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Developing Algebra Skills. Session Four – 12 March 2014. A Quick Recap …. Multiply out the brackets in (x + 5)(x – 3) Grid = x 2 + 5 x - 3 x – 15 = x 2 + 2 x – 15. x 2. +5 x. -3 x. -15. A Quick Recap …. Multiply out the brackets in (x + 5)(x – 3) FOIL ( x + 5)( x – 3)

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developing algebra skills

Developing Algebra Skills

Session Four – 12 March 2014

a quick recap
A Quick Recap …
  • Multiply out the brackets in (x + 5)(x – 3)
  • Grid

= x2 + 5x - 3x – 15

= x2 + 2x – 15

x2

+5x

-3x

-15

a quick recap3
A Quick Recap …
  • Multiply out the brackets in (x + 5)(x – 3)
  • FOIL
  • (x + 5)(x – 3)

= x2 - 3x + 5x – 15

= x2 + 2x – 15

F First

O Outside

I Inside

L Last

a quick recap4
A Quick Recap …
  • Using either method calculate (b + 3)(b - 6)

= b2 + 3b - 6b -18

= b2 - 3b - 18

b2

+3b

- 6b

-18

multiplying expressions
Multiplying Expressions
  • What is the calculation to find the area of this rectangle
  • (x - 3)(x - 5) = x2 – 5x – 3x + 15
  • = x2 – 8x + 15

X

5

X

3

a quick recap6
A Quick Recap …
  • Putting the brackets back in
  • Eg a2 – 9a + 20 = (a )(a )
  • Find two numbers that multiply to give 20 and add to give – 9
  • = (a – 5)(a – 4)

This came from

This times by this

a quick recap7
A Quick Recap …
  • Factorise y2 – 7 + 12
  • Two numbers which multiply to give 12 and add to give -7
  • So y2 – 7 + 12 = (y – 3)(y – 4)
solving quadratics by factorisation
Solving Quadratics by Factorisation
  • y2 – 7 + 12 = 0
  • (y – 3)(y – 4) = 0
  • Therefore y – 3 = 0 or y – 4 = 0
  • So y = 3 or y = 4

For this to be true, one of the brackets must be 0

solving linear equations
Solving Linear Equations
  • What is a linear equation ?
  • How do we solve …
  • 3 + x = 7
  • We can solve this only if there is one unknown value
simultaneous linear equations
Simultaneous Linear Equations
  • If we have two unknowns we must have two equations ….
  • a + 2b = 12
  • a – b = -3
  • There are 2 methods
    • Substitution
    • Addition / subtraction
simultaneous linear equations11
Simultaneous Linear Equations
  • Method 1 - Substitution
  • a + 2b = 12
  • a – b = -3

Rearrange one equation to be either

a = something or b = something

Lets rearrange to be a = b – 3

1

2

2

simultaneous linear equations12
Simultaneous Linear Equations

1

  • We can now substitute a = b – 3 into
  • a + 2b = 12
  • So b – 3 + 2b = 12
  • Rearrange and solve for b, as usual
  • i.e. 3b = 15
  • b = 5 Substitute back in
  • a = 5 – 3
  • a = 2

a = 2, b = 5 is the ONLY solution which satisfies both equations

simultaneous linear equations13
Simultaneous Linear Equations
  • Method 2 – Addition or subtraction
  • 2a + b = 10
  • a - b = -1
  • Add the equations to remove the b terms …
  • 2a + b = 10
  • a - b = -1
  • 3a = 9 so a = 3
  • Substitute back in to find b
  • a – b = -1 so b = 4

