Mathematics (0580) Core Standard cheat sheet

    Mathematics (0580) · CAIE · Core

    Standard
    5 pages
    28 formulas, 42 concepts
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    Number

    • Percentage Change
      ChangeOriginal Value×100%\frac{\text{Change}}{\text{Original Value}} \times 100\%
      Use to find the percentage increase or decrease between two values.
    • Simple Interest
      I=P×R×T100I = \frac{P \times R \times T}{100}
      Calculates interest (I) based on the principal (P), rate (R), and time (T).

    Key concepts: **Prime Number**: A natural number greater than 1 that has exactly two factors: 1 and itself (e.g., 2, 3, 5, 7, 11)., **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 and have non-repeating, non-terminating decimals (e.g., $\pi$, $\sqrt{2}$)., **Standard Form**: A way of writing very large or small numbers in the form $A \times 10^n$, where $1 \le A < 10$ and n is an integer., **Limits of Accuracy**: The range within which the true value of a measurement lies, determined by the degree of rounding. For a number rounded to the nearest 10, the bounds are $\pm 5$.

    Exam tips

    • When asked to estimate, round all numbers to 1 significant figure before performing any calculations.
    • For questions involving fractions, decimals and percentages, it is often easiest to convert all values to the same format (e.g., all to decimals) before comparing or ordering them.

    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=amn(a^m)^n = a^{mn}
      When raising a power to another power, multiply the indices.
    • 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: **Like Terms**: Terms that have the exact same variables raised to the same powers. Only like terms can be added or subtracted., **Factorising**: The process of writing an expression as a product of its factors, which is the reverse of expanding brackets. Start by finding the highest common factor., **Solving Linear Equations**: To find the value of the unknown variable, perform inverse operations to both sides of the equation until the variable is isolated., **nth Term of a Linear Sequence**: The rule for a sequence, often in the form $an+b$, where 'a' is the common difference and 'b' is the 'zeroth' term., **Substitution**: Replacing variables (letters) in an expression or formula with their given numerical values.

    Exam tips

    • Be careful with negative signs when expanding brackets, for example, $-2(x - 5)$ becomes $-2x + 10$.
    • When solving simultaneous equations, always check your answer by substituting both values back into the original equations.

    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 of a Line
      m=riserun=y2−y1x2−x1m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}
      Calculates the steepness of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$.

    Key concepts: **Cartesian Coordinates**: A pair of numbers $(x, y)$ that locate a point on a grid. The x-coordinate is the horizontal position and the y-coordinate is the vertical position., **Gradient (m)**: Measures the steepness of a line. A positive gradient slopes up from left to right; a negative gradient slopes down., **Y-intercept (c)**: The point where a line crosses the vertical y-axis. Its coordinates are always $(0, c)$., **Parallel Lines**: Lines that never intersect and have the same gradient (m)., **Horizontal and Vertical Lines**: Horizontal lines have the equation $y=k$ and a gradient of 0. Vertical lines have the equation $x=k$ and an undefined gradient.

    Exam tips

    • When reading coordinates, remember the rule 'along the corridor, then up the stairs' (x-axis first, then y-axis).
    • To find the gradient from a graph, draw a right-angled triangle using grid lines to clearly identify the 'rise' and 'run'.

    Geometry

    • Angles on a Straight Line
      Sum of angles=180∘\text{Sum of angles} = 180^\circ
      Angles that form a straight line add up to 180 degrees.
    • Angles in a Triangle
      Sum of angles=180∘\text{Sum of angles} = 180^\circ
      The three interior angles of any triangle add up to 180 degrees.
    • Angles at a Point
      Sum of angles=360∘\text{Sum of angles} = 360^\circ
      Angles around a central point add up to 360 degrees.
    • Interior Angles of a Polygon
      Sum=(n−2)×180∘\text{Sum} = (n-2) \times 180^\circ
      Calculates the sum of the interior angles of a 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 $180^\circ$., **Similarity**: Two shapes are similar if one is an enlargement of the other. Corresponding angles are equal and corresponding sides are in the same ratio., **Bearing**: An angle measured clockwise from the North direction, always written with three figures (e.g., $045^\circ$)., **Congruence**: Two shapes are congruent if they are identical in size and shape. One can be fitted exactly over the other by rotation, reflection or translation., **Nets**: A 2D shape that can be folded to make a 3D solid. You must be able to draw and recognise nets for common solids like cuboids and prisms.

    Exam tips

    • In angle problems, you must state a reason for each step of your working (e.g., 'angles in a triangle sum to 180'). Marks are awarded for correct reasons.
    • For construction questions, always leave your construction arcs visible. Do not rub them out as they show your method.

    Mensuration

    • 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 and 'h' is the perpendicular height.
    • Area of a Circle
      A=πr2A = \pi r^2
      Calculates the area of a circle with radius 'r'.
    • Circumference of a Circle
      C=2πrC = 2\pi r or C=πdC = \pi d
      Calculates the distance around the edge of a circle with radius 'r' or diameter 'd'.
    • Volume of a Cylinder
      V=πr2hV = \pi r^2 h
      Calculates the volume of a cylinder with radius 'r' and height 'h'.
    • Volume of a Cuboid
      V=l×w×hV = l \times w \times h
      Calculates the volume of a cuboid with length 'l', width 'w', and height 'h'.

