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www.mathsrevision.com S3.3 Algebraic Operations Removing Brackets of the type 2(x-1) Pair of brackets (x+3)(x+4) Removing Brackets of the type y(3+a) Removing Brackets of the type -2(x+1) Equations and brackets 3(x-4) = 2(x-1) More equations (x+2) 2 =x 2 +64 4-May-15Created by Mr. Lafferty@mathsrevision.com Squaring Brackets (x+4) 2

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4-May-15 Starter Questions www.mathsrevision.com Q1.Is the following true or false (a)(-3a) x 5a = -15a (a) (-6x) x (-7y) =-42xy Q2.Are the 2 answers the same ? (a)(-3h) 2 =(b)-(3d) 2 = S3.3 Created by Mr. Lafferty@mathsrevision.com Q3.Spilt £64 pounds into the ratio 7:9 Q4.Explain why 0.0675 is equal to 6.75 x 10 -2

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4-May-15 www.mathsrevision.com Learning Intention Success Criteria 1.To show how to multiply out (remove) a single bracket with a positive number outside the bracket. 1.Understand the keypoints of multiplying out an expression with a single bracket and a positive number outside the bracket. S3.3 2.Be able multiply out a expression with a single bracket. Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket

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4-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Algebra Simplifying Algebraic Expressions Reminder ! We can only add and subtract “ like terms “

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4-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Algebra Simplifying Algebraic Expressions Reminder ! Multiplying terms ( NOT 2b ) ( NOT 8m )

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www.mathsrevision.com S3.3 3(b + 5) =3b + 15 Example 1 4(w - 2) =4w - 8 Example 2 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket

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www.mathsrevision.com S3.3 2(y - 1) =2y - 2 Example 3 7(w - 6) =7w - 42 Example 4 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket

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www.mathsrevision.com S3.3 8(x + 3) =8x + 24 Example 5 4(3 -2m) =12 - 8m Example 6 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket

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www.mathsrevision.com S3.3 Example 7 : Write down an expression for the amount indicate below in two difference ways. 3(x + 3)=3x + 9 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket x+3

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www.mathsrevision.com S3.3 Example 8 : Write down an expression for the amount indicate below in two difference ways. 5(y - 2)= 5y - 10 4-May-15 Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket y-2

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4-May-15Created by Mr. Lafferty@mathsrevision.com Now try Ex 2.1 Ch3 MIA (page 43) www.mathsrevision.com S3.3 Removing a Single Bracket

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4-May-15 Starter Questions www.mathsrevision.com Q1. Explain why the following are True or False (a)-3y x 5y =-30y (b)-6q x (-4q) = 24q 2 Q2. Is the following true -2a( b – a) = -2ab +4a S3.3 Created by Mr. Lafferty@mathsrevision.com Q3.Write down the two numbers that multiply to give 8 and subtract to give 2.

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4-May-15 www.mathsrevision.com Learning Intention Success Criteria 1.To show how to multiply out (remove) a single bracket with a letter outside the bracket. 1.Understand the keypoints of multiplying out an expression with a single bracket with a letter outside. S3.3 2.Be able multiply out a expression with a single bracket with a letter outside the bracket. Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket

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www.mathsrevision.com S3.3 a(b + 5) =ab + 5a Example 1 p(w - 2) =pw - 2p Example 2 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket

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www.mathsrevision.com S3.3 y(y - 1) =y2y2 - y Example 3 7w(w - 3) =7w 2 - 21w Example 4 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket

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www.mathsrevision.com S3.3 a(a 2 + 3b) =a3a3 + 3ab Example 5 y 2 (3y -2c) =3y 3 - 2cy 2 Example 6 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket

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4-May-15Created by Mr. Lafferty@mathsrevision.com Now try Ex 2.2 Ch3 MIA (page 44) www.mathsrevision.com S3.3 Removing a Single Bracket

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4-May-15 Starter Questions www.mathsrevision.com Q1.Multiply out. (a)3y(x - y) =(b)6q 2 (2 - 4q) = Q2.Explain your working to show that if we split 24 into the ratio 1:5 the answer is 4:20 S3.3 Created by Mr. Lafferty@mathsrevision.com Q3.Writing out in full 5.2 x 10 -3 to get 0.052 Is the correct?

