Transcript Document

Algebraic Operations
S3
Credit
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Factors / HCF
Common Factors
Difference of Squares
Factorising Trinomials (Quadratics)
Factor Priority
18-Jul-15
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Starter Questions
S3
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Q1.
Multiply out
(a)
a (4y – 3x)
(b)
(2x-1)(x+4)
Q2. True or false
5 1 4
1
  1
6 3 3
3
Q3. Write down all the number that divide
into 12 without leaving a remainder.
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Factors
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Using Factors
Learning Intention
1. To explain that a factor
divides into a number
without leaving a
remainder
2. To explain how to find
Highest Common Factors
18-Jul-15
Success Criteria
1. To identify factors using
factor pairs
2. Find HCF for two numbers
by comparing factors.
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Factors
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Factors
Example :
Find the factors of 56.
Numbers that divide into 56 without leaving a remainder
F56 = 1 and 56
2 and 28
4 and 14
7 and
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8
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Factors
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Highest Common Factor
Highest
Common
Factor
Largest
Same
Number
We need to write out all factor pairs in order
to find
the Highest Common Factor.
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Factors
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Highest Common Factor
Example :
Find the HCF of 8 and 12.
F8 = 1 and 8
F12 = 1 and 12
2 and 4
2 and 6
3 and 4
HCF = 4
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Factors
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Highest Common Factor
Example :
Find the HCF of 4x and x2.
F4x = 1, and 4x
2 and 2x
4 and
Example :
Fx2 = 1 and x2
x and x
x
HCF = x
Find the HCF of 5 and 10x.
F5 = 1 and 5
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F10x = 1, and 10x
2 and 5x
HCF = 5
5 and 2x
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10 and x
Factors
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Highest Common Factor
Example :
Find the HCF of ab and 2b.
F ab = 1 and ab
F2b = 1 and 2b
a and b
Example :
Find the HCF of 2h2 and 4h.
F 2h2 = 1 and 2h2
2 and h2 , h and 2h
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2 and
F4h = 1 and 4h
HCF = 2h
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2 and 2h
4 and h
b
HCF = b
Factors
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Find the HCF for these terms
(a)
16w and 24w
8w
(b)
9y2 and 6y
3y
(c)
4h and 12h2
4h
(d)
ab2 and a2b
ab
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Factors
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Now try Ex 2.1 & 3.1
First Column in each
Question
Ch5 (page 86)
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Starter Questions
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Q1.
Expand out
(a)
a (4y – 3x) -2ay
(b)
(x + 5)(x - 5)
Q2. Write out in full
4.85 10
3
Q3. True or False all the factors of 5x2 are
1, x, 5
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Factorising
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Using Factors
Learning Intention
1. To show how to factorise
terms using the Highest
Common Factor and one
bracket term.
18-Jul-15
Success Criteria
1. To identify the HCF for
given terms.
2. Factorise terms using the
HCF and one bracket term.
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Check by multiplying
Factorising
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Example
out the bracket to get
back to where you
Factorise 3x +started
15
1.
Find the HCF for 3x and 15
2.
HCF goes outside the bracket
3.
To see what goes inside the bracket
divide each term by HCF
3x ÷ 3 = x
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15 ÷ 3 = 5
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3
3(
)
3( x + 5 )
Check by multiplying
Factorising
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Example
out the bracket to get
back to where you
Factorise 4x2 –started
6xy
1.
Find the HCF for 4x2 and 6xy
2.
HCF goes outside the bracket
3.
To see what goes inside the bracket
divide each term by HCF
4x2 ÷ 2x =2x
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6xy ÷ 2x = 3y
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2x
2x(
)
2x( 2x- 3y )
Factorising
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Factorise the following :
(a)
3(x + 2)
3x + 6
Be
careful !
2x(2y – 1)
(b)
4xy – 2x
(c)
6a + 7a2
a(6 + 7a)
(d)
y2 - y
y(y – 1)
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Factorising
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Now try Ex 4.1 & 4.2
First 2 Columns only
Ch5 (page 88)
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Starter Questions
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Q1.
In a sale a jumper is reduced by 20%.
The sale price is £32.
Show that the original price was £40
Q2. Factorise
3x2 – 6x
Q3. Write down the arithmetic operation
associated with the word ‘difference’.
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Difference of
Two Squares
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Learning Intention
1. To show how to factorise
the special case of the
difference of two squares.
Success Criteria
1. Recognise when we have a
difference of two squares.
2. Factorise the difference of
two squares.
