Mathematics (0580) Core Comprehensive cheat sheet

    Mathematics (0580) · CAIE · Core

    Comprehensive
    5 pages
    34 formulas, 56 concepts
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    All Mathematics (0580) Core cheat sheets

    Number

    • Percentage Change
      Percentage Change=New Value−Original ValueOriginal Value×100\text{Percentage Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100
      Use to find the percentage increase or decrease between two values.
    • Simple Interest
      I=PRT100I = \frac{PRT}{100}
      Calculates simple interest, where P is principal, R is rate, and T is time in years.
    • Standard Form
      A×10nA \times 10^n
      A way of writing very large or very small numbers, where 1 ≤ A < 10 and n is an integer.

    Key concepts: **Types of Numbers**: Natural: Counting numbers (1, 2, 3...). Integers: Whole numbers and their negatives (...-1, 0, 1...). Prime: Divisible only by 1 and itself (2, 3, 5, 7...). Square: A number multiplied by itself (1, 4, 9...). Cube: A number multiplied by itself twice (1, 8, 27...)., **HCF and LCM**: Highest Common Factor (HCF) is the largest number that divides into two or more numbers. Lowest Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers., **Rational vs Irrational**: Rational numbers can be written as a fraction (e.g., 0.5, 7, 1/3). Irrational numbers cannot be written as a fraction and have non-repeating, non-terminating decimals (e.g., π, √2)., **Set Notation**: ∈ means 'is an element of'. ∪ means 'union' (all elements in both sets). ∩ means 'intersection' (elements common to both sets). n(A) is the number of elements in set A., **Fractions, Decimals, Percentages**: These are different ways to represent parts of a whole. Be able to convert between them (e.g., 1/4 = 0.25 = 25%)., **Limits of Accuracy**: The upper and lower bounds of a number rounded to a certain degree of accuracy. For a number rounded to the nearest 10, the bounds are ±5., **Ratio**: Compares parts to parts (e.g., 2:3). To share an amount in a ratio, find the total number of parts and divide the amount by it, then multiply by each part of the ratio., **Proportion**: Direct proportion means as one quantity increases, the other increases at the same rate. Inverse proportion means as one quantity increases, the other decreases.

    Exam tips

    • Always show your working, even for calculator questions, to get method marks.
    • When estimating, round each number to one significant figure before calculating.
    • Read questions carefully to see if the answer should be a fraction, decimal, or percentage.

    Algebra and graphs

    • Index Law (Multiplication)
      am×an=am+na^m \times a^n = a^{m+n}
      When multiplying powers with the same base, add the indices.
    • Index Law (Division)
      am÷an=am−na^m \div a^n = a^{m-n}
      When dividing powers with the same base, subtract the indices.
    • Index Law (Power of a Power)
      (am)n=am×n(a^m)^n = a^{m \times n}
      When raising a power to another power, multiply the indices.
    • Zero Index
      a0=1a^0 = 1
      Any non-zero number raised to the power of zero is 1.
    • Negative Index
      a−n=1ana^{-n} = \frac{1}{a^n}
      A negative index indicates the reciprocal of the base raised to the positive index.

    Key concepts: **Simplifying Expressions**: Collect 'like terms' - terms with the same variable and power (e.g., 3x + 2y - x + 5y simplifies to 2x + 7y)., **Expanding Brackets**: Multiply the term outside the bracket by every term inside the bracket (e.g., 3(x + 4) = 3x + 12)., **Factorising**: The reverse of expanding. Find the highest common factor of all terms and place it outside the bracket (e.g., 6x + 9 = 3(2x + 3))., **Solving Linear Equations**: Use inverse operations to isolate the unknown variable (letter) on one side of the equation., **Changing the Subject**: Rearrange a formula to make a different variable the subject, using the same principles as solving equations., **Sequences**: A list of numbers following a rule. An arithmetic sequence has a common difference between consecutive terms., **Interpreting Graphs**: Understand what the axes represent. For travel graphs, the gradient represents speed and a horizontal line means stationary.