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4-May-15 www.mathsrevision.com Learning Intention Success Criteria 1.To show how to multiply out (remove) a single bracket with a negative outside the bracket. 1.Understand the keypoints of multiplying out an expression with a single bracket with a negative outside. S3.3 2.Be able multiply out a expression with a single bracket with a negative outside the bracket. Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket

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www.mathsrevision.com S3.3 -(x + y) =-x - y Example 1 -5(a - 7) =-5a + 35 Example 2 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket

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www.mathsrevision.com S3.3 7 + 3(4 - y) =7 + 12 Example 3 9 - 3(8 - y) =9 - 24 Example 4 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket - 3y = 19 - 3y Tidy Up + 3y Tidy Up = -15 + 3y

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www.mathsrevision.com S3.3 A = 10(x – 2) Example 9 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket Find my Area (x - 2) 10 Find my Area x 12 A = 12x A = L x BA = L x B

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www.mathsrevision.com S3.3 Example 9 : Find area of the orange border 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing a Single Bracket (x - 2) 10 x 12 Area =Big Rec – Small Rec = 12x – 10(x - 2) = 12x – 10x + 20 = 2x + 20

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4-May-15Created by Mr. Lafferty@mathsrevision.com Now try Ex 3.1 Ch3 MIA (page 44) www.mathsrevision.com S3.3 Removing a Single Bracket

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4-May-15 Starter Questions www.mathsrevision.com Q1.Find the area of the second shape given the first has area 7w - 14 Q2.Split 48 into the ratio 1:3 S3.3 Created by Mr. Lafferty@mathsrevision.com Q3.In standard form 18294000 is 1.8x10 7 is the correct? (w - 2) 7 (w – 5) 3

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4-May-15 www.mathsrevision.com Learning Intention Success Criteria 1.To show how to multiply out (remove) and simplify an expression that has two single brackets. 1.Understand the keypoints of multiplying out an expression with two single brackets. S3.3 2.Be able multiply out a expression with two single brackets and the simplify. Created by Mr. Lafferty@mathsrevision.com Removing Two Single Brackets

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www.mathsrevision.com S3.3 4(m - 3) - (m + 2) =4m - 12 - m - 2 Example 1 Example 2 4-May-15Created by Mr. Lafferty@mathsrevision.com = 3m - 14 Tidy Up 7(y - 1) - 2(y + 4) =7y - 7 - 2y - 8 = 5y - 15 Tidy Up Removing Two Single Brackets

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www.mathsrevision.com S3.3 3 x 14 – 8 = 34 A = 4(y – 5) Example 3 : Find an expression for the difference in the areas. 4-May-15Created by Mr. Lafferty@mathsrevision.com (y - 5) 47 A = 7(y- 4) (y- 4) Difference = 7(y - 4) - 4(y – 5) = 7y - 28 - 4y + 20 = 3y - 8 Calculate difference if y = 14 Removing Two Single Brackets

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www.mathsrevision.com S3.3 2 x 4 + 20 =28 A = 7(x + 1) Example 3 : Find an expression for the difference in the areas. 4-May-15Created by Mr. Lafferty@mathsrevision.com (x + 1) 79 A = 9(x + 3) x + 3 Difference = 9(x + 3) - 7(x + 1) = 9x + 27 – 7x - 7 = 2x + 20 Calculate difference if x = 4 Removing Two Single Brackets

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4-May-15Created by Mr. Lafferty@mathsrevision.com Now try Ex 3.2 Ch3 MIA (page 46) www.mathsrevision.com S3.3 Removing Two Single Brackets

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4-May-15 Starter Questions www.mathsrevision.com Q1.Solve for x (a)x + 3 = 8(b)2x – 14 = 50 Q2.Is this statement true (x – 1) – 3(x + 1) = -2x S3.3 Created by Mr. Lafferty@mathsrevision.com Q3.

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4-May-15 www.mathsrevision.com Learning Intention Success Criteria 1.To show how to solve equations that have bracket terms. 1.Be able to multiply out brackets and solve equations. S3.3 Created by Mr. Lafferty@mathsrevision.com Equations and brackets

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www.mathsrevision.com S3.3 5(x - 3) = 25 5x Example 1 4-May-15Created by Mr. Lafferty@mathsrevision.com - 15 = 25 Multiply out the bracket first and then solve. Equations and brackets 5x = 25 + 15 x = 40 ÷ 5 = 40 = 8

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www.mathsrevision.com S3.3 6(x - 2) = 3(x + 1) 6x Example 2 4-May-15Created by Mr. Lafferty@mathsrevision.com - 12 = 3x Equations and brackets = 3 + 3 6x – 3x -12 3x -12 = 3 3x = 3 + 12 = 15 x = 15 ÷ (3) = 5 Solve as normal Substitute value into original equation to check answer 18

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www.mathsrevision.com S3.3 -3(2 + 2y) = 6 – (y + 2) -6 Example 3 4-May-15Created by Mr. Lafferty@mathsrevision.com - 6y = 6 Equations and brackets = 4 - y - y - 2 -6 – 6y -6 - 6y + y = 4 -6 – 5y = 4 – 5y = 10 y = 10 ÷ (-5) = -2 Tidy up RHS Solve as normal Substitute value into original equation to check answer 6 6

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4-May-15Created by Mr. Lafferty@mathsrevision.com Now try Ex 4.1 & Ex 4.2 Ch3 MIA (page 47) www.mathsrevision.com S3.3 Equations and brackets

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4-May-15 Starter Questions www.mathsrevision.com Q1.Does 579300 = 5.793x10 5 Q2.Explain why the answer to 4(w + 2) = 6(w + 1) is w = 1 S3.3 Created by Mr. Lafferty@mathsrevision.com Q3.

