FROM QUADRATIC FACTORISATION TO QUADRATIC FACTORISATION ...

Summary of today's maths lesson (14/9/10)

For today, we recapped that:

• (a+b)² = a²+2ab+b²
• (a-b)² = a²-2ab+b²
• (a+b)(a-b)=a²-b²
• x²+y² is NOT equals to (x+y)²
• x²+y² is equal to (x+y)²-2xy

As for today, we learnt that:

• 201² -402+1 = (201)² -2(201)(1)+(1)²

= (201-1)²

= 200²

=40 000

Note: Before you start doing anything, you need to look out for the relationship between each numbers, and make sure it follows exactly as the quadratic expressions that you learn by heart. In this case, 402 is twice of 201. Since you need to make it exactly the same, there has to be a squared for the 1. Hence, you can square the 1 as the result is still the same.

• 823² -177² = (800-23)² -(800-3)² Do not do this method.

Instead, do this method:

823² -177² = (800-+177)(800-177)

because it is easier to solve. The first method is not encouraged to do as it will take up a very long time to solve.

• All the above sums are called factorisation.
• In factorisation, there are four different methods (but we only learnt two of them today) — By common factors and perfect square.

An example for the common factors method is:

b²-3bc = b(b-3c)

Note: You are actually taking out the HIGHEST common factors, which means ALL the factors must have the same common factors. NOT only one or two of the factors.

Summary of Linear Equations

• What we had learnt from today's maths lesson (for the first 3, let p & q the x-intercepts)

1. The roots of a parabola are the x-intercepts and right them in the proper way like (p,0)
2. To find the y-intercept of the parabola, the person must find the result without the x in it such as from equation (x-p)(x-q) and the result from equation without the x in it is pq so the y-intercept would be (0,pq).
3. To find the line of symmetry of the parabola, we must find the centre of the x-intercepts and the equations would be p+q/2.
4. The way to expand equations, let the integer in a bracket multiply with the equation in the other bracket. eg. (2-k)(4+k)= 2x4, 2xk= -k(8+2k)=-kx4, -kxk= 8+2k-4k-k square= 8-2k-k square
5. Equations needed to remember

• (a+b)square=a square + 2ab+ b square
• (a-b)square= a square - 2ab +b square
• (a+b)(a-b)= a square - b square

Note: (a+b) square is not equal to a square + b square

Question: There are U & n shaped parabola going upwards or downwards so is there such thing as a C shaped parabola?

If there is such thing, is it called parabola or anything else?

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

Summary of Maths Lesson(27 August 2010)

-Recap on linear graphs:

 We learnt that:

• convert all equations to y = mx + c
• Visualise the gradient; increasing/decreasing
• X-intercept= -c/m
• Y-intercept is when x=o
• Do working step-by-step to prevent error and careless mistakes
• Substitute the given values of x and y values into y = mx + c if two variables are unknown.
• Gradient

 = m 

 = rise/run

• It is the CHANGE IN Y-AXIS/ CHANGE IN X-AXIS
• Look at the slope; increasing or decreasing slope.
• Expansion of Quadratic Expressions:

We learnt that:

- expand a certain term

• write in alphabetical order
• convince that (a-b)2 = (a-b)(a-b) using expansion
• answers can leave in improper fraction unless state the co-efficient of the term.
• reduce to simpler form
• decimal is also possible
• did (a+b)2, (a+b)3 , (a+b)4 and so on as practice

Pascal Triangle:

• a shortcut way of attaining answers for quadratic expansion.
• Co-efficient of the number.

Question of the day:

How do you determine that the Pascal triangle is always correct and does it always give the exact co-efficient for all expansion of quadratic expressions? Is there a particular occasion when the Pascal triangle did not meet the co-efficient needed? In what circumstances? Examples is required.

new summary-tan kein shuen

1.we discovered that if x/a + y/b, then a and b CANNOT be zero.
2.the product of two linear factors will be the y intercept of its parabola.
3.we learnt that the line of symmetry is (a+b)/2.
4.question: why must both the two lines be parallel to each other to form a parabola?

Summary of Math Lesson (24/08/10)

1. With the given values in the questions (values of m, x, y and c), we can substitute to find out the part of the equation that we do not know.

