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# Add Math Form 4 Question

Submitted By asyrafzaman
Words 2371
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MATHEMATICS
MODULE 1

FUNCTIONS
Organized by
Jabatan Pelajaran Pulau Pinang 2006

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CHAPTER 1 : FUNCTIONS
Contents
1.1 Concept map

Page
2

1.2 Determine domain , codomain , object, image and range of relation

3

1.3 Classifying the types of relations

3

2.1 Recognize functions as a special relation.
2.2 Expressing functions using function notation. 2.3 Determine domain , object , image and range 4-5

3.0 Composite Functions

6 -9

4.0 SPM Questions

9 – 10

5.0 Assessment test

11 – 12

13 – 14

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CONCEPT MAP

FUNCTIONS

Relations

Object images ………….,,
……………
……………

Functions

Function
Notation

Type of relation

y
Or
………………

Composite
Functions

Inverse
Functions

f: x

One to one Many to one ………..

fg ( x ) = …………….

Object

f(x)=y
 ………………

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1.1 Functions
Express the relation between the following pairs of sets in the form of arrow diagram, ordered pair and graph.
Arrow diagram
Ordered pair
Graph
a ) Set A =
 Kelantan, Perak ,
Selangor 
Set B =  Shah Alam
, Kota Bharu ,Ipoh 
Relation: ‘ City of the state in Malaysia ‘ b )Set A
=  triangle,rectangle, pentagon 
Set B =  3,4,5 
Relation : ‘ Number of
Sides’

1.2 Determine domain , codomain , object, image and range of relation.
List down the domain , codomain , objects , images and the range of the following relation
.

3

9

2

5

1

4

-2

3

-3

1

Set P

Diagram 1

Set Q

Domain =  ……………………………………… 
Codomain =  ……………………………………… 
Object
=……………………
Image
=……………………
Range
=…………………...

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1.3 Classifying the types of relations
State the type of the following relations
a)

x
2

x

3

4

2

9

16

4

x
X

4

2

-2
-3

36

6

……………………………………………..
c)

x

b)

x

X2

………………………………………….
Type of number

d)
4
9

3
4

2
-3

Prime
Even

-3

9

……………………………………………..

……………………………………………

2.0 Functions
2.1 Recognize functions as a special relation.
2.2 Expressing functions using function notation.
2.3 Determine domain , object , image and range
2.1 Identify each of the following relations is a function or not.

a)

A

B

A

B

b)

p

1

q

2

r

3

A

B

c) p a

a

p

q

b

b

q

r

c

c

r

d

……………………………

……………………………

……………………………

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2.2 Express each of the following functions using function notation.

a)

A
2

b)

B

A

B

4

2

3

6

4

8

c)

B

4

1

5

3

9

2

7

4

Function notation f : x  …………….. or f ( x ) = ……………

A

16

3

9

Function notation g : x  …………….. or g ( x ) = ……………

Function notation h : x  …………….. or h ( x ) = ……………

2.3 a)Find the image for each of the following functions.
( i ) f : x  2x + 9

( ii ) f : x 

f (5 ) =

5x  3
2

x
+6
5 find h ( -2 )

iii) h : x 

f (-3 ) =

…………………………

…………………………

…………………………..

b ) Find the object for each of the following functions. i )f : x  2x – 3 , find the object when the image is 5.

2x  8
, find the
3
object when the image is 3.

ii )f : x 

6
– 7 , find the x object when the image is -5.

i )f : x 

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c ) Find the value of x for each of the following function.
8
2x  1 for which g (x) = 4

ii )

i ) f ( x ) = 2x + 7 for which f ( x ) = 3

g(x)=

iii)

x 3
2
for which h ( x ) = x

h(x) =

3.0 Composite Functions

g

f

a

b

c

fg

f: a  b g: b  c gf : a  c

3.1 ( a ) Find the value for each of the following composite functions

i ) f( x ) = x + 2 and g ( x) = 5x + 3 find fg ( 2 ) =

ii ) g ( x ) = 2 +5x

iii) f(x) = 3x+

find g2(4)
=

1 and g ( x )
2

1
.Find fg ( 1 ) x 1

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( b ) Find the following composite function.

