Maths Rules and Equations
- Created by: faholly21
- Created on: 27-05-15 14:52
Standard Form Rules
Multiplying numbers in standard form
- Rearrange so powers of 10 are together
-Multiply the number parts
-Add the Powers
-Rewrite in standard form if neccessary
Dividing numbers in standard form
– Rearrange so powers of 10 are together
- Divide the number parts
- Subtract the powers
-Rewrite answer if necc.
Surd Rules
√ab = √a x √b
√a/b = √a / √b
Rationalising the denominator
Multiply the top and bottom of the fraction by the surd part in the denominator.
Working out the nth term
Write the difference between each term e.g 4
Find the 0th term of the sequence
nth term= difference x n + zero term.
Factorising
Look for the largest factor you can take out of every term in the expression if it is x(y -or+ z)
x^2 + bx + c - find tow numbers which add up to b that are the same as two numbers that multiply to c
Both numbers are positive if b and c are positive
Bigger number is positive and the small number is negative if b is positive and c is negative
Smaller number is positive and the big number is positive if b is negative and c is positive
Both numbers are negative if b and c are negative
if ax^2 + bx + c means on of the brackets must have one ax or two numbers in front of x that multiply to a.
x^2-c means the equation must be (x-a) (x+a) Difference of two squares.
Linear equations
If letters are on both sides of the equation - have to get it on one side - collect the like terms on one side.
If there are brackets within equations- multiply them out.
Equations w/ fractions - Get rid of the fractions before solving – multiply all the terms by the lowest common multiple of the denominators ( Look over in book)
Straight Line Graphs
Gradient - Change in y / change in x
Y intercept- use equation y=mx + c and gradient value to rearrange to find the intercept.
Parralel lines have a gradient of 1; Perpendicular lines gradients multiply to -1
3D co-ordinates
They have a z axis (x, y, z)
To find missing co-ordinates look at x co-ordinate at face and the y and z co-ordinates of the line adjecent to the point.
Rearranging Formulae
Letter appearing twice
- Group all terms with that letter on one side of the formula and all other terms on the other side
- Factorise so the leeter only appears once
- Divide everuthing in the bracket to get the letter on its own.
Inequalities
Open circle for < and >
Closed circle for ≤ and ≥
On graphs show x ≥2 by doing a solid line on x=2 and do an arrow in the right direction
Graphs of k/a and a^x
y= k/x are reiprocal graphs
y=a^ x or y=a^-x are Exponential graphs
Simultaneous Equations
Algebraic solution
IF NECCESSARY, multiply equations so co-effecients of one unknown are the same
Add/ subtract to eliminate unknown
Once one unknown found use subsititution
Check the answer by substituting both values in the orignal equations.
Quadratic equations
ax^2 +bx+c = 0 - if not in this form, should rearrange before starting the calculating.
Factorise the left hand side to find the answers w/o a calculator
Completing the Square
when written as (x+p)^2 + q in completed square form.
Use the formulae x^2 + 2bx + c = (x+b)^2 - b^2 + c
Simultaneous Equations 2
If there is x^2 or y^2 in a pair of simultaneous quations need to use Substitution of one equation into another to make one letter the subject. Then factorise to find solutions. Then subsitute to find unknown value.
