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S3 Credit Algebraic Operations 30-May-14Created by Mr. Difference of Squares Factors / HCF Common Factors.

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Presentation on theme: "S3 Credit Algebraic Operations 30-May-14Created by Mr. Difference of Squares Factors / HCF Common Factors."— Presentation transcript:

1 www.mathsrevision.com S3 Credit Algebraic Operations 30-May-14Created by Mr. Lafferty@mathsrevision.com Difference of Squares Factors / HCF Common Factors Factorising Trinomials (Quadratics) Factor Priority

2 Starter Questions 30-May-14Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com Q1.Multiply out (a)a (4y – 3x)(b)(2x-1)(x+4) Q2.True or false S3 Credit Q3.Write down all the number that divide into 12 without leaving a remainder.

3 30-May-14Created by Mr. Lafferty@mathsrevision.com Learning Intention Success Criteria 1.To identify factors using factor pairs 1.To explain that a factor divides into a number without leaving a remainder 2.To explain how to find Highest Common Factors 2.Find HCF for two numbers by comparing factors. www.mathsrevision.com Factors Using Factors S3 Credit

4 30-May-14Created by Mr. Lafferty@mathsrevision.com Factors www.mathsrevision.com Factors Example :Find the factors of 56. F56 =1 and 56 2 and 28 S3 Credit Numbers that divide into 56 without leaving a remainder 4 and 14 7 and 8

5 Factors 30-May-14Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com Highest Common Factor We need to write out all factor pairs in order to find the Highest Common Factor. Highest Common Factor Largest Same Number S3 Credit

6 F8 =1 and 8 2 4 Factors 30-May-14Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com Example :Find the HCF of 8 and 12. HCF = 4 F12 = 1 and 12 2 and 6 3 4 Highest Common Factor S3 Credit

7 2 and 5x F4x =1, and 4x Factors 30-May-14Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com Example :Find the HCF of 4x and x 2. HCF = x Fx 2 = 1 and x2x2 Highest Common Factor F5 = 1 and 5 Example :Find the HCF of 5 and 10x. HCF = 5 F10x = 1, and 10x S3 Credit 5 and 2x 10 and x 2 and 2x 4 and x x and x

8 F ab =1 and ab a and b Factors 30-May-14Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com Example :Find the HCF of ab and 2b. HCF = b F2b = 1 and 2b 2 and b Highest Common Factor F 2h 2 = 1 and 2h 2 2 and h2 h2, h 2h Example :Find the HCF of 2h 2 and 4h. HCF = 2h F4h = 1 and 4h 2 and 2h 4 and h S3 Credit

9 30-May-14Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com Factors Find the HCF for these terms (a)16w and 24w (b) 9y 2 and 6y (c) 4h and 12h 2 (d)ab 2 and a 2 b 8w 3y 4h ab S3 Credit

10 30-May-14Created by Mr. Lafferty Now try Ex 2.1 & 3.1 First Column in each Question Ch5 (page 86) www.mathsrevision.com Factors S3 Credit

11 Starter Questions 30-May-14Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com Q1.Expand out (a)a (4y – 3x)-2ay(b)(x + 5)(x - 5) Q2.Write out in full S3 Credit Q3.True or False all the factors of 5x 2 are 1, x, 5

12 30-May-14 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.To identify the HCF for given terms. 1.To show how to factorise terms using the Highest Common Factor and one bracket term. 2.Factorise terms using the HCF and one bracket term. www.mathsrevision.com Factorising Using Factors S3 Credit

13 30-May-14Created by Mr. Lafferty@www.mathsrevision.com Factorising www.mathsrevision.com Example Factorise 3x + 15 1.Find the HCF for 3x and 153 2.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 S3 Credit

14 30-May-14Created by Mr. Lafferty@www.mathsrevision.com Factorising www.mathsrevision.com 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 S3 Credit

15 30-May-14Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com 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 ! S3 Credit

16 30-May-14 Created by Mr. Lafferty@www.mathsrevision.com Now try Ex 4.1 & 4.2 First 2 Columns only Ch5 (page 88) www.mathsrevision.com Factorising S3 Credit

17 Starter Questions 30-May-14Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com Q1.In a sale a jumper is reduced by 20%. The sale price is £32. Show that the original price was £40 Q2.Factorise 3x 2 – 6x S3 Credit Q3.Write down the arithmetic operation associated with the word difference.

