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maths (algebra (algebraic expressions (terms (operations (multiplication…

- - - - a set of terms combined using the four operations
    - - a statement with an equals sign that says that two expressions are equal in value
    - - the individual parts of expressions
      - collecting like terms
        
        gather the terms that are the same
      - operations
        
        addition and subtractions
        
        like terms
        
        multiplication and division
        
        indices
        
        putting the terms together
    - - single brackets
        
        multiply all terms by the outside terms
      - double brackets
        
        multiply each term in the first bracket by each term in the second bracket
    - - reverse of expanding brackets
      - remove common factors
      - quadratics
        
        double brackets
    - - a mathematical proof is a sequence of statements that follow on logically from each other that shows something is always true
      - odd and even
        
        2n
  - - - a calculation for a specific purpose
        
        fahrenheit to celsius
      - always true no matter what values are put in
      - can be calculated by substituting values
      - can be written for real life situations
    - - links an input value to an output value
  - - - a way of writing rules using flow diagrams
    - - an equation will remain balanced as log as the same operations are done to both sides
  - - - eliminating one letter
      - multiply one or both to create a coefficient
  - - - putting into double brackets
    - - b=or-Vb^2-4ac / 2a
    - - gives us coordinates
      - expanding
  - - - < >
        
        open circles
      - =< =>
        
        closed circles
  - - - increase or decrease by the same amount each time
    - - numbers within sequences
      - term to term rule
        
        describes how sequences progress
      - position to term rule
        
        shows how to use a general formula to find any term in a sequence
        
        nth term
        
        any term can be found by substitution
    - - a sequence that contains an n^2
      - the first differences between terms will not match but second differences will
    - - a sequence in which the next term is found by multiplying or dividing by a common ratio
  - - - shown as (x,y)
      - plotted going along x axis then up or down y axis
    - - useful for plotting
      - substituting values
    - - should have at least three pairs to ensure that it is correct
    - - m
        
        gradient
        
        change in y / change in x
        
        two lines are parallel if they have the same gradient
        
        two lines are perpendicular if they meet at a right angle
        
        product of gradients = -1
      - c
        
        y intercept
  - - - plotting
  - - - b,c,d can be zero
      - a cannot be zero
    - - can have more than one change in direction
  - - - y=mx+c
  - - - y=f(x)
      - translated parallel to the x or y axis
    - - can be reflected in x or y axus
      - if f(x) is a function
        
        -f(x)
        
        function reflected in the x axis
        
        f(-x)
        
        function reflected in the y axis
  - - - not total distance traveled
- - - - right angle
        
        90°
        
        represented by square angle notation
      - acute angle
        
        less than 90°
      - obtuse angle
        
        greater than 90°
        
        less than 180°
      - straight line
        
        180°
      - reflex angle
        
        greater than 180°
        
        less than 360°
      - circle
        
        360°
        
        sum of angles around a point
      - where two straight lines meet
    - - always the same distance apart
      - never meet
      - angles
        
        corresponding angles
        
        F
        
        equal
        
        alternate angles
        
        Z
        
        equal
        
        co-interior angles
        
        C
        
        add up to make 180°
        
        vertically opposite angles
        
        X
        
        equal
    - - scalene
        
        no equal sides
        
        no equal angles
      - isosceles
        
        two equal sides
        
        equal base angles
      - equilateral
        
        three equal sides
        
        angles = 60°
      - interior angles
        
        180°
    - - four sided shapes
      - interior angles
        
        180°
    - - regular
        
        equal sides and angles
      - irregular
        
        unequal sides and angles
      - angles
        
        exterior angles
        
        360° / sides
        
        interior angles
        
        (n-2)x180°
        
        n is the number of sides
    - - a shape has a line of symmetry if it can be folded in half with each side of the fold matching perfectly
      - a shape has rotational symmetry when it can be rotated and it still looks the same
        
        how many times a shape can be rotated in 360°
  - - - a path formed by a point which moves according to a rule
    - - used in conjunction with the four compass points
        
        north, south, east, west
      - measured from north line
      - measured in a clockwise direction
      - contain three digits
  - - - length, width, height
    - - flat surfaces
    - - where two faces meet
    - - corner
    - - cross section x depth
      - measured in cm^3
      - the amount of space or capacity it occupies
    - - total area of individual faces
    - - two dimensional drawings of three dimensional shapes
  - - - a line that joins two points on the circumference of a circle and goes through the centre
    - - goes from the centre of a circle to the circumference
      - half the diameter
    - - a line that joins two parts of a circumference
    - - area of a circle made from the circumference and a chord
    - - a line that touches the outside of a circle in one place only
  - - - four sided shape drawn inside a circle
      - opposite angles or a cyclic quadrilateral add up to 180°
  - - - identical size and shape
    - - moves a shape up, down, or side to side
      - size and orientation of the shape remain unchanged
      - a vector is needed
        
