### 6 AM Quiz (a): Do We Meet?

Do Parallel Lines Meet?
"Yes" and "No".
Why?

### 6 AM Quiz (b): Gradients? Do We Have?

We describe a straight line with gradient and y-intercept.
We can therefore express them in the form of y = m x + c, where m is the gradient while c is the y-intercept.

We describe a parabola with the following generic form: y = ax² + bx + c.
So, does a parabola have any gradient?

### Lesson Summary (28th September 2010)

Points to note:
1. We cannot assume that a line is straight or that there is a right angle unless they provide us that information.
2. We can only conclude that lines are parallel if they draw arrows on the line indicating they are parallel.
3. Constructing is NOT the same as drawing or sketching.
Angle BXF = Angle DYF (Corr. angles; AB//CD) [corresponding angles]
Angle AXF + Angle CYE = 180° (int. angles; AB//CD) [interior angles]
Angle DYF = Angle CYE (vert. opp. angles) [vertically opposite angles]

Constructing a perpendicular bisector:
• Use a compass
• Cut above
• Cut below
• Equal distance from both points
Every point along the constructed line has an equal distance to the two points at the ends.

The triangle formed between a point on the constructed line and the points on the original given line would be an isosceles triangle.

To find the area of this triangle, use the length from where the constructed line cuts the base of the triangle to the point on the constructed line as the height. The base would be the length from one point of the original given line to another point on the same line. The area would be the product of both these lengths divided by half.

### Key Takeaways from Math Lesson

For Linear Inequalities:

1. A
2. AB therefore A
3. A>C, BA

Points to take note:

Where there is a fraction, round it off to 3 Significant Figures

Always draw a number line

When they are two inequalities, solve each inequality and the unknown (x) has to be between a range from both equation.

For Angles and Parallel Lines:

Definitions: Point - Position, No

Lines - Infinite number of points, No width, Determine by 2 points, Can be curved

Segment - Finite length, part of a line

(Usually not tested)Ray - One end fixed, the other infinite

Plane - Any flat surface, no thickness

Angles:

Complementary angle - Comp <

Supplementary angle - Supp < / <>

<>

Angle at a point - <>

Acute angle : 0 degree <>

Right angle : x = 90 degree

Obtuse angle : 90 degree <>

Straight angle : 180 degree

Reflex angle : x > 180 degree

### Lesson Notes (Wednesday)

Today, we recapped the method TRIAL & ERROR for factorization and we started on linear inequalities.

Inequalities are basically statements about the size or order of 2 objects, OR about whether they are the same or not.
They will mostly look like this : a < b or a > b

The other stuff is in the notes, and for the FINAL ANSWER of each question about inequalities,
It should look like this :

NOTE: A number line MUST be drawn after the statement. A white circle (empty) shows that the particular object is smaller than the other, while a black circle (filled) shows that the object is either more/less than or equal to the other.

A common mistake most people make is that when

X<2 OR X>4 (theres supposed to be a line under the >)

4 <(plus a line) X < 2
meaning that 2 is bigger than X which is bigger than 4. It's WRONG.

Following X-Code,

if ( C > 0 )
{ ac > bc ;
a/c > b/c;
}

The C MUST be positive.

Lastly, the FINAL answer MUST be X ONLY. No negative X is allowed.

### Factorization( tuesday )

Factorization:
Rules:

1. Common Factors

3a-12=3(a-4)

3a^2-5a=a(3a-5)

3a^2-12a=3a(a-4)

Rule 2: Perfect Square & Difference of 2 squares

Perfect Square:

1.(a+b)^2=a^2+2ab+b^2

2.(a-b)^2=a^2-2ab+b^2

Difference of 2 squares:

1.(a)^2-(b)^2=(a+b)(a-b)

x^2+4x+4=(x)^2+2(x)(2)+(2)^2 (type 1.)

x^2+4x+3=(x)^2+2(x)(2)+(2)^2 -1

square^^^^^^^^^^

=(x+2)^2-1^2

=(x+2+1)(x+2-1)

=(x+3)(x+1)

**(every 1 mark, you have about 1.5minutes to do the question. So if you get a 3 mark question you only have 4.5minutes to do the question

### Notes for today, 14-09-10

Factorization
1. Take out common factors
2. Perfect squares (a+b)^2
Difference of 2 squares (b)^2-(a)^2
The main key takeaway today is to recognise which line to use for the equation and to make sure that you will not end up in the “game over” mode. We will have to make sure that we must be able to apply the line in our factorization process.