2. Substitution is NOT the y intercept. The values of x and y given is a coordinate that lies on the line.

3. x/a + y/b=1. a is the y intercept, while b is the y intercept. Why?

when a=x intercept, y=0.

=x/a + 0/b=1

=1+0

=1

4. Parabola consists of 2 root/x-intercepts.

5. If 2 lines share the same x intercept as the parabola, with substitution of the equation, it causes one of the equation of the parabola to be 0, thus y=0 regardless of the 2nd equation

6. We only did one type of parabola today: We can only tell the min value and not the max.

7. Why is x/a + 0/b always 1? (Point 3)

8. Is it possible for the parabola to have not two equations, but one?

Yan Jin

Summary for the lesson (19/08/10)

Sorry that i posted at this late hour..... Just came back from YOG....
1) If (a,b) lies at y=mx+c, then b=m(a)+c
2) Gradient is = rise divided by run = change in y coordinate divided by change in the x coordinate.
THE ORDER IS IMPORTANT
3) y - 1x +c (Coordinates 1,5 and -3,1)
To find c, you have to find x. It can go two ways;
Either you substitute the coordinates (1,5) and get
5=1(1)+c
  =1+c
c = 4#
OR
1 = 1(-3)+c
   = -3+c
.: c = 4
4) To satisfy an equation means to balance both sides of an equation.

Summary of Linear Equations

Casandra Ong- 18Aug

1) Discuss if there is an equation, when c is not the y intercept.
2) At y-intercept, x is 0
3) c is not always the intercept. For example, y= 1÷x + c. Therefore, the value of y is undefined when x=0 and c would not be the y- intercept.
4) x= -c÷m
5) m= gradient
= (change in y-coordinate) ÷ (change in x-coordinate)

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 by Iskandar B Dzulkarnain

A square has all of its sides equal, thus it is a rhombus. A rhombus, on the other hand, DOES NOT have perpendicular lines, hence, it is NOT a square.

Question 4 by Iskandar b Dzulkarnain

This square here, as indicated at BCD, has a 90º angle which is one of the characteristics of a square. But a parallelogram on the other hand, does not have any perpendicular lines, thus, it is not a square.

Question 5 by Iskandar B Dzulkarnain

ABCD, as it is a parallelogram, it will have all sides equal to the opposite's. Taking over from ABCD, BFDE will also have this characteristics thus proving that it is a parallelogram.

Question 5 by Niloy Faiyaz

It is a parallelogram since each side is parallel to the opposite side.

Question 4 by Niloy Faiyaz

I do not agree with the statement since for squares, its a must for every corner to have a 90 degree angle while a parallelogram can have any degree.

Question 2 by Niloy Faiyaz

The answer is D, all of the above.

The reason is that:

(a) a square and a parallelogram are quadrilaterals since they all have 4 sides.

(b) The opposite sides are parallel since they are facing the same direction.

(c) The trapezoid has one pair of parallel lines since one pair faces in different directions while the other pair faces at the same side.

so all are true.

Question 1,4,5

Question 1:

A square is a rombus as it has four equal lines,which two of them are parallel to each other but a rombus is not a square as the lines should be perpendicular to each other.

Question 4:

i disagree with the statement,as a parallelogram  does not have its sides perpendicular to each other.

Question 5:

Since E and F are midpoints,BF and ED should have the same distance,so BFDE should be a parallelogram.

 

Questions 1, 4 and 5 by Gavin Lim

5)If both e and f are midpoints, there is the same distance from AE and FC so BE and FD are parallel, which makes BEFD a parallelogram.

4)No. The sides of a square must all be perpendicular to each other while the sides of a parallelogram may not be perpendicular.

1) A square is a rhombus as it has 2 pairs of parallel lines of equal length but a rhombus is not a square as the sides of a square must all be perpendicular to each other while the sides of a rohmbus may not be perpendicular.

Problem with posts below

The images bay not be seen in my post below, as if i sent the text version, all my images would have ended up in the wrong question. Just try reloading, and it should work soon

Thanks,

Azeem Arshad Vasanwala
(Class S1 - 03)
School of Science and Technology, Singapore

Question 1,2,4 by Gregory Chew

1. The statement is justified as a rhombus can be a square since a square has two pair of parallel line and all the lines are of equal length, but a rhombus is not a square since a square must have right angles but a rhombus does not.

