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Nat 5
Algebraic OperationsAlgebraic Operations
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Difference of Squares
Factors / HCF
Common Factors
Factorising Trinomials (Quadratics)Factor Priority
See Quadratic Theory for Exam Questions
Starter QuestionsStarter Questions
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.comw
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.com Q1. Remove the brackets
(a) a (4y – 3x) (b) (2x-1)(x+4) Q2. Calculate
Nat 5
Q3. Write down all the number that divide into 12 without leaving a remainder.
5 1
6 3
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. To identify the HCF for To identify the HCF for given terms.given terms.
1. We are learning how to factorise terms using the Highest Common Factor and one bracket term.
2.2. Factorise terms using the Factorise terms using the HCF and one bracket term.HCF and one bracket term.
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FactorisingFactorisingUsing FactorsNat 5
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
FactorisingFactorisingw
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Example Factorise 3x + 15
1. Find the HCF for 3x and 15 3
2. HCF goes outside the bracket 3( )
3. To see what goes inside the bracketdivide each term by HCF
3x ÷ 3 = x 15 ÷ 3 = 5 3( x + 5 )
Check by multiplying out the bracket to get back to where
you started
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21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
FactorisingFactorisingw
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Example
1. Find the HCF for 4x2 and 6xy 2x
2. HCF goes outside the bracket 2x( )
3. To see what goes inside the bracketdivide each term by HCF
4x2 ÷ 2x =2x 6xy ÷ 2x = 3y 2x( 2x- 3y )
Factorise 4x2 – 6xy
Check by multiplying out the bracket to get back to where
you started
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21 Apr 202321 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.comw
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FactorisingFactorising
Factorise the following :
(a) 3x + 6
(b) 4xy – 2x
(c) 6a + 7a2
(d) y2 - y
3(x + 2)
2x(2y – 1)
a(6 + 7a)
y(y – 1)
Be careful !
Nat 5
21 Apr 202321 Apr 2023 Created by Mr. LaffertyCreated by Mr. Lafferty
Now try
N5 TJ Ex 7.1Ch7 (page 65)
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FactorsFactorsNat 5
Starter QuestionsStarter Questions
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.comw
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.com Q1. In a sale a jumper is reduced by 20%.
The sale price is £32. What is the original price of the
jumper.
Q2. Factorise 3x2 – 6x
Nat 5
Q3. Write down the arithmetic operationassociated with the word ‘difference’.
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Recognise when we Recognise when we have a difference of have a difference of two squares.two squares.
1. We are learning how to factorise the special case of the difference of two squares.
2.2. Factorise the difference of Factorise the difference of two squares.two squares.
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Difference of Difference of Two SquaresTwo SquaresNat 5
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
w.m
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.com 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
Secondsquare term
Difference
Difference of Difference of Two SquaresTwo SquaresNat 5
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
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First square term
Secondsquare term
Difference
This factorises to
( a + b )( a – b )
Two brackets the same except for + and a -
Check by multiplying out the bracket to get back to where
you started
Difference of Difference of Two SquaresTwo SquaresNat 5
21 Apr 202321 Apr 2023 Created by Mr. LaffertyCreated by Mr. Laffertyww
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Format a2 – b2
Always the difference sign -
( a + b )( a – b )
Difference of Difference of Two SquaresTwo SquaresNat 5
21 Apr 202321 Apr 2023 Created by Mr. LaffertyCreated by Mr. Laffertyww
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Factorise using the difference of two squares
(a) x2 – y2
(b) w2 – z2
(c) 9a2 – b2
(d) 16y2 – 100k2
(x + y )( x – y )
( w + z )( w – z )
( 3a + b )( 3a – b )
( 4y + 10k )( 4y – 10k )
Difference of Difference of Two SquaresTwo SquaresNat 5
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
Now try N5 TJ Ex 7.2 Q1
Ch7 (page 66)
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Difference of Difference of Two SquaresTwo SquaresNat 5
21 Apr 202321 Apr 2023 Created by Mr. LaffertyCreated by Mr. Laffertyww
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Trickier type of questions to factorise.Sometimes we need to take out a commonAnd the use the difference of two squares.
