Counting
Counting means saying one number for each thing in a group, in order, to find out how many there are. The last number you say is the total.
Learning Wiki
Maths
Numbers, shapes, measuring and data, from counting to Pythagoras.
Around ages 3–7
First steps
Around ages 5–11
Primary
Addition
Addition is putting amounts together to find a total. The sign is +, and the answer is called the sum or total, so 4 + 3 = 7.
2D shapes
2D shapes are flat shapes with length and width but no thickness, like circles, triangles, squares and hexagons. You can draw them on paper.
Number bonds
Number bonds are pairs of numbers that add up to a given total, like 3 and 7 making 10. Knowing them by heart makes adding and subtracting much faster.
Odd and even numbers
Even numbers can be split into two equal groups or pairs with none left over (2, 4, 6…). Odd numbers always leave one left over (1, 3, 5…).
Tally charts
A tally chart is a table that counts things with little marks. Each mark is one, and every fifth mark crosses the four before it, so you can count in fives.
Pictograms
A pictogram shows data using small pictures or symbols. A key tells you what each picture is worth, such as one apple standing for 2 apples. In the US it is often called a picture graph.
Subtraction
Subtraction is taking one amount away from another, or finding the difference between two numbers. The sign is −, so 9 − 4 = 5.
Telling the time
To tell the time on a clock, read the short hand for the hour and the long hand for the minutes. When the long hand points to 6, it is half past the hour.
Comparing numbers
Comparing numbers means deciding which is greater, which is less, or if they are equal, and showing it with >, < or =. For example, 52 > 25 because 5 tens is more than 2 tens.
Length and measuring
Length is how long, tall or far something is. It is measured with a ruler or tape in units such as millimetres, centimetres, metres and kilometres (or inches, feet and miles).
Money
Money maths means counting coins and notes, adding up prices and working out change. Most currencies split one unit into 100 smaller ones, like 100 pence in a pound.
Number line
A number line is a straight line with numbers marked at equal steps, smallest on the left and biggest on the right. It helps you count, compare, add and subtract.
3D shapes
3D shapes are solid shapes with length, width and height, like cubes, spheres, cylinders and pyramids. They have faces, edges and vertices.
Capacity
Capacity is how much a container can hold, usually measured in millilitres (ml) and litres (l). A large bottle of water often holds 2 litres, which is 2000 ml.
Fractions
A fraction is part of a whole that has been split into equal parts. In ¾, the bottom number says how many equal parts and the top says how many you have.
Mass and weight
Mass is how much stuff is in an object, measured in grams and kilograms. Weight is the pull of gravity on that mass. In everyday life people say weight for both.
Arrays
An array is a set of objects arranged in equal rows and columns. 3 rows of 4 dots is an array that shows 3 × 4 = 12, so arrays help children see multiplication.
Place value
Place value means a digit's value depends on its position in a number. In 352, the 3 is worth 300, the 5 is worth 50 and the 2 is worth 2.
Division
Division is sharing into equal groups, or finding how many groups fit. The sign is ÷, so 12 ÷ 3 = 4. The answer is called the quotient.
Multiplication
Multiplication is adding equal groups quickly. 4 × 3 means 4 groups of 3 (or 3 groups of 4), which makes 12. The answer is called the product.
Times tables
Times tables are lists of the multiplication facts for a number, such as 3 × 1 = 3 up to 3 × 12 = 36. Knowing them by heart speeds up almost all later maths.
Bar charts
A bar chart shows data using bars of different lengths. The longer the bar, the bigger the number. In the US it is usually called a bar graph.
Faces, edges and vertices
Faces are the flat surfaces of a 3D shape, edges are the lines where two faces meet, and vertices are the corners. A cube has 6 faces, 12 edges and 8 vertices.
Estimation
Estimating means making a sensible guess at an answer, usually by rounding the numbers first. It tells you roughly what to expect and helps you spot mistakes.