There would also be 2 solutions
Proportionality formulae
A statement of proportionailty y is directly proportional to x = y=kx
k= constant of proportonality
y is directly proportional to the square of x = y= kx^2
y is directly proportional to the cube of x = y= kx^3
y is directly proportional to the square root of x = y= k√x
y is inversley proportional to x = y= k/x
y is inversley proportional to the square of x = y= k/ x^2
Transformations
y=f(x) + a MOVE UP A UNITS
y= f(x) - a MOVE DOWN A UNITS
y=f (x+a) MOVE LEFT A UNITS
y=f (x-a) MOVE RIGHT A UNITS
y= af(x) STRETCH IN THE VERTICAL DIRECTION, SCALE FACTOR A (X VALUE STAYS THE
SAME)
y= f(ax) STRETCH IN HOROZONTIAL DIRECTION, SCALE FACTOR 1/A (Y VALUE STAYS THE
SAME)
y= -f(x) REFLECTION IN THE X AXIS
y= f(–x) REFLECTION IN THE Y AXIS
Graphs of sin and cos
Graphs of y= cos x intercept at 1 and pass through x=0 at 90 and 270
Graphs of y= sin x intercept at 0 and pass through x= 0 at 180 and 360Sine graph
Cosine graph
Algebraic Fractions
To add/ subtract algebaric fractions with diff denominators
Find a common denominator; Add or subtract the numerators; Don't change denominator and Simplify
To Multiply A fractions
multiply numerators/ denominaters and Simplify
To Divide A Fractions
Change second fraction to be reciprocal; Change / to x; Multiply fractions and simplify.
Angle Properties
Corresponding Angles are equal
Alternate Angles are equal
Opposite Angles are equal
Allied Angles add up to 180
The Exterior angle of a triangle = sum of interior angles of other two vertices
Opposite angles of a parallelogram are equal
Reasons in Angle Problems
- Angles on a straight line add up to 180
- Angles around a point add up to 360
- Opposite angles are equal
- Corresponding angles are equal
- Alternate angles are equal
- Angles in a triangle add up to 180
- Angles in a quad. add up to 360
- Base angles of an isoceles triangle are equal.
Angles in Polygons
n= number of sides
Sum of interior angles = 180 x (n–2)
Sum of exterior angles = 360 to find regular exterior angle 360/n
Perimeter and Area
Area of a Triangle = 1/2 * b * h
Area of Parallelogram= bh
Area of a Trapezium = 1/2 (a+b) h
To calculate areas and perimeters of more complicated shapes split them into parts.
Prisms
Volume of a Cubiod = length x width x height
Volume of a Prisim = area of cross section x length
Surface area of shapes = add together all the areas of faces.
Circles and cylinders
Circumference of a Circle- 2πr or πd
Area of a Circle- πr^2
Volume of a Cylinder- πr^2h
Surface Area of a Cylinder- 2πr^2 + 2πrh
Sectors of A Circle
x= angle; r= radius
Area of a sector= x/360 x πr^2
Arc length= x/360 x 2πr
Rearrange first equation to find angle
Volumes of 3D shapes
Cubiod- length x width x height (lwd)
Pyramid 1/3 x area of base x vertical height (1/3 Ah)
Cone: 1/3 x area of base x vertical height (1/3 πr^2 h)
Sphere: 4/3 πr^3
Pythagoras
a^2 + b^2 = c^2 Must be a right angled triangle. c= hypotense
Surface Area
Curved Surface Area of Cone= π x radius x slant height (πrl)
Whole Surface Area = πr^2 (area of base)+ πrl
Surface Area of a Sphere- 4πr^2
Surface Area of a Hemisphere (1/2 x 4πr^2)+πr^2 (area of base)
Converting Units
1km = 1000m 1m= 100cm 1cm= 10mm
1 tonne= 1000kg 1kg= 1000g 1g= 1000mg
1 litre = 100cl 1000 litres= 1m^3 1cl= 100ml
Metric Units Imperial Units
1kg 2.2 pounds (lb)
1 litre 1.75 pints
4.5 litres 1 gallon
8km 5 miles
30cm 1 foot
Area and Volume conversions
Area conversions
1cm^2= 100mm^2
1m^2= 10,000cm^2
1km^2= 1,000,000m^2
Volume Conversions
1cm^3 = 1000mm^3
1m^3 = 1,000,000cm^3
1 litre= 1000cm^3
1ml= 1cm^3
Speed
Speed = distance/ time
Distance = speed * time
Time= Distance/ Speed
Units = m/s (metres per second) kilometres per hour (km/h) and mph (miles per hour) DEPENDS ON WHAT YOU USE IN THE FORMULAE.
