Nat 5 Algebraic Operations 16-Oct-15Created by Mr. Difference of Squares Factors / HCF Common Factors Factorising Trinomials (Quadratics) Factor Priority See Quadratic Theory for Exam Questions

Starter Questions 16-Oct-15Created by Mr. 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.

16-Oct-15 Created by Mr. Learning Intention Success Criteria 1.To identify the HCF for given terms. 1.We are learning how to factorise terms using the Highest Common Factor and one bracket term. 2.Factorise terms using the HCF and one bracket term. Factorising Using Factors Nat 5

16-Oct-15Created by Mr. Factorising Example Factorise 3x Find the HCF for 3x and HCF goes outside the bracket3( ) 3.To see what goes inside the bracket divide each term by HCF 3x ÷ 3 = x15 ÷ 3 = 53( x + 5 ) Check by multiplying out the bracket to get back to where you started Nat 5

16-Oct-15Created by Mr. Factorising Example 1.Find the HCF for 4x 2 and 6xy2x 2.HCF goes outside the bracket2x( ) 3.To see what goes inside the bracket divide each term by HCF 4x 2 ÷ 2x =2x6xy ÷ 2x = 3y2x( 2x- 3y ) Factorise 4x 2 – 6xy Check by multiplying out the bracket to get back to where you started Nat 5

16-Oct-15Created by Mr. Factorising Factorise the following : (a)3x + 6 (b) 4xy – 2x (c) 6a + 7a 2 (d)y 2 - y 3(x + 2) 2x(2y – 1) a(6 + 7a) y(y – 1) Be careful ! Nat 5

16-Oct-15Created by Mr. Lafferty Now try N5 TJ Ex 7.1 Ch7 (page 65) Factors Nat 5

Starter Questions 16-Oct-15Created by Mr. 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 3x 2 – 6x Nat 5 Q3.Write down the arithmetic operation associated with the word ‘difference’.

16-Oct-15 Created by Mr. Learning Intention Success Criteria 1.Recognise when we have a difference of two squares. 1.We are learning how to factorise the special case of the difference of two squares. 2.Factorise the difference of two squares. Difference of Two Squares Nat 5

16-Oct-15Created by Mr. 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 a 2 – b 2 First square term Second square term Difference Difference of Two Squares Nat 5

16-Oct-15Created by Mr. a 2 – b 2 First square term Second square 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 Two Squares Nat 5

16-Oct-15Created by Mr. Lafferty Keypoints Formata 2 – b 2 Always the difference sign - ( a + b )( a – b ) Difference of Two Squares Nat 5

16-Oct-15Created by Mr. Lafferty Factorise using the difference of two squares (a)x 2 – y 2 (b) w 2 – z 2 (c) 9a 2 – b 2 (d)16y 2 – 100k 2 (x + y )( x – y ) ( w + z )( w – z ) ( 3a + b )( 3a – b ) ( 4y + 10k )( 4y – 10k ) Difference of Two Squares Nat 5

16-Oct-15 Created by Mr. Now try N5 TJ Ex 7.2 Q1 Ch7 (page 66) Difference of Two Squares Nat 5

16-Oct-15Created by Mr. Lafferty Trickier type of questions to factorise. Sometimes we need to take out a common And the use the difference of two squares. ExampleFactorise2a ( a + 3 )( a – 3 ) Difference of Two Squares Nat 5 First take out common factor 2(a 2 - 9) Now apply the difference of two squares

16-Oct-15Created by Mr. Lafferty Factorise these trickier expressions. (a)6x 2 – 24 (b) 3w 2 – 3 (c) 8 – 2b 2 (d) 27w 2 – 12 6(x + 2 )( x – 2 ) 3( w + 1 )( w – 1 ) 2( 2 + b )( 2 – b ) 3(3 w + 2 )( 3w – 2 ) Difference of Two Squares Nat 5

16-Oct-15 Created by Mr. Now try N5 TJ Ex 7.2 Q2 Ch7 (page 66) Difference of Two Squares Nat 5

Starter Questions 16-Oct-15Created by Mr. Q1.Multiple out the bracket and simplify. (a)y ( y + 6 ) -7y Q2.Factorise 49 – 4x 2 Nat 5 Q3.Write in scientific notation

16-Oct-15 Created by Mr. Learning Intention Success Criteria 1.Understand the steps of the St. Andrew’s Cross method. 2.Be able to factorise quadratics using SAC method. 1.We are learning how to factorise trinomials ( quadratics) using St. Andrew's Cross method. Nat 5 Factorising Using St. Andrew’s Cross method

