# 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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