algebraic operations
DESCRIPTION
Algebraic Operations. Removing Brackets of the type 2(x-1). 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 =x 2 +64. S3.3. - PowerPoint PPT PresentationTRANSCRIPT
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S3.3
Algebraic Algebraic OperationsOperations
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 =x2 +64Apr 22, 2023Apr 22, 2023 Created by Mr. Created by Mr.
[email protected]@mathsrevision.com
Squaring Brackets (x+4)2
Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questionsww
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aths
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sion.
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
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Q3. Spilt £64 pounds into the ratio 7:9
Q4. Explain why 0.0675 is equal to 6.75x10-2
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m Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To show how to multiply out (remove) a single bracket with a positive number outside the bracket.
1.1. Understand the keypoints Understand the keypoints of multiplying out an of multiplying out an expression with a single expression with a single bracket and a positive bracket and a positive number outside the number outside the bracket.bracket.
S3.3
2.2. Be able multiply out a Be able multiply out a expression with a single expression with a single bracket.bracket.
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Removing aRemoving aSingle BracketSingle Bracket
22 Apr 202322 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.www.
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mAlgebraAlgebra
Simplifying Algebraic Expressions
Reminder !We can only add and subtract “ like terms “
x x x x 4x9 7p p 2p
10 6 1w 3 6a b a b
2 2 2x x x
2 7a b9 6w
23x
22 Apr 202322 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.www.
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mAlgebraAlgebra
Simplifying Algebraic Expressions
Reminder !Multiplying terms
7 a 7a5w 5w
2 4m m b b 2b
28m( NOT 2b )
( NOT 8m )
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S3.3
3(b + 5) =3b + 15Example 1
4(w - 2) =4w - 8Example 2
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Removing aRemoving aSingle BracketSingle Bracket
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S3.3
2(y - 1) =2y - 2Example 3
7(w - 6) =7w- 42Example 4
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Removing aRemoving aSingle BracketSingle Bracket
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S3.3
8(x + 3) =8x+ 24Example 5
4(3 -2m) = 12 - 8mExample 6
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Removing aRemoving aSingle BracketSingle Bracket
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S3.3
Example 7 : Write down an expressionfor the amount indicate belowin two difference ways.
3(x + 3) =3x + 9
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Removing aRemoving aSingle BracketSingle Bracket
x+3x+3
x+3
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S3.3
Example 8 : Write down an expressionfor the amount indicate belowin two difference ways.
5(y - 2) = 5y - 10
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Removing aRemoving aSingle BracketSingle Bracket
y-2
y-2 y-2
y-2 y-2
22 Apr 202322 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Now try Ex 2.1
Ch3 MIA (page 43)
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S3.3
Removing aRemoving aSingle BracketSingle Bracket
Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questionsww
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sion.
com Q1. Explain why the following are True or False
(a) -3y x 5y =-30y (b) -6q x (-4q) = 24q2
Q2. Is the following true
-2a( b – a) = -2ab +4a
S3.3
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Q3. Write down the two numbers that multiplyto give 8 and subtract to give 2.
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m Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To show how to multiply out (remove) a single bracket with a letter outside the bracket.
1.1. Understand the keypoints Understand the keypoints of multiplying out an of multiplying out an expression with a single expression with a single bracket with a letter bracket with a letter outside.outside.
S3.3
2.2. Be able multiply out a Be able multiply out a expression with a single expression with a single bracket with a letter bracket with a letter outside the bracket.outside the bracket.
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Removing aRemoving aSingle BracketSingle Bracket
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S3.3
a(b + 5) =ab + 5aExample 1
p(w - 2) =pw - 2pExample 2
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Removing aRemoving aSingle BracketSingle Bracket
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S3.3
y(y - 1) = y2 - yExample 3
7w(w - 3) =7w2 - 21wExample 4
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Removing aRemoving aSingle BracketSingle Bracket
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S3.3
a(a2 + 3b) =a3 + 3abExample 5
y2(3y -2c) =3y3 - 2cy2
Example 6
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Removing aRemoving aSingle BracketSingle Bracket
22 Apr 202322 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Now try Ex 2.2
Ch3 MIA (page 44)
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S3.3
Removing aRemoving aSingle BracketSingle Bracket
Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questionsww
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sion.
com Q1. Multiply out.