    Exam tips

    • When solving equations, whatever you do to one side, you must do to the other.
    • Be careful with negative signs when expanding brackets or simplifying expressions.
    • Check your answer by substituting it back into the original equation.

    Coordinate geometry

    • Equation of a Straight Line
      y=mx+cy = mx + c
      Represents a straight line, where m is the gradient and c is the y-intercept.
    • Gradient (m)
      m=riserun=change in ychange in xm = \frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}}
      Calculates the steepness of a line from two points on the line.

    Key concepts: **Cartesian Coordinates**: A pair of numbers (x, y) that locate a point on a grid. The x-coordinate is the horizontal position ('along the corridor') and the y-coordinate is the vertical position ('up the stairs')., **Midpoint of a Line Segment**: The coordinates of the midpoint are the average of the x-coordinates and the average of the y-coordinates of the endpoints., **Gradient**: Measures the steepness of a line. A positive gradient slopes upwards from left to right, a negative gradient slopes downwards., **Y-intercept (c)**: The point where the line crosses the vertical y-axis. Its coordinate is (0, c)., **Drawing Linear Graphs**: Create a table of values for x and y, plot at least three points, and join them with a straight line using a ruler., **Parallel Lines**: Parallel lines have the same gradient (the same 'm' value in y = mx + c).

    Exam tips

    • Always read the scale on the axes carefully, as they might not go up in ones.
    • Use a ruler for all straight line graphs.
    • Remember that horizontal lines have the equation y = a and vertical lines have the equation x = b.

    Geometry

    • Angles on a Straight Line
      a+b=180∘a + b = 180^\circ
      Angles on a straight line add up to 180 degrees.
    • Angles at a Point
      a+b+c=360∘a + b + c = 360^\circ
      Angles around a point add up to 360 degrees.
    • Angles in a Triangle
      a+b+c=180∘a + b + c = 180^\circ
      The interior angles of a triangle add up to 180 degrees.
    • Angles in a Quadrilateral
      a+b+c+d=360∘a + b + c + d = 360^\circ
      The interior angles of a quadrilateral add up to 360 degrees.
    • Interior Angle of a Regular Polygon
      (n−2)×180∘n\frac{(n-2) \times 180^\circ}{n}
      Calculates one interior angle of a regular polygon with 'n' sides.

    Key concepts: **Parallel Line Angles**: Alternate angles (Z-shape) are equal. Corresponding angles (F-shape) are equal. Co-interior angles (C-shape) add up to 180°., **Types of Triangles**: Equilateral (3 equal sides, 3 equal angles of 60°). Isosceles (2 equal sides, 2 equal base angles). Scalene (no equal sides or angles). Right-angled (one 90° angle)., **Types of Quadrilaterals**: Square, Rectangle, Parallelogram, Rhombus, Trapezium, Kite. Know their properties regarding sides, angles, and diagonals., **Circle Vocabulary**: Radius (centre to edge), Diameter (edge to edge through centre), Circumference (perimeter), Chord (line connecting two points on the edge), Tangent (line that touches the circle at one point), Arc (part of circumference), Sector (pizza slice shape)., **Symmetry**: Line symmetry is when a shape can be folded onto itself. Rotational symmetry is how many times a shape fits onto itself during a 360° turn., **Bearings**: Measured from the North line, in a clockwise direction, and always written as a three-figure number (e.g., 045°)., **Nets**: A 2D shape that can be folded to make a 3D solid. Be able to recognise nets for cubes, cuboids, prisms, and pyramids.

    Exam tips

    • When asked to calculate an angle, you must give a reason for your answer (e.g., 'angles on a straight line').
    • Use a pencil, ruler, and protractor for accurate geometrical constructions and drawings.
    • For bearings, always draw a North line at the point you are measuring from.