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Difference of
Two Squares
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When an expression is made up of
the difference of two squares
then it is simple to factorise
The format for the difference of two squares
a2 – b2
First
square term
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Difference
Second
square term
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Difference of
Two Squares
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2 by multiplying
a2 – out
bCheck
the bracket to get
First
square term
back to where you
Second
Difference started
square term
This factorises to
( a + b )( a – b )
Two brackets the same except for + and a 18-Jul-15
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Difference of
Two Squares
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Keypoints
Format
a2 – b2
Always the difference sign ( a + b )( a – b )
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Difference of
Two Squares
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Factorise using the difference of two squares
(a)
x2 – 72
(x + 7 )( x – 7 )
(b)
w2 – 1
( w + 1 )( w – 1 )
(c)
9a2 – b2
(d)
16y2 – 100k2
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( 3a + b )( 3a – b )
( 4y + 10k )( 4y – 10k )
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Difference of
Two Squares
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Trickier type of questions to factorise.
Sometimes we need to take out a common factor
and then use the difference of two squares.
Example
Factorise
2a2 - 18
First take out common factor
2(a2 - 9)
Now apply the difference of two squares
2( a + 3 )( a – 3 )
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Difference of
Two Squares
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Factorise these trickier expressions.
(a)
6x2 – 24
6(x + 2 )( x – 2 )
(b)
3w2 – 3
3( w + 1 )( w – 1 )
(c)
8 – 2b2
(d)
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27w2 – 12
2( 2 + b )( 2 – b )
3(3 w + 2 )( 3w – 2 )
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Difference of
Two Squares
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Now try Ex 5.1 & 5.2
First 2 Columns only
Ch5 (page 90)
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Starter Questions
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Q1.
True or false
y ( y + 6 ) -7y = y2 -7y + 6
Q2. Fill in the ?
49 – 4x2 = ( ? + ?x)(? – 2?)
Q3. Write in scientific notation 0.0341
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Factorising
Using St. Andrew’s Cross method
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Learning Intention
1. To show how to factorise
trinomials ( quadratics)
using
St. Andrew's Cross method.
18-Jul-15
Success Criteria
1. Understand the steps of the
St. Andrew’s Cross method.
2. Be able to factorise
quadratics using SAC method.
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Factorising
Using St. Andrew’s Cross method
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There are various ways of factorising
trinomials (quadratics)
e.g. The ABC method, FOIL method.
We will use the
St. Andrew’s cross method
to factorise trinomials / quadratics.
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Removing
Double Brackets
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A LITTLE REVISION
Multiply out the brackets and Simplify
(x + 1)(x + 2)
1.
Write down
2.
Tidy up !
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F
O
I
L
x2 + 2x + x + 2
x2 + 3x + 2
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Factorising
Using St. Andrew’s Cross method
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We use the SAC method to go the opposite way
FOIL
(x + 1)(x + 2)
(x + 1)(x + 2)
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SAC
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x2 + 3x+ 2
x2 + 3x+ 2
Factorising
Using St. Andrew’s Cross method
S3
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Strategy for factorising quadratics
Find two numbers that
multiply to give last number (+2)
and
Diagonals sum to give middle value +3x.
x2 + 3x + 2
x
+2
x
+1
(
)(
18-Jul-15
)
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(+2) x( +1) = +2
(+2x) +( +1x) = +3x
Factorising
Using St. Andrew’s Cross method
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Strategy for factorising quadratics
x2 + 6x + 5
Find two numbers that
multiply to give last number (+5)
and
Diagonals sum to give middle value +6x
x
+5
x
+1
(
18-Jul-15
)(
)
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(+5) x( +1) = +5
(+5x) +( +1x) = +6x
Factorising
One number
Using St. Andrew’s Cross method
must be +
and one -
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Strategy for factorising quadratics
x2 + x - 12
Find two numbers that
multiply to give last number (-12)
and
Diagonals sum to give middle value +x.
x
+4
x
-3
(
18-Jul-15
)(
)
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(+4) x( -3) = -12
(+4x) +( -3x) = +x
Factorising
Both numbers
Using St. Andrew’s Cross method
must be -
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Strategy for factorising quadratics
x2 - 4x + 4
Find two numbers that
multiply to give last number (+4)
and
Diagonals sum to give middle value -4x.