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4-May-15 www.mathsrevision.com Learning Intention Success Criteria 1.To show how to multiply out a pair of brackets using a multiplication table. 1.Understand the keypoints of multiplying out paired brackets. Int 2 2.Be able to multiply out paired brackets using a multiplication table. Created by Mr. Lafferty@mathsrevision.com Removing Double Brackets

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www.mathsrevision.com S3.3 Below is a floor plan of the down stairs of a house. 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing Double Brackets y 5 y 3 Tidy up ! y 2 + 8y + 15 Living Room Kitchen Bathroom Bedroom What is the total area of the downstairs house? y 2 5y 3y15

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www.mathsrevision.com S3.3 Multiplication table for brackets 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing Double Brackets (y + 3)(y + 5)y+ 3y+ 5 Double brackets are used in AREA calculations +5y +15+3y y 2 Tidy up ! y 2 + 8y + 15

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www.mathsrevision.com S3.3 Example 2 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing Double Brackets (2x - 1)(x + 3)2x- 1x+ 3 Be careful with the negative signs +6x -3 -x 2x 2 Tidy up ! 2x 2 + 5x - 3

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www.mathsrevision.com S3.3 Example 3 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing Double Brackets (x + 4)(x 2 + 3x + 2) x+ 4 x2x2 + 3x Just a bigger Multiplication Table +3x 2 +12x +4x 2 x 3 Tidy up ! x 3 + 2 +2x +8 + 7x 2 + 14x+ 8

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4-May-15Created by Mr. Lafferty@mathsrevision.com Now try Ex 5.1 First column in each Question Ch3 MIA (page 49) www.mathsrevision.com S3.3 Removing Double Bracket

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4-May-15 Starter Questions www.mathsrevision.com Q1.0.00897 is in scientific notation is this statement True or False? Q2.Why does (w + 2)(w + 1) = w 2 + 3w + 2 S3.3 Created by Mr. Lafferty@mathsrevision.com Q3.

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4-May-15 www.mathsrevision.com Learning Intention Success Criteria 1.To show how to multiply out a pair of brackets using FOIL. 1.Understand the keypoints of multiplying out double brackets using FOIL. Int 2 2.Be able to multiply out double brackets using FOIL. Created by Mr. Lafferty@mathsrevision.com Removing Double Brackets

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www.mathsrevision.com S3.3 We can use the F O I L method to multiply out DOUBLE brackets. 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing Double Brackets Simply remember the word FOIL Multiply First 2 Multiply Last 2 Multiply Outside 2 Multiply Inside 2 FOIL Method

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www.mathsrevision.com S3.3 (x + 1)(x + 2) x2x2 + 2x Example 1 : Multiply out the brackets and Simplify 4-May-15Created by Mr. Lafferty@mathsrevision.com 1.Write down F O I L + x + 2 2.Tidy up ! x 2 + 3x + 2 Removing Double Brackets Try again using multiplication table

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www.mathsrevision.com S3.3 (x - 1)(x + 2) x2x2 + 2x Example 2 : Multiply out the brackets and Simplify 4-May-15Created by Mr. Lafferty@mathsrevision.com Removing Double Brackets 1.Write down F O I L - x - 2 2.Tidy up ! x 2 + x - 2 Try again using multiplication table

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www.mathsrevision.com S3.3 FOIL method is very limited therefore from now on we will use the multiplication table method. 4-May-15Created by Mr. Lafferty@mathsrevision.com Squaring Brackets

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www.mathsrevision.com S3.3 4-May-15Created by Mr. Lafferty@mathsrevision.com (x + 1)(x - 2) Removing Double Brackets (x - 1)(x - 2) (x + 3)(x + 2) (x - 3)(x + 2) (x + 3)(x - 2) x 2 - x - 2 x 2 - 3x + 2 x 2 + 5x + 6 x 2 - x - 6 x 2 + x - 6

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4-May-15Created by Mr. Lafferty@mathsrevision.com Now try Ex 5.2 First column in each Question Ch3 MIA (page 50) www.mathsrevision.com S3.3 Removing Double Brackets

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4-May-15 Starter Questions www.mathsrevision.com Q1.Does 2 + 3 x 4 = 20 Explain ? Q2.Multiply out using a multiplication table (w + 2)(w 2 - 2w + 1) S3.3 Created by Mr. Lafferty@mathsrevision.com Q3.Which sums are true

