Using and applying mathematics

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Using and applying mathematics

• Sequences Formulae
• Year 10

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• The following series of lessons will equip you
  with the necessary skills to complete a complex
  investigation at the end of the unit.
• The initial lessons may seem tedious, but bear
  with us

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At the end of the unit you will be presented with
this problem
What size must the cut-out corners be to give the
maximum volume for the open box?
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To attempt this problem you need to be able to
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Objective
Monday 21st February

• To be able to simplify algebraic expressions

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About 500 meteorites strike Earth each year.
A meteorite is equally likely to hit anywhere on
earth.
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Objective
Wednesday 23rd February

• To be able to solve algebraic equations.

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Take the number of the month of your birthday
Multiply it by 5
Add 7
Multiply by 4
Add 13
Multiply by 5
Add the day of your birth
Subtract 205
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What have you got?
Why does this work?
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Objective
Thursday 24th February

• To be able to formulate expressions and formulae.

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The length of a rectangular field is a metres.
The width is 15m shorter than the length. The
length is 3 times the width.
a
a - 15
Write down an equation in a and solve it to find
the length and width of the field.
Length 22.5m Width 7.5m
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Imagine a triangle
Choose a length for its base. Call it z
Make the vertical height 3 units longer than the
base
Work out the area of your triangle
Write down an equation in z that satisfies your
conditions
Give it to your partner to solve for the base
length of your triangle (z).
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Problems involving quadratic equations
A rectangle has a length of ( x 4) centimetres
and a width of ( 2x 7) centimetres.
If the perimeter is 36cm, what is the value of x?
X 7
If the area of a similar rectangle is 63cm2 show
that 2x2 x 91 0 and calculate the value of x
X 6.5
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Monday 28th February
Objective Formulate equations and solve by trial
and improvement.
Level 6 / 7
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The length of a rectangular field is a metres.
The width is 15m shorter than the length. The
length is 3 times the width.
a
a - 15
Write down an equation in a and solve it to find
the length and width of the field.
a 3 (a 15)
Length 22.5m Width 7.5m
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Formulating quadratic equations
a) Write down a quadratic equation in x
b) Solve the equation to find Joan's age.
x ( x 25 ) 306
x2 25x 306
x2 25x 306 0
How can we solve this?
Factorisation?
Graphically
Formula
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x2 25x 306 0
This example factorises
( x 34 )( x 9 ) 0
Either x 34 0
Or, x 9 0
x -34
x 9
Since Joan cannot be 34 years old, she must be
9.
Some quadratic equations do not factorise exactly.
Solving some equations (i.e. cubic ) by a
graphical method is not very accurate.
A more accurate method is trial and improvement
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Solving equations by trial and improvement.
E.g. 1 A triangle has vertical height 3 cm longer
than its base. Its area is 41 cm2. What is the
length of its base to 1 d.p?
x 3
x ( x 3) 41 x 2 82
x2 3x 82 0
x
Too small
Try x 7 72 (3 x 7) 82 -12
Too big
Try x 8 82 (3 x 8) 82 6
Too small
Try x 7.6 7.62 ( 3 x 7.6) 82 - 1.44
Try x 7.7 7.72 ( 3 x 7.7) 82 0.39
Too big
Too small
Try x 7.65 7.652 ( 3 x 7.65) 82 -0.5275
x 7.7 to 1 d.p
Base 7.7cm to 1 d.p
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• 5x2 12x 5 0 For x gt 1
• x2 5x 1 0 For x gt 0
• 2x2 2x 3 0 For x gt 0
• 5x2 9x 6 0 For x gt 1

• 1.9
• 5.2
• 1.8
• 0.5

To 1 d.p
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Wednesday 2nd March
Transposition of formula
Objective To be able to accurately rearrange
formula for a given subject.
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Here are some questions and answers (by students
A and B) on rearranging formulae. Decide which
answers to tick (correct) and which to trash
(incorrect). You must give reasons for your
decision.
Question 1. Make x the subject of the following
Trash
Trash
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Rearrangement of formulae
When doing these sort of problems, remember these
things
a) Whatever you do to one side of the formula,
you must also do the same to the other side
To rearrange the following formula making x the
subject
Add y to both sides of the formula giving
As (y y 0) and (2y y 3y) we can say
Now subtract x from both sides leaving
So to get x we can now divide both sides by 2
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b) When you are dealing with more complicated
formulae, try to strip off the outer layers first.
First get rid of the square root, by squaring
both sides
Now get rid of the division bar, by multiplying
both sides by x
To leave you with x on one side, divide both
sides by g2
c) When you want to get rid of something in a
formula, remember to do the opposite (inverse) to
it.
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One last example
Make u the subject of the following
Multiply both sides by (u v)
Expand the bracket
Collect the u terms on one side
Factorise the LHS to isolate u
Divide both sides by (f v)