We need to add or subtract to remove either the a or the b terms

simultaneous linear equations14
Simultaneous Linear Equations
  • Solve the equations a + b = 9 and 4a – b =1
  • By substitution …
  • a = 9 – b
  • So 4(9 – b) – b = 1
  • Expand brackets and collect like terms
  • 36 – 4b – b = 1
  • So 35 = 5b b = 7 and
  • a = 9 – b a = 2
simultaneous linear equations15
Simultaneous Linear Equations
  • Solve the equations a + b = 9 and 4a – b =1
  • By addition …
  • a + b = 9
  • 4a – b = 1
  • 5a = 10 So a = 2
  • Substitute back to give
  • 2 + b = 9 So b = 7
formulae
Formulae
  • e.g Formulae to find perimeter of rectangle
  • P = 2l + 2w
  • If the length is 23cm and the width is 14cm, what is the perimeter ?
  • Substitute into formulae
  • P = 2(23) + 2(14)
  • P = 46 + 28
  • P = 74 cm
formulae18
Formulae
  • The area of a trapezium is given by …
  • A = ½ (a + b)h
  • Where a and b are the lengths of the parallel sides and h is the distance between them
  • Find A when a = 2.7, b = 6.9 and h = 4.2
  • NB – Use your calculator
formulae19
Formulae
  • A = ½ (a + b)h
  • A = ½ (2.7 + 6.9)4.2
  • A = ½ (9.6)4.2
  • A = ½ (40.32)
  • A = 20.16 cm
formulae20
Formulae
  • The connection between temperature in °C and °F is given by the formula …
  • C = (F – 32)
  • Find C when F = 68
  • C = (68 – 32)
  • C = (36) So C = 20°
formulae21
Formulae
  • What if we have the temperature in °C and what to convert it to °F ?
  • We need to transpose the formula to make F the subject…
  • C = (F – 32)
  • C = F – 32 So, C + 32 = F
transposition of formulae
Transposition of Formulae
  • Changing the subject of a formula
  • e.g Formulae to find perimeter of rectangle
  • P = 2l + 2w
  • Rearrange the formula to make l the subject
  • = l + w
  • So - w = l

Remember:

Whatever we do to one side we must do to the other side to maintain the equality

transposition of formulae23
Transposition of Formulae
  • Rearrange the formula to make y the subject
  • 3x + 2y = 7
  • 2y = 7 – 3x
  • y = 7 – 3x

2

Subtract 3x from both sides

Divide both sides by 2

transposition of formulae24
Transposition of Formulae
  • The area of a trapezium is given by A = ½ (a + b)h
  • Find the height when the Area = 72m2 and the lengths of the parallel sides are 8m and 10m
  • First rearrange the formula to make h the subject
transposition of formulae25
Transposition of Formulae
  • A = ½ (a + b)h
  • 2A = (a + b)h
  • So 2A = h

(a + b)

  • h = 2 x 72

(8 + 10) So h = 8m

Multiply both sides by 2

Divide both sides by (a + b)

Now substitute in the given values

trial and improvement
Trial and Improvement
  • This is a method that involves making an intelligent guess and then getting closer to the answer by improving the guess
  • Eg Find √115 to 2 d.p.
  • So, first guess is 10.5 10.52 = 110.25
  • Improved guess is 10.75 10.752 = 115.5625
  • Improved guess is 10.7 10.72 = 114.49
  • Improved guess is 10.73 10.732 = 115.1329
  • Improved guess is 10.72 10.722 = 114.9184
  • So √115 = 10.72 (2 d.p)

We know 102 = 100 and 112 = 121

trial and improvement28
Trial and Improvement
  • The equation x3 – 2x2 + 3x – 9 = 0 has a solution between x = 2 and x = 3. Find the solution correct to 2 d.p.
  • First guess x = 2.5 equation = 1.625
  • Improved guess x = 2.3 equation = -0.513
  • Improved guess x = 2.4 equation = 0.504
  • Improved guess x = 2.35 equation = -0.00171
  • Improved guess x = 2.37 equation = 0.18825
  • Improved guess x = 2.36 equation = 0.08506
  • So x3 – 2x2 + 3x – 9 = 0 when x = 2.35 (2 d.p)
trial and improvement29
Trial and Improvement
  • Try Exercise …
session summary
Session Summary
  • Simultaneous Equations
  • Transposition of Formulae
  • Trial and Improvement