( i ) f( x ) = 2x + 3 g( x ) = 1 – x fg ( x ) =

( ii ) f( x ) = 2x + 3 g( x ) = 2 + 5 x2 gf ( x ) =

( iii ) f ( x ) = 1 g(x)=

x
2

4 x fg ( x ) =

( c) Find the value of x for each of the following composite function.
3
x g ( x ) = 2x + 1

i) f ( x ) =

fg ( x ) = 5

ii )f( x ) = 2x + 4

iii) f ( x ) = 1 -

g ( x ) = x -2

g (x ) =

fg ( x ) = 2

4 x fg ( x ) = -1

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x
,
2

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( d ) Solve i )Given the function f: x  4x + k , g : x x – 2 fg : x  mx + 8

ii )Given the function f: x  9 – 2x , g : x  ax + b and fg: x  1 – 6x

iii)Given the function f: x  2x – 1 , g : x  4x and gf : x  ax + b

Find the value of k and m

Find the value of a and b.

Find the value of a and b

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4.0 SPM QUESTIONS
SPM 2004 ( paper 1, question no 1 )
1. Diagram 1 shows the relation between set P and set Q

 w

d

x

e

y

f

z
Set P

Set Q
Diagram 1

State
( a ) the range of the relation,
( b ) the type of relation.

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[2 marks]

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SPM 2004 ( paper 1, question no 3)
1. Given the function h ( x ) =

6
, x  0 and the composite function hg ( x ) = 3x , find x (a)g(x)
( b ) the value of x when gh ( x ) = 5.

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[ 4 marks]

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SPM 2005 Question
1. In Diagram 1 , the function h maps x to y and the function g maps y to z. x y

h

g

z

8
5
2

Diagram 1

Determine
( a ) gh ( 2 )

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5.0 Assessment( 30 minutes)
1)
P

a

q

b

r

c

s
A
Diagram 1

B

The diagram above shows the relation between set A and set B. State
a) the type of relation
b) the range of relation

2) Given that f : x  2x + 7 find the object when image is 3.

3) Given that f ( x ) = 10 – kx and f ( 2 ) = 4 ( k constant ) . Find the value of k.

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4) Given that f ( x ) = 4x -1 and g ( x ) = 2x + 3 .
Find
i ) fg ( x ) ii ) fg ( - 2 )

5 ) Given the function f : x  px + 2 and g : x  qx + 3 . If the composite function fg is such that fg ( x ) = 8x + 8 , find the values of p and q.

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1.1
Arrow diagram

Ordered pair
KB

Kel

SA

Sel

(kel, kb), (Sel,
S.Alam), (
Perak,Ipoh) 

Ipoh

Per

A

Graph
KB
SA
Ipoh

B

Kel

3

Tri

(Tri, 3), (rec,
4), ( Pen,5) 

3
4

rec

Per

5
4

5

Pen

Sel

3
A

B

Pen

rec

Tri

1.3

Domain =  -3,-2,1,2,3  , codomain =  1,3,4,5,9  , Object =1,2,3,-2,-3
Image = 1,3,4,5,9 , Range =  1,4,9 
( a ) one to one ( b ) One to many( c ) many to one( d ) many to many

2.1

( a ) function

2.2

(a) 2x (b) x2 (c) 2x + 3

2.3

a) ( I ) 19

(ii) -6 (iii ) 3

3.1

a) ( i) 15

(ii ) 102

b) (i) 5 – 2x

(ii) 20x2+60x+47

1.2

( b ) Not function

1
(ii) 1
5
d) (i) k=16,m=4

c) (i) -

4.0 SPM QUESTIONS
SPM 2004( P1,Q1) a)
SPM 2004( P1,Q3)
SPM 2005( P1,Q1)

( c ) function

b) (i) 2 (ii)

1
1
(iii) 3 c) (i) -2 (ii)
(iii) -3
2
2

( iii) 2
(iii) 1-

2 x (iii) 1
(ii) a=3,b=4

(iii) a=8,b= - 4

range =  x,y

(a) g(x) =

2
, x 0
3

b )many to one

( b) x = 15

8

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5.0 Assessment( 30 minutes)
1)

( a)many to many

2)

(i) 8x+11
(ii) -5
1
p = , q = 16
2

5)

p,q,r

k=3

4)

-2

3)

( b ) range =

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MATHEMATICS
MODULE 2

FUNCTIONS

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CHAPTER 1 : FUNCTIONS
Contents

