Maths Year 9

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Indices

a

a is the base number

   is the index number

2 x 2 = 2                                     4 -- 4 = 4                              a = 1             

(a )  = a                                        (2x )  = 8x                            2  = 1/2

2  = 1/8 or 1/2                              3  = 1/3                                (1/2)  = 2  or 8

A  = 1/A                                       1/27  = 3                              2A  = 2/A

2/x  = 2x                                      2/5 A  = 2/5A                       27  =   27 or 3

(125/27)  = 125  / 27  = 5/3            (16/81)  = (81/16)  = 81  /16  = 3/2

8  = 8  x 8  = (8  )   = (   8)  = 4                                                9  = (9  )  = (   9)  = 27                                           

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Standard form

Used to write very big/ very small numbers

  • Big number- positive power (decimal to the right)
  • 7 x 10  = 7 000
  • 3.5 x 10  = 35 000
  • Small number- negative power (decimal to the left)
  • 6.29 x 10   = 0.00629

Adding/ Subtracting- expand numbers before working out

  • Multiplication- (4.8 x 10  ) -- (6 x 10  ) = 0.8 x 10   = 8 x 10   (subtract powers)
  • Division- (2 x 10  ) x (6 x 10  ) = 12 x 10  x 10  = 12 x 10   (add powers)
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Recurring Decimals

eg. 0.15 = 0.15151515......

0.15 as a fraction:

     x = 00.15151

                                                                                                                                         100x = 15.15151                                                                                                                     -    x = 00.15151

 99x = 15/99 = 5/33

1 number = 10x  ,  2 numbers = 100x  ,  3 numbers = 1000x

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Surds

Simplifying:

  • Find the highest square number factor:

    27 =    9 x   3 =     3

Rule:    a x   b =    ab

Multiplying:

Rule:    a x    a = a

  18 x   32 =     2 x     2 = 3 x 4 x    2 x   2 = 3 x 4 x 2 = 24

Dividing:

Rule: 

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Surds (Part 2)

Dividing:

Expanding brackets:

  3 ( 2 -    3) =      3 - 3 

                                                                                                                                              (    3 + 2) (    3 - 5) = 3 -     3 +    3  - 10 = -7 -     3

Rationalising the denominator:

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Circle theorems

Angle in a semi circle is a right angle:

  • Line AC has to be the diameter.
  • Angle ABC is a right angle

Alternate Angles:

  • Angle ABC is the same as CAD

Angle at the centre is x2 the angle                                                                                          at the circumference:

  • Angle ACB is half angle AOB
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Circle theorems (Part 2)

Angles in the same segment are the same:

  • Angle ABD and ACD are the same

Opposite angle add up to 180 (cyclic quad):

  • Angle ABC and ADC add up to 180
  • A, B, C and D must touch the edges of the circle

The radius always meets the tangent at 90 :

  • The angle OAB is 90  
  • The radius meets tangent at 90  
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Angles

  • Angles in a triangle add up to 180
  • Angles in a square add up to 360
  • Angles on a straight line add up to 180
  • Angles around a point add up to 360
  • Exterior angles add up to 360
  • Exterior angle: 360/ number of sides
  • Interior angle: (number of sides - 2) x 180                                                                              
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Percentages

  • Interest/ profit: increase  ,  Depreciation/ loss: decrease
  • Use a multiplier (decimal equivalent) when increasing/ decreasing:
  • Eg. Increase 480 by 5% (5% as a decimal: 100% + 5%) = 480 by 1.05 = 504

Compound:

  • Original no. x (multiplier)
  • Eg. $480 with interest at 5% for 6 years = 480 x (1.05)  = $643.25

Percentage Increase/ Decrease:

  • (Difference/ Original) x 100
  • Eg. Percentage increase from 25 to 42 = 17/25 x 100 = 68% increase

Find the Original:

  • Original x multiplier = Current price  ,   Current price -- multiplier = original price
  • Eg. Reduced in price by 15%, current price $56
  • Multiplier: 100 - 15% = 0.85
  • Original price = 56 -- 0.85 = $65.88
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Simultaneous Equations

x + 2y = 9                                                                                                                               x -   y = 3

  x + 2y = 9                                                                                                                                                - x -   y = 3                                                                                                                                    3y = 6                                                                                                                                       y = 2

Substitute: x + 4 = 9  ,  x = 5

6x - 2y = 1                                                                                                                                                4x + 7y = 9

  12x -   4y  = 2                                                                                                                                               - 12x + 21y = 27                                                                                                                               -25y = -25

y = 1  ,  x = 0.5

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Pythagoras' Theorem

We use this to find an unknown side or angle in right angled triangles

a  + b  = c

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Trigonometry

Have to be in right angled triangles

To find side x:

  • Label sides
  • Choose function (sin, cos, tan)
  • Choose the formula
  • Solve

To find angle   :

  • Use inverse functions: sin   , cos   , tan   
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Bearings In Trigonometry

  • Create a right angled triangle
  • Always measure angle from North ( clockwise)
  • Use Trig in the same way (sin in this case)
  • O = sin64 x 20(km) = 18.0
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Sine Rule

Angles and sides are opposite each other

Use for non right angled triangles

To find an angle: sinA / a  =  sinB / b  =  sinC / c

                                                                                                                                            To find a side: a / sinA = b / sinB = c / sinC

  • Eg. To find side a:
  • a = a / sinA = c / sinC
  • a = a / sin37 = 7.9 / sin83
  • a = 7.9 x sin37 / sin83
  • a = 4.79
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Sine Rule (Part 2)

  • To find angle x:
  • x = sinx/6.5 = sin76/8.1
  • x = sinx = sin76 x 6.5/8.1
  • x = sin  (sin76 x 6.5 / 8.1)
  • x = 51.1
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Cosine Rule

a  = b  + c  = - 2bc CosA

Used when you have one angle between two sides

  • Label the angle 'A'
  • To find the side opposite 'A'
  • a  = 8  + 14  - (2 x 8 x 14 x Cos106)
  • a  = 260 - (-61.742) (Only round at the end)
  • a  = 321.74
  • a =   321.74
  • a = 17.9
  • To find an angle:
  • Use the formula: CosA = b  + c  - a   / 2bc
  • CosA = 10.5  + 15  - 12 / 2 x 10.5 x 15
  • CosA = 0.607 (Must be lesss than 1)
  • A = cos  (0.607)
  • A = 52.6
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Area of a triangle

Using sine:

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