S3_3_Algebra - Algebraic Operations S3.3 Removing Brackets...

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Unformatted text preview: Algebraic Operations S3.3 Removing Brackets of the type 2(x-1) www.mathsrevision.com 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) Pair of brackets (x+3)(x+4) Squaring Brackets (x+4)2 More equations (x+2)2 =x2 +64 Feb 13, 2012 Created by Mr. [email protected] S3.3 Starter Questions Q1. Is the following true or false (3a) x 5a = 15a (a) (6x) x (7y) =42xy Are the 2 answers the same ? (a) (3h)2 = (b) (3d)2 = Q3. Q4. Feb 13, 2012 www.mathsrevision.com (a) Q2. Spilt 64 pounds into the ratio 7:9 Explain why 0.0675 is equal to 6.75x102 Created by Mr. [email protected] S3.3 Removing a Single Bracket Learning Intention Success Criteria 1. Understand the keypoints of multiplying out an expression with a single bracket and a positive number outside the bracket. 2. Be able multiply out a expression with a single bracket. 1. To show how to multiply out (remove) a single bracket with a positive number outside the bracket. www.mathsrevision.com Feb 13, 2012 Created by Mr. [email protected] Simplifying Algebraic Expressions Algebra www.mathsrevision.com We can only add and subtract " like terms " Reminder ! x + x + x + x = 4x 9p -7p = 2p 3a + b - a + 6b = 2a + 7b 10 + 6 w - 1 = 9 + 6w 2 2 2 2 x + x + x = 3x 13 Feb 2012 Created by Mr. Lafferty Maths Dept. Simplifying Algebraic Expressions Algebra www.mathsrevision.com Reminder ! Multiplying terms 7 a = 7a w 5 = 5w 2 b b = b ( NOT 2b ) 2 2m 4m = 8m ( NOT 8m ) 13 Feb 2012 Created by Mr. Lafferty Maths Dept. S3.3 Removing a Single Bracket Example 1 www.mathsrevision.com 3(b + 5) = 3b + 15 Example 2 4(w 2) = 4w 8 Feb 13, 2012 Created by Mr. [email protected] S3.3 Removing a Single Bracket Example 3 www.mathsrevision.com 2(y 1) = Example 4 2y 2 7(w 6) = 7w 42 Feb 13, 2012 Created by Mr. [email protected] S3.3 Removing a Single Bracket Example 5 www.mathsrevision.com 8(x + 3) = 8x + 24 Example 6 4(3 2m) = Feb 13, 2012 Created by Mr. [email protected] 12 8m S3.3 Removing a Single Bracket Example 7 : Write down an expression for the amount indicate below in two difference ways. x+3 www.mathsrevision.com x+3 x+3 Feb 13, 2012 3(x + 3) =3x + 9 Created by Mr. [email protected] S3.3 Removing a Single Bracket Example 8 : Write down an expression for the amount indicate below in two difference ways. y2 www.mathsrevision.com y2 y2 Feb 13, 2012 y2 y2 5(y 2) = 5y 10 Created by Mr. [email protected] S3.3 Removing a Single Bracket Now try Ex 2.1 Ch3 MIA (page 43) www.mathsrevision.com 13 Feb 2012 Created by Mr. [email protected] S3.3 Starter Questions Q1. Explain why the following are True or False www.mathsrevision.com (a) Q2. 3y x 5y = 30y (b) 6q x (4q) = 24q2 Is the following true 2a( b a) = 2ab +4a Q3. Write down the two numbers that multiply to give 8 and subtract to give 2. Created by Mr. [email protected] Feb 13, 2012 S3.3 Removing a Single Bracket Learning Intention Success Criteria 1. Understand the keypoints of multiplying out an expression with a single bracket with a letter outside. 2. Be able multiply out a expression with a single bracket with a letter outside the bracket. 