(sorry for the little notes...)

### REMEDIAL SESSION ...

Good morning everyone,
This is to confirm that we'll be having our remedial this THURSDAY (9th SEPTEMBER) at 0930 ... we'll meet in the LEARNING OASIS ... bring loads of writing paper and your calculator ... you would also need your LEARNING DEVICE in order to download the worksheet ...

Come prepared to work ... there is some issue with the notion of GRAPHS, its GRADIENT and the ALGEBRAIC PROBLEM SOLVING ...

JASON INGHAM

### Lesson Notes- 27 August

Some things to note:
1) (a+b)(a+b) = (axb)(axa)+(bxa)(bxb)
2) a^2 + b^2 IS NOT ( a+b)^2
3) Do the expansion systematically
4) Write in alphabetical order so as to avoid a x b NOT= b x a
5) Write in order of power and u get =) Examiners

Notes for the topic: Linear Equations
1) Convert all equations to y=mx+c
2) Rise/Run -> Order matters
3) Drawings/sketches are a TOOL and NOT for ANSWERS
4) Example question:
y= 1/2x+c
so one has to substitute in (6,0) into the equation
so : 0 = -1/2(6) + c
ans: c= 3
y= 1/2x+3

Practice Question:
Line A cuts through the y-axis at (0,6) and it cuts the x-axis at (-6,0). Find the equation of the Line A.

Note: Sorry for being late in posting this up by a day. :(
Done by: Jonathan Soh- S101

### Lesson Notes 26.8.2010

In order to let y=0, the x in the following equations have to be:
1. y=2x-2
x=1
2. y=x+2
x=-2
3. y=(2x-2)(x+2)
y=(2x1-2)(-2+2)
=0x0
Thus, y=0
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
y=(3x-5)(x-4)
y=20
because (-5)(-4)=20
y=x^2-x-2
=(x ?*) (x ?~)
Steps to find the values of “?*” and “?~”
---> The product of ?* and ?~ has to be -2.
1. List down all the possibilities (The product of ?* and ?~ values is -2)
2. In this case, the sum of ?* and ?~ has to be -1
3. Choose the correct possibility
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
If the equation is y=x^2 -6x+8
= (x ?*) (x ?~)
-----> The product of ?* and ?~ has to be 8
------> The sum of ?* and ?~ has to be -6
1. List down all the possibilities (The product of ?* and ?~ values is -2)
2. In this case, the sum of ?* and ?~ has to be -6
3. Choose the correct possibility (so that the product of ?* and ?~ values is -2 &  the sum of ?* and ?~ is -6
Ans: -2 & -4

### Summary of Math lesson (24 August)

In today's Math Lesson, we learnt about x-intercept, turning point and line of symmetry.

1. Quadratic equations only occurs when two linear equations are multiplied with each other, provided that x²  is the greatest power in the equation.
e.g y = (x+2)(x-4)
= x²  - 2x - 8

2. Quadratic equations are symmetrical. Line of symmetry lies on the turning point.

3. The roots of the resultant quadratic equation are the same as the individual linear equations

4. Calculate turning point:
x = turning point = (root) + (root) / 2

e.g x = 1 = 4 + (-2) / 2
Refer to pic above

Why, for point number 4, must the (root)+(root) be divided by 2?

### Summary of Maths Lesson(20 August 2010)

What we have learnt?

1) Revision and extension of previous lesson

2) Double Intercept formula

Shortcut for checking answers- rewriting an equation (y=mx+c) into (x/a+y/b=1)

Note. The a is the x intercept and the b is the y intercept.

i.e. y= 1/2x-3

3= 1/2x-y

1= x/6-y/3

1=x/6+y/-3

To make this in the form of y=mx+c, cross multiply.

i.e. x/-1+y/5=1

5x-y/-5=1

5x-y= -5

5x-y-5=0

y= 5x-5

Why is the x intercept (a, 0) and the y intercept (0, b)?

