Mr Gerry working MYP students
Gerry老師正在給中學部的學生上課
In Year 8 our recent focus has been on Index Notation, a form of representing numbers with wide application in Mathematics and in other disciplines. It never ceases to surprise how students struggle to understand that 『72』 and 『73』, ( read as 『seven to the power two』 and 『seven to the power three』) , represents 『7 x 7』 and 『7 x 7 x 7』 respectively, and not 7 x 2 and
Decoding the Index Notation
分解指數計記數法
7 x 3. This must be one of the most prevalent misconceptions in all Mathematics! More confusion is caused to our students by learning that there are three different words used in the English language to mean the small, raised numbers which signify Index Notation : 『index』, ( even more confusion as its plural is indices!), 『power』 and 『exponent』.
最近，八年級正在重點學習指數記數法，這是一種廣泛應用於數學和其他領域中的數字表達式。學習過程中，學生總是感到驚訝，因為「72」和「73」（讀作「7的二次方」和「7的三次方」）分別表示「7 x 7」和「7 x 7 x 7」，而不是「7 x 2」和「7 x 3」，這通常是數學中最常見的誤解之一！在英語中，有三個不同的詞語表示小的、右上角的指數符號：「Index指數」（其複數形式為indices，非常具有迷惑性！）、「power開方」和「exponent冪」，學生容易混淆。
Once our Year 8 students had learned to decode and encode simple Index Notation representations, we moved on to learn about the first two so-called Laws of Indices. These laws deal with the multiplication and division of numbers in Index Notation which have the same base number, for instance, 『72 x 73』. In this example 『7』 is the base number. Once more misconceptions abound about the correct method for solving this problem. Intuitively many learners believe they should separately multiply the 『7s』 ( 7 x 7) and the 『indices』 ( 2 x 3) to obtain a product of 『496』.
當八年級學生們學會了分解與構建簡單的指數表達式後，我們繼續學習指數第一和第二定律。這兩條定律適用於有相同底數的指數乘除法，例如「72 x 73」，這裡的「7」是底數。對於這個問題的正確解法，又有許多誤解。許多學生在直覺上認為應該把底數「7」和「指數」分別相乘，即7 x 7和2 x 3，從而得到「496」的結果。
However from learning to decode Index Notation our Year 8 students have learned that this multiplication, 『72 x 73』, is the same as 『(7 x 7) x (7 x 7 x 7)』. We explicitly write out this
The Importance of Visualisation
視覺形象化的重要性
decoded version of the sum on large pieces of paper [photo]. This helps us to see that this is actually the same as 『75』, in other words 『7』 multiplied by itself 『5』 times. (A recurring theme in many inquiries is how 『visual』 Mathematics can be). Many students then go on to see that 『75』 would be the sum if you added the indices together in the original 『72 x 73』, and leave the base numbers untouched. Next we learn that we can represent this First Law of Indices by using the following rule or 『generalisation』, as we like to call rules in Mathematics : 『ab x ac = ab+c』 . This is how we say in Mathematics that when base numbers are the same (represented by 『a』) we can complete a multiplication involving indices by simply adding them together. Welcome to inquiry Mathematics!
但是，隨著分解指數的學習，八年級的學生們理解了72 x 73其實與(7 x 7) x (7 x 7 x 7)相同。我們把這一指數分解後的結果清晰寫在大白紙上（如圖所示），藉由它我們看出實際上這與75相同，換言之是5個7相乘。（數學的「視覺形象化」是許多研究中反覆出現的主題）。接著許多學生發現，在不改變底數的情況下，將72 x 73中的指數相加，就會得到75的結果。然後，學生學到可以用下面這條規律歸納表示指數第一定律，也就是我們常說的數學公式：「ab x ac = ab+c」。在數學中，同底數（用「a」表示）相乘，底數不變，指數相加。這就是數學探究！
It will not be lost on readers how counter-intuitive this First Law of Indices may appear, requiring 『adding』 in order to multiply! The Second Law of Indices deals with division and is in fact the inverse (meaning the opposite in Mathematics) of this First Law. So to divide indices we subtract so long as the powers are attached to the same base numbers. Our students quickly learn that the Second Law 『generalisation』 can be represented as follows: ! This is also confusing for our learners because they are conditioned into believing this structure represents a 『fraction』 and not a 『division』. This Second Law tells us that, for instance, , in other words, you subtract the indices to solve a division problem involving the same bases. So for our learners this once more represents a very counter-intuitive process but one that is completely logical when the problem is decoded to.
