Maths rules

How can we calculate the total number of ways two tasks can be done?

m x n when there are m ways of doing something and n ways of doing something

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When we multiply powers what do we do to the indices?

When we multiply powers we add the indices X^m x X^n = X^m+n

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When we divide powers what do we do to the indices?

When we divide powers we subtract the indices X^m / X^n = X^m-n

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When we raise a power to another power what do we do to the indices?

When we raise a power to another power, we multiply the indices. eg. (a^n)^m = a^nxm

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What does any number to the power of 0 equal?

Anything to the power of 0 = 1 eg. 4^0 = 1

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What happens when a^-1?

a^1 = 1 / a

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What happens when a^-m

a^-m = 1 / a^m

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What happens when a^1/2

a^1/2 = √a

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What happens when a^1/m

a^1/m = m√a

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What happens when a^m/n

a^m/n = (n√a)^m

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How do we write a number in standard form?

A number in standard format A x 10^n, where A is a number between 1 and 10 and n is an integer

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How do we rationalise the denominator?

To rationalise the denominator, multiply the denominator and the numerator by the surd in the denominator and simplify.

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How do we simplify a surd?

Identify any factors that are square numbers

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What happens when a^1/n

a^1/n = n√a

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What is an identity?

When two sides of a relation such as 2(x + 5) = 2x + 10 are equal for all values of x it is called an identity

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How do we expand a bracket eg. x(y+z)

x(y+z) = xy+xz

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How do we calculate the HCF?

We find the products of prime factors for both of the numbers you are finding the HCF from. To find the HCF, we find any common factors of the two numbers and multiply the numbers together to find the HCF.

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How do we calculate the LCM?

We find the products of prime factors for both numbers, using a prime factor tree. To find the LCM we multiply the HCF and the left over numbers together.

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Find the HCF and LCM of 24 and 180?

prime factors of 24 = 2 x 2 x 2 x 3 180 = 2 x 2 x 3 x 3 x 5 HCF = 2 x 2 x 3 = 12 LCM = 12 x 3 x 2 x 5 = 360

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What is an expression?

An expression contains letter and number terms but no equals signs

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What is an equation?

An equation has an equals sign, terms in one letter and numbers

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What is a formula?

A formula has an equals sign and letters to represent different quantities

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What is an arithmetic sequence?

In an arithmetic sequence the terms increase or decrease by a fixed number = common difference

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What do we do when an arithmetic sequence has a common difference of d? Let the common difference be p

When an arithmetic sequence with a common difference d is input into this function machine, the output sequence has a common difference p x d

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What happens in a Fibonacci sequence?

In a Fibonacci sequence, the next number is found by adding the previous two numbers together

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What happens in a geometric sequence?

In a geometric sequence the terms increase or decrease by a constant multiplier eg. the nth term is ar^n

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How can we work out the nth term of a quadratic sequence?

1. work out the second difference 2. halve the second difference to get a an^2 term 3. subtract the sequence an^2, you may need to add a constant or find the nth term of the remaining terms.

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What is a frequency polygon?

A frequency polygon is a graph made by joining the midpoints of the bars in a bar chart with straight lines.

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What is the modal class?

The modal class has the highest frequency

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How do we estimate the mean from a grouped frequency table?

Add together the products of class midpoints and their frequencies, and divide by the total frequency

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How do we find the group where the median lies in a grouped frequency table? let the total frequency be n

The median lies in the group containing the n + 1 / 2 th item of data

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What is a interpolation?

Using a line of best fit to predict data within the range of data

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What is a extrapolation?

Using a line of best fit to predict data outside the range of data given

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How can we write the reciprocal of a number?

The reciprocal of the number n is 1/n or this can be written as n^-1

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How do we turn a mixed number fraction into an improper fraction?

1. multiply the whole number part with the denominator of the fraction 2. add that to the numerator 3. then write the result on top of the denominator

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When are two quantities in direct proportion?

When two quantities are in direct proportion, as one is multiplied by a number, n, so is the other, their ratio stays the same as they either increase or decrease

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How can we share a quantity in a given ratio?

Work out how much one part is worth and then multiply this by the number of parts each person receives.

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How can we calculate percentage change? What is the formula for percentage change?

percentage change = actual change/original amount x 100

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How can we calculate percentage loss (or profit)?

percentage loss (or profit = actual loss (or profit) / original amount x 100

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What does depreciate mean?

Loss in value

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What does per annum mean?

Per year

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What do we multiply a reoccurring decimal by if a. 1 decimal place recurs b. 2 decimal places reoccurs c. 3 decimal places reoccur

a. x10 b. x100 c. x1000

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Convert the reoccurring decimal into a fraction. 1.042424242...

x = 1.0424242... 1000x = 1042.42424242... 10x = 10.424242... 1000x - 10x = 990x 1042.424242 - 10.424242 = 1032 so 990x = 1032 fraction = 1032/990

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Angles on a straight line add up to?