1. To show how to multiply out (remove) a single bracket with a letter outside the bracket. www.mathsrevision.com Feb 13, 2012 Created by Mr. [email protected] S3.3 Removing a Single Bracket Example 1 www.mathsrevision.com a(b + 5) = ab + 5a Example 2 p(w 2) = pw 2p Feb 13, 2012 Created by Mr. [email protected] S3.3 Removing a Single Bracket Example 3 www.mathsrevision.com y(y 1) = Example 4 y2 y 7w(w 3) = 7w2 21w Feb 13, 2012 Created by Mr. [email protected] S3.3 Removing a Single Bracket Example 5 www.mathsrevision.com a(a2 + 3b) = a3 + 3ab Example 6 y2(3y 2c) = Feb 13, 2012 Created by Mr. [email protected] 3y3 2cy2 S3.3 Removing a Single Bracket Now try Ex 2.2 Ch3 MIA (page 44) www.mathsrevision.com 13 Feb 2012 Created by Mr. [email protected] S3.3 Starter Questions Q1. Multiply out. (a) 3y(x y) = (b) 6q2 (2 4q) = www.mathsrevision.com Q2. Explain your working to show that if we split 24 into the ratio 1:5 the answer is 4:20 Q3. Feb 13, 2012 Writing out in full 5.2 x 103 to get 0.052 Is the correct? Created by Mr. [email protected] S3.3 Removing a Single Bracket Learning Intention Success Criteria 1. Understand the keypoints of multiplying out an expression with a single bracket with a negative outside. 2. Be able multiply out a expression with a single bracket with a negative outside the bracket. 1. To show how to multiply out (remove) a single bracket with a negative outside the bracket. www.mathsrevision.com Feb 13, 2012 Created by Mr. [email protected] S3.3 Removing a Single Bracket Example 1 www.mathsrevision.com (x + y) = Example 2 x y 5(a 7) = 5a + 35 Feb 13, 2012 Created by Mr. [email protected] S3.3 Removing a Single Bracket Example 3 Tidy Up www.mathsrevision.com 7 + 3(4 y) = Example 4 7 + 12 3y = 19 3y Tidy Up 9 3(8 y) = Feb 13, 2012 Created by Mr. [email protected] 9 24 + 3y = 15 + 3y S3.3 Removing a Single Bracket Example 9 www.mathsrevision.com 12 10 Find my Area x Find my Area (x 2) A = L x B A = 12x Feb 13, 2012 A = L x B A = 10(x 2) Created by Mr. [email protected] S3.3 Removing a Single Bracket Example 9 : Find area of the orange border 12 Area =Big Rec Small Rec (x 2) www.mathsrevision.com x 10 = 12x 10(x 2) = 12x 10x + 20 = 2x + 20 Feb 13, 2012 Created by Mr. [email protected] S3.3 Removing a Single Bracket Now try Ex 3.1 Ch3 MIA (page 44) www.mathsrevision.com 13 Feb 2012 Created by Mr. [email protected] S3.3 Starter Questions Q1. Find the area of the second shape given the first has area 7w 14 (w 2) (w 5) 7 3 www.mathsrevision.com Q2. Q3. Feb 13, 2012 Split 48 into the ratio 1:3 In standard form 18294000 is 1.8x107 is the correct? Created by Mr. [email protected] S3.3 Removing Two Single Brackets Learning Intention Success Criteria 1. Understand the keypoints of multiplying out an expression with two single brackets. 