GCSE Mathematics (Higher) OCR

This subject is broken down into 99 topics in 8 modules:

  1. Number 10 topics
  2. Surds 12 topics
  3. Algebra 16 topics
  4. Graphs 13 topics
  5. Ratio, Proportions and Rates of Change 6 topics
  6. Geometry and Measures 17 topics
  7. Pythagoras and Trigonometry 7 topics
  8. Probability and Statisitcs 18 topics
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  • 36,200
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This page was last modified on 28 September 2024.

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Mathematics (Higher)

Number

BODMAS and types of number

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BODMAS and types of number

BODMAS Rule

  • BODMAS stands for Brackets, Orders (i.e., Powers and Square Roots, etc.), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
  • This rule dictates the order in which operations should be carried out to solve an expression.
  • Brackets always come first in this hierarchy. Compute everything inside brackets first.
  • Orders or indices come next: any calculations involving powers or square roots are done at this stage.
  • Division and Multiplication come next and are equal in the hierarchy. These operations are performed from left to right.
  • Finally, Addition and Subtraction are also equal in hierarchy. These operations are performed from left to right.

Types of Number

Prime Numbers

  • A prime number is any number that has only two distinct natural number divisors: 1 and itself.
  • Examples are 2, 3, 5, 7, 11, 13. Note, 1 is not a prime number.

Composite Numbers

  • A composite number is any number that is greater than one and is not a prime number.
  • In other words, a composite number has more than two distinct natural number divisors.
  • Examples include 4, 6, 8, 9, 10.

Rational Numbers

  • A rational number can be expressed as the quotient or fraction of two integers, where the denominator is not zero.
  • Examples are -7, 0, 1/2, -4/9.

Irrational Numbers

  • An irrational number cannot be expressed as a fraction or quotient of two integers. Their decimal expansion is non-repeating and non-terminating.
  • Notable examples include the square root of 2 and π.

Real Numbers

  • Real numbers include all the counting numbers, fractions, terminating and recurring decimals, and irrational numbers.
  • In other words, if a number is either rational or irrational, it is a real number.

Square Numbers and Cube Numbers

  • A square number is the result when a number has been multiplied by itself, while a cube number is a number multiplied by itself twice.
  • For instance, 4 (2x2) and 9 (3x3) are examples of square numbers, while 8 (2x2x2) and 27 (3x3x3) are examples of cube numbers

Course material for Mathematics (Higher), module Number, topic BODMAS and types of number

Mathematics (Higher)

Graphs

Gradients of Real-Life Graphs

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Gradients of Real-Life Graphs

Gradients of Real-Life Graphs

Definitions

  • Gradient refers to the slope of a line on a graph. It can demonstrate rate of change or correlation in datasets.
  • The gradient of a line can be positive, negative, zero or undefined.
  • Positive gradient implies an increasing trend.
  • Negative gradient indicates a decreasing trend.
  • Zero gradient signifies a constant, or unchanging, value.
  • Undefined gradient occurs when a vertical line is drawn on the graph.

Calculating Gradient

  • To calculate the gradient of a straight line, use the formula (change in y)/(change in x) or rise/run.
  • Select two points on the line, preferably as far apart as possible for accuracy, and apply the formula.
  • If the line is a curve, find the gradient at a specific point by drawing a tangent to the curve at that point and calculating its gradient.

Linear Graphs

  • Linear graphs have a constant gradient. The graph will be a straight line.
  • Positive gradients slope upwards from left to right, representing direct proportionality. When one variable increases, so does the other.
  • Negative gradients slope downwards from left to right, showing inverse proportionality. When one variable increases, the other decreases.

Non-Linear Graphs

  • Non-linear graphs do not have a constant gradient. The lines curve either upwards or downwards.
  • For these graphs, the gradient varies at different points. For any point, the instantaneous gradient is given by the gradient of the tangent at the point.

Understanding Gradients

  • Gradient is a powerful tool in interpreting real-life situations through graphs. It allows us to see at a glance increasing and decreasing trends in datasets.
  • For example, in a graph plotting a car's journey, the gradient of the graph at any point is equal to the car's speed at that moment.

Practice

  • To build a solid understanding on gradients, solve problems from different real-life situations. Try plotting your own graphs and determining the gradients at multiple points. Pay attention to what the gradient conveys about the situation modelled by the graph.

Course material for Mathematics (Higher), module Graphs, topic Gradients of Real-Life Graphs

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