Density
Density = mass/ volume
Volume= mass/ density
Mass= Volume x Density
Units of Density = g/cm^3 (grams per cubic centimetre); kg/m^3( kilograms per cubic metre)
Congruent Triangles
Congruent if they have exactly the same size and shape. To prove, show one of these conditions are true
SSS (three sides are equal)
AAS( two angles and a corresponding side are equal )
SAS ( two sides and the included angle are equal)
RHS ( right angle, hypotenuse and a side are equal)
Similar Shapes
All three angles are equal;
All three pairs of sides are in the same ratio
Two sides are in the same ratio and the included angle is equal
To Calculate a length AB/FG= AE/FJ input values and rearrange to find answer
Scale factor =k
Enlarged surface area = k^2 x original surface area
Enlarged volume = k^3 x original volume
Enlarged mass = k^3 x original mass
Constructions
Construct a triangle- Draw and label one side with a ruler, use a compass to find the other vertex by extending it to the size of the other sides.
Construct a angle bisector- Mark two equal distanced from angle points on the arms. Then use arcs to find third point which equi-distant from these points
Construct a perpindicular bisector- Use compass to draw intersecting arcs with centres A and B
Construct a perpendicular line that passes through a certain point Use compass to mark two points on the line equidistant from the point. widen compass and draw arcs with their centres at these points
Construct a perpendicular line that passes through a point off the line Use compass to mark two points equidistant from P. Draw two arcs using this points
Line Segments
Use Pythagoras to find the length of a line segment
Trigonometry 1
SOH CAH TOA
sin x = opposite/ hypotenuse; cos x= adjecent / hypotenuse; tan x = opposite/ adjacent
To find the size of an angle. Use tan/sin/cos ^–1 to get the angle on its own
Trigonometry 2
To find a length of another side in a right- angled triangle - you need to have the length of another side and the size of one of the acute angles.
Subsititute it into the equation e.g sin40 = a/10 a=10 x sin40 = 6.43 cm
Pythagoras in 3D
d^2 = a^2 +b^2+c^2
Triangles and segments
Area of triangle if you only have two sides and the angle between them = 1/2 ab sin C
Area of segments= Area of whole sector – Area of triangle
Sine Rule
Applies to any triangle. To find a missing side :
a/ sinA = b/ sinB = c/ sinC
To find a missing angle:
sinA/ a = sinB/ b = sinC/ c
To use this rule, must know either two angles and a side (ASA) or two sides and a non included angle (SSA)
Cosine Rule
Applies to any triangle To find a missing side:
a^2 = b^2 + c^2 –2bc cos A
To find a missing angle:
cos A = b^2 + c^2 –a2/ 2bc
Use if you have two sides and the included angle to find a missing side or three sides and are looking for a missing angle.
Use sine when a problem involves two sides and two angles
Use cosine when a problem involves three sides and one angle
Circle facts
The angle between a radius and a tangent is 90 degress
Two tangents which meet at a point outside a circle are the same length
A triangle which has one vertex at the centre of the circle and two angles on the circumference is an ISOSCELES TRIANGLE.
Circle Theorems
A perpendicular line from a chord to the centre of the circle bisects the chord.
The angle at the centre of the circle is twice the angle on the circumference
Angles in the same segment are equal
The angle in a semicircle is 90 degrees
Opposite angles in a cyclic quadrilateral add up to 180
Angle between a tangent and a chord is equal to the angle in the alternate segment - ALTERNATE SEGMENT THEOREM
Collecting Data
Add response boxes
Don't ask embarrasing/ personal questions
Make sure response boxes are numerical and dont overlap
Don't ask biased questions- Statement and response agree, disagree neither
Data Collection sheets: Tally and Frequency columns
Stratified Sampling
The equation is( sample/ whole population) x amount of people from group
e.g How many Year 7 boys will be sampled out of the population of 382 and a sample of 85
Year 7
Boys 50 85/382 * 50 = 11.12 -> 11 students
Mean, Median and Mode
Mean- Add up all the values; Divide by total number of values
Median- Write the values smallest to largest; Count the number of values; Odd no is middle value, Even is half way between two middle values
Mode- Look for the mose common value
Frequency table averages
To find the median: Find the total frequency; divide by two- that value is the median.