2. Be able multiply out a expression with two single brackets and the simplify. www.mathsrevision.com 1. To show how to multiply out (remove) and simplify an expression that has two single brackets. Feb 13, 2012 Created by Mr. [email protected] Removing Two Single Brackets S3.3 Example 1 www.mathsrevision.com Tidy Up 4(m 3) (m + 2) = Example 2 4m 12 m 2 = 3m 14 Tidy Up 7(y 1) 2(y + 4) = Feb 13, 2012 Created by Mr. [email protected] 7y 7 2y 8 = 5y 15 Removing Two Single Brackets S3.3 Example 3 : Find an expression for the difference in the areas. www.mathsrevision.com (y 5) 7 A = 7(y 4) A = 4(y 5) 4 (y 4) Calculate 3 x 14 8 difference if y = = 34 14 Feb 13, 2012 Difference = 7(y 4) 4(y 5) = 7y 28 4y + 20 = 3y 8 Created by Mr. [email protected] Removing Two Single Brackets S3.3 Example 3 : Find an expression for the difference in the areas. www.mathsrevision.com (x + 1) 9 A = 9(x + 3) A = 7(x + 1) 7 x + 3 Calculate 2 x 4 + 20 difference if x = =28 4 Feb 13, 2012 Difference = 9(x + 3) 7(x + 1) = 9x + 27 7x 7 = 2x + 20 Created by Mr. [email protected] S3.3 Removing Two Single Brackets www.mathsrevision.com Now try Ex 3.2 Ch3 MIA (page 46) 13 Feb 2012 Created by Mr. [email protected] S3.3 Starter Questions Q1. Solve for x (a) Q2. x + 3 = 8 (b) 2x 14 = 50 www.mathsrevision.com Is this statement true (x 1) 3(x + 1) = 2x Q3. Feb 13, 2012 1 1 3 -1 2 3 Created by Mr. [email protected] S3.3 Equations and brackets Learning Intention Success Criteria 1. Be able to multiply out brackets and solve equations. www.mathsrevision.com 1. To show how to solve equations that have bracket terms. Feb 13, 2012 Created by Mr. [email protected] Equations and brackets S3.3 Multiply out the bracket first and then solve. www.mathsrevision.com Example 1 Feb 13, 2012 5(x 3) = 25 5x 15 = 25 5x = 25 + 15 = 40 x = 40 5 = 8 Created by Mr. [email protected] Equations and brackets S3.3 Example 2 www.mathsrevision.com Substitute value into original equation to check answer 18 6(x 2) = 6x 12 6x 3x 12 3x 12 18 3(x + 1) = 3x + 3 = 3 = 3 Solve as normal 3x = 3 + 12 3x = 15 Feb 13, 2012 Created by Mr. [email protected] x = 15 (3) = 5 Equations and brackets S3.3 Example 3 www.mathsrevision.com Substitute value into original equation to check answer 6 3(2 + 2y) = 6 6y 6 6y 6 6y + y 6 6 (y + 2) = 6 y 2 = 4 y = 4 Tidy up RHS Solve as normal 6 5y = 4 5y = 10 y = 10 (5) = 2 Feb 13, 2012 Created by Mr. [email protected] S3.3 Equations and brackets www.mathsrevision.com Now try Ex 4.1 & Ex 4.2 Ch3 MIA (page 47) 13 Feb 2012 Created by Mr. [email protected] S3.3 Starter Questions Q1. Does 579300 = 5.793x105 Explain why the answer to 4(w + 2) = 6(w + 1) is w = 1 www.mathsrevision.com Q2. Q3. 1 7 2 3 4 Created by Mr. [email protected] Feb 13, 2012 Int 2 Removing Double Brackets Learning Intention Success Criteria 1. Understand the keypoints of multiplying out paired brackets. 2. Be able to multiply out paired brackets using a multiplication table. 1. To show how to multiply out a pair of brackets using a multiplication table. www.mathsrevision.com Feb 13, 2012 Created by Mr. [email protected] S3.3 Removing What is the total area of the downstairs house? Double Brackets Below is a floor plan of the down stairs of a house. www.mathsrevision.com y 5 Bedroom Kitchen y 3 Feb 13, 2012 Living 2 Room y 5y Bathroom 3y 15 Tidy up ! y2 + 8y + 15 Created by Mr. [email protected] S3.3 are used in AREA Multiplication table for brackets www.mathsrevision.com Removing Double Brackets Double brackets y(y + 3)(y + 5) + 3 y + 5 y2 +5y calculations Tidy up ! +3y +15 Feb 13, 2012 Created