18 30-May-14 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Recognise when we have a difference of two squares. 1.To show how to factorise the special case of the difference of two squares. 2.Factorise the difference of two squares. www.mathsrevision.com Difference of Two Squares S3 Credit

19 30-May-14Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.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 a 2 – b 2 First square term Second square term Difference Difference of Two Squares S3 Credit

20 30-May-14Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com 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 S3 Credit

21 30-May-14Created by Mr. Lafferty www.mathsrevision.com Keypoints Formata 2 – b 2 Always the difference sign - ( a + b )( a – b ) Difference of Two Squares S3 Credit

22 30-May-14Created by Mr. Lafferty www.mathsrevision.com Factorise using the difference of two squares (a)x 2 – 7 2 (b) w 2 – 1 (c) 9a 2 – b 2 (d)16y 2 – 100k 2 (x + 7 )( x – 7 ) ( w + 1 )( w – 1 ) ( 3a + b )( 3a – b ) ( 4y + 10k )( 4y – 10k ) Difference of Two Squares S3 Credit

23 30-May-14Created by Mr. Lafferty www.mathsrevision.com Trickier type of questions to factorise. Sometimes we need to take out a common factor and then use the difference of two squares. ExampleFactorise2a 2 - 18 2( a + 3 )( a – 3 ) Difference of Two Squares S3 Credit First take out common factor 2(a 2 - 9) Now apply the difference of two squares

24 30-May-14Created by Mr. Lafferty www.mathsrevision.com 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 S3 Credit

25 30-May-14 Created by Mr. Lafferty@www.mathsrevision.com Now try Ex 5.1 & 5.2 First 2 Columns only Ch5 (page 90) www.mathsrevision.com Difference of Two Squares S3 Credit

26 Starter Questions 30-May-14Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com Q1.True or false y ( y + 6 ) -7y = y 2 -7y + 6 Q2.Fill in the ? 49 – 4x 2 = ( ? + ?x)(? – 2?) S3 Credit Q3.Write in scientific notation 0.0341

27 30-May-14 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Understand the steps of the St. Andrews Cross method. 2.Be able to factorise quadratics using SAC method. 1.To show how to factorise trinomials ( quadratics) using St. Andrew's Cross method. www.mathsrevision.com S3 Credit Factorising Using St. Andrews Cross method

28 30-May-14 Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com S3 Credit There are various ways of factorising trinomials (quadratics) e.g. The ABC method, FOIL method. We will use the St. Andrews cross method to factorise trinomials / quadratics. Factorising Using St. Andrews Cross method

29 www.mathsrevision.com S3 Credit (x + 1)(x + 2) x2x2 + 2x A LITTLE REVISION Multiply out the brackets and Simplify 30-May-14Created by Mr. Lafferty@mathsrevision.com 1.Write down F O I L + x + 2 2.Tidy up ! x 2 + 3x + 2 Removing Double Brackets

30 www.mathsrevision.com S3 Credit (x + 1)(x + 2)x2x2 + 3x We use the SAC method to go the opposite way 30-May-14Created by Mr. Lafferty@mathsrevision.com + 2 FOIL (x + 1)(x + 2) x2x2 + 3x + 2 SAC Factorising Using St. Andrews Cross method

31 www.mathsrevision.com S3 Credit + 1 + 2+ 2 30-May-14Created by Mr. Lafferty@mathsrevision.com x 2 + 3x + 2 Strategy for factorising quadratics Factorising Using St. Andrews 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

32 www.mathsrevision.com S3 Credit + 5 + 1 + 5 30-May-14Created by Mr. Lafferty@mathsrevision.com x 2 + 6x + 5 Strategy for factorising quadratics Factorising Using St. Andrews 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

33 www.mathsrevision.com S3 Credit - 3 + 4+ 4 - 3 30-May-14Created by Mr. Lafferty@mathsrevision.com x 2 + x - 12 Strategy for factorising quadratics Factorising Using St. Andrews 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

