Algebra: Revision (Algebraic Fraction)

posted by Mr Johari

ALGEBRAIC FRACTION

Revision:

Solving Equa

Video 1: Solving Multi-Step Equations with Fractions


Video 2: Solving Equations with Fractions


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Linear Graph (worksheet 12B)

posted by Mr Johari

LINEAR GRAPH

Solution to WORKSHEET 12B − GRADIENTS & LINEAR EQUATIONS


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linear Graph resource

posted by Mr Johari

LINEAR GRAPH

reference of Linear Equation. (click here)

• Focuses on
• What is linear equation?
• Plotting and Table of Values
• Slope / Gradient of a Line
• Slope and y-intercept (how to form a Linear Equation)

linear Graph worksheet (Equation) solution

Linear Graph (Equation of Line)

posted by Mr Johari

Worksheet focuses on the following concept:

• gradient of line
• intersection with both axes (i.e. x-intercept and y-intercept)
• sketch of line

Linear Graph worksheet (Gradient) solution

Linear Graph (Gradient of a Line)

posted by Mr Johari

reference on Linear Equation, Gradient, intercepts.

Slideshare on Linear Equation

Supplementary Worksheet on Linear Graph

This worksheet focuses on the concept of gradient

Please refer to hard copy of the worksheet given to you earlier.

Level Test

Level Test Term 3

posted by Mr Johari

Level Test

As mentioned at the beginning of the term, the Mathematics level test will be conducted in week 10.

Detail will be as follows:

Duration: 40 minutes

Topics tested:

• Introduction to Algebra [chapter 4]
• Algebraic Manipulation [chapter 5]
• Simple Equations in one unknown [chapter 6]
• Coordinates and Linear Graph (sketching of graph, concept of equation of line and gradient) - [chapter 12]

linear Graph part 2

Linear Graph (Chapter 12)

posted by Mr Johari

source: http://www.math.com/school

This is a supplementary note that focuses on linear equation:

Part 2: Graphing Linear Equation (Exercise)

Complete the 2 exercises, state your answers as comments. Where possible justify your answers with simple working.

Question 1: focuses on the linear graph of the form y = mx + c
Question 2: focuses on the characteristics of the linear graph.

[Coordinate, gradient/slope, intercept, equation]

Linear Graph part 1

Linear Graph (Chapter 12)

posted by Mr Johari

source: http://www.math.com/school

This is a supplementary note that focuses on plotting of a linear equation:

Part 1: Graphing Linear Equation

Method 1: Given linear equation, plot the graph.

Technique: Using plotting points and coordinates. (i.e. identify any points for x and find corresponding values of y for plotting)

Method 2: Given linear equation, plot the graph.

Technique: Using gradient and y-intercept

Form the general linear equation ie. y=mx+c, where m is the gradient and c is the y-intercept.

Exercise: (for self practice - answer provided)

Given a linear line, find the equation of the line.

Questions 1, 2, 3, 4 and 5 by Victor Ang

Question 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'.
Ans: A square is a rhombus but a rhombus is not a square. A square is a special type of rhombus, it is a quadrilateral which has all four sides the same length and opposite sides are parallel while it is a polygon with all four sides closed up and each of its four internal angles are right angles. So a square is a type of rhombus just like a banana is a type of fruit.

Question 2
All the above
A square and a parallelogram are quadrilaterals. They both have four sides.
Opposite sides of a square and a parallelogram are parallel. They have one pair of parallel side if not they are not considered trapezoid any more.
A trapezoid has one pair of parallel sides. A trapezium has only one pair of side which are parallel and the other two are not.

Question 3
It is a trapezium. The two opposite sides that are not equal are parallel and a pair of opposite angles are supplementary.

Question 4
I do not agree with the statement. Parallelograms are like a fruit and squares are banana. A banana is a fruit but a fruit is not part of that banana, just like a square is a parallelogram but a parallelogram is not a square.

Question 5
The reason is that BF and ED are of equal length and are parallel.

Activity 3 By Johanan Teo

Question 1:

A square is a rhombus but a rhombus is not a square

A square has 4 equal sides so it can be called a rhombus. However, a rhombus does not have four right angles thus it cannot be called 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

The answer I have chose is D. (All of the sentences are correct)

A square and a parallelogram are in the group called quadrilaterals which has four sides and their inside angles add up to 360 degrees. 