ExampleFactorise 2a2 - 18
2( a + 3 )( a – 3 )
Difference of Difference of Two SquaresTwo SquaresNat 5
First take out common factor 2(a2 - 9)
Now apply the difference of two squares
21 Apr 202321 Apr 2023 Created by Mr. LaffertyCreated by Mr. Laffertyww
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Factorise these trickier expressions.
(a) 6x2 – 24
(b) 3w2 – 3
(c) 8 – 2b2
(d) 27w2 – 12
6(x + 2 )( x – 2 )
3( w + 1 )( w – 1 )
2( 2 + b )( 2 – b )
3(3 w + 2 )( 3w – 2 )
Difference of Difference of Two SquaresTwo SquaresNat 5
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
Now try N5 TJ Ex 7.2 Q2
Ch7 (page 66)
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Difference of Difference of Two SquaresTwo SquaresNat 5
Starter QuestionsStarter Questions
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.comw
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.com Q1. Multiple out the bracket and simplify.
(a) y ( y + 6 ) -7y
Q2. Factorise 49 – 4x2
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Q3. Write in scientific notation 0.0341
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Understand the steps of Understand the steps of the St. Andrew’s Cross the St. Andrew’s Cross method.method.
2.2. Be able to factorise Be able to factorise quadratics using SAC quadratics using SAC method.method.
1. We are learning how to factorise trinomials ( quadratics) using
St. Andrew's Cross method.
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Nat 5
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
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Nat 5
There 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.
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
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(x + 1)(x + 2)
x2 + 2x
A LITTLE REVISIONMultiply out the brackets and Simplify
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
1. Write down F O I L+ x + 2
2. Tidy up !x2 + 3x + 2
RemovingRemovingDouble BracketsDouble Brackets
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(x + 1)(x + 2) x2 + 3x
We use SAC method to go the opposite way
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
+ 2FOIL
(x + 1)(x + 2) x2 + 3x + 2SAC
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
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+ 1
+ 2+ 2
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
x2 + 3x + 2
Strategy for factorising quadratics
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
x
x + 1
Find two numbers that multiply to give last number (+2)
and Diagonals sum to give middle value +3x.
( ) ( )
x
x
(+2) x( +1) = +2
(+2x) +( +1x) = +3x
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+ 5
+ 1
+ 5
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
x2 + 6x + 5
Strategy for factorising quadratics
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
x
x + 1
( ) ( )
x
x
Find two numbers that multiply to give last number (+5)
and Diagonals sum to give middle value +6x.
(+5) x( +1) = +5
(+5x) +( +1x) = +6x
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Nat 5
- 3
+ 4+ 4
- 3
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
x2 + x - 12
Strategy for factorising quadratics
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
x
x
( ) ( )
x
x
One number must be + and one -
Find two numbers that multiply to give last number (-12)
and Diagonals sum to give middle value +x.
(+4) x( -3) = -12
(+4x) +( -3x) = +x
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Nat 5
- 2
- 2- 2
- 2
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
x2 - 4x + 4
Strategy for factorising quadratics
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
x
x
( ) ( )
x
x
Both numbers must be -
Find two numbers that multiply to give last number (+4)
and Diagonals sum to give middle value -4x.
(-2) x( -2) = -4
(-2x) +( -2x) = -4x
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- 3- 3
+ 1
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
x2 - 2x - 3
Strategy for factorising quadratics
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
x
x + 1
( ) ( )
x
x
One number must be + and one -
Find two numbers that multiply to give last number (-3)
and Diagonals sum to give middle value -2x
(-3) x( +1) = -3
(-3x) +( x) = -2x
21 Apr 202321 Apr 2023 Created by Mr. LaffertyCreated by Mr. Laffertyww
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Factorise using SAC method
(a) m2 + 2m +1
(b) y2 + 6m + 5
(c) b2 – b -2
(d) a2 – 5a + 6
(m + 1 )( m + 1 )
( y + 5 )( y + 1 )
( b - 2 )( b + 1 )
( a - 3 )( a – 2 )
Nat 5
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
Now try N5 TJ Ex 7.3 Q1 & 2Ch7 (page 67)
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Difference of Difference of Two SquaresTwo SquaresNat 5
Starter QuestionsStarter Questions
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.comw
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Q1. Cash price for a sofa is £700. HP terms are 10% deposit the 6 months equal payments of £120. How much more do you pay with
HP.