(a) 3y(x - y) = (b) 6q2 (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
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Q3. Writing out in full 5.2 x 10-3 to get 0.052Is the correct?
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m Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To show how to multiply out (remove) a single bracket with a negative outside the bracket.
1.1. Understand the keypoints Understand the keypoints of multiplying out an of multiplying out an expression with a single expression with a single bracket with a negative bracket with a negative outside.outside.
S3.3
2.2. Be able multiply out a Be able multiply out a expression with a single expression with a single bracket with a negative bracket with a negative outside the bracket.outside the bracket.
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Removing aRemoving aSingle BracketSingle Bracket
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S3.3
-(x + y) = -x - yExample 1
-5(a - 7) =-5a + 35Example 2
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Removing aRemoving aSingle BracketSingle Bracket
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S3.3
7 + 3(4 - y) =7 + 12Example 3
9 - 3(8 - y) =9 - 24Example 4
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Removing aRemoving aSingle BracketSingle Bracket
- 3y= 19 - 3y
Tidy Up
+ 3y
Tidy Up
= -15 + 3y
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S3.3
A = 10(x – 2)
Example 9
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Removing aRemoving aSingle BracketSingle Bracket
Find my Area (x - 2)
10Find my Areax
12
A = 12xA = L x B A = L x B
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S3.3
Example 9 : Find area of the orange border
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Removing aRemoving aSingle BracketSingle Bracket
(x - 2)10x
12Area =Big Rec – Small Rec
= 12x – 10(x - 2)= 12x – 10x + 20= 2x + 20
22 Apr 202322 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Now try Ex 3.1
Ch3 MIA (page 44)
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S3.3
Removing aRemoving aSingle BracketSingle Bracket
Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questionsww
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sion.
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
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Q3. In standard form 18294000 is 1.8x107
is the correct?
(w - 2)
7
(w – 5) 3
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m Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To show how to multiply out (remove) and simplify an expression that has two single brackets.
1.1. Understand the keypoints Understand the keypoints of multiplying out an of multiplying out an expression with two expression with two single brackets.single brackets.
S3.3
2.2. Be able multiply out a Be able multiply out a expression with two single expression with two single brackets and the simplify.brackets and the simplify.
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Removing Two Single Removing Two Single BracketsBrackets
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S3.3
4(m - 3) - (m + 2) =4m - 12 - m - 2Example 1
Example 2
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= 3m - 14
Tidy Up
7(y - 1) - 2(y + 4) =7y - 7 - 2y - 8 = 5y - 15
Tidy Up
Removing Two Single Removing Two Single BracketsBrackets
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S3.3
3 x 14 – 8= 34
A = 4(y – 5)
Example 3 : Find an expression for the difference inthe areas.
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(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 Removing Two Single BracketsBrackets
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S3.3
2 x 4 + 20=28
A = 7(x + 1)
Example 3 : Find an expression for the difference inthe areas.
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(x + 1)
79 A = 9(x + 3)
x + 3Difference = 9(x + 3) - 7(x + 1) = 9x + 27 – 7x - 7
= 2x + 20Calculate
difference if x = 4
Removing Two Single Removing Two Single BracketsBrackets
22 Apr 202322 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Now try Ex 3.2
Ch3 MIA (page 46)
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S3.3
Removing Two Single Removing Two Single BracketsBrackets
Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questionsww
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sion.
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
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Q3.1 13 12 3
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m Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To show how to solve equations that have bracket terms.
1.1. Be able to multiply out Be able to multiply out brackets and solve brackets and solve equations.equations.
S3.3
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Equations and Equations and bracketsbrackets
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S3.3
5(x - 3) =25 5x
Example 1
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- 15= 25
Multiply out the bracket first and then solve.