16-Oct-15 Created by Mr. 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. Factorising Using St. Andrew’s Cross method

Nat 5 (x + 1)(x + 2) x2x2 + 2x A LITTLE REVISION Multiply out the brackets and Simplify 16-Oct-15Created by Mr. 1.Write down F O I L + x Tidy up ! x 2 + 3x + 2 Removing Double Brackets

Nat 5 (x + 1)(x + 2)x2x2 + 3x We use SAC method to go the opposite way 16-Oct-15Created by Mr. + 2 FOIL (x + 1)(x + 2) x2x2 + 3x + 2 SAC Factorising Using St. Andrew’s Cross method

Nat Oct-15Created by Mr. x 2 + 3x + 2 Strategy for factorising quadratics Factorising Using 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

Nat Oct-15Created by Mr. x 2 + 6x + 5 Strategy for factorising quadratics Factorising Using 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

Nat Oct-15Created by Mr. x 2 + x - 12 Strategy for factorising quadratics Factorising Using 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

Nat Oct-15Created by Mr. x 2 - 4x + 4 Strategy for factorising quadratics Factorising Using 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

Nat Oct-15Created by Mr. x 2 - 2x - 3 Strategy for factorising quadratics Factorising Using 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

16-Oct-15Created by Mr. Lafferty Factorise using SAC method (a)m 2 + 2m +1 (b) y 2 + 6m + 5 (c) b 2 – b -2 (d)a 2 – 5a + 6 (m + 1 )( m + 1 ) ( y + 5 )( y + 1 ) ( b - 2 )( b + 1 ) ( a - 3 )( a – 2 ) Nat 5 Factorising Using St. Andrew’s Cross method

16-Oct-15 Created by Mr. Now try N5 TJ Ex 7.3 Q1 & 2 Ch7 (page 67) Difference of Two Squares Nat 5

Starter Questions 16-Oct-15Created by Mr. 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 – x 2 Nat 5

16-Oct-15 Created by Mr. Learning Intention Success Criteria 1.Be able to factorise trinomials / quadratics using SAC. 1.To show how to factorise trinomials ( quadratics) of the form ax 2 + bx +c using SAC. Nat 5 Factorising Using St. Andrew’s Cross method

Nat x Oct-15Created by Mr. 3x 2 - x - 4 Strategy for factorising quadratics Factorising Using 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

Nat x - 3 2x Oct-15Created by Mr. 2x 2 - x - 3 Strategy for factorising quadratics Factorising Using 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

Nat 5 4x 2 - 4x - 3 Two numbers that multiply to give last number (-3) and Diagonals sum to give middle value (-4x) 4x 16-Oct-15Created by Mr. Factorising Using St. Andrew’s Cross method x ( ) Keeping the LHS fixed Can we do it ! one number is + and one number is - Factors 1 and and 3

Nat x - 3 4x 2 - 4x - 3 2x 16-Oct-15Created by Mr. Factorising Using St. Andrew’s Cross method 2x ( ) Find another set of factors for LHS Factors 1 and and 3 Repeat the factors for RHS to see if it factorises now + 1 2x

Nat 5 8x 2 +22x+15 Find two numbers that multiply to give last number (+15) and Diagonals sum to give middle value (+22x) 8x 16-Oct-15Created by Mr. Factorising Using 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 Factors 1 and 15 3 and 5

Nat x 2 +22x+15 4x 16-Oct-15Created by Mr. Factorising Using St. Andrew’s Cross method 2x ( ) x 2x Find another set of factors for LHS Factors 3 and 5 1 and 15 Repeat the factors for RHS to see if it factorises now

16-Oct-15 Created by Mr. Now try N5 TJ Ex 7.3 Q1 & Q3 Ch7 (page 68) Difference of Two Squares Nat 5

Starter Questions 16-Oct-15Created by Mr. Q1.Multiple out the brackets and simplify. (a)( 2x – 5 )( x + 5 ) Nat 5 Q3.Factorise 3ab – b 2 Q2.After a 20% discount a watch is on sale for £240. What was the original price of the watch.

16-Oct-15 Created by Mr. Learning Intention Success Criteria 1.Be able use the factorise priorities to factorise various expressions. 1.To explain the factorising priorities. Nat 5 Summary of Factorising

16-Oct-15 Created by Mr. Summary of Factorising Nat 5 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.Factorise any quadratic expression left. Only TWO terms THREE terms

16-Oct-15 Created by Mr. Now try N5 TJ Ex 7.3 Q4 Ch7 (page 68) Difference of Two Squares Nat 5 If you can successfully complete this exercise then you have the necessary skills to pass the algebraic part of the course.