Opposite sides of a square and a parallelogram are parallel.

And a trapezoid has one pair of parallel sides. ( In the picture )

Question 4 :

All parallelograms are square. Do you agree with this statement?

No. I do not agree with this statement as a square has to have four right angles but a parallelogram may or may not have four right angles and thus not all parallelograms are square.

Question 1:     
Read through the conversation between a mentor and a student. 
Mentor:     Does anyone recall or know what we call it when 2 lines run side-by-side and never cross?
Student:    Yes. Lines like that are called parallel lines.
Mentor:     Great! We've already learned that quadrilaterals have how many sides? 
Student:    Four.
Mentor:     That's right and we call quadrilaterals with parallel sides parallelograms.
Student:    But, how can all the sides be parallel if a quadrilateral is a polygon and is all closed off? 
Mentor:      Great thinking! I guess what I should have said is that a parallelogram has two pairs of opposite sides that are parallel, like this:

Student:     Oh, so the top is parallel to the bottom and the sides are parallel to each other. I understand now!

Mentor:      Good. Now I want to tell you about a special kind of parallelogram. It's called a rhombus. A rhombus is a parallelogram, but all four sides have the same length.
Student:     So a rhombus is a type of parallelogram just like a banana is a type of fruit.
Mentor:      Right, we should not say that all parallelograms are rhombi, just like we don't say that all fruits are bananas.

   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'.

I believe that this statement is true. A rhombus is a figure that has opposite sides that are parallel to each other. A square happens to fall in that category. However a square must have all internal angles at 90 degrees before it can be called a square. However, not all rhombuses have 90 degree corners, so not all rhombuses can be 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

The correct statements are: D

A is correct as both the square and the parallelogram have 4 sides, and a quadrilateral figure has 4 sides. Thus, this statement is correct. 

B is also correct as it is a fact that the opposite sides of a square and parallelogram are parallel. 

C is also correct as it is also a fact that a trapezoid has a pair of parallel sides. 

Since all the statements are correct the correct statement is D. 

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 supplementary¹. Identify this figure, and justify your answer with reasons.

This figure is a trapezium. A trapezium fits the description. It could have a pair of opposite sides that are equal in length, and it also could have another pair which are not equal in length. The other quadrilateral figures do not fit this description as all of the figure's opposite sides are equal in length, with the exception of the kite. The trapezium also has a pair of opposite angles that are supplementary. Since the trapezium fits this description the best, the figure should be the trapezium. 

Done by: Lee YuChong S1-09 (17)

Question 1,2,5 by Michelle Dapito

Question 1:

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

A square has 4 equal sides so it is considered a rhombus. However, a rhombus does not have 90º right angles so it is not considered 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 choose D.

A: A shape with four sides is called a quadrilateral, therefore, a square and a parallelogram have 4 sides so they are quadrilaterals.

B: Squares and parallelograms both have two pairs of parallel lines which makes the opposite sides of these quadrilaterals parallel.

C: For a shape to be a trapezoid, it has two have one pair of parallel lines.

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 is a parallelogram in which the two longer parallel lines are the segment of a midpoint of the line and a point of the line, so therefore, they should be equal in length. The two shorter parallel lines are segments of two points, a midpoint of the line and a point, therefore, they should be equal in length too. BFDE has two pairs of parallel lines so therefore it is a parallelogram.

Done by Michelle Dapito.

Question 1, 2, and 4 by Davina Ling

Question 1:
‘A square is a rhombus but a rhombus is not a square’.
Rhombus is a quadrilateral whose four sides all have the same length. The angles of a rhombus need not be a perfect angle.

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.
A)Squares and parallelograms all have 4 sides.
B)All four sides are of equal length, and all four angles are right angles. An equivalent condition is that opposite sides are parallel. Rhombus is a quadrilateral whose four sides all have the same length.
C)The four-sided figure with one pair of parallel sides is referred to as trapezoid
I) As a/b/c are all correct so is D.


Question 4:
‘All parallelograms are squares?’ Do you agree with this statement?
Justify your answer with example/s.


No, I do not agree with this statement. The angles on a parallelogram does not necessarily need to be 90º. Unlike a square the angle have to be exactly 90º.

