飯後的兩堂呀!我只隨意寫兩道題著他們想想。當然只有寥寥小貓理會。當中有學生反而問我以下問題:

Select 3 letters from a set {A,A,B,B,B,C,C}. How many different 3-letter ‘words’ can be formed? (e.g. AAB, CBA etc.)

學生給的答案是

\frac{P^7_3}{2!3!2!}

計出 (more…)

2012/02/13

Core Math 某題:概率

Filed under: NSS,Teaching — johnmayhk @ 11:14 上午
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In a lucky draw of a car, only 1 key out of 10 can open the door of the car. Chris, Rachel and Mike take turns to draw a key at random without replacement. The person who can open the car door will get the car. Find the probabilities of the following events happening.

(a) Chris will get the car.
(b) Rachel will get the car.
(c) Mike will get the car.

在堂上計算 (a) 時,我寫 (more…)

2012/02/07

Integration by parts

Filed under: NSS,Pure Mathematics,Teaching — johnmayhk @ 9:29 上午

幸好中學數學仍有延伸部份,同學可以接觸積分(integration)並當中的技巧:部分積分(integration by parts)。

對於應用 integration by parts 求

\int u(x)v(x)dx

(其中 u(x) 是多項式,v(x) 是容易計算積分的函數)

的題目,可用「列表法」簡化運算,例子:

Evaluate \int x^2\cos xdx.

先把多項式 u(x) 寫出:

對它求導(find derivatives) (more…)

2011/10/18

Chain rule

Filed under: HKALE,NSS,Pure Mathematics,Teaching — johnmayhk @ 4:29 上午
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早前,中七的 Thomas 問:「Chain rule 個 proof 錯咩?有冇 counter-example?」

先把教科書的東西記下:

以下是記在教科書內 Chain rule 的證明: (more…)

2011/09/23

second fundamental theorem of calculus

Filed under: HKALE,Pure Mathematics,Teaching — johnmayhk @ 11:42 上午

教到 second fundamental theorem of calculus:

\frac{d}{dx}\int_a^xf(t)dt=f(x)

(其中 f 在開區間 I 內連續,a\in I 是常數。)

時,同學或有諸如以下的疑惑:

\frac{d}{dx}\int_0^x(x-t)^2dt 是甚麼? (more…)

2011/08/19

某數算題

Filed under: Additional / Applied Mathematics,HKALE,NSS,Teaching — johnmayhk @ 6:34 下午

給學生 98 道「暑期」(注 1)概率題目,這是第 77 題:

If three tickets are chosen at random without replacement from a set of 6n tickets numbered respectively 1, 2,…, 6n, what is the probability that the sum of the numbers on the numbers on the chosen tickets is 6n?

現在講解一下。 (more…)

2011/08/17

還是數算

Filed under: Additional / Applied Mathematics,HKALE,NSS,Teaching — johnmayhk @ 11:29 下午

暑假補課(注 1)時,學生問:

There are 10 empty boxes. 5 balls are going to put one by one into a randomly selected box. Find the probability that two of the boxes each contains 2 balls.

習題給的解是: (more…)

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