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Algebraic Operations

Algebraic Operations. Removing Brackets. Pairs of Brackets. Factors. Common Factors. Difference of Squares. Factorising Trinomials (Quadratics). Factor Priority. Int 2. Q1. Calculate (a) -3 x 5 = (b) -6 x -7 =. Starter Questions. Q2. Calculate (a) w x w = (b) -2a x 4a =.

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Algebraic Operations

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  1. Algebraic Operations Removing Brackets Pairs of Brackets Factors Common Factors Difference of Squares Factorising Trinomials (Quadratics) Factor Priority

  2. Int 2 Q1. Calculate (a) -3 x 5 = (b) -6 x -7 = Starter Questions Q2. Calculate (a) w x w = (b) -2a x 4a = Q3. Find the gradient of the line if (3, 7) and (12, 34)

  3. Removing a Single Bracket Int 2 Learning Intention Success Criteria • To show how to multiply out (remove) a single bracket. • Understand the keypoints of multiplying out a expression with a single bracket. • Be able multiply out a expression with a single bracket.

  4. Removing a Single Bracket Example 1 + 15 3(b + 5) = 3b Example 2 - 8 4(w - 2) = 4w

  5. Removing a Single Bracket Example 3 - a a(y - 1) = ay Example 4 - 6p p(w - 6) = pw

  6. Removing a Single Bracket Example 5 + 3x x(x + 3) = x2 Example 6 - 6mq 3q(3q -2m) = 9q2

  7. Be careful with negatives !! Removing a Single Bracket Example 7 - 10 -2(h + 5) = -2h Example 8 + 9 -(g - 9) = -g

  8. Removing a Single Bracket Example 9 (x + 4) Find my Area 6 + 24 6(x + 4) = 6x

  9. Be careful only multiply everything inside the bracket Removing a Single Bracket Example 10 8 +2(h + 3) = 8 + 2h + 6 Now tidy up ! = 2h + 14

  10. Be careful only multiply everything inside the bracket Removing a Single Bracket Example 10 -2(y - 1) + 4 = -2y + 2 + 4 Now tidy up ! = -2y + 6

  11. Be careful only multiply everything inside the bracket Removing a Single Bracket Example 11 y - (4 - y) = y - 4 + y Now tidy up ! = - 4

  12. Removing a Single Bracket Example 12 Find the area of the picture frame. (x + 6) (x + 4) x 4 x(x + 6) – 4(x + 4) Area =

  13. Removing a Single Bracket Example 12 x(x + 6) – 4(x + 4) Area = x2 + 6x - 4x - 16 Now tidy up ! x2 + 2x - 16

  14. Removing a Single Bracket Example 13 x(x - 3) + 2(x - 3) x2 - 3x + 2x - 6 Now tidy up ! x2 - x - 6

  15. Removing a Single Bracket Int 2 Now try Exercise 1 Ch5 MIA (page 48)

  16. Int 2 Q1. Calculate (a) -3y x 5y = (b) -6q x (-4q) = Starter Questions Q2. Calculate (a) a(b - c) = (b) -2a( b – a) = Q3. Write down the gradient and were the line cuts the y – axis. y = 5 – 3x

  17. Removing Double Brackets Int 2 Learning Intention Success Criteria • To show 2 methods for multiplying out brackets • Understand the keypoints of multiplying out double brackets. • Be able multiply out double brackets using 2 methods.

  18. Removing Double Brackets There two methods we can use to multiply out DOUBLE brackets. First Method Simply remember the word F O I L Multiply Last 2 Multiply First 2 Multiply Outside 2 Multiply Inside 2

  19. Created by Mr. Lafferty@mathsrevision.com Removing Double Brackets Example 1 : Multiply out the brackets and Simplify (x + 1)(x + 2) 1. Write down F O I L x2 + 2x + x + 2 2. Tidy up !

  20. Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket Example 2 : Multiply out the brackets and Simplify (x - 1)(x + 2) 1. Write down F O I L x2 + 2x - x - 2 2. Tidy up !

  21. Removing Double Brackets (x + 1)(x - 2) x2 - x - 2 (x - 1)(x - 2) x2 - 3x + 2 (x + 3)(x + 2) x2 + 5x + 6 (x - 3)(x + 2) x2 - x - 6 (x + 3)(x - 2) x2 + x - 6

  22. Removing a Single Bracket Int 2 Now try Exercise 2 Q1 Ch5 MIA (page 50)

  23. Removing Double Brackets We have Multiplication Table “the wee table method” (y + 2)(y + 5) y + 2 y + 5 Tidy up ! y2 +5y +2y +10 y2 + 7y +10

  24. Removing Double Brackets Be careful with the negative signs Example 2 (2x - 1)(x + 3) 2x - 1 x + 3 Tidy up ! +6x 2x2 2x2 + 5x - 3 -3 -x

  25. Removing Double Brackets Just a bigger Multiplication Table Example 3 (x + 4)(x2 + 3x + 2) x2 + 3x + 2 x + 4 Tidy up ! +3x2 x3 +2x x3 + 7x2 + 14x + 8 +12x +8 +4x2

  26. Removing a Single Bracket Int 2 Now try Exercise 2 Ch5 MIA (page 50)

  27. Int 2 Q1. Remove the brackets (a) a (4y – 3x) = (b) (2x-1)(x+4) = Starter Questions Q2. Calculate The interest on £20 over 5 years @ a compound interest of 7% per year. Q3. Write down all the number that divide into 12 without leaving a remainder.