Symmetry
A shape has symmetry when one half is a mirror image of the other (line symmetry), or when it looks the same after being turned (rotational symmetry).
Sequences
A sequence is a list of numbers or shapes that follows a rule, like 3, 7, 11, 15 (add 4 each time). Each item is called a term.
Comparing fractions
To compare fractions, check whether they share a denominator or a numerator, or rewrite them with a common denominator. The bigger fraction is the one with more of the whole.
Equivalent fractions
Equivalent fractions look different but are the same size, like 1/2, 2/4 and 4/8. You make them by multiplying or dividing the top and bottom by the same number.
Remainders
A remainder is what is left over when one number does not divide exactly into another. In 17 ÷ 5 = 3 remainder 2, the 2 is left over after making three groups of 5.
Metric units
Metric units measure length, mass and capacity in steps of 10, 100 and 1,000. Millimetres, metres, grams, kilograms and litres are all metric units.
Nets of 3D shapes
A net is a flat 2D pattern that folds up into a 3D shape. A cube's net is 6 squares joined edge to edge, and a cube has 11 different nets.
Parallel and perpendicular lines
Parallel lines run side by side and never meet, always the same distance apart. Perpendicular lines cross or meet at a right angle of exactly 90 degrees.
Perimeter
Perimeter is the total distance all the way around the outside of a flat shape. For a rectangle 5 cm long and 3 cm wide, it is 5 + 3 + 5 + 3 = 16 cm.
Polygons
A polygon is a flat shape made only of straight sides that close up completely. Triangles, squares, pentagons and hexagons are polygons; a circle is not.
Quadrilaterals
A quadrilateral is any flat shape with four straight sides. Squares, rectangles, rhombuses, parallelograms, trapezia and kites are all quadrilaterals, and their angles add up to 360 degrees.
Roman numerals
Roman numerals write numbers with seven letters: I, V, X, L, C, D and M. Letters are added, and a smaller letter before a bigger one is subtracted, so IV is 4 and XI is 11.
Rounding
Rounding means replacing a number with a simpler one that is close to it, such as 47 rounded to the nearest ten being 50. It makes numbers easier to work with.
Types of triangles
Triangles are named by their sides (equilateral, isosceles, scalene) or by their angles (acute, right-angled, obtuse). The three angles of any triangle add up to 180 degrees.
Parts of a circle
A circle has a centre, a radius, a diameter and a circumference. The radius runs from the centre to the edge, and the diameter is twice the radius.
Angles
An angle measures the amount of turn between two lines that meet at a point. Angles are measured in degrees (°); a right angle is 90° and a full turn is 360°.
Adding and subtracting fractions
To add or subtract fractions, first give them the same denominator. Then add or subtract only the numerators and keep the denominator. For example, 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
Decimals
Decimals are numbers with a decimal point. Digits after the point show tenths, hundredths and so on, so 2.5 means two and five tenths.
Improper fractions and mixed numbers
An improper fraction has a numerator bigger than its denominator, like 7/4. A mixed number shows the same amount as a whole number plus a fraction, like 1 3/4.
Long multiplication
Long multiplication multiplies a big number by splitting the second number into ones, tens and hundreds, finding each part, then adding the results together.
Simplifying fractions
To simplify a fraction, divide the top and bottom by the same number until you cannot go any further. For example, 12/18 simplifies to 2/3 when you divide both by 6.
The 24-hour clock
The 24-hour clock counts the hours from 00:00 at midnight to 23:59, so there is no am or pm. For afternoon times, add 12: 3 pm is 15:00.
Divisibility rules
Divisibility rules are shortcuts that tell you whether a number divides exactly by 2, 3, 4, 5, 6, 8, 9, 10, 11 or 12 without doing the division, by looking at its last digits or its digit sum.
Area
Area is the amount of flat space inside a shape, measured in square units like cm² or m². A rectangle 4 cm by 3 cm has an area of 4 × 3 = 12 