Equations and Equations and bracketsbrackets
5x = 25 + 15 x = 40 ÷ 5
= 40= 8
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S3.3
6(x - 2) = 3(x + 1) 6x
Example 2
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- 12 = 3x
Equations and Equations and bracketsbrackets
= 3 + 3
6x – 3x -123x -12 = 3
3x = 3 + 12 3x = 15 x = 15 ÷ (3)
= 5
Solve as normalSubstitute
value into original
equation to check answer
18
18
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S3.3
-3(2 + 2y) =6 – (y + 2) -6
Example 3
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- 6y = 6
Equations and Equations and bracketsbrackets
= 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
22 Apr 202322 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Now try Ex 4.1 & Ex 4.2
Ch3 MIA (page 47)
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S3.3
Equations and Equations and bracketsbrackets
Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questionsww
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sion.
com Q1. Does 579300 = 5.793x105
Q2. Explain why the answer to 4(w + 2) = 6(w + 1) is w = 1
S3.3
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Q3.1 372 4
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m Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To show how to multiply out a pair of brackets using a multiplication table.
1.1. Understand the keypoints Understand the keypoints of multiplying out paired of multiplying out paired brackets.brackets.
Int 2
2.2. Be able to multiply out Be able to multiply out paired brackets using a paired brackets using a multiplication table.multiplication table.
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Removing Removing Double BracketsDouble Brackets
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S3.3
Below is a floor plan of the down stairs of a house.
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RemovingRemovingDouble BracketsDouble Brackets
y 5
y
3Tidy up !
y2+ 8y + 15
Living Room
KitchenBathroom
Bedroom
What is the total area of the
downstairs house?
y2 5y
3y 15
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S3.3
Multiplication table for brackets
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RemovingRemovingDouble BracketsDouble Brackets
(y + 3)(y + 5) y+ 3 y+ 5
Double brackets are used in AREA calculations
+5y+15+3y
y2 Tidy up !
y2+ 8y + 15
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S3.3
Example 2
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RemovingRemovingDouble BracketsDouble Brackets
(2x - 1)(x + 3) 2x - 1 x+ 3
Be careful with the negative
signs
+6x
-3-x
2x2 Tidy up !
2x2+ 5x - 3
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S3.3
Example 3
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RemovingRemovingDouble BracketsDouble Brackets
(x + 4)(x2 + 3x + 2) x+ 4 x2 + 3x
Just a biggerMultiplicatio
n Table
+3x2
+12x+4x2
x3 Tidy up !
x3
+ 2
+2x
+8 + 7x2+ 14x+ 8
22 Apr 202322 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Now try Ex 5.1First column in each
Question
Ch3 MIA (page 49)
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S3.3
Removing Removing Double BracketDouble Bracket
Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questionsww
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sion.
com Q1. 0.00897 is in scientific notation is this
statement True or False?
Q2. Why does (w + 2)(w + 1) = w2 + 3w + 2
S3.3
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Q3.1 254 7
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m Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To show how to multiply out a pair of brackets using FOIL.
1.1. Understand the keypoints Understand the keypoints of multiplying out double of multiplying out double brackets using FOIL.brackets using FOIL.
Int 2
2.2. Be able to multiply out Be able to multiply out double brackets using double brackets using FOIL.FOIL.
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Removing Removing Double BracketsDouble Brackets
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S3.3
We can use the F O I L method to multiply out DOUBLE brackets.
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RemovingRemovingDouble BracketsDouble Brackets
Simply remember the word
F O I LMultiply First 2
Multiply Last 2
Multiply Outside 2
Multiply Inside 2
FOIL Method
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S3.3
(x + 1)(x + 2)
x2 + 2x
Example 1 : Multiply out the brackets and Simplify
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1. Write down F O I L+ x + 2
2. Tidy up ! x2 + 3x + 2
RemovingRemovingDouble BracketsDouble Brackets
Try again using multiplication
table
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S3.3
(x - 1)(x + 2)
x2 + 2x
Example 2 : Multiply out the brackets and Simplify
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Removing Removing Double BracketsDouble Brackets
1. Write down F O I L- x - 2
2. Tidy up ! x2 + x - 2
Try again using multiplication
table
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S3.3
FOIL method is very limited
therefore from now on we will use
the multiplication table method.
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Squaring Squaring BracketsBrackets
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S3.3
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(x + 1)(x - 2)
RemovingRemovingDouble BracketsDouble Brackets
(x - 1)(x - 2) (x + 3)(x + 2) (x - 3)(x + 2) (x + 3)(x - 2)
x2 - x - 2 x2 - 3x + 2 x2 + 5x + 6 x2 - x - 6 x2 + x - 6
22 Apr 202322 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Now try Ex 5.2First column in each
Question
Ch3 MIA (page 50)
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S3.3
RemovingRemovingDouble BracketsDouble Brackets
Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questionsww
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sion.
com Q1. Does 2 + 3 x 4 = 20 Explain ?