        (x/y)
        
        x is horizontal movement
        
        y is vertical movement
    - - making a shape bigger or smaller
      - scale factor
        
        describes how many times bigger or smaller the shape has become
      - a centre of enlargement is needed
      - negative enlargement
        
        changes the enlargements position as well as size
        
        -1 = rotation 180°
    - - creates a mirror image of the original shape
      - line of reflection
        
        line of symmetry
        
        mirror line
        
        x=y
    - - spins shapes around
      - centre of rotation
      - direction of turn
      - angle of turn
  - - - c is the hypotenuse
  - - - one dimensional
      - cm
      - m
    - - two dimensional
      - cm^2
      - m^2
    - - three dimensional
      - cm^3
      - m^3
      - litres
    - - how fast something is travlling
      - distance / time
    - - a way of comparing how compact different materials are
      - mass / volume
    - - the price for a fixed amount
      - helpful when trying to identify the best value between different sized packets of the same item
      - price / volume
    - - imperial
        
        length
        
        feet
        
        yards
        
        miles
        
        mass
        
        ounces
        
        pounds
        
        stone
        
        volume
        
        gallons
      - metric
        
        length
        
        metres
        
        mass
        
        kilograms
        
        volume
        
        cubic metres
  - - - sin = opposite / hypotenuse
      - cos = adjacent / hypotenuse
      - tan = opposite / adjacent
- - - - used by calculators
      - the order of operations
    - - below zero
      - for powers on calculator use brackets
  - - - terminate
        
        do not recur, stop
      - recur
        
        do not terminate, keep going infinitely
  - - - five or above goes up four or below goes down
      - significant figures
        
        most important number
        
        furthest numbers to the left
        
        first non zero number
    - - estimating without rounding
      - miss off all numbers after a point
    - - upper and lower bounds
      - error intervals
  - - - divide numerator and denominator by any common factor
    - - whole numbers and fractions mixed together
    - - numerator is bigger than denominator
      - top heavy fractions
    - - addition and subtraction
        
        tepee method
      - multiplication
        
        multiply both sides with each other
        
        numerator with numerator
        
        denominator with denominator
      - division
        
        Keep Flip Change
  - - - integers that divide exactly into a number
      - highest common factor
        
        the largest factor that goes into two or more numbers
      - prime factors
        
        factors that are prime numbers
    - - the number multiplied by an integer
      - lowest common multiple
        
        the smallest number that is a common multiple of two or more numbers
    - - have two factors
        
        one and itself
    - - the amount of times a number has been multiplied by itself
  - - - reciprocal
    - - denominator root
      - numerator power
  - - - a number between one and ten
      - multiplied by a power of ten
        
        can be positive or negative
    - - addition and subtraction
        
        convert into normal numbers, do the sum, put back into standard form
      - multiplication
        
        multiply the numbers and add the indices together
      - division
        
        divide the numbers and subtract the indices for each other
  - - - multiplication and division
        
        by itself
        
        one
        
        can be simply multiplied or divided
    - - removing a surd from the denominator of the fraction
      - multiply by the surd over itself
  - - - paid by employer to employees in return for time worked
      - hourly rate or annual salary
    - - how much a person earns over a year
        
        gross pay
        
        how much is made before deductions are made
        
        net pay
        
        how much is made after deductions
        
        take home pay
    - - income tax
      - national insurance
      - pension contributions
    - - money made after all expenses are paid
      - if expenses are more than money received this is a loss
    - - a record of the money in and out of an account
      - over a month
    - - automatic payments that come out of a bank account
      - often for variable amounts
        
        mobile phone billd
        
        amount billed changes each month
    - - automatic payments
      - fixed amounts
    - - Value Added Tax
      - purchase tax paid on goods and services
      - some products are zero rated
        
        food
        
        essential items
      - 20% on top of cost of goods and services
- - - - fractions
      - decimals
      - percentages
      - ratios
    - - all calculations written out
    - - they can be set in real life context
      - a frame work can be used to tackle these problems
  - - - common graphs
        
        travel
        
        temperature
        
        conversion
      - a frame work can be used to tackle graphical problems
  - - - triangles
      - quadrilaterals
      - circles
    - - questions will often involve knowing angle and length properties
      - e.g.
        
