掛羊頭賣狗肉？

琴日係 Instagram Story 道問左一條問題：

唔知各位同學識唔識計呢？

其實，題目既 expression 可以寫成 

，

而呢兩個 square root 正正就係 distance formula，代表某點 (x, y) 同 (0, 0) 之間既 distance 同埋 (x, y) 同 (5, -12) 之間既 distance 既總和。

所以呢題問題可以轉化成為 minimizing the sum of the distance of a point (x, y) to (0, 0) and to (5, -12)。我地畫番個圖就會見到以下既樣：

係個圖裡面，大家會見到 C 係 AB 上面既其中一點，所以 AD + BD ≥ AB (by triangle inequality) = AC + BC，所以無論 D 係成個 plane 既邊一個位置都好，(AD + BD) 都唔會短過 (AC + BC)，所以堂 D 呢點移到去 C 既時候，(AD + BD) 就已經 minimize 左，所以個 minimum value 就係同 (AC + BC) 一樣，亦即 AB 既長度 13。

所以，Final answer 就係 13 喇！

P.S. 1：好似冇乜人答到啊，要繼續努力啊！！！

P.S. 2：計到呢題既同學可以試下計呢題：Minimize [sqrt(x^2 + y^2) + sqrt(x^2 + y^2 - 10x + 24y + 169) + sqrt(x^2 + y^2 - 42x + 441)] ~~~

DSE Further Mathematics Curriculum Proposal Part 2

今次我地就正式進入 Foundation Area 喇！

Foundation Area 既第一個 Topic 係 More about functions，內容主要係 M2 課程現時有既 Odd and even functions 同埋 CE Additional Mathematics 既 Absolute values 既部分。

Learning Unit 1 – More about functions

Teaching time: 5 hours

Learning objective	Remarks
1.1  Understand the concept of odd and even functions 1.2  Understand the definition of absolute values 1.3  Solve equations involving absolute values 1.4  Sketch simple graphs involving absolute values	The concept that |x - y| represents the distance between x and y should be introduced. Examples include y = |x + 1| + 2, y = |x + 2| - |x| and y = x|x| + 2x. Students also need to learn to sketch the graphs of y = f(|x|) and y = |f(x)| from the graph of y = f(x).

係現行既 M1 同埋 M2 既課程裡面，absolute values 只係會係 integration 突然出現，現後就會話「將所有負數變成正數」就得，但連 absolute values 本身既 formal definition 同 meaning 都唔知，咁樣之後遇上其他問題既時候（比如 Learning Unit 15 既 Analytic geometry 及 Algebra module 裡面 Learning Unit 18 既 Vectors）就會不知所措，所以係整個課程開首就加上關於 absolute values 既內容以令整個課程更為完整及更有系統。

不過，此 proposed syllabus 並未有包括有關 absolute values 既 inequalities，因為呢類 inequalities 同其他課既既內容未有太大關連。

Learning Unit 2 – The principle of Mathematical Induction

Teaching time: 5 hours

Learning objective	Remarks
2.1  Understand the principle of mathematical induction	Only the First Principle of Mathematical Induction is required. Applications to proving propositions related to the summation of a finite sequence and inequalities are included. Proving propositions involving divisibility is not required.

呢個部分同現行 M2 既課程相若，但加左有關 inequalities 既 statements，增加有關 proving statements related to inequalities 可以令同學對 inequalities 既了解加深（因為呢個世界好多時時候見到既都唔係等式），亦都可以增加有關 logic 既訓練（而亦不需要到達舊制 AL PMaths Inequalities 既難度），為日後既課題作好準備。

下次就到講兩個關於 Polynomials and rational functions 既 learning units - The binomial theorem 同埋 More about rational functions 喇！