by Mr. [email protected] y2 + 8y + 15 S3.3 Example 2 www.mathsrevision.com Removing Double Brackets Be careful with 2x 1 x + 3 (2x 1)(x + 3) 2x2 x +6x 3 Tidy up ! the negative signs 2x2 + 5x 3 Feb 13, 2012 Created by Mr. [email protected] S3.3 Removing Double Brackets Just a bigger Example 3 Multiplication Table www.mathsrevision.com (x + 4)(x22 + 3x + 2) x + 4 x + 3x + 2 x3 +4x2 Feb 13, 2012 +3x2 +12x +2x +8 Tidy up ! x3 + 7x2 + 14x + 8 Created by Mr. [email protected] S3.3 Removing Double Bracket First column in each Question www.mathsrevision.com Now try Ex 5.1 Ch3 MIA (page 49) 13 Feb 2012 Created by Mr. [email protected] S3.3 Starter Questions Q1. 0.00897 is in scientific notation is this statement True or False? Q2. Why does (w + 2)(w + 1) = w2 + 3w + 2 www.mathsrevision.com Q3. Feb 13, 2012 1 2 5 4 7 Created by Mr. [email protected] Int 2 Removing Double Brackets Learning Intention Success Criteria 1. Understand the keypoints of multiplying out double brackets using FOIL. 2. Be able to multiply out double brackets using FOIL. 1. To show how to multiply out a pair of brackets using FOIL. www.mathsrevision.com Feb 13, 2012 Created by Mr. [email protected] S3.3 Removing Double Brackets FOIL Method We can use the F O I L method to multiply out DOUBLE brackets. www.mathsrevision.com Simply remember the word Multiply First 2 Feb 13, 2012 F O IL Multiply Outside 2 Created by Mr. [email protected] Multiply Last 2 Multiply Inside 2 S3.3 Removing Try again using multiplication table Double Brackets Example 1 : Multiply out the brackets and Simplify www.mathsrevision.com (x + 1)(x + 2) 1. 2. Feb 13, 2012 Write down F Tidy up ! Created by Mr. [email protected] x2 + 2x + x + 2 x2 + 3x + 2 O I L S3.3 Removing Try again using multiplication table Double Brackets Example 2 : Multiply out the brackets and Simplify www.mathsrevision.com (x 1)(x + 2) 1. 2. Feb 13, 2012 Write down Tidy up ! F Created by Mr. [email protected] x2 + 2x x 2 x2 + x 2 O I L Squaring Brackets S3.3 www.mathsrevision.com FOIL method is very limited therefore from now on we will use the multiplication table method. Feb 13, 2012 Created by Mr. [email protected] S3.3 Removing Double Brackets (x + 1)(x 2) (x 1)(x 2) (x + 3)(x + 2) (x 3)(x + 2) (x + 3)(x 2) Feb 13, 2012 Created by Mr. [email protected] x2 x 2 x2 3x + 2 x2 + 5x + 6 x2 x 6 x2 + x 6 www.mathsrevision.com S3.3 Removing Double Brackets First column in each Question www.mathsrevision.com Now try Ex 5.2 Ch3 MIA (page 50) 13 Feb 2012 Created by Mr. [email protected] S3.3 Starter Questions Q1. Does 2 + 3 x 4 = 20 Explain ? Q2. Multiply out using a multiplication table (w + 2)(w2 2w + 1) Which sums are true www.mathsrevision.com Q3. 1 1 1 = 2 2 4 Feb 13, 2012 Created by Mr. [email protected] 1 y 1 1 = 2 y y Int 2 Squaring Brackets Learning Intention Success Criteria 1. Understand the keypoints of multiplying out squared brackets using Multiplication Table. 