Question 1,2&4 by Rayner Tan Xue Wen

Q1)

Ans: The statement: ‘A square is a rhombus but a rhombus is not a square’ is correct. A rhombus is a parallelogram with 4 equal sides but the angles inside the rhombus need not all be equal. A square is a type of rhombus that has equal angles inside of the square. So a square can be a rhombus but a rhombus can’t be a square.

Q2)

Ans: D, all of the above statements are correct. A quadrilateral is a shape with 4 sides. Both a square and parallelogram have 4 sides so they are quadrilaterals, statement A is therefore correct. A parallelogram is a quadrilateral which has two sets of parallel lines opposite each other and a square is a type of parallelogram so statement B is correct. A trapezoid is a quadrilateral with only one pair of parallel lines so statement C is correct.

Q4)

Ans: I disagree with the statement. A square is a type of parallelogram with 4 equal sides and 4 equal inner angles. A quadrilateral need not have 4 equal sides and 4 equal inner angles to be a parallelogram. So the statement is wrong.

Rayner Tan(21)

S1-09

Question 1 by Chan Jun Wei

The above statement is partially justified because:

A square is a rhombus but not all rhombuses are squares because a square's adjacent sides are perpendicular to each other, opposite sides are parallel and all sides are of equal lengths. Rhombuses' opposite sides are parallel and all sides are of equal lengths but its adjacent sides may not be perpendicular to each other.

Question 5 by Chan Jun Wei

BFDE must be a parallelogram as BC and AD are parallel to each other and are of equal lengths and E and F are their midpoints.

Question 2 by Chan Jun Wei

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

All of the above is correct. A square and a parallelogram are quadrilaterals as they have four sides. Opposite sides of a square and a parallelogram are parallel because their adjacent sides are of equal length. And the trapezoid has a pair of parallel sides because the parallel sides have different lengths.

e-Learning Math Activity 3

These are the answers and explanations to the questions 1, 2 and 4.

Q1: The statement about squares being rhombuses but rhombuses not being squares is true.

A rhombus has 4 sides of equal length, with two pairs of parallel lines. A square fulfills this rule for rhombuses. However, a in order for a quadrilateral to be a square, it must have 4 sides of equal length, two pairs of parallel lines and all angles must be 90 degrees. A rhombus  need not have angles of 90 degrees. Thus, a rhombus is not a square.

Q2: A) " A square and parallelogram are quadrilaterals " is true. A quadrilateral has 4 sides. All angles in a quadrilateral add up to 360 degrees.  A parallelogram fulfills this by having 4 sides, and opposite angles which add up to 180 degrees, and both sets of opposite angles added up is 360 degrees. A square also fulfills this by having 4 sides, and four angles each being 90 degrees, all adding up to 360 degrees.

B) " Opposite sides of a square and a parallelogram are parallel " is true. In order for angles to have parallel opposite sides, it must have adjacent angles that add up to 180 degrees. A square's adjacent angles are all 90 degrees and 90 degrees, adding up to 180 degrees. A parallelogram also has adjacent angles that add up to 180 degrees, though not 90 and 90(in degrees).

C) " A trapezium has one pair of parallel sides " is true. If a quadrilateral has one pair of parallel sides, it must have sets of adjacent angles that add up to 180 degrees. The trapezium does have adjacent angles that add up to 180 degrees, so it can have one pair of adjacent sides. But as the sets of adjacent angles are not equal in degrees, the trapezium must have only one pair. 

D) " All of the above " is true as all the above statements are true.

Thus the statement D is correct.

Q4: " All parallelograms are squares?" is false. In order to be a square, a quadrilateral has four sides, two pairs of parallel lines, all being equal in length, and all angles must be 90 degrees.   

  A parallelogram has 4 sides and is a quadrilateral. Also, it has two pairs of parallel lines. This causes adjacent angles in a parallelogram to add up to 180 degrees. However, it does not have all sides of equal length. A parallelogram also does not any angles of 90 degrees. Its adjacent angles might add up to 180 degrees like a square, but are not equal. 

  Thus, no parallelograms are squares. Though, vice versa is true.

 Tim Yap

   

Activity 3 by Jonan Ling

Activity 3

Question 1: A square is a rhombus but a rhombus is not a square.
This statement is wholly justified. A rhombus is a figure which has 4 lines of equal length and whose top is parallel to the bottom and whose side is parallel to its opposite. A square is a figure that has 4 lines of equal length, whose top is parallel to the bottom and whose side is parallel to its opposite. A square also has 4 angles of which all equal 90 degrees.
Thus, since a square fulfills all the requirements of being a rhombus, a square is also a rhombus but since a rhombus does not fulfill all the requirements of being a square, not all rhombi are squares.