    Mensuration

    • Area of a Rectangle
      A=length×widthA = \text{length} \times \text{width}
      Calculates the area of a rectangle.
    • Area of a Triangle
      A=12×base×heightA = \frac{1}{2} \times \text{base} \times \text{height}
      Calculates the area of a triangle, using the perpendicular height.
    • Area of a Trapezium
      A=12(a+b)hA = \frac{1}{2}(a+b)h
      Calculates the area of a trapezium, where 'a' and 'b' are the parallel sides.
    • Circumference of a Circle
      C=πdC = \pi d or C=2πrC = 2\pi r
      Calculates the distance around the outside of a circle.
    • Area of a Circle
      A=πr2A = \pi r^2
      Calculates the area of a circle.
    • Volume of a Cuboid
      V=l×w×hV = l \times w \times h
      Calculates the volume of a rectangular prism.
    • Volume of a Prism
      V=Area of cross-section×lengthV = \text{Area of cross-section} \times \text{length}
      Calculates the volume of any prism, such as a triangular prism or cylinder.
    • Volume of a Cylinder
      V=πr2hV = \pi r^2 h
      A specific application of the volume of a prism formula.

    Key concepts: **Metric Unit Conversions**: Length: 1km = 1000m, 1m = 100cm, 1cm = 10mm. Mass: 1kg = 1000g. Volume/Capacity: 1 litre = 1000ml, 1ml = 1cm³., **Perimeter**: The total distance around the outside of a 2D shape. It is found by adding up the lengths of all the sides., **Area**: The amount of surface a 2D shape covers. Units are squared (e.g., cm², m²)., **Volume**: The amount of space a 3D object occupies. Units are cubed (e.g., cm³, m³)., **Surface Area**: The total area of all the faces of a 3D object. Imagine unfolding the net and finding its total area., **Arc Length**: A fraction of the circumference. Calculated as (angle/360) x πd., **Sector Area**: A fraction of the area of a circle. Calculated as (angle/360) x πr².

    Exam tips

    • Check the units given in the question and make sure your answer is in the correct units.
    • Use the π button on your calculator for more accurate answers, unless told to use an approximation like 3.14.
    • For compound shapes, split them into simpler shapes you know how to calculate the area or volume of.

    Trigonometry

    • Pythagoras' Theorem (finding hypotenuse)
      a2+b2=c2a^2 + b^2 = c^2
      In a right-angled triangle, used to find the longest side (c) when the two shorter sides (a, b) are known.
    • Pythagoras' Theorem (finding a shorter side)
      a2=c2−b2a^2 = c^2 - b^2
      In a right-angled triangle, used to find a shorter side when the hypotenuse and one other side are known.
    • Sine Ratio (SOH)
      sin⁡(x)=OppositeHypotenuse\sin(x) = \frac{\text{Opposite}}{\text{Hypotenuse}}
      Relates an angle to the opposite side and the hypotenuse in a right-angled triangle.
    • Cosine Ratio (CAH)
      cos⁡(x)=AdjacentHypotenuse\cos(x) = \frac{\text{Adjacent}}{\text{Hypotenuse}}
      Relates an angle to the adjacent side and the hypotenuse in a right-angled triangle.
    • Tangent Ratio (TOA)
      tan⁡(x)=OppositeAdjacent\tan(x) = \frac{\text{Opposite}}{\text{Adjacent}}
      Relates an angle to the opposite and adjacent sides in a right-angled triangle.

    Key concepts: **Right-Angled Triangle**: A triangle with one angle of exactly 90°. Trigonometry (SOH CAH TOA) and Pythagoras' Theorem only apply to right-angled triangles., **Hypotenuse**: The longest side of a right-angled triangle, always opposite the right angle., **Opposite and Adjacent**: Relative to a specific angle (not the right angle). The 'Opposite' side is across from the angle. The 'Adjacent' side is next to the angle (and is not the hypotenuse)., **Finding a Side**: Use SOH CAH TOA when you know an angle and one side, and you want to find another side., **Finding an Angle**: Use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) when you know two sides and want to find an angle.

    Exam tips

    • Make sure your calculator is in Degrees (DEG) mode before starting any trigonometry questions.
    • Always start by labelling the sides of the triangle as Opposite (O), Adjacent (A), and Hypotenuse (H) relative to the angle you are using.
    • Remember to square root your answer at the end of a Pythagoras' Theorem calculation to find the length.