2. Statement D is correct.  Statement A is correct because both squares and parallelograms have four sides and a quadrilaterals has four sides.  Statement B is correct because a square is a parallelogram.  Since they both share the same properties that lines of the opposite sides are parallel, this statement is correct as well.   Statement C is correct because a trapezoid has one pair of parallel lines.  Since all 3 statements are correct, statement D is correct.

4. I do not agree with this statement as a parallelogram does not have to have for sides of equal lengths but a square has.  Also, a parallelogram may may not have all angles to be 90 degrees but in a square, all the angles are right angles. 

Question 1, 2 and 4 by Azeem Arshad Vasanwala

Question 1:

Question 2:

Question 4:

Thanks,

Azeem Arshad Vasanwala

(Class S1 - 03)

School of Science and Technology, Singapore

** CONFIDENTIALITY: If this email has been sent to you by mistake, please notify the sender and delete it immediately. As it may contain confidential information, the retention or dissemination of its contents may be an offence under the Official Secrets Act.

Question 1, 2 and 4 by Lucas Chia

Question 1, 2 and 4 by Lucas Chia

Question 1, 2 and 5 - Ho Yan Jin

Question 1, 2 and 5 - Ho Yan Jin

Questions 1,4 and 2 by Brandon Yeo

Question 1:

'A square is a rhombus but a rhombus is not a square'.

-A square has 4 equal sides and all opposite sides are parallel to each other, just like a rhombus.

-However, A rhombus has 

4 equal sides and all opposite sides are parallel to each other BUT does not have 4 right angles like a square. 

That is why a 

square is a rhombus but a rhombus is not a square.

Question 2:

D ) All the above

Question 4:

'All parallelograms are squares?' Do you agree with this statement?

No. The parallelogram's two pairs of opposites sides are not equal to each other.  When a square is slanted to the side it forms a rhombus instead of a parallelogram.

Trapezium and Kite

Questions 2, 3 & 4 by Nur Nadiah [Final]

Question 1,2,5 by Nadiah

Trapezoid & Parallelogram- Casandra Ong

Trapezoid & Parallelogram

Question 2, 3 & 4 by Nur Nadiah

kein shuen (both activity 2&3)

Question 1,2,5 by Nadiah

NEW Q1,2,5- Casandra Ong

Question 1, 2, and 4 by Goh Jin Hao

Question 1
The statement is not justified as although all the sides of both the rhombus and the square are equal, the corners of a square must be exactly 90 degrees so a rhombus cannot be a square which makes the statement unjustified.

Question 2
D ) All of the above
Both squares and parallelograms are quadrilaterals as they both have four sides.
Both the squares' and the parallelograms' opposite sides are parallel as the other two sides are of equal length.
A trapezoid has a pair of parallel sides as it is made out of a rectangle and a triangle and a rectangle has a pair of parallel sides.

Question 4
I do not agree with the statement. A parallelogram cannot be a square as all corners of the square must be 90 degrees and all of the sides must be equal.

Casandra Ong- Q1,2,5

Question 1, 2, 4 by Justin Ong

Question 1
"A square is a rhombus but a rhombus is not a square".

A rhombus is a quadrilateral whose four sides all have the
same length.
However, a square has four equal sides and four equal right
angles at the corners.
If a square is turned into a rhombus, it loses its four
right angle corners. Therefore, it cannot be counted as a
square. However, if the rhombus is turned into a square, it
retains its length and still counts as a rhombus.

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

A ) A square and a parallelogram are quadrilaterals.
Correct, because a square has 4 sides and 4 corners, and so
does a parallelogram.

B ) Opposite sides of a square and a parallelogram are
parallel.
For the square, it is correct as 4 corners are supposed to
be 90°.
For the parallelogram, it is also correct because even
though it does not have 4 equal angles, it still has 2 sets
of parallel lines.
href='http://img269.imageshack.us/i/trapezium.png/'><img
src='http://img269.imageshack.us/img269/5461/trapezium.png'
border='0'/></a>

C ) A trapezoid has one pair of parallel sides.
Correct as well, refer to diagram below.
href='http://img237.imageshack.us/i/squareparallelogram.png/'><img
src='http://img237.imageshack.us/img237/9694/squareparallelogram.png'
border='0'/></a>

D ) All the above

Question 4
"All parallelograms are squares?" Do you agree with this
statement?