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4-May-15 www.mathsrevision.com Learning Intention Success Criteria 1.To show how to multiply out squared brackets using Multiplication Table. 1.Understand the keypoints of multiplying out squared brackets using. 1.Understand the keypoints of multiplying out squared brackets using Multiplication Table. Int 2 2.Be able to multiply out squared brackets using. 2.Be able to multiply out squared brackets using Multiplication Table. Created by Mr. Lafferty@mathsrevision.com Squaring Brackets

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www.mathsrevision.com S3.3 = (x + 1)(x + 1) Example 1 : (x + 1) 2 Multiply out the brackets and Simplify 4-May-15Created by Mr. Lafferty@mathsrevision.com Squaring Brackets (x + 1)(x + 1)x+ 1x+ 1 + x + 1+ x x 2 Tidy up ! x 2 + 2x + 1

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www.mathsrevision.com S3.3 Example 2 : (x + y) 2 Multiply out the brackets and Simplify 4-May-15Created by Mr. Lafferty@mathsrevision.com Squaring Brackets = (x + y)(x + y) (x + y)(x + y)x+ yx+ y +xy + y 2 +xy x 2 Tidy up ! x 2 + 2xy +y 2

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www.mathsrevision.com S3.3 4-May-15Created by Mr. Lafferty@mathsrevision.com (x - 1) 2 (x - 2) 2 (x - 4) 2 (2x – 1) x 2 - 2x + 1 x 2 - 4x + 4 x 2 - 8x + 16 4x 2 - 4x + 1 Squaring Brackets

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4-May-15Created by Mr. Lafferty@mathsrevision.com Now try Ex 6.1 & 6.2 First column in each Question Ch3 MIA (page 50) www.mathsrevision.com S3.3 Squaring Brackets

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4-May-15 Starter Questions www.mathsrevision.com Q1.Does (x + 3) 2 = x 2 + 9 Q2.The sun is 92 million mile away from the earth. Write this in standard form. S3.3 Created by Mr. Lafferty@mathsrevision.com Q3.Is it true

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4-May-15 www.mathsrevision.com Learning Intention Success Criteria 1.To show how to solve equations that have paired or squared brackets. 1.Be able to solve equations that contain paired or squared brackets. S3.3 Created by Mr. Lafferty@mathsrevision.com Equations and brackets

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www.mathsrevision.com S3.3 (x + 2) 2 = x 2 + 8 2 (x + 2)(x + 2) Example 1 4-May-15Created by Mr. Lafferty@mathsrevision.com = x 2 + 64 Multiply out the bracket first and then solve. Equations and brackets x 2 + 4x + 4 4x = 60 x = 60 ÷ 4 = 15 X table = x 2 + 64 4x + 4 = 64 Substitute value into original equation to check answer 289

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www.mathsrevision.com S3.3 m(m + 2) = m 2 + 10 m 2 + 2m Example 2 4-May-15Created by Mr. Lafferty@mathsrevision.com = m 2 + 10 Multiply out the bracket first and then solve. Equations and brackets 2m m = 10 ÷ 2 = 5 = 10 Substitute value into original equation to check answer 35

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www.mathsrevision.com S3.3 6(x - 2) = 3(x + 1) 6x Example 3 4-May-15Created by Mr. Lafferty@mathsrevision.com - 12 = 3x Equations and brackets = 3 + 3 6x – 3x -12 3x -12 = 3 3x = 3 + 12 = 15 x = 15 ÷ (3) = 5 Solve as normal Substitute value into original equation to check answer 18

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www.mathsrevision.com S3.3 -3(2 + 2y) = 6 – (y + 2) -6 Example 4 4-May-15Created by Mr. Lafferty@mathsrevision.com - 6y = 6 Equations and brackets = 4 - y - y - 2 -6 – 6y -6 - 6y + y = 4 -6 – 5y = 4 – 5y = 10 y = 10 ÷ (-5) = -2 Tidy up RHS Solve as normal Substitute value into original equation to check answer 6 6

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www.mathsrevision.com S3.3 Solve in the usual way A = (x + 7)(x – 2) Example 5 : The two areas are equal. Find the value of x. 4-May-15Created by Mr. Lafferty@mathsrevision.com (x + 7) (x – 2) (x + 2) A = (x + 2) 2 (x + 2) 2 = (x + 7) (x - 2) x 2 + 4x + 4 x = 18 X table Equations and brackets = x 2 + 5x - 14 4x + 4= 5x - 14 4= x - 14 Substitute value into original equation to check answer 400

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4-May-15Created by Mr. Lafferty@mathsrevision.com Now try Ex 7.1 & 7.2 Ch3 MIA (page 55) www.mathsrevision.com S3.3 Equations and brackets

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