Q2. Factorise 2 – 3x – x2
Nat 5
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Be able to factorise Be able to factorise trinomials / quadratics trinomials / quadratics using SAC.using SAC.
1. To show how to factorise trinomials ( quadratics) of the form ax2 + bx +c using SAC.
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Nat 5
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
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- 43x - 4
+ 1
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
3x2 - x - 4
Strategy for factorising quadratics
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
3x
x + 1
( ) ( )
x
One number must be + and one -
Find two numbers that multiply to give last number (-4)
and Diagonals sum to give middle value -x
(-4) x( +1) = -4
(3x) +( -4x) = -x
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- 32x
- 32x
+ 1
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
2x2 - x - 3
Strategy for factorising quadratics
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
x + 1
( ) ( )
x
One number must be + and one -
Find two numbers that multiply to give last number (-3)
and Diagonals sum to give middle value -x
(-3) x( +1) = -3
(-3x) +( +2x) = -x
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4x2 - 4x - 3Two numbers that multiply to give
last number (-3)and
Diagonals sum to give middle value (-4x)
4x
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
x
( ) ( )
Keeping the LHS fixed
Can we do it !
one number is + and
one number is -
Factors1 and -3-1 and 3
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- 3
+ 1
2x
- 3
4x2 - 4x - 3
2x
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
2x
( ) ( )
Find another set of factors for LHS
Factors1 and -3-1 and 3
Repeat the factors for RHS to see if it factorises now
+ 12x
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Nat 5
8x2+22x+15Find two numbers that
multiply to give last number (+15)and
Diagonals sum to give middle value (+22x)8x
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
x
( ) ( )
Keeping the LHS fixed
Can we do it !
Both numbers
must be +
Find all the factors of (+15) then try and
factorise
Factors1 and 153 and 5
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+ 5
8x2+22x+15
4x
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
FactorisingFactorisingUsing St. Andrew’s Cross methodUsing St. Andrew’s Cross method
2x
( ) ( )
+ 5
+ 3+ 3
4x2x
Find another set of factors for LHS
Factors3 and 5
1 and 15
Repeat the factors for RHS to see if it factorises now
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
Now try N5 TJ Ex 7.3 Q1 & Q3Ch7 (page 68)
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Difference of Difference of Two SquaresTwo SquaresNat 5
Starter QuestionsStarter Questions
Apr 21, 2023Apr 21, 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.comw
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(a) ( 2x – 5 )( x + 5 )
Nat 5
Q3. Factorise 3ab – b2
Q2. After a 20% discount a watch is on sale for £240. What was the original price of the watch.
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Be able use the factorise Be able use the factorise priorities to factorise priorities to factorise various expressions.various expressions.
1. To explain the factorising priorities.
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Nat 5
Summary ofSummary ofFactorisingFactorising
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
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Summary ofSummary ofFactorisingFactorisingNat 5
When we are asked to factorise there is priority wemust do it in.
1. Take any common factors out and put them outside the brackets.
2. Check for the difference of two squares.
3. Factorise any quadratic expression left.
Only TWO terms
THREE terms
21 Apr 202321 Apr 2023 Created by Mr. [email protected] by Mr. [email protected]
Now try N5 TJ Ex 7.3 Q4
Ch7 (page 68)
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Difference of Difference of Two SquaresTwo SquaresNat 5
If you can successfully complete this exercise
then you have the necessary skills to pass the algebraic part of the
course.