180

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For any polygon interior + exterior angle =?

180

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In a triangle, the exterior angle is equal to the two triangles at the..

other two vertices

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What is the formula to calculate the sum of the interior angles in a regular polygon?

n = number of sides (n-2) x 180

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What is the formula to calculate the sum of exterior angles in a regular polygon?

360/n = exterior angles in a regular polygon (where n = number of sides)

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What is Pythagoras' theorem?

a^2 + b^2 = c^2 and c^2 = a^2 + b^2 (c is always the hypotenuse)

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What is the hypotenuse opposite to in a right angles triangle?

The right angle

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What sides do we label opposite and adjacent in a right angled triangle?

side opposite theta = opposite angle / side next to theta = adjacent

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sinϑ =?

sinϑ = opposite / hypotenuse

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cosϑ =?

cosϑ = adjacent / hypotenuse

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tanϑ =?

tanϑ = opposite / adjacent

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When do we use inverse sin/cos/tan? (sin-1/cos-1/tan-1)

When we want to calculate a missing angle (ϑ)

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What is the a. the angle of elevation? b. the angle of depression?

a. the angle measured upwards from the horizontal b. the angle measured downwards from the horizontal

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What is the equation for a straight line graph?

y = mx + c where m = the gradient and c = the y-intercept

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What do parallel lines have in common?

parallel lines have the same gradient

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How can we find a. the y-intercept b. the gradient

a. find the y-co ordinate where x = 0 b. find the x-co ordinate where y = 0

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Equation for average speed?

speed = total distance/total time

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Equation for acceleration? (velocity time graph)

acceleration = change in velocity / time

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How do we find distance travelled from a velocity time graph?

the area under a velocity time graph is the distance travelled

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Equation for the co-ordinates of the midpoints?

(X1 + X2 / 2 , Y1 + Y2 / 2)

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When two lines are perpendicular the product of the gradients is?

-1

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When a graph has a gradient of m, what is the gradient of a line perpendicular to it?

-1/m

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What is the equation of a circle with centre (0,0) and radius r

x^2 + y^2 = r^2

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Formula for the area of a trapezium?

1/2(a+b)h

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How do we convert a. cm^2 to mm^2 b. mm^2 to cm^2

a. x 100 b. /100

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What is volume measured in?

mm^3 cm^3 or m^3

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How to calculate volume of a prism?

= area of cross section x length

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Circumference formula?

πd or 2πr

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area of a circle formula?

πr^2

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Volume of a cylinder?

πr^2h

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Surface area of cylinder?

2πr + 2πrh

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Surface area of a sphere?

4πr^2

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Volume of a sphere?

4/3πr^3

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Volume of pyramid?

1/3area of base x vertical height

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Volume of cone?

1/3 area of base x vertical height (1/3πr^2h)

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Curved surface area of cone?

πrl (r = radius / l = slanted length)

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Total surface area of cone?

πrl + πr^2

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What is the a. plan b. front elevation c. side elevation

a. the view from above the object b. the view of the front of the object c. the view of the side of the object

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What are the four types of transformations?

reflections, rotations, translations and enlargements

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What do you need to describe a rotation?

direction of turn (clockwise or anticlockwise), angle of turn and centre of rotation

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How do we describe an enlargement?

centre of enlargement and the scale factor (to find COE join corresponding points of the object to the image)

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What happens when a scale factor is negative?

The shape is the opposite side of the COE

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In a vector what does the top and the bottom number represent?

top number = right or left (left is negative) bottom number = up or down (down is negative)

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What is a bearing and how do we measure a bearing correctly? How many digits must you write a bearing in?

A bearing is an angle in degrees, clockwise from north. A bearing is always written using three digits.

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What does a. perpendicular bisector do b. angle bisector do

a. perpendicular bisectors cuts a line in half at a right angle b. angle bisector cuts an angle exactly in half

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What is the quadratic formula?

x= -b + or - √b^2 - 4ac / 2a (all over 2a)

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Completing the square formula?

(x + b/2)^2 - (b/2)^2 + c

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How can we find the co-ordinates where two graphs intersect?

By using simultaneous equations

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What does a. empty circle b. filled circle represent on an inequality number line

a. the value is not included b. the value is included

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What is the set notation for an inequality?

( x : x > n)

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What should you do to the inequality signs when you divide or multiply by a negative number?

You should reverse the inequality signs

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When an inequality has an upper and lower limit, you should..

Solve the two sides separately

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What happens if two events are mutually exclusive?

