Mastering Mathematics Smartly by Wee Wen Shih

Forum

Dear ASM401 students,

Do feel free to use the discussion forum at: forums.delphiforums.com/weews

Week 1

Lesson summary:
1. Introductions

2. Sequences and conjectures
- Conjectures are obtained by means of inductive reasoning.

3. Mathematical induction
- We use mathematical induction to prove that a result is true for all cases.

- There are 5 steps:
(i) State the proposition.
(ii) Show that the proposition is true for the first value of n.
(iii) Assume that the proposition is true for n = k.
(iv) Show that the proposition is true for n = k + 1.
(v) State the conclusion.

4. Recurrences, arithmetic and geometric progressions
- In a recurrence relation, the successor term in the sequence is obtained by means of its predecessor term.

- For an arithmetic progression, $\text{T}_n = \text{T}_{n-1} + d$ where d is the common difference. We can use inductive reasoning to arrive at $\text{T}_n = a + (n-1) \times d$ where a is the first term.

- For a geometric progression, $\text{T}_n = r \times \text{T}_{n-1}$ where r is the common ratio. We can use inductive reasoning to arrive at $\text{T}_n = a \times r^{n-1}$ where a is the first term.

5. Homework problems for topics 1 and 2
Apologies, please note these typo errors:
1. In Andy's method: 3rd term should have been 18 = 3 x 6 = 3 x (3 + 3).
2. In Andy's method: 4th term should have been 28 = 4 x 7 = 4 x (4 + 3).
3. In Aaron's method: 4th term should have been 28 = 5 x 6 - 2.

Week 2

Lesson plan:
1. Some word problems cannot be solved by the model approach.

2. In algebraic problem solving, what is being defined as x matters to clarity of the solution. Translating the information systematically into mathematical expressions (either via line-by-line or tabulation) is also a crucial process.

3. We apply the addition and multiplication properties to solve equations.

4. Here is a set of further examples for in-class discussion.

5. Solution for further example 3:
Let x be the number of students who play badminton before the trial period.

	Badminton	Squash	Table Tennis
Before	$x$	$\frac{x}{4} \times 3$	$x + 36$
After	$(x + 12) - \frac{1}{4}(x + 36)$	$\frac{3x}{4} - 12$	$\frac{5}{4}(x + 36)$

Relationship: $\frac{\frac{3x}{4}-12}{\frac{5}{4}(x + 36)} = \frac{3}{10}$ , from which we obtain $x = 68$ .

Weeks 3 & 4

No lessons in week 3.

Geometer SketchPad session will be held at Fourier Lab.

Week 5

E-learning assignment is at: http://math.nie.edu.sg/shutler

Week 6

Homework:

Please attempt questions 6, 7 and 8 in Tutorial 4 as well as 3 questions on page 26 in this document.

Helpful link on platonic solids: http://www.mathsnet.net/geometry/solid/nets.html

Preparation for Quiz 1:

Attempt these questions to boost your confidence. Jiayou!

Hints to selected questions:
Q1(c) What happens when we add 2 rows, 3 rows, 4 rows?

Q2 Consider a - d, a, a + d.

Q3(b) Use trial and error.

Q4(c) Consider the equation "(a) - (b) = 60".

Q5 Given B + C -> 4 units, A -> 5 units.
Let C -> x units.
Try to form the equation (4 - x)/5 + x/3 = 5/6.

I wish all muslim students a very happy Hari Raya Adil Fitri!

Weeks 8 & 9

1. List of examples for chapters 5 and 6

2. Lecture slides

3. Excel file on regression lines

Week 10 and 11

1. Summary of points

2. Steps to draw a boxplot in Excel

3. When we compare boxplots, we may comment on:
(i) Range as shown by whiskers;

(ii) IQR;

(iii) Type of distribution (i.e. symmetrical, skewed to the left or skewed to the right);

(iv) Any other notable observations.

Read this example.

4. Questions to be submitted for tutorial 6: Handout exercises 1, 2 and 3.

Comparison of Class A's and Class B's boxplots in Exercise 1:
(i) Class A has a larger range than Class B (25 versus 20).

(ii) Class B's IQR is about twice that of than Class A (13 versus 7).

(iii) Class A has a symmetrical distribution, whereas Class B has a distribution that is skewed to the left.

5. Solutions to Tutorial 6 questions:
Q11
The vertical axis has no label.

Q12
Consider the list of scores given by 0, 0, 10, 100. The mean of 27.5 and median of 5 are not representative of the scores at all.

Q13
Doubling the radius means quadrupling of sales, which is misleading when sales had doubled in reality.

Q15
The line graph has no explanatory title.
Both axes do not have numbers that are equally spaced.
The horizontal axis has no label.

6. Quiz 2 consultation: 4 Nov (Wed) between 10am and 2pm at NIE Library Cafe.