34 www.mathsrevision.com S3 Credit - 2 - 2- 2 - 2 30-May-14Created by Mr. Lafferty@mathsrevision.com x 2 - 4x + 4 Strategy for factorising quadratics Factorising Using St. Andrews 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

35 www.mathsrevision.com S3 Credit - 3- 3 + 1 30-May-14Created by Mr. Lafferty@mathsrevision.com x 2 - 2x - 3 Strategy for factorising quadratics Factorising Using St. Andrews 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

36 30-May-14Created by Mr. Lafferty www.mathsrevision.com Factorise using SAC method (a)m 2 + 2m + 1 (b) y 2 + 6y + 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 ) S3 Credit Factorising Using St. Andrews Cross method

37 30-May-14 Created by Mr. Lafferty@www.mathsrevision.com Now try Ex6.1 Ch5 (page 93) www.mathsrevision.com S3 Credit Factorising Using St. Andrews Cross method

38 Starter Questions 30-May-14Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com 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 2 + x – x 2 S3 Credit

39 30-May-14 Created by Mr. Lafferty@www.mathsrevision.com 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. www.mathsrevision.com S3 Credit Factorising Using St. Andrews Cross method

40 www.mathsrevision.com S3 Credit - 4 3x - 4 + 1 30-May-14Created by Mr. Lafferty@mathsrevision.com 3x 2 - x - 4 Strategy for factorising quadratics Factorising Using St. Andrews 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

41 www.mathsrevision.com S3 Credit - 3 2x - 3 2x + 1 30-May-14Created by Mr. Lafferty@mathsrevision.com 2x 2 - x - 3 Strategy for factorising quadratics Factorising Using St. Andrews 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

42 www.mathsrevision.com S3 Credit 4x 2 - 4x - 3 Two numbers that multiply to give last number (-3) and Diagonals sum to give middle value (-4x) 4x 30-May-14Created by Mr. Lafferty@mathsrevision.com Factorising Using St. Andrews Cross method x ( ) Keeping the LHS fixed Can we do it ! one number is + and one number is - Factors 1 and -3 -1 and 3

43 www.mathsrevision.com S3 Credit - 3 + 1 2x - 3 4x 2 - 4x - 3 2x 30-May-14Created by Mr. Lafferty@mathsrevision.com Factorising Using St. Andrews Cross method 2x ( ) Find another set of factors for LHS Factors 1 and -3 -1 and 3 Repeat the factors for RHS to see if it factorises now + 1 2x

44 www.mathsrevision.com S3 Credit 8x 2 +22x+15 Find two numbers that multiply to give last number (+15) and Diagonals sum to give middle value (+22x) 8x 30-May-14Created by Mr. Lafferty@mathsrevision.com Factorising Using St. Andrews 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

45 www.mathsrevision.com S3 Credit + 5 8x 2 +22x+15 4x 30-May-14Created by Mr. Lafferty@mathsrevision.com Factorising Using St. Andrews Cross method 2x ( ) + 5 + 3+ 3 4x 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

46 30-May-14 Created by Mr. Lafferty@www.mathsrevision.com Now try Ex 7.1 First 2 columns only Ch5 (page 95) www.mathsrevision.com S3 Credit Factorising Using St. Andrews Cross method

47 Starter Questions 30-May-14Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com Q1. S3 Credit Q3.True or false 3a 2 b – ab 2 =a 2 b 2 (3b – a) Q2.After a 20% discount a watch is on sale for £240. What was the original price of the watch. Use a multiplication table to expand out (2x – 5)(x + 5)

48 30-May-14 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Be able use the factorise priorities to factorise various expressions. 1.To explain the factorising priorities. www.mathsrevision.com S3 Credit Summary of Factorising

49 30-May-14 Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com Summary of Factorising S3 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.Factorise any quadratic expression left.

50 30-May-14 Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com Summary of Factorising S3 Credit Take Out Common Factor 2 squares St. Andrews Cross method Difference

51 30-May-14 Created by Mr. Lafferty@www.mathsrevision.com Now try Ex 8.1 Ch5 (page 97) www.mathsrevision.com S3 Credit Summary of Factorising If you can successfully complete this exercise then you have the necessary skills to pass the algebraic part of the course.


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