### Summary of Maths Lesson(19 August 2010)

What have we done in today Maths lesson?

1. We had revised that gradient is equal to rise over run. Gradient is also the difference in y value over the difference in x value.

2. The order of the coordinates is also important. We can write in the form of y2-y1/x2-x1, but not y2-y1/x1-x2, as the order is all wrong.

3. The points that coincide with the vertical line, or the y axis, has a gradient of zero(or infinite values) as they all have the same x coordinates. On the other hand, the points that coincide with the horizontal line, or the x axis, has a gradient of zero(or infinite values) as they all have the same y coordinates.

4. We also tried to do questions which asked us to find the value of c, or the constant. We first had to find m, the gradient, before substituting the coordinates into the equation to find the x and y value.

5. We also concluded that all parallel lines have the same gradient. The lines that are vertical or horizontal have a gradient of zero.

Here is a question related to what we have been discussing today.

AB is a line. If A is (-2, 2) and B is (5, 4), what is the gradient of AB?
What is the equation of AB then?

### SUMMARY OF LINEAR EQUATIONS ...

ANONYMOUS - 17th August 2010
1. Today we reconsidered the linear equation & the general form of a linear equation is y = mx + c
2. where by the m refers to the GRADIENT
3. where c is the Y-INTERCEPT (or the value of y when x = 0)
4. m is POSITIVE when as x increases, y also increases
5. m is NEGATIVE when as x increases, y will decrease
6. m is ZERO when as x changes, y is constant (HORIZONTAL LINE)
7. m is undefined when x is constant and y changes (VERTICAL LINE)
8. X-INTERCEPT would be equal to -c / m ... (or the value of x when y = 0)
NOTE - c (or the constant in all the equation) is always the y-intercept, even for equations such as y = ax^2 + bx + c ...
Why do you think this is so? (HINT - look at point (3) above)

Prove point (8) for yourself ...
Question 1 : ‘A square is a rhombus but a rhombus is not a square’.

This is true as a square has two sets of parallel sides like a rhombus. But the corners of the rhombus are not 90 degrees like a square.

Question 2 : Which of the given statements is correct? Justify your answer/s with examples.

A) A square and a parallelogram are quadrilaterals.
B) Opposite sides of a square and a parallelogram are parallel.
C) A trapezoid has one pair of parallel sides.
D) All the above

i think it's D.

a)Both a square and a parallelogram have 4 sides
b) The opposite lines will never meet
c)one pair of lines will meet but another will never meet

Question 3: 'All parallelograms are squares

I do not agree with this statement as some corners of the parallelograms are not 90 degrees so they are not squares.

### Question 1, 2, 4 By Chong Guang Jun

Question 1 : ‘A square is a rhombus but a rhombus is not a square’.

This statement is correct as a rhombus has 2 pairs of opposite sides that are parallel, and all sides are equal, so does a square. Thus, a square is a rhombus. A rhombus is not a square as it does not have 4 right angles on each side where the lines meet like a square, although it has 2 pairs of opposite sides that are parallel, and all sides are equal, so does a square.

Question 2 : Which of the given statements is correct? Justify your answer/s with examples.

A) A square and a parallelogram are quadrilaterals.
B) Opposite sides of a square and a parallelogram are parallel.
C) A trapezoid has one pair of parallel sides.
D) All the above

I agree with statement D.
The reason for A is that both a square and a parallelogram have 4 sides closed up, with 4 angles, which qualifies them as quadrilaterals.
B is also true as the opposite sides of both a square and a parallelogram will never meet no matter how long it is.
C is correct as one pair of sides will not meet, while the other pair will meet up if it is drawn longer.

Question 4 : ‘All parallelograms are squares?’
I disagree with the statement above. The reason being that a square requires 4 EQUAL sides and all 4 angles of 90 degrees. The statement mentioned that ALL parallelograms are squares, but most parallelograms have 2 pairs of equal sides and 2 pairs of the same degree, which does not qualify as a square.

### Question 2, 3, 4 By Lai Ziying

Question 2:
Which of the given statements is correct? Justify your answer/s with examples.
A ) A square and a parallelogram are quadrilaterals.
B ) Opposite sides of a square and a parallelogram are parallel.
C ) A trapezoid has one pair of parallel sides.
D ) All the above

Justification:

A: A square and a parallelogram are formed by 4 straight lines.