        each angle in an equilateral triangle is 60°
    - - show working out on diagram
      - label each step to the side of the diagram
    - - knowledge of shape properties is essential
      - a frame work can be used to tackle these problems
  - - - most will be like this
    - - two sides are the same
      - two angles are the same
      - angles in a triangle add up to 180°
    - - particularly
        
        angles
        
        lengths
    - - problems often include a mix of
        
        algebra
        
        number
        
        geometry
      - a frame work can be used to tackle these problems
  - - - finding the mean, median, mode or range of some data
      - extracting and using information given in a statistical chart or diagram
    - - can include algebra
    - - diagrams may include information which needs to be extracted
      - a frame work can be used to tackle these problems
- - - - x:y
    - - the same way as fractions
  - - - the difference between the amounts divided by the original amount multiplied by one hundred
    - - simple interest
        
        the percentage amount multiplied by the number of periods
      - compound interest
        
        each time interest is paid onto an amount, the added interest also receives an interest from then on
    - - work out the original value of an item after it has been increased or decreased
      - an item always begins at 100%
  - - - used to show how different quantities of amounts relate to one another
      - can be used to work out if the product is good value
      - scale factor
        
        can be used to calculate scale factors of enlargement
        
        shows how many times bigger or smaller each side length of a shape is
        
        original length / image length
      - multiple of proportion
        
        k
    - - when one amount is a multiple of another
      - x=ky
    - - where the value of one item increases as the other decreases
      - x=k/y
      - x=1/y
- - - - data that can only be written in words, not numbers
      - e.g.
        
        the colours of cars in a car park
    - - data that can be written in numbers
      - e.g.
        
        the heights of children
      - discrete data
        
        numerical data that cannot be shown in decimals
        
        e.g.
        
        the number of children in a classroom
      - continuous data
        
        numerical data that can be shown in decimals
        
        e.g.
        
        the weights of ten babies
    - - data that has been collected from the original source for a specific purpose
      - e.g.
        
        if a school wanted to know what their students thought of the school canteen service they would question the pupils directly
    - - data that is not originally collected by a group for a specific purpose
      - e.g.
        
        finding out the average cost of cars in a car park by using national statistics
  - - - the height of each bar represents the frequency of the group
      - the modal category is always the highest bar
      - comparative bar charts
        
        dual bar charts
        
        useful when comparing two sets of data
        
        e.g.
        
        boys and girls
        
        a key is needed to compare data
    - - line charts
      - plotting the mid points of each category and joining them with straight lines
    - - shows the frequency of each category with a diagram
      - a key is used to show the quantity that each diagram represents
    - - shows the frequency as a sector of the circle
      - the sizes of the angles of the sectors are in proportion to the frequency
    - - show the relationship between two variables
        
        known as correlation
      - a line of best fit should travel through the points as closely as possible
    - - cumulative frequency diagrams
        
        show the running total of the frequencies
        
        can be used to find median and quartile values
      - box plots show
        
        the lowest and highest values
        
        the upper and lower quartiles
        
        the median point
    - - the area if the bar shows the frequency
      - used when the intervals or class groupings are not equal
      - the distance between each category is called class width
  - - - mean
        
        adding up all of the data and dividing by the number of items of data
      - median
        
        the value of the middle item of data when all the data is arranged in order
        
        if there are two middle items of data
        
        the median is halfway between the two values
      - mode
        
        the item of data with the highest frequency
        
        in a set of data there can be more than one mode or no mode
    - - shows the spread of data
      - the smaller the range
        
        the more consistent the data
      - the largest value - the smallest value
    - - shows the spread of the central 50% of the data
      - upper quartile - lower quartile