Page

1.0 Inverse Functions ( concept map )

2–4

2.0 Absolute Function

4-6

3.0 SPM Questions

7–8

4.0 Assessment ( 30 minutes )

9 – 10

11 – 13

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CHAPTER 1 : FUNCTIONS

FUNCTIONS
Ordered pairs
Arrow diagram
Relations

Functions

graph

Object images Domain
Codomain ,
Range

Function
Notation

Type of relation

f: x

y

Or f(x) = y

One to one Many to one image

Inverse
Functions

Composite
Functions

fg ( x ) = f [ g(x) ]

Object

One to many Many to many

f(x)=y
 f-1( y ) = x

1.0 Inverse Functions
1.1 Determine the object by inverse mapping
1.2 Determine the inverse functions a ) Find the inverse function of each of the following functions. x ii) f ( x ) =
i)f(x)=x+3
5

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iii ) f ( x ) =

3x  1
2

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iv ) f ( x ) = 7 – 5x

v)f(x)=

3 x
4

vi ) f( x ) =

5  4x
3

b ) Find the inverse function of each of the following functions in terms of p and q

i ) f ( x ) = px - q

ii ) f ( x ) =

x p q http://mathsmozac.blogspot.com
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iii) f ( x ) = px +

1 q http://sahatmozac.blogspot.com

c ) Given the function f : x 

x2
, x  1 and g(x) =2x -6 , find f-1 g . x 1

d ) Inverse function f is define by f-1 : x 

x 5
1
, x  . Find f ( 2 )
2x 1
2

2.0 Absolute Function
1. Sketch the graph of each of the following functions

a)f(x)=x

b)f(x)=x+1

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c) f (x ) = | x |

d)f(x)=|x+1|

e ) f (x) = |x| + 1

f)f(x)=|x|-1

2.Sketch the graph of each the following functions and state the corresponding range.
a) f : x  2x – 3 for 0  x  4

b) f : x  |2x – 3| for 0  x  4

f(x)

f(x)

Range :……………………………….

Range : …………………………………………

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c) f : x  | 5 – 2x |for -1  x  4

d) f : x  |9 – 2x| for 0  x  6

f(x)

Range :……………………………….

Range :……………………………….

e) f : x  |2x| – 1 for -1  x  3

f ) f : x  | 3x | for -2  x  2

Range :……………………………….

Range :……………………………….

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3.0 SPM QUESTIONS
SPM 2004 Question.
1. Given the functions h : x  4x + m and h-1 : x  2kx +

5
, where m and k are
8

constants, find the value of m and of k.
[ 3 marks]

SPM 2005 ( Paper 1, Question 1 )
2. In Diagram 1 , the function h maps x to y and the function g maps y to z. x y

h

g

z

8
5
2

Diagram 1

Determine
( a ) h-1 ( 5 )

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SPM 2005 ( Paper 1, Question 2 )
1.The function w is defined as w ( x ) =
( a ) w-1 ( x ),
( b ) w-1 ( 4 ).

5
, x  2.
2 x

[ 3 marks]

SPM 2005 ( Paper 1, Question 3 )
1.The following information refers to the functions h and g.

h: x g: x

 2x – 3
 4x - 1

Find gh-1 ( x ).

[ 3 marks]

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4.0 Assessment ( 30 minutes )
1.A function f is defined by f: x  6 Find
a)f(x)

1 x 2

b ) f-1 ( 5 )

2. Inverse function f is defined by f-1 : x 

5  4x
3

find f ( 2 ).

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3.Given the function f : x  2x - m and inverse function f-1 : x  nx +

7
3

Find the value of m and n.

4. Sketch the graph of the function f ( x ) = |2x – 5 | for 0  x  6. Hence , state the corresponding range.

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1.1

2x  1
7x
5  3x
(iv)
(v) 3 – 4x (vi)
3
5
4
xq x 1
(b)
(ii ) xq + p (iii) p p pq
2x  4
7
( c)
, x
7  2x
2
( d ) -1

( a )( i) x- 3 ( ii) 5x (iii)

Absolute function f(x) a)

f(x)

b)

_1
|
-1

x

f(x)

c)

x

f(x)

d)

_1
|
-1

x

f(x)

e)

x

f(x)

f)

_1
|
-1

|
-1

x

x

|
1

_ -1

|
1

|
2 x

2.
a)

f(x)

b)

_8

_

.

f(x)
_5

|
1

|
2 x

-3

 3  f ( x)  8

0  f(x)  5

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c)

f(x)
_7

f(x)

d)

_9

|
2

|
4

|
3 x

0  f(x)  7

0  f(x)  9

f(x)
_5

e)

|
5 x

f(x)
_6

f)