B: This statement is true as the the opposite sides of both the shapes will never meet.

C: The top side and the bottom side of a trapezoid will will never meet after they are extended. But the lines of the left and the right side will meet. Thus it has only 1 pair of parallel sides.

Question 3:
A quadrilateral is drawn on a piece of paper. It has one pair of opposite sides equal in length, the other pair not equal in length, and a pair of opposite angles that are supplementary1Identify this figure, and justify your answer with reasons.
sum of two angles equals 1800

The lines of the left and right sides of a trapezium can be equal in length while the lines of the top and bottom sides are different in length. And a trapezium suits all the conditions given in the question.

Question 4:
‘All parallelograms are squares?’ Do you agree with this statement?

Answer: I do not agree with the statement. The reason is simply because all parallelograms do not suit the conditions of being squares. First, the 4 angles of a square are of the same angle of 90 degrees; Second, the diagonals of a square are perpendicular with each other; Third, the 4 sides of a square are of the same length.

### Question 1, 2, 3 By Harsh Seth

1) 'A square is a rhombus but a rhombus is not a square’.

I agree with this statement as square has 4 equal sides and the opposite angles do add up to 180° but a rhombus can is not a square as all angles in a square are 90° but rhombus's angle need not be 90°. It can be more or less.

2) Which of the given statements is correct? Justify your answer/s with examples.
A) A square and a parallelogram are quadrilaterals.
B ) Opposite sides of a square and a parallelogram are parallel.
C ) A trapezoid has one pair of parallel sides.
D ) All the above

I agree with statement D.
Why A? They both have 4 sides. They also have have 4 corners and 4 angles, all adding up to 360°.

Why B? A square and Parallelogram do have opposite parallel sides or else they would not the shape.

Why C? A trapezoid has one pair of parallel line as the others are sloped towards each other. they are not parallel.

3) A quadrilateral is drawn on a piece of paper. It has one pair of opposite sides equal in length, the other pair not equal in length, and a pair of opposite angles that are supplementary1. Identify this figure, and justify your answer with reasons. 1 sum of two angles equals 180°.

This quadrilateral is an trapezium as it has one pair of equal sides and another pair not equal sides.The two opposite angles of the trapezium are equal to 180°

### Question 2,4,5 by Calvin Heng

Question 2)Which of the given statements is correct? Justify your answer/s with examples.

A ) A square and a parallelogram are quadrilaterals.

B ) Opposite sides of a square and a parallelogram are parallel.

C ) A trapezoid has one pair of parallel sides.

D ) All the above

B and C are correct.

The property of a parallelogram is that opposite sides are parallel as the square has an equal length and angle of all sides, its opposite sides are also equal. Thus, B is correct.

C is also correct as the trapezoid has two opposite lines at the top of the figure and at the bottom which is parallel. The two sides of the figure is not parallel. Thus there is only one pair of parallel lines. Thus, C is correct.

Question 3

It is a parallelogram. Firstly opposite angles of this figure adds up to 180 degrees. The opposite lengths are equal within this figure. Its pair of opposite angles are also supplementary. Thus, the quadrilateral is a parallelogram.

 Question 5: ABCD is a parallelogram. If E is midpoint of AD and F is midpoint of BC show, with reasons, that BFDE must be a parallelogram.

BFDE must be a parallelogram. Firstly it has two pairs of parallel lines, this is one of the properties of a parallelogram. THe opposite sides are equal in this figure. Which is also a property in the parallelogram. Thus we can conclude that it must be a parallelogram

### Questions 1, 2 and 4 Oh Sher Li

Question 1:

“A square is a rhombus but a rhombus is not a square.”

This statement is true as the definition of a rhombus is that it has 4 equal sides and the diagonally opposite sides are of equal angles. A square has 4 equal sides, and since all the angles are 90Âº, a square can be considered a rhombus. However, in order to be a square, all 4 angles needs to be 90Âº, and not all rhombuses have all 4 angles of 90Âº, thus not all rhombuses are squares.