大家一定感受到了需要通過「相加」達成相乘的指數第一定律有多麼地違反直覺。指數第二定律與除法有關，是指數第一定律的反向運用。故涉及同底數的指數除法，我們只需要將指數相減。學生很快學到第二定律可以歸納為：！但這也讓學生們感到困惑，因為他們習慣性認為這樣的結構代表的是分數而非除法。指數第二定律是指，同底數相除，底數不變，指數相減，例如。因此對學生而言，這又是一個違反直覺的過程，但將問題分解為後，顯然是合乎邏輯的。
More Decoding
更多講解示例
There were a number of other Laws of Indices which we touched on in this inquiry including 『raising a power to a power』, the Third Law of Indexes, for instance ( 33 )4 . The 『generalisation』 for this type is 『abc』, in other words, we multiply the indices together to remove the brackets. Once more this can be very confusing for learners.
我們還學習了解了包括指數第三定律在內的其他定律，指數第三定律為「指數相乘」，例如( 33 )4，可以歸納為abc，即為指數相乘，脫去括號。這同樣會使學生感到不解。
The highlight for our students was learning how to solve equations featuring numbers in Index Notation with different base numbers and including variables in the index, for instance 910 = 3X. [photos]This really provided an opportunity for our students to bring together all the new knowledge and understanding developed during our inquiry.
學生的學習重點在於解開底數不同且指數中含有未知數的方程，例如910 = 3X。（如圖所示）這有利於學生總結學到的新知識和學習中形成的理解。
We used 『a scrambled jigsaw』 approach [Photos available] to facilitate our solution to this challenging problem. This meant our students had each step of the solution in 『cut up/scrambled』 form and their task was to order the steps. This process was facilitated by our learners already knowing from their existing understanding of algebraic manipulation that the final solution would be in the form of 『x = ….』 . They quickly found the solution and were able to work back step-by-step to the opening line of the problem. This helped them to see that the first step in solving the problem was to make the base numbers the same which was possible in this example.
我們使用「亂序拼圖」的方式（如圖所示）輔助解決這道難題。這意味著學生們將為剪開並打亂順序的解題步驟排序。學生們學習過代數運算，他們知道最終的結果會是「x = …」的形式。他們很快找到了解法，並且能夠按步驟推導回問題。這個過程幫助學生了解到，解答這類問題的第一步是使等號兩邊的底數相同，在例題中是可以實現的。
Ordering the Scrambled Problem
排序
This method was effective in helping our students see the process for solving the penultimate stage of the problem: 320 = 3X . They could see that only this line could lead to the solution of x = 20 and that this was achieved by removing the base numbers. Our students could also see that it was permissible to remove the base number, 『3』, because it was common to both indices. In this way our students learned some very advanced Algebra which consolidated their knowledge of The Laws of Indices. It was also very gratifying from a teacher’s perspective that the student response to this success was to demand and correctly solve more similar problems without the use of a 『scambled jigsaw』!
這種方法有效地幫助學生推導出答案的前一步：320 = 3X。進展到這一步後，他們看出去掉底數，可得答案x = 20，並且發現能夠去掉底數3是因為等號兩邊的底數相同。通過這種方式，學生學習了高等代數，並鞏固了指數定律的知識。在成功解題後，學生要求做更多類似的題目，並且在不藉助「亂序拼圖」的幫助下正確進行了作答，從老師的角度來看，這非常令人欣慰！
Most importantly as a teacher it is clear that this inquiry has left our students with a very strong foundation to their understanding of a very important area of Mathematics which has widespread application in other disciplines, notably in the world of finance.
最重要的是，從教師角度，這樣的探究式學習過程顯著地為學生在指數記數和計算這一重要的數學領域打下了堅實基礎，它已廣泛應用於其他領域，特別是在金融業中。
Mr Gerry
MYP Mathematics Teacher