x
-2
x
-2
(
18-Jul-15
)(
)
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(-2) x( -2) = +4
(-2x) +( -2x) = -4x
Factorising
One number
Using St. Andrew’s Cross method
must be +
and one -
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Strategy for factorising quadratics
x2 - 2x - 3
Find two numbers that
multiply to give last number (-3)
and
Diagonals sum to give middle value -2x
x
-3
x
+1
(
18-Jul-15
)(
)
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(-3) x( +1) = -3
(-3x) +( x) = -2x
Factorising
Using St. Andrew’s Cross method
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Factorise using SAC method
(a)
m2 + 2m + 1
(m + 1 )( m + 1 )
(b)
y2 + 6y + 5
( y + 5 )( y + 1 )
(c)
b2 – b - 2
( b - 2 )( b + 1 )
(d)
a2 – 5a + 6
( a - 3 )( a – 2 )
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Factorising
Using St. Andrew’s Cross method
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Now try Ex6.1
Ch5 (page 93)
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Starter Questions
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Q1.
Cash price for a sofa is £700.
HP terms are 10% deposit the 6 months
equal payments of £120.
Show that you pay £90 using HP terms.
Q2. Factorise
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2 + x – x2
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Factorising
Using St. Andrew’s Cross method
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Learning Intention
1. To show how to factorise
trinomials ( quadratics) of
the form ax2 + bx +c using
SAC.
18-Jul-15
Success Criteria
1. Be able to factorise
trinomials / quadratics
using SAC.
Created by Mr. [email protected]
Factorising
One number
Using St. Andrew’s Cross method
must be +
and one -
S3
Credit
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Strategy for factorising quadratics
3x2 - x - 4
Find two numbers that
multiply to give last number (-4)
and
Diagonals sum to give middle value -x
3x
-4
x
+1
(
18-Jul-15
)(
)
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(-4) x( +1) = -4
(3x) +( -4x) = -x
Factorising
One number
Using St. Andrew’s Cross method
must be +
and one -
S3
Credit
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Strategy for factorising quadratics
2x2 - x - 3
Find two numbers that
multiply to give last number (-3)
and
Diagonals sum to give middle value -x
2x
-3
x
+1
(
18-Jul-15
)(
)
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(-3) x( +1) = -3
(-3x) +( +2x) = -x
Factorisingone number is +
Using St. Andrew’s Cross method
and
one number is -
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Two numbers that multiply to give
last number (-3)
and
Diagonals sum to give middle value (-4x)
4x2 - 4x - 3
4x
Keeping
the LHS fixed
Factors
1 and -3
-1 and 3
x
(
18-Jul-15
)(
)
Can we do it !
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Factorising
Using St. Andrew’s Cross method
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Credit
Find another set of
factors for LHS
4x2 - 4x - 3
2x
-3
2x
+1
(
18-Jul-15
)(
Repeat the factors for
RHS to see if it
factorises now
)
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Factors
1 and -3
-1 and 3
Factorising
Both numbers
Using St. Andrew’s Cross method
must be +
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Find two numbers that
multiply to give last number (+15)
and
Diagonals sum to give middle value (+22x)
8x2+22x+15
8x
Keeping
the LHS fixed
Factors
1 and
15 factors
Find
all the
3 andtry
5 and factorise
of (+15) then
x
(
18-Jul-15
)(
)
Can we do it !
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Factorising
Using St. Andrew’s Cross method
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Credit
Find another set of
factors for LHS
8x2+22x+15
4x
+5
2x
+3
(
18-Jul-15
)(
Repeat the factors for
RHS to see if it
factorises now
)
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Factors
3 and 5
1 and 15
Factorising
Using St. Andrew’s Cross method
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Now try Ex 7.1
First 2 columns only
Ch5 (page 95)
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Starter Questions
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Credit
Q1. Use a multiplication table to expand out
(2x – 5)(x + 5)
Q2. After a 20% discount a watch is on sale
for £240.
What was the original price of the watch.
Q3. True or false
18-Jul-15
3a2 b – ab2 =a2b2(3b – a)
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Summary of
Factorising
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Learning Intention
1. To explain the factorising
priorities.
18-Jul-15
Success Criteria
1. Be able use the factorise
priorities to factorise
various expressions.
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Summary of
Factorising
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Credit
When we are asked to factorise there is priority we
must do it in.
1.
Take any common factors out and put them
outside the brackets.
2.
Check for the difference of two squares.
3.
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Factorise any quadratic expression left.
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Summary of
Factorising
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St. Andrew’s Cross method
2
Difference
squares
Take Out Common Factor
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If youof
can successfully
Summary
complete this exercise
then you have the
Factorising
necessary skills to pass
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the algebraic part of the
course.
Now try Ex 8.1
Ch5 (page 97)
18-Jul-15
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