Question 2:

The answer is D, all of the above. A square and a parallelogram are 4-sided figures, thus they are quadrilaterals. The rule of squares and parallelograms states that the opposite sides must be parallel. A trapezoid has one pair of parallel sides, the top and bottom or sometimes the left and the right.

Question 4:

“All parallelograms are squares.”

I do not agree with this statement. For a figure to be a square, there must be 4 equal parallel sides and 4 angles of 90Âº. A parallelogram does not need to have 4 equal sides, only the opposite sides need to be equal. It also does not need to have 4 angles of 90Âº, as long as apposite angles are equal. It is true that some parallelograms are squares and all squares are parallelograms, but not all parallelograms are squares.

### Question 1, 2 and 4 Lim Zhi Qi, Crimson

1.Based on the above conversation discuss, with examples and justification whether the following statement is justified.
‘A square is a rhombus but a rhombus is not a square’.

This statement is true.
The properties of a rhombus is that the lines must of equal length and the opposite sides must be parallel while the properties of a square is that all line must be of equal length, opposite sides must be parallel and that every line must be perpendicular to another.
A square can be a rhombus as it fulfills all the properties of a rhombus. However, the rhombus cannot be a square as the angles in the rhombus can be of any angle as long as the opposite angle is the same and the top and bottom angle add up to 180 degrees.

Question 2:
Which of the given statements is correct? Justify your answer/s with examples.
A ) A square and a parallelogram are quadrilaterals.
B ) Opposite sides of a square and a parallelogram are parallel.
C ) A trapezoid has one pair of parallel sides.
D ) All the above

A square and a parallelogram are quadrilaterals as they have 4 sides.

The opposite sides of a square and a parallelogram are parallel as they will never meet even if you extend the lines.

A trapezoid has one pair of parallel sides as it has only one pair parallel sides as the two angles of the vertical lines add up to 180 degrees.

4.‘All parallelograms are squares?’ Do you agree with this statement?
I do not agree with this statement. Parallelograms do not have equal sides unlike squares. The opposite angles of a square are supplementary while a parallelogram does not.

### Question 2, 4 and 5 by Soh Fan

Question 2:

Which of the given statements is correct? Justify your answer/s with examples.

A ) A square and a parallelogram are quadrilaterals.

B ) Opposite sides of a square and a parallelogram are parallel.

C ) A trapezoid has one pair of parallel sides.

D ) All the above

Ans: D) All of the above. A square and a parallelogram are quadrilaterals as they have four sides and the sides are straight lines.The o

pposite sides of a square and a parallelogram are parallel and a
trapezoid has one pair of parallel sides. These are some of the properties of these quadrilaterals.

Question 4:
‘All parallelograms are squares?’ Do you agree with this statement?
Ans: Not all parallelograms are square. Parallelograms are only square if all sides are of equal length and the interior angles are 90 degree.

Question 5:

ABCD is a parallelogram. If E is midpoint of AD and F is midpoint of BC show, with reasons, that BFDE must be a parallelogram.

Ans: Length BF and ED are equal as they cut off at on the sides parallel to each other. Thus the lines connected to them would also be parallel with equal lengths and thus BFDE must be a parallelogram.

### Question 1,2 and 4. Grace Tan Soo Woon

Q1. Based on the above conversation discuss, with examples and justification whether the following statement is justified.

‘A square is a rhombus but a rhombus is not a square’.

Answer: The properties of a rhombus is that the lines must of equal length and the opposite lines must be parallel. The properties of a square is that all line must be of equal length, opposite sides must be parallel and that all the angles in the square must be equal (90 degrees every corner).

A square is a type of rhombus as it fulfills all the properties needed. On the other hand, the rhombus cannot be a square as the angles in the rhombus can be of any angle as long as the opposite angle is the same and the top and bottom angle has to add up to 180 degrees.

Q2. Which of the given statements is correct? Justify your answer/s with examples.

A ) A square and a parallelogram are quadrilaterals.

B ) Opposite sides of a square and a parallelogram are parallel.

C ) A trapezoid has one pair of parallel sides.

D ) All the above

Quadrilaterals are four sided figures. Both the square and the parallelogram have 4 sides. Thus the square and parallelogram are quadrilaterals.

