Maths Year 9
- Created by: carmen.lau
- Created on: 09-06-15 16:37
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
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)
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
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:
Surds (Part 2)
Dividing:
Expanding brackets:
3 ( 2 - 3) = 3 - 3
( 3 + 2) ( 3 - 5) = 3 - 3 + 3 - 10 = -7 - 3
Rationalising the denominator:
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
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
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
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
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
Pythagoras' Theorem
We use this to find an unknown side or angle in right angled triangles
a + b = c
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
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
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
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
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
Area of a triangle
Using sine:
Related discussions on The Student Room
- *Predict your GCSES* (YEAR 11s) »
- predict your gcse grades!! (2023-2024) »
- GCSE Results: Post your results »
- GCSE Results Day Countdown - 24th August 2023 TODAY - #BeResultsProud! »
- Which GCSE result are you most proud of? »
- Is further maths gcse worth it? »
- Predict Your gcse Grades! »
- Mock results vs Actual GCSEs »
- Official Year 11 Chat 2023-24 »
- I've been stuck on a grade 5 for 3 years :/ »
Comments
No comments have yet been made