_1
|
-1

|
_ -1 1

|
3x

-1  f(x)  5

|
-2

|
2

0  f(x)  6

SPM 2004 ( P1,Q2)
1
5
K= , m=8
2
SPM 2005 ( P1,Q2)
(a) 2
SPM 2005 ( P1,Q2)
2x  5
3
(a)
, x 0 ( b ) x 4
SPM 2005 ( P1,Q3)
2x+ 5
Assessment ( 30 minutes )
1)( a) 12 – 2x ( b ) 8
1
2) 4
1
3) n = , m=7
2
4)

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x

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a)

f(x)
_7
_5

|
2

|
3

|
6

x

The corresponding range of f(x ) = 0  f(x)  7

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...teaching GCSE maths re-sits has changed during the course. My thinking about teaching GCSE maths re-sits has changed in that I’m more focused on improving the learning of my students. Simplistically, before it may have been a case of “What do they need to know?” and then relaying what the student needs to know for that particular subject, in a clear and concise manner, which may have been “got” by most students, but then they would have forgot the method shortly afterwards when it came to a formative or a summative assessment. Now I’m more determined to help students learn in a way that is going to be enjoyable to them and help them remember what they need to for the exam. This will involve doing more kinaesthetic and visual activities as a lot of learners learn by doing. Realistic Mathematics Education (RME)  The course has introduced me to Hodder Education’s range of books called ‘Making Sense of Maths’ for KS3 and KS4 (http://www.hoddereducation.co.uk/makingsenseofmaths). Mr Gough, a maths teacher and one of the authors of the aforementioned book(s) states the following: “My experience of teaching GCSE Foundation Tier is that by the time they get to KS4 they’ve already covered most of the content and they’re having the same content repeated in the same way that they found difficult in the first place so my experience of using this approach is that it seemed very different to them and it reinvigorated their interest in maths and they were very positive about......

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#### Economics

...national level in the United States economy. It is recommended that students take ECON 201 before ECON 202. MATH 105 is highly| |recommended but not required. (F,W,S). | |ECON 201 - Prin: Macroeconomics | |Together with ECON 202, this course serves to introduce the student to the basic ideas and concepts of modern economic analysis, and applies| |them to current economic problems, policies and issues. The focus of this course is on macroeconomics: income and wealth, employment, and | |prices at the national level in the United States economy. It is recommended that students take ECON 201 before ECON 202. MATH 105 is highly| |recommended but not required. (F,W,S). | Together with ECON 202, this course serves to introduce the student to the basic ideas and concepts of modern economic analysis, and applies them to current economic problems, policies and issues. The focus of this course is on macroeconomics: income and wealth, employment, and prices at the national level in the United States economy. It is recommended that students take ECON 201 before ECON 202. MATH 105 is highly recommended but not required. (F,W,S). 3.000 Credit hours It is the sole......

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#### Trignometry

...Goal: Students will use the trigonometric ratios (Sine, Cosine, and Tangent) to measure the real life height of various objects. How: Students will work with one partner, however they both will have to turn in individual projects. Step One Create a clinometer, by following these steps. ▪ Cut out the photocopied protractor. ▪ Glue it to your piece of cardboard. ▪ Cut out the cardboard so that it is the shape of your protractor ▪ Cut a 4-inch piece of string, tape it to the middle of the protractor by your mark. It is important that you tape your string before you tape your straw. ▪ Tape your straw to the top flat end of your protractor. ▪ At the end of the string, tape a couple pennies to it. Your clinometer should look like this: [pic] Step Two Measure the height of the wall using sine, cosine, tangent, and your clinometer. ▪ In order to measure the height of the wall, complete the Clinometer worksheet that is attached to this project sheet. ▪ After you measure the height of the wall, you will measure one more object at school, and one more at home for homework. Step Three Draw a story/cartoon that shows how you used trigonometry to solve the problem......