Question 2:
The answer is (D) and my justification is shown below:
Quadrilaterals are figures which have 4 sides, angles and vertices. A square has 4 angles, sides and vertices. A parallelogram is a figure that has 4 sides, 4 angles and 4 vertices. Thus, squares and parallelograms are quadrilaterals.
A square has 2 pairs of parallel lines, top-bottom and side-side. A parallelogram has 2 pairs of parallel lines, to-bottom and side-side. Thus, opposite sides of squares and parallelograms are parallel.
A trapezoid is a figure with 1 pair of parallel lines and 1 pair of non-parallel lines.

Question 4:
All parallelograms are squares.

I do not agree with this statement. Squares are figures with 4 sides, 4 angles that are 90º each and 2 pairs of parallel lines. Parallelograms have 4 sides, 4 angles that do not need to be 90º each and 2 pairs of parallel lines. Thus, this statement is false.

Activity 3 eLearning - Benjamin Fheng

Question 1 

"A square is a rhombus but a rhombus is not a square"

I agree with the statement as a square has four equal sides just like a rhombus. But however a rhombus cannot be a square as a rhombus does not have angles that are all 90 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

I choose D as a square and a parallelogram are quadrilaterals which means the shape has four sides. The opposite sides of a square and parallelogram are parallel to one another and a trapezoid is a four sided figure with a pair of parallel sides.

Question 4

'All parallelograms are squares?' Do you agree with this statement?
Justify your answer with example/s.

I disagree with the statement as not ALL parallelograms are squares. Parallelograms are squares only if all four sides are of the same length and all interior angles are 90^o.

Question 1, 2, 4 by Stacey Yip

Question 1:    

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

This statement is true as every square has 4 equal sides like the rhombus, but not all rhombuses have right angles like the 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
Statement A and B is correct. A quadrilateral has four sides which a  square and parallelogram has. Both a square and a parallelogram has two pairs of parallel lines.

Question 4:
'All parallelograms are squares?' Do you agree with this statement?
Justify your answer with example/s.
I disagree that all parallelograms are squares. A square has to have right angles while not all parallelograms have a right angle. However, a square can be a parallelogram as it has 2 pairs of parallel lines. 

Questions 1,2 and 4 by Naveena Menon

1. Mentor:     Does anyone recall or know what we call it when 2 lines run side-by-side and never cross?
Student:    Yes. Lines like that are called parallel lines.
Mentor:     Great! We've already learned that quadrilaterals have how many sides? 

Student:    Four.
Mentor:     That's right and we call quadrilaterals with parallel sides parallelograms.
Student:    But, how can all the sides be parallel if a quadrilateral is a polygon and is all closed off? 
Mentor:      Great thinking! I guess what I should have said is that a parallelogram has two pairs of opposite sides that are parallel, like this:

Student:     Oh, so the top is parallel to the bottom and the sides are parallel to each other. I understand now!

Mentor:      Good. Now I want to tell you about a special kind of parallelogram. It's called a rhombus. A rhombus is a parallelogram, but all four sides have the same length.
Student:     So a rhombus is a type of parallelogram just like a banana is a type of fruit.
Mentor:      Right, we should not say that all parallelograms are rhombi, just like we don't say that all fruits are bananas.
 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’.Ans:This statement is justified because though both shapes are 4-sided polygons, the angles at the corners of a square are always 90° but the angles at the corners of a rhombus are not always at 90°.
________________________________________________________

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) A square and a parallelogram are quadrilaterals. - This statement is correct as the quadrilaterals are any 4-sided polygon and both a square and a parallelogram are 4-sided polygons.

B) Opposite sides of a square and a parallelogram are parallel. - This statement is true because opposite sides of a square and a parallelogram will never meet.

C) A trapezoid has one pair of parallel sides. -This statement is true.

Ans: Hense, D(All the above) is the answer.

___________________________________________________

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

Justify your answer with example/s.

Ans: I do not agree with the statement as the angles at the corners of a square will always be 90° but the angles at the corners of a parallelogram are not always 90°.