Q2. Multiply out using a multiplication table
(w + 2)(w2 - 2w + 1)
S3.3
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Q3. Which sums are true1 1 12 2 4 2
1 1 1y y y
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m Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To show how to multiply out squared brackets using Multiplication Table.
1.1. Understand the keypoints Understand the keypoints of multiplying out squared of multiplying out squared brackets using brackets using Multiplication Table..
Int 2
2.2. Be able to multiply out Be able to multiply out squared brackets using squared brackets using Multiplication Table..
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Squaring Squaring BracketsBrackets
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S3.3
= (x + 1)(x + 1) Example 1 : (x + 1)2Multiply out the brackets and Simplify
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Squaring Squaring BracketsBrackets
(x + 1)(x + 1) x+ 1 x+ 1
+ x+ 1+ x
x2 Tidy up !
x2+ 2x + 1
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S3.3
Example 2 : (x + y)2Multiply out the brackets and Simplify
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Squaring Squaring BracketsBrackets
= (x + y)(x + y) (x + y)(x + y) x+ y x+ y
+xy+ y2+xy
x2 Tidy up !
x2+ 2xy +y2
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S3.3
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(x - 1)2 (x - 2)2
(x - 4)2 (2x – 1)
x2 - 2x + 1 x2 - 4x + 4 x2 - 8x + 16 4x2 - 4x + 1
Squaring Squaring BracketsBrackets
22 Apr 202322 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com
Now try Ex 6.1 & 6.2
First column in each Question
Ch3 MIA (page 50)
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S3.3
Squaring Squaring BracketsBrackets
Apr 22, 2023Apr 22, 2023
Starter QuestionsStarter Questionsww
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sion.
com Q1. Does (x + 3)2 = x2 + 9
Q2. The sun is 92 million mile away from the earth. Write this in standard form.
S3.3
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Q3. Is it true1 1 52 3 6
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m Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To show how to solve equations that have paired or squared brackets.
1.1. Be able to solve equations Be able to solve equations that contain paired or that contain paired or squared brackets.squared brackets.
S3.3
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Equations and Equations and bracketsbrackets
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S3.3
(x + 2)2 =x2 + 82
(x + 2)(x + 2)Example 1
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= x2 + 64
Multiply out the bracket first and then solve.
Equations and Equations and bracketsbrackets
x2 + 4x + 4
4x = 60x = 60 ÷ 4 = 15
X table= x2 + 64
4x + 4= 64
Substitute value into original
equation to check answer
289 289
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m(m + 2) =m2 + 10 m2 + 2m
Example 2
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= m2 + 10
Multiply out the bracket first and then solve.
Equations and Equations and bracketsbrackets
2mm = 10 ÷ 2 = 5
= 10Substitute value into original
equation to check answer
35
35
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6(x - 2) = 3(x + 1) 6x
Example 3
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- 12 = 3x
Equations and Equations and bracketsbrackets
= 3 + 3
6x – 3x -123x -12 = 3
3x = 3 + 12 3x = 15 x = 15 ÷ (3)
= 5
Solve as normalSubstitute
value into original
equation to check answer
18
18
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-3(2 + 2y) =6 – (y + 2) -6
Example 4
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- 6y = 6
Equations and Equations and bracketsbrackets
= 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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S3.3
Solve in the usual
way
A = (x + 7)(x – 2)
Example 5 : The two areas are equal. Find the value of x.
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(x + 7)(x – 2)(x + 2)A = (x + 2)2
(x + 2) (x + 2)2 = (x + 7) (x - 2) x2 + 4x + 4
x = 18
X table
Equations and Equations and bracketsbrackets
= x2 + 5x - 14 4x + 4= 5x - 14
4= x - 14
Substitute value into original
equation to check answer
400 400
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Now try Ex 7.1 & 7.2
Ch3 MIA (page 55)
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Equations and Equations and bracketsbrackets