2. Be able to multiply out squared brackets using Multiplication Table. brackets using www.mathsrevision.com 1. To show how to multiply out squared brackets using Multiplication Table. Feb 13, 2012 Created by Mr. [email protected] Squaring Brackets S3.3 Example 1 : (x + 1)2 www.mathsrevision.com Multiply out the brackets and Simplify = (x + 1)(x + 1) x + 1 x + 1 (x + 1)(x + 1) x2 + x + x + 1 Tidy up ! x2 + 2x + 1 Feb 13, 2012 Created by Mr. [email protected] Squaring Brackets S3.3 Example 2 : (x + y)2 www.mathsrevision.com x(x + y)(x + y) + y x + y x2 +xy Multiply out the brackets and Simplify = (x + y)(x + y) Tidy up ! +xy + y2 Feb 13, 2012 x2 + 2xy +y2 Created by Mr. [email protected] Squaring Brackets S3.3 (x 1)2 www.mathsrevision.com x2 2x + 1 x2 4x + 4 x2 8x + 16 4x2 4x + 1 Created by Mr. [email protected] (x 2)2 (x 4)2 (2x 1) Feb 13, 2012 S3.3 Squaring Brackets Now try Ex 6.1 & 6.2 First column in each Question www.mathsrevision.com Ch3 MIA (page 50) 13 Feb 2012 Created by Mr. [email protected] S3.3 Starter Questions Q1. Does (x + 3)2 = x2 + 9 Q2. The sun is 92 million mile away from the earth. Write this in standard form. www.mathsrevision.com Q3. Is it true 1 1 5 = 2 3 6 Created by Mr. [email protected] Feb 13, 2012 S3.3 Equations and brackets Learning Intention Success Criteria 1. Be able to solve equations that contain paired or squared brackets. www.mathsrevision.com 1. To show how to solve equations that have paired or squared brackets. Feb 13, 2012 Created by Mr. [email protected] Equations and brackets S3.3 Multiply out the bracket first 289 289 and then solve. www.mathsrevision.com (x + 2)2 = x2 + 82 X table (x + 2)(x + 2) = x2 + 64 Substitute value into original equation to x2 + 4x + 4 = x2 + 64 check answer 4x + 4 = 64 4x = 60 x = 60 4 = 15 Feb 13, 2012 Created by Mr. [email protected] Example 1 Equations and brackets S3.3 Multiply out the bracket first and then solve. www.mathsrevision.com 35 35 Example 2 Substitute value into original equation to check answer m(m + 2) = m2 + 10 m2 + 2m = m2 + 10 2m = 10 m = 10 2 = 5 Created by Mr. [email protected] Feb 13, 2012 Equations and brackets S3.3 Example 3 www.mathsrevision.com Substitute value into original equation to check answer 18 6(x 2) = 6x 12 6x 3x 12 3x 12 18 3(x + 1) = 3x + 3 = 3 = 3 Solve as normal 3x = 3 + 12 3x = 15 Feb 13, 2012 Created by Mr. [email protected] x = 15 (3) = 5 Equations and brackets S3.3 Example 4 www.mathsrevision.com Substitute value into original equation to check answer 6 3(2 + 2y) = 6 6y 6 6y 6 6y + y 6 6 (y + 2) = 6 y 2 = 4 y = 4 Tidy up RHS Solve as normal 6 5y = 4 5y = 10 y = 10 (5) = 2 Feb 13, 2012 Created by Mr. [email protected] Equations and brackets S3.3 Example 5 : The two areas are equal. Find the value of x. www.mathsrevision.com (x + 2) A = (x + 2) 2 A = (x + 7)(x 2) 400 (x 2) X table (x + 7) 400 (x + 2) (x + 2)2 = (x + 7) (x 2) x2 + 4x + 4 = x2 + 5x 14 4x + 4 = 5x 14 4 = x 14 x = 18 Created by Mr. [email protected] Substitute value into original equation to check answer Feb 13, 2012 Solve in the usual way S3.3 Equations and brackets www.mathsrevision.com Now try Ex 7.1 & 7.2 Ch3 MIA (page 55) 13 Feb 2012 Created by Mr. [email protected] ...
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This note was uploaded on 02/13/2012 for the course MAT 116 A 211 taught by Professor Skinner during the Spring '11 term at University of Phoenix.

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