The opposite sides of the square and parallelogram are parallel as their ends do not meet.

A trapezoid has one pair of parallel sides as the properties of the trapezoid are that the two angles on the same side of the trapezium always add up to 180 degrees. Other than that, there are only one pair of parallel lines, unlike squares and parallelograms which have two pairs of parallel lines.

Thus the answer is All of the Above.

Q4. ‘All parallelograms are squares?’ Do you agree with this statement?

Answer: No i do not agree with this statement. The properties of a square are that there are two paris of parallel lines, the angles in the square are all 90 degrees and the lines are all of equal length. Parallelograms have two pairs of parallel lines but not all the angles are of equal degrees and that the length of the lines of one pair would be longer then the other as if they were the same, it would result in a rhombus.

### Question 1, 2 and 4 - Loh Cheng Ngee

1) ‘A square is a rhombus but a rhombus is not a square’.
This statement is true. The properties of a square are that there are 2 pairs of parallel lines and there are 4 right angles while the properties of a rhombus are that no defined angles for each of the 4 and that there are 2 pairs of parallel lines. Since a rhombus can have 4 right angles, a square is a rhombus in that situation. A rhombus however cannot be a square as a square has defined 4 right angles while there are no given defined angles for the rhombus as the angles of the 2 pairs of parallel lines will be different.

2) D is correct. A square and a parallelogram are quadrilaterals as they both have 4 sides to them, as they have 4 angles in the square and parallelogram. Opposite sides of a square and a parallelogram are parallel as they both have 2 pairs of parallel lines, so each ‘pair’ of a side in the 2 quadrilaterals will definitely be parallel. A trapezoid has only one pair parallel sides as the two angles of the vertical lines add up to 180 degrees. If it were in any other way, they would not add up to 180 degrees.

4) I do not agree with the statement. Parallelograms are not squares as although the opposite sides are parallel, the length of the opposite sides are different. They must be different as the angles of the parallelogram will change, and a square has the same length for all 4 sides, therefore parallelograms cannot be squares.

### Question 1, 3 and 4: Tan Jianhui

Q1: A square fulfils all the requirements of a rhombus; 2 pairs of parallel sides and all sides are of equal length. But none of a rhombus’ sides meet at 90°.

Q3: The figure is a rhombus. A rhombus has a pair of opposite sides equal in length, the other pair is not equal in length and one of the properties of a rhombus is that a pair of opposite angles must add up to 180°, therefore the figure is a rhombus.

Q4: No, I do not agree. The first reason is because the interior angles of a parallelogram are not 90°. Second reason is that all four sides of a parallelogram are not equal.

### Maths Question- Sue Lun

1) ‘A square is a rhombus but a rhombus is not a square’.

A square must have four sides of equal length and four right angles. The angles must be 90 degrees. The opposite sides are also parallel.

A rhombus must have four sides of equal length. The opposite sides are also parallel. The opposite angles of a rhombus are the same. However, the angles must not only be 90 degrees but can differ.

Hence, a square can be a rhombus as all it’s properties fulfil the ones of a rhombus. However, a rhombus is not a square as the angles can differ. It might not be 90 degrees, which is needed to fulfil the requirements to be a square.

2) Which of the given statements is correct? Justify your answer/s with examples.

A ) A square and a parallelogram are quadrilaterals.

B ) Opposite sides of a square and a parallelogram are parallel.

C ) A trapezoid has one pair of parallel sides.

D ) All the above

1. Properties of a square- A square must have four sides of equal length and four right angles. The angles must be 90 degrees. The opposite sides are also parallel.

Properties of a parallelogram- A parallelogram has two pairs of parallel sides. The opposite angles are the same and the opposite sides are of the same length.

Properties of a trapezium- A trapezium have a pair of parallel sides.

Firstly, quadrilaterals are four sided figures that are made out of four straight lines. A square and a parallelogram both have four sides that are straight lines. Hence, they are quadrilaterals.

Next, the properties of a square and a parallelogram both include ‘opposite sides are parallel’. Hence, statement B is true.

Besides that, the properties of a trapezium also state that it must have one pair of parallel sides. Hence, statement C is true.

Question 4:

‘All parallelograms are squares?’ Do you agree with this statement?