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#### Tessellations: Mathematical Art

...nature. Tessellations: Mathematical Art What is a term used for the tiling a surface without gaps or overlaps? The term is Tessellation. The Math Forum states that “ a tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps” (“What is a Tessellation?”, n.d) Early cultures used tessellations to cover the floors and ceilings of buildings, many of its artistic elements can be found in many early cultures (Hoopes-Myers, 2010). Tessellations are also found in the nature. A perfect example of nature’s tessellation is the honeycomb of the honeybee; there are no gaps or overlaps in its hexagonal shapes. In Ireland, a volcanic episode created tessellations in the landscape of The Giant’s Causeway (“Giant’s Causeway” n.d.). Artists like M. C Escher use tessellations to create fascinating works of art. In his works Escher concentrated on tessellations and repeated forms. Mathematicians and scientists embraced Escher’s works because they involved the concepts of “geometry, logic, space and infinity” (M.C. Escher Biography, n.d.). One doesn’t have to be a mathematician to tessellate but knowing how shapes will fit together helps in creating beautiful tiled images. Tessellations had functionality in ancient cultures, in mathematics, and in today’s real world it is considered a form of art. Tessellation History Historically the art of tessellation can be traced back as far as 4000 BC. The Sumerians decorated their homes......

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#### Management

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#### Research Methodology

...methods 1. What is Educational Research? (uploaded 7.17.09) 2. Writing Research Questions (uploaded 7.20.09) 3. Experimental Design (uploaded 7.20.09) ------------------------------------------------- Experimental Design The basic idea of experimental design involves formulating a question and hypothesis, testing the question, and analyzing data. Though the research designs available to educational researchers vary considerably, the experimental design provides a basic model for comparison as we learn new designs and techniques for conducting research. Note: This review is similar to the overview of significance testing, so you will see some of the introductory material on the scientific method repeated in both places. Part I: The Scientific Method We start with familiar territory, the scientific method. To illustrate, we’ll look at a basic research question: How does one thing (variable A) affect another (variable B)? You may have seen variable A referred to as the treatment, or independent, variable, and variable B as the outcome, or dependant, variable. Let’s call variable A parental involvement and lets call variable B a test score. The traditional way to test this question involves: Step 1. Develop a research question. Step 2. Find previous research to support, refute, or suggest ways of testing the question. Step 3. Construct a hypothesis by revising your research question: Hypothesis | Summary | Type | H1: A = B | There is no relationship......

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#### Greek Influence

...Gretchel M. Quinones HUMA 101 WORK SHOP 4 Essay Ricardo Serano Greek science and math the influence: Development of Science Long time ago, people lacked knowledge on why certain things happened. Without scientific answers, like we have today, the Ancient Greeks created their own answers about the world and an individual’s place in it. By doing the research for this essay I had learn a lot of the Greeks contribution in science and math methods. Science in Ancient Greece was based on logical thinking and mathematics. It was also based on technology and everyday life. The arts in Ancient Greece were sculptors and painters. The Greeks wanted to know more about the world, the heavens and themselves. People studied about the sky, sun, moon, and the planets. The Greeks found that the earth was round. Many important people contributed to Greek scientific thought and discoveries. Biology, a very vast and interesting topic, was studied by Hippocrates, Aristotle, Theophrastus, Dioscorides, Pliny, and Galen. These men were among the main researchers of Greek biology who contributed many ideas, theories, and discoveries to science. Some of their discoveries were observations, descriptions, and classifications of the various forms of plants and animal life. Other discussions in biology were natural selection and zoology. All living things were the basic concern of biology. Greek biologists were interested in how living things began, how they developed, how they functioned, and...

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#### Math Helps

...assemble a cake or build a house is important. The same holds true for evaluating any expression in math. We call this the Order of Operations. In your own words, explain the Order of Operations. • Give an example of an expression to fit this situation in math and an example in real life. • Please share any trick or mnemonic device to help you recall this order or how to use it. A mnemonic device is a memory trick such as using “The quick brown fox jumped over the lazy dog” to learn all the keys on the keyboard. Professor and Class-Good Morning! The Order of Operations- A set of rules for the order in which to solve mathematical problems. The order goes: * and /, + and -. If there are parentheses then work inside them first. Example: Correct: 1+2*3=7. Not Correct: 1+2*3=9 Rule 1: First perform any calculations inside parentheses. Rule 2: Next perform all multiplications and divisions, working from left to right. Rule 3: Lastly, perform all additions and subtractions, working from left to right. (http://www.mathgoodies.com/lessons/vol7/order_operations.html) The Order of Operations is very much like baking a cake, or driving a car, you cant change much in order to have the same outcome. The Mnemonic Device that i use it this : Please Excuse My Dear Aunt Sally-- Meaning.... Parentheses, Exponents, Multiplication, Division, Addition, Subtraction Unit 4 Linear relationships between two quantities can be described by an equation or a graph.......

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