I do not agree with this statement. A parallelogram refers
to a figure with 2 sets of parallel lines. However, a square
has 2 sets of parallel lines, 4 sides of the same length AND
must have the corner angles equal to 90° each. A rectangle
has 2 sets of parallel lines. However, it does not have 4
equal sides. Another example is a rhombus. A rhombus has 2
sets of parallel lines. However, it again does not have 4
corner angles, making it not qualify as a square.

Re: Question 4 by Looi Wei Chern

These two are both parallelograms but only Fig. 35 is a square but Fig. 34 is not.

On Fri, Aug 13, 2010 at 2:40 PM, Wei Chern ,Jeremy <weichern97@gmail.com> wrote:

My answer would be No as Parallelograms are squares only if all four sides are of the same length and all interior angles are 90^o.

Question 4 by Looi Wei Chern

My answer would be No as Parallelograms are squares only if all four sides are of the same length and all interior angles are 90^o.

Question 1 by Looi Wei Chern

 
Question for discussion
 
 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'.
 

A rhombus is a quadrilateral whose four sides are all congruent.    

A square is a quadrilateral whose four sides are all congruent and whose angles are all right angles.  

 In other words, a square is a rhombus that is also a rectangle.  

The above two quadrilateral are both rhombuses but the four sided, with vertex of 90 degrees is a square, not a rhombus:

Question 3 by Looi Wei Chern

 
Question for discussion
 
 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'.
 

A rhombus is a quadrilateral whose four sides are all congruent.    

A square is a quadrilateral whose four sides are all congruent and whose angles are all right angles.  

In other words, a square is a rhombus that is also a rectangle.  

The above two quadrilateral are both rhombuses but the four sided, with vertex of 90 degrees is a square, not a rhombus:

Question 3 by Looi Wei Chern

Question for discussion

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'.

A rhombus is a quadrilateral whose four sides are all congruent    A square is a quadrilateral whose four sides are all congruent and whose angles are all right angles  In other words,    A square is a rhombus that is also a rectangle  These are both rhombuses:    +--------+   |        |        +--------+   |        |       /        /   |        |      /        /   |        |     /        /   +--------+    +--------+  But only this is a square:    +--------+   |        |   |        |   |        |   |        |   +--------+ 

Question 2 by Looi Wei Chern (Images)

1st Image is for Evidence for A

2nd Image is for Evidence for B

3rd Image is for Evidence for C

Question 2 by Looi Wei Chern

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

D is Correct.

Evidence for A) A square and a parallelogram are both quadrilaterals as the definition of a quadrilateral is a polygon with four sides and four vertices or corners. Thus, the following are all quadrilaterals, which includes squares and parallelograms:

                                       

 

Evidence for B) Parallel lines are two lines lying in the same plane but never meeting no matter how far extended as it can be clearer seen below in the pictures of the square and the parallelogram.

                                       

Evidence for C) Like I said above, the definition of parallel lines is two lines lying in the same plane but never meeting no matter how far extended thus from the diagram below, only DC and AB is parallel to one another but AD and BC are not parallel to one another thus the definition of a trapezoid simply means a four-sided figure with one pair of parallel sides.

103's CLASSIFICATION OF QUADRILATERALS

Summary of what we have learnt - T3 W03 (12 July - 16 July)

• When we are factorising expressions by grouping we need to take care of the signs ... for instance ... 1 + 2t - k - 2kt = (1+2t) - k(1 + 2t) ... the LEFT hand term (1 + 2t) is the same as 1(1 + 2t) ... hence, when we factorise the expression, there is a COMMON factor of (1+2t) ... leaving us with ... (1+2t)(1 - k) ... some of you have forgotten the (1) in the term (1 - k) !!!
• When we work with algebraic fractions, make sure that you FACTORISE both the NUMERATOR and DENOMINATOR !!!

Term 3 Week 2 - 5th to 9th July 2010

Most of the class can understand what was being taught today, but some still have doubts about the equation "undefined" .

QUESTION -

Why 2/0 is undefined when 2÷0=0?

Well, firstly, note that 2/0 is the SAME as 2÷0 & hence, it would still be UNDEFINED!