    Key concepts: **Perimeter**: The total distance around the outside of a 2D shape. For a circle, this is called the circumference., **Area**: The amount of surface a 2D shape covers, measured in square units (e.g., cm², m²)., **Volume**: The amount of space a 3D object occupies, measured in cubic units (e.g., cm³, m³)., **Surface Area**: The total area of all the faces of a 3D object., **Arc Length and Sector Area**: An arc is part of the circumference and a sector is part of the area of a circle. They are calculated as a fraction of the full circle, e.g., Sector Area = $\frac{\theta}{360} \times \pi r^2$.

    Exam tips

    • Double-check if you are given the radius or the diameter of a circle. If given the diameter, remember to halve it to find the radius before using formulas like $A = \pi r^2$.
    • Pay close attention to units. If a question uses different units (e.g., metres and centimetres), convert them all to the same unit before you start calculating.

    Trigonometry

    • Pythagoras' Theorem
      a2+b2=c2a^2 + b^2 = c^2
      In a right-angled triangle, finds a missing side when two sides are known. 'c' must be the hypotenuse.
    • 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 containing one angle of exactly $90^\circ$. Pythagoras' Theorem and SOHCAHTOA only work in right-angled triangles., **Hypotenuse**: The longest side of a right-angled triangle, always located opposite the right angle., **Opposite and Adjacent**: The names of the other two sides in a right-angled triangle, defined relative to the angle you are working with., **SOHCAHTOA**: A mnemonic used to remember the three trigonometric ratios: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.

    Exam tips

    • Always check that your calculator is in Degrees (DEG) mode before starting a trigonometry question.
    • When using Pythagoras' theorem to find a shorter side (a or b), you must rearrange the formula to subtract: $a^2 = c^2 - b^2$.

    Transformations and vectors

    • Translation Vector
      (xy)\begin{pmatrix} x \\ y \end{pmatrix}
      Describes a translation. The top number moves the shape right (positive) or left (negative), the bottom number moves it up (positive) or down (negative).

    Key concepts: **Reflection**: A transformation that 'flips' a shape over a mirror line. The image is the same size and shape, but reversed., **Rotation**: A transformation that 'turns' a shape around a fixed point (the centre of rotation) by a certain angle and direction., **Translation**: A transformation that 'slides' a shape to a new position without changing its size, shape or orientation. It is described by a vector., **Enlargement**: A transformation that changes the size of a shape. It is described by a scale factor and a centre of enlargement.

    Exam tips

    • When describing a transformation, you must give all the required details: Reflection (mirror line), Rotation (centre, angle, direction), Translation (vector), Enlargement (scale factor, centre).
    • Tracing paper is an extremely useful tool for performing rotations and reflections accurately in an exam.

    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.
    • Complementary Events
      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 number=P(event)×Number of trials\text{Expected number} = P(\text{event}) \times \text{Number of trials}
      Estimates how many times you expect an event to occur in a certain number of trials.

    Key concepts: **Probability Scale**: Probability is always a number between 0 and 1. A probability of 0 means the event is impossible, and a probability of 1 means it is certain., **Relative Frequency**: An estimate of probability based on experimental results. It is calculated by dividing the number of successful trials by the total number of trials., **Sample Space**: The set of all possible outcomes of an experiment. It can be shown as a list, a table, or a tree diagram., **Mutually Exclusive Events**: Events that cannot happen at the same time. For these events, $P(A \text{ or } B) = P(A) + P(B)$.

    Exam tips

    • Probability answers can be given as fractions, decimals, or percentages. Fractions are often best as they do not require rounding.
    • When a question asks for an 'estimate' of probability, it usually refers to using the relative frequency from an experiment.

    Statistics

    • Mean
      Mean=Sum of all data valuesNumber of data values\text{Mean} = \frac{\text{Sum of all data values}}{\text{Number of data values}}
      Calculates the mean average of a set of discrete data.
    • Range
      Range=Highest value−Lowest value\text{Range} = \text{Highest value} - \text{Lowest value}
      Measures the spread of a set of data.
    • Pie Chart Angle
      Angle=FrequencyTotal Frequency×360∘\text{Angle} = \frac{\text{Frequency}}{\text{Total Frequency}} \times 360^\circ
      Calculates the angle for a sector in a pie chart.

    Key concepts: **Mode**: The value that appears most often in a set of data. A dataset can have more than one mode or no mode., **Median**: The middle value in a set of data that has been arranged in order of size. If there are two middle values, the median is their average., **Frequency**: The number of times a particular data value occurs., **Correlation**: Describes the relationship between two variables on a scatter diagram. It can be positive (as one increases, the other increases), negative, or have no correlation., **Data Display**: Data can be represented in various ways, including bar charts, pie charts, pictograms, and stem-and-leaf diagrams.

    Exam tips

    • Always remember to order the data set from smallest to largest before you attempt to find the median.
    • When asked to compare two sets of data, make one comparison using an average (mean or median) and one using the spread (range).