They cannot happen at the same time

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How can we calculate P(not A) for mutually exclusive events?

1 - P(A)

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How can we calculate the expected number of outcomes?

Expected number of outcomes = number of trials x probability

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How can we calculate relative frequency?

Relative frequency = frequency / total number of trials

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What happens if two events are independent?

If one happening does not affect the probability of the other, they are independent

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How do we find the probability of two independent events?

Multiply their probabilities eg. P(a) x P(b) = P(a and b)

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What does a. P(A∩B) b. P(AUB) c. A' d. P(A ∩B I B)

a. A intersection B (all elements in A and B) b. A union B (all elements in A or B or both) c. Elements not in A d. The probability of A and B given B

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What is compound interest?

The interest earned each year added to money in the account and earns interest the next year

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Formula for total interest?

Total interest = amount in the account at the end of the investment - amount invested

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How can we calculate compound interest?

initial amount x (100+interest rate/ 100)^n (where n = number of years)

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Density formula? Pressure formula?

Density = mass / volume Pressure = force / area

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When x and y are in direct proportion k =?

k = y/x as y =kx (where k is the gradient of the graph)

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What happens when x and y are in inverse proportion?

y is proportional to 1/x as one doubles the other halves and xy = k so y = k/x

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What are congruent shapes?

Shapes that have the same size/ shape and angles

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What are the rules of congruency?

SSS, SAS, AAS, RHS

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When are shapes similar?

When one shape is an enlargement of the other

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When the linear scale factor is k: a. lengths are multiplied by ... b. area is multiplied by c. volume is multiplied by

a. k b. k^2 c. k^3

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When does a. a sine graph first hit 0 and repeat b. a cosine graph first hit 0 and repeat c. when does a tangent graph repeat

a. first hits zero at 180 and hits zero in intervals of 180, repeats every 360 b. hit zero first at 90 and hits 0 every 180, repeats every 360 in both directions c. tangent graph repeats every 180 in both directions

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What is the formula for the area of a triangle using sin?

1/2absinC

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What is the formula for the sine rule to calculate an unknown side?

a/sinA = b/sinB = c/sinC

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What is the formula for the sine rule to calculate a unknown angle?

sinA/a = sinB/b = sinC/c

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What is the formula for the cosine rule to calculate an unknown side?

a^2 = b^2 + c^2 - 2bcCosA

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What is the formula for the cosine rule to calculate an unknown angle?

CosA = b^2 + c^2 - a^2 / 2bc

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The graph of y = -f(x) is the refection of the graph y =f(x) in the..

x - axis

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The graph of y = f(-x) is the reflection of the graph y =f(x) in the...

y - axis

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The graph y = -f(-x) is a reflection in both the ...

x and y - axis and are equivalent to a rotation of 180 degrees from the origin

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The graph y = f(x+a) is a vertical stretch of the graph y = f(x) by

a scale factor of a, parallel to the y- axis

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The graph of y = f(ax) is a horizontal stretch of the graph of y - f(x) with a scale factor of

1/a parallel to the x-axis

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What is a. census b. sample c. stratified sample d. strata

a. a survey of the whole population b. a smaller number os items from a population c. a number of people taken from each group proportional to the group size d. a group of a population

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How can we calculate a. the median b. lower quartile c. upper quartile d. interquartile from a cumulative frequency diagram

a. n/2th value b. n/4th value c. 3n/4th value d. UQ - LQ = IQ

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For a set of n data values how do we calculate the a. Lower quartile b. Upper quartile

a. (n+1)/4 b. 3(n+1)/4

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Formula for frequency density?

frequency density = frequency / class width

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The angle between a tangent and radius =

90 degrees

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Tangents drawn from a point outside the circle are..

equal in length

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What is a chord?

A straight line connecting two points on a circle

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An angle at the centre of a circle is...

twice the angle at the circumference when both are subtended from the same arc

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The angle in a semicircle is...

at a right angle

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Angles subtended at the circumference by the same arc are..

equal

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Angles in the same segment are...

equal

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What is a cyclic quadrilateral?

A quadrilateral with all four vertices on the circumference of a circle

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Opposite angles in a cyclic quadrilateral...

add up to 180

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An exterior angle of a cyclic quadrilateral is equal to..

to the opposite interior angle

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What is the alternate segment rule?

The angle between the tangent and the chord is equal to the angle in the alternate segment

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When we expand three brackets what do we do first?

expand two of the brackets first

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How can we calculate the turning point?

By completing the square

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How can we find the roots of an equation?

By expanding the brackets

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How can we calculate the y-intercept from the solutions in the brackets in a cubic equation?

multiplying all the numbers in the brackets together gives the y-intercept for a cubic graph

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