Study guides

Short, illustrated explanations of the ideas behind school maths. Each one has plain words for the learner, a worked example, and notes for the grown-up helping them.

Pascal's triangleYears 9 to 13 · A triangle of numbers where each one is the two above it added together, and the rows give the numbers in a binomial expansion.
Prime numbers and factor treesYears 5 to 9 · Every number can be built from prime numbers, and a factor tree shows how.
Place valueYears 3 to 6 · The position of a digit tells you how much it is worth.
Equivalent fractionsYears 3 to 7 · Different fractions can name the same amount, and multiplying or dividing top and bottom by the same number keeps the size.
Multiplication as groups and arraysYears 3 to 6 · Multiplication means equal groups, and a grid of dots shows why the order does not matter.
Negative numbers and the number lineYears 4 to 8 · Numbers below zero live on the left of the number line, and adding and subtracting are moves along it.
PercentagesYears 5 to 11 · A percentage is a number out of 100, and it links to fractions and decimals.
Angles in a triangleYears 5 to 9 · The three angles in any triangle always add up to 180°.
Pythagoras' theoremYears 9 to 11 · In a right-angled triangle, the square on the longest side equals the sum of the squares on the other two.
Parts of a circleYears 6 to 11 · The radius, diameter and circumference, and how they are linked.
Gradient and straight-line graphsYears 8 to 11 · The gradient of a line tells you how steep it is, and y = mx + c describes it.
Quadratic graphs and their rootsYears 10 to 12 · A quadratic makes a U-shaped curve, and its roots are where it crosses the x-axis.
Probability tree diagramsYears 8 to 13 · A tree diagram shows every outcome of two events and how to multiply along the branches.
Differentiation and the tangentYears 12 and 13 · The derivative gives the gradient of a curve at any point, which is the gradient of its tangent.
Averages: mean, median, mode and rangeYears 6 to 11 · Different ways to describe a typical value, and how spread out the data is.
Telling the timeYears 2 to 4 · Read an analogue clock: the short hand shows the hour and the long hand shows the minutes.
Area of rectangles and trianglesYears 4 to 9 · Area is the space inside a shape. A rectangle is length × width, and a triangle is half of that.
Sequences and the nth termYears 4 to 10 · A sequence follows a rule. The nth term is a formula that gives any term from its position.
Powers and indicesYears 6 to 10 · A power tells you how many times to multiply a number by itself, and the index laws make working with powers quick.
Standard formYears 9 to 11 · Standard form writes very big or very small numbers as a number between 1 and 10 times a power of ten.
Trigonometry: sine, cosine and tangentYears 9 to 11 · In a right-angled triangle, sin, cos and tan link an angle to the ratio of two sides.
Binary numbersYears 5 to 11 · Computers count with only 0 and 1, and every number can be built from doubling place values.
Algorithms and loopsYears 3 to 9 · An algorithm is a list of exact steps, and a loop repeats steps so you do not have to write them again and again.
Sorting and searchingYears 4 to 11 · Computers sort lists by comparing neighbours, and search sorted lists by halving them again and again.
How machines learn from examplesYears 6 to 13 · A machine learning program finds patterns in labelled examples, then uses them to decide about something new.
Neural networksYears 9 to 13 · An artificial neuron multiplies inputs by weights, adds them up, and fires if the total is big enough. Networks join many together.
Fair and unfair dataYears 7 to 13 · An AI learns from its data, so missing or lopsided data can make it work worse for some people.
Secret codes and ciphersYears 9 to 13 · A Caesar cipher shifts every letter along the alphabet. Remainders make the alphabet wrap round, and modern codes use the same idea with huge numbers.
Quantum ideas: packets and probabilitiesYears 10 to 13 · At the scale of atoms, energy comes in packets, and measurements give results with probabilities.

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