  28. Factors Int 2 Using Factors Learning Intention Success Criteria • To identify factors using factor pairs • To explain that a factor divides into a number without leaving a remainder • To explain how to find Highest Common Factors • Find HCF for two numbers by comparing factors.

  29. Factors Int 2 Factors Example : Find the factors of 56. Always divide by 1 and find its pair F56 = 1 and 56 From 2 find other factors and their pairs 2 and 28 4 and 14 7 and 8

  30. Factors Int 2 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.

  31. Factors Int 2 Highest Common Factor Example : Find the HCF of 8 and 12. F8 = 1 and 8 2 and 4 F12 = 1 and 12 2 and 6 3 and 4 HCF = 4

  32. Factors Int 2 Highest Common Factor Example : Find the HCF of 4x and x2. F4x = 1, and 4x , 2 and 2x 4 and x Fx2 = 1 and x2 x and x HCF = x Example : Find the HCF of 5 and 10x. F5 = 1 and 5 F10x = 1, and 10x 2 and 5x , 5 and 2x 10 and x HCF = 5

  33. Factors Int 2 Highest Common Factor Example : Find the HCF of ab and 2b. F ab = 1 and ab a and b Fx2 = 1 and 2b 2 and b HCF = b Example : Find the HCF of 2h2 and 4h. F 2h2 = 1 and 2h2 2 and h2 , h and 2h F4h = 1 and 4h 2 and 2h 4 and h HCF = 2h

  34. Factors Int 2 Find the HCF for these terms 8w • (a) 16w and 24w • 9y2 and 6y • (c) 4h and 12h2 • (d) ab2 and a2b 3y 4h ab

  35. Factors Int 2 Now try Exercise 3 Q3 and Q4 Ch5 (page 52)

  36. Int 2 Q1. Remove the brackets (a) a (4y – 3x) = (b) (x + 5)(x - 5) = Starter Questions Q2. For the line y = -x + 5, find the gradient and where it cuts the y axis. Q3. Find the highest common factor for p2q and pq2.

  37. Factorising Int 2 Using Factors Learning Intention Success Criteria • To identify the HCF for given terms. • To show how to factorise terms using the Highest Common Factor and one bracket term. • Factorise terms using the HCF and one bracket term.

  38. Check by multiplying out the bracket to get back to where you started Factorising Int 2 Factorise 3x + 15 Example 1. Find the HCF for 3x and 15 3 2. HCF goes outside the bracket 3( ) • To see what goes inside the bracket • divide each term by HCF 3x ÷ 3 = x 15 ÷ 3 = 5 3( x + 5 )

  39. Check by multiplying out the bracket to get back to where you started Factorising Int 2 Factorise 4x2 – 6xy Example 1. Find the HCF for 4x2 and 6xy 2x 2. HCF goes outside the bracket 2x( ) • To see what goes inside the bracket • divide each term by HCF 4x2 ÷ 2x =2x 6xy ÷ 2x = 3y 2x( 2x- 3y )

  40. Factorising Int 2 Factorise the following : 3(x + 2) • (a) 3x + 6 • 4xy – 2x • 6a + 7a2 • (d) y2 - y Be careful ! 2x(y – 1) a(6 + 7a) y(y – 1)

  41. Factorising Int 2 Now try Exercise 4 Start at Q2 Ch5 (page 53)

  42. Int 2 Q1. Remove the brackets (a) a (8 – 3x + 6a) = Starter Questions Q2. Factorise 3x2 – 6x Q3. Write down the first 10 square numbers.

  43. Difference of Two Squares Int 2 Learning Intention Success Criteria • Recognise when we have a difference of two squares. • To show how to factorise the special case of the difference of two squares. • Factorise the difference of two squares.

  44. Difference of Two Squares Int 2 When we have the special case that 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 Second square term Difference

  45. Difference of Two Squares Int 2 Check by multiplying out the bracket to get back to where you started a2 – b2 First square term Second square term Difference This factorises to ( a + b )( a – b ) Two brackets the same except for + and a -

  46. Difference of Two Squares Int 2 Keypoints Format a2 – b2 Always the difference sign - ( a + b )( a – b )

  47. Difference of Two Squares Int 2 Factorise using the difference of two squares (x + y )( x – y ) • (a) x2 – y2 • w2 – z2 • 9a2 – b2 • (d) 16y2 – 100k2 ( w + z )( w – z ) ( 3a + b )( 3a – b ) ( 4y + 10k )( 4y – 10k )

  48. Difference of Two Squares Int 2 Trickier type of questions to factorise. Sometimes we need to take out a common And the use the difference of two squares. Example Factorise 2a2 - 18 2(a2 - 9) First take out common factor Now apply the difference of two squares 2( a + 3 )( a – 3 )

  49. Difference of Two Squares Int 2 Factorise these trickier expressions. 6(x + 2 )( x – 2 ) • (a) 6x2 – 24 • 3w2 – 3 • 8 – 2b2 • (d) 27w2 – 12 3( w + 1 )( w – 1 ) 2( 2 + b )( 2 – b ) 3(3 w + 2 )( 3w – 2 )

  50. Difference of Two Squares Int 2 Now try Exercise 5 Ch5 (page 54)

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