Anyway, if 2 ÷ 0 is possible, we will be able to see that

letting 2 ÷ 0 = x

0 ( 2 ÷ 0) = 0( x ) .................................................................... convince yourself that 3( x ÷ 3) = x

2 = 0

This is a CONTRADICTION and hence we can see that 2 / 0 is not POSSIBLE ...

Which means that if we have 1÷x, this would be UNDEFINED if x = 0.

QUESTION -

If we have 1÷(x+1), for what value of x would render this undefinable?

For 1÷(x+1), it will be UNDEFINABLE when (x + 1) = 0 ... that means that solving it would make x = -1 ... hence, when x = -1, the function will be undefinable ... Consider the GRAPH of the function y = 1÷(x+1) (using grapher), what do you notice at x = -1?

What about 1÷(2x-3)? Hence, what would be general concept be that would make the expression 1÷(ax+b), undefinable? Does this change if the numerator was something else?

Hence, for functions of the form y = 1÷(ax+b), the DENOMINATOR can never be equal to ZERO ... and this will happen when x = -b ÷ a ... CHECK TO SEE THAT YOU KNOW WHY???

QUESTION -

In order to solve 2x - 5 = 3, we need to find the value of x, that SATISFIES (we discussed this in class today) the equation. Hence, we will have to leave x alone on the left-hand side. That means, we need to remove the constant (-5) and the coefficient (2) ...

How can this be done? I believe that Zhi Qi has a slight issue with this.

Well, when 2x - 5 = 3, we need to REMOVE the 5 on the LEFT ... hence,

2x - 5 + 5 = 3 + 5 ... NOTE that we have ADDED 5 on both sides ... leaving us with 2x = 8 ...

removing the coefficient 2 requires a DIVISION

2x ÷ 2 = 8 ÷ 2 ... leaving us with x = 4 ...

Hence, the EQUATION is SATISFIED when x = 4

CHECKING ... 2 (4 ) - 3 = 5 when is the same as the VALUE as provided in the question!

QUESTION -

When you're trying to SOLVE equations involving algebraic FRACTIONS, what is the general principal involved for questions of this form?

The general principal is to ensure that only 1 fraction is present on both sides of the equal sign!

Welcome Back for a New Semester

Dear students,

Welcome back to school after your June Holidays.

Let us start the new semester with this task.

Under the comment section, post up

1) 1 interesting thing you have done / see during the June Holidays.

2) 1 interesting knowledge that you have learn during the June Holidays.

3) The expectation that you are going to set for yourself in the learning of Mathematics.

Chapter 9.2 : Average Rate (Lesson 1)

Rate is a ratio between two quantities with different units of measurement.

Rate allows us to express a quantity as a proportion of another quantity thus enable us to make comparison between different quantity.

Examples of rate being used in our daily life are:
1) Speed of a car, where the distance is measured against time (Kilometer per Hour or Meter per Second)

2) Buying of food and drink, where the price is measured against the weight or volume (Dollars per Kilograms or Dollars per Litres)

3) Frequency of Buses (Number of buses in operation per Hour)

4) Heart Rate (Number of beat per Minute)

The examples of rate in our daily life in countless.....

Thus give 2 examples of the use of Rate in your life and briefly describe how you can make use of these information to help you make better decisions in your life.

Please also refer to your Textbook 1B from Pg 9 to 11 and your Ace - Learning Portal for more materials and examples.

Chapter 5.1 : Like Terms and Unlike Terms

Dear 103s,

We are back into our study of Algebra....

Like English & Chinese, Mathematics is another form of communication between people and Algebra is part of this big family....

Thus, let us now get to find out more about the Algebraic Language...

The Algebraic Language

How to Understand the Vocabulary Of Algebra

Chapter 9.1 : Ratio

The Golden Ratio in Human Body

The Golden Ratio in Architecture

Chapter 16 : Data Handling Lesson 4

Dear 103s,

Welcome back from Common Test.

We are still in our study of Statistics.

Often we hear others compare a group of data using the MEAN, the MEDIAN and the MODE.

But what exactly is MEAN, MEDIAN, MODE?

Do an online search to find out about the meaning of MEAN, MEDIAN and MODE.
Post your findings under the Comment Section.
Please also include in examples on how you determine the MEAN, MEDIAN and MODE of a data set.