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```LESSON PLAN
School:
Teacher:
Date:
Time: 10:00am-10:40am Class: Secondary Three
Subject: Mathematics
Topic: Indices
LEARNING PREREQUISITES:
(a) Definition of a n , where n is a positive integer.
(b) The fact that a 0 = 1, for a ≠ 0.
(c) The index laws for positive integral indices:
(i)
(ii)
(iii)
a m  a n  a m n
am
= a mn (m  n)
n
a
(a m ) n  a mn
BEHAVIOURAL OBJECTIVES:
Students will be able to
(a) convert any number with an index which is a negative integer into one with a
positive index, and
(b) evaluate any number with an index which is a negative integer.
Definition of a p , for rational p , a  0
Graph of y  a x (a  0) .
Definition of a 1 n , for
positive integral n
Meaning of a 2
Definition of a  n , for
positive integral n
a 0  1 for a ≠ 0
a m  a n  a m n
am
= a mn (m  n)
n
a
Definition of a n , for positive integral n
-1-
(a m ) n  a mn
TEACHING STRATEGIES:
The students in the class are not very bright. They may have difficulties in handling
abstract symbols. This constitutes the major challenge in this lesson. To deal with this,
Furthermore, a  n 
1
is not deduced in the formal way. Rather, we try to come up
an
with such a result by comparison and analogy, with the use of the index laws in
between.
PRESENTATION:
Time Teacher-student interaction
1. Set
(Pre-assessment)
5min The teacher recalls the learning of a n (where n is a positive integer) and
of the index laws in the previous period.
Numbers like 35, a 5 and a m are written on the board for the students to
evaluate.
Students are also asked to simplify expressions like 35  32, a 4  a 2 and
a6
.
a3
(Motivation)
The number 3-2 is deliberately mixed with the numbers above and it is
expected that no one can find this answer.
2. Development
5min In response to the query made by the students, the teacher cleans the board,
leaving only 3-2 and promises to investigate such numbers in greater detail.
The teacher then asks two students to simplify 35  3-2 and 35 
1
32
respectively, which are put side by side on the board. (The teacher may
prompt the students to use the index laws as far as they think reasonable.)
The teacher polishes the answer, inserting in-between steps, if necessary:
35  3-2 = 35-2 = 33 =27.
35 
1
33333
=
= 3  3  3 = 27.
2
3
33
By comparing the results, the conjecture that 3-2 =
1
32
5min A number of similar problems are worked out to reinforce the plausibility
of the conjecture.
-2-
5min Teacher checks the validity of the conjecture:
35  32 352 33
333
1
3 =
 5  5 
 2
5
33333 3
3
3
3
Teacher points out that the above ‘proof’ also works if 2 in the power is
replaced by any other positive integer, e.g. by 4:
-2
35  34 354 31
3
1
3 =
 5  5 
 4
5
33333 3
3
3
3
1
Hence the result that 3  n = n is generalized.
3
-4
5min The teacher asks the students to guess a formula for a  n .
1
The teacher confirms the result a  n  n .
a
7min The students are asked to do a number of exercises on the above formula.
3. Closure
(Conclusion)
5min The teacher reminds the students of the fact a 0 = 1, for a ≠ 0.
The teacher proves the formula:
a n  a n
an n 0a 1
a 
 n  n
an
a
a
an
The formula is stated once again.
n
(Post-assessment)
A few problems involving negative integral indices are asked orally.
(Consolidation)
3min The students are requested to do a number of problems for homework,
including the completion of the following table, which will be used in
plotting the graph of y  2 x later on.
x
-3 -2 -1
0
1
2
3
2x
The students are asked to search for negative indices (from pages /// to ///)
in their physics textbook (kg  m -3, ms -1, ms -2, etc.).
1-2 challenging problems are given. The working of them is optional.
-3-

1. 必須有詳細之教案，教材不一定列出，但應詳列每步的教學法。
2. 整課應有連貫的策略與計劃。
3. 語調與板書均須清晰。
4. 應有學生活動，不應全堂講授。
5. 有效使用發問，引發學生思考，避免自問自答。
6. 著重概念之瞭解，而非枯燥的運算。
7. 善用各種教具及資訊科技，須配合教學，不應為用而用。
8. 重視學習之趣味性及與學生的溝通。
9. 注意時間分配。
10. 堂課時應巡視及作個別輔導。

１． 本課的學習重點是＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿
＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿
２．（a）你預期本課學生最難明白之處為：＿＿＿＿＿＿＿＿＿＿＿＿
＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿
２．（b）處理辦法：＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿
＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿
３．（a）你預期本課學生感到最沉悶之處為：＿＿＿＿＿＿＿＿＿＿＿
＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿
３．（b）處理辦法：＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿
＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿
４．（a）本課內容於＿＿＿＿年級＿＿＿＿＿＿＿＿＿＿課有相疊之處。
４．（b）處理辦法：＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿
＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿
-4-
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