    Transformations and vectors

    • Translation Vector
      (xy)\begin{pmatrix} x \\ y \end{pmatrix}
      Describes a translation. The top number (x) is the horizontal movement (right is positive) and the bottom number (y) is the vertical movement (up is positive).

    Key concepts: **Reflection**: A 'flip' of a shape in a mirror line. To describe a reflection, you must state the equation of the mirror line (e.g., reflect in the line x=2)., **Rotation**: A 'turn' of a shape. To describe a rotation, you must state the centre of rotation, the angle of rotation, and the direction (clockwise or anticlockwise)., **Translation**: A 'slide' of a shape. To describe a translation, you must give the translation vector., **Enlargement**: Changes the size of a shape. To describe an enlargement, you must state the centre of enlargement and the scale factor., **Scale Factor**: In an enlargement, this determines the new size. A scale factor > 1 makes the shape bigger. A scale factor between 0 and 1 makes the shape smaller.

    Exam tips

    • Use tracing paper to help you perform rotations accurately. Trace the shape and the centre of rotation.
    • When describing a transformation, make sure you give all the required pieces of information (e.g., for rotation, you need centre, angle, and direction).
    • For enlargements, draw lines from the centre of enlargement through the vertices of the original shape to find the new vertices.

    Probability

    • Probability of an Event
      P(event)=Number of favourable outcomesTotal number of possible outcomesP(\text{event}) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}
      Calculates the theoretical probability of a single event occurring.
    • Probability of 'Not' an Event
      P(not A)=1−P(A)P(\text{not A}) = 1 - P(A)
      The probability of an event not happening is 1 minus the probability that it does happen.
    • Expected Frequency
      Expected Frequency=P(event)×Number of trials\text{Expected Frequency} = P(\text{event}) \times \text{Number of trials}
      Calculates how many times you would expect an event to occur in a certain number of trials.

    Key concepts: **Probability Scale**: Probability is always a value between 0 and 1. 0 means the event is impossible, 1 means it is certain., **Relative Frequency**: An estimate of probability based on the results of an experiment. It is calculated as (number of times event occurs) / (total number of trials)., **Sample Space**: The set of all possible outcomes of an experiment. Can be shown as a list or in a diagram., **Mutually Exclusive Events**: Events that cannot happen at the same time. The probability of one OR the other occurring is found by adding their individual probabilities.

    Exam tips

    • Answers for probability can be given as a fraction, decimal, or percentage. Fractions are often the easiest.
    • Make sure your fractions are simplified unless the question says otherwise.
    • Expected frequency is an estimate, so the actual result in an experiment may be different.

    Statistics

    • Mean
      Mean=Sum of all valuesNumber of values\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}
      Calculates the mean average of a set of data.
    • Range
      Range=Highest value−Lowest value\text{Range} = \text{Highest value} - \text{Lowest value}
      Measures the spread of a set of data.

    Key concepts: **Averages**: Mean: The sum of the data divided by the count. Median: The middle value when the data is in order. Mode: The most frequently occurring value., **Frequency Table**: A table that shows data and how many times each value (the frequency) occurs. To find the mean from a frequency table, add an 'fx' column (frequency x value)., **Bar Chart**: Used to display discrete data. The bars should be of equal width and have gaps between them., **Pie Chart**: Used to show proportions. To calculate the angle for a category: (frequency / total frequency) x 360°., **Stem-and-Leaf Diagram**: A way of organising data to show its distribution. It must have a key., **Scatter Diagram**: A graph that plots pairs of numerical data to show the relationship between them., **Correlation**: Describes the relationship shown on a scatter diagram. It can be positive (as one variable increases, so does the other), negative (as one increases, the other decreases), or no correlation.

    Exam tips

    • Always arrange data in order before finding the median.
    • When drawing charts or diagrams, remember to include a title and label your axes.
    • Don't confuse the different types of averages; read the question carefully to see which one is required.