To calculate the mean: Multiply the frequency by the value. Add these together to find the total frequency. total value/ total frequency = mean
If it has class interval values and it asks to calculate the mean - find the midpoint of the value and multiply that by the frequency.
Interquartile Range
Interquartile range is between the Lower quartile(Q1) and the Upper quartile (Q3). (Q3-Q1)
To find the Lower quartile n+1 /4 th value, n= number of data values
To find the Upper quartile 3(n+1)/4 th value
Frequency Polygons
Plot the midpoints of each class interval
Draw points with straight lines
Histograms
Frequency Density = Frequency/ Class Width
Vertical axis is labelled Frequency Density
Area of each bar proportional to frequency
Culmulative frequency
Plot 0 at the beginning
Plot each value at the upper end of the class interval
Join points as a SMOOTH CURVE
Find Lower and upper quartile values by using the equations and looking where the value lies on the graph.
Box Plots
When comparing box plots use the range, interquartile range and the largest and smallest values and the median. It is always better to compare measures of spread rather than medians or end points.
Scatter Graphs
Negative Correlation - Points almost on a negative straight line
Positive Correlation- Points almost on a positive straight line
No Correlation - Completely random points.
Lines of best fit- Straight line which is as close as possible to all points; X need to go thru 0,0 ; Drawn with a ruler; ignores isolated points.
Highest Common Factor and Lowest Common Multiple
HCF - all prime factors that are common in both numbers multiplied together
LCM multiply all prime factors from both numbers
Indices Rules
a^m x a^n = a^m+n
a^m/ a^n = a^m-n
(a^m)^n= a^mn
a^0= 1
Fractions
To divide fractions, turn the second fraction upside down and multiply the two fractions together
To convert the fraction into a decimal- divide the numerator by the denominator
To convert recurring decimals into fractions - write decimal as 'n' and multiply by ( 10 if one recurring digit; 100 if two recurring digits; 1000 if three recurring digit). Then subtract n and divide both sides by 9, 99 or 999 to get a fraction - simplify . IF there is one unrecurring digit e.g 0.4737373etc let 99n be 46.9 and multiply by ten to get 469/990
Rounding and Estimation
5 or more - round up
less than 5 - round down
to estimate - round to one sig figure
Upper Bounds and Lower Bounds
Add Subtract Multiply Divide
Greatest value UB+UB UB–LB UB*UB UB/LB
Lowest Value LB+ LB LB–UB LB*LB LB/UB
LB= lower bound
UB= Upper bound
Fractions Percentages and Decimals
Find fraction of amount: e.g 2/5 of 90 - divide by denominater and multiply by numerator 90/5 =18*2= 36
Write one quantity as a percentage of another e.g 7 out of 20 - Divide first value by second then multiply by 100 7/20 * 100=35%
Find % of an amount 45% of 200 divide percentage by 100 then multiply by value 45/100 *200= 90 Without a calculator use multiples of 1% and 10%
% change : Find percentage of amount and subtract the decrease or find % of original amount e.g 280 is reduced by 25% 0.75 x 280= 210
Calculate increase/decrease Work out the amount of increase/decrease- write as a percentage of the original amount e.g 60 now 39, 60-39=21, 21/60= 35%
Reverse Percentage: Divide by the multipler e.g find orginal price of 132.88 before a 12% decrease 132.88/0.88
Compound Intrest: (starting amount) x (multiplier)^n n= number of years
Indices 2
a^–n = 1/ a^n
a^–1 =1/a (reciprocal)
(a/b)^n= a^n / b^n
(a/b)^–n = b^n / a^n (upside down and negative to positive power )
a^1/2 = 2√a a^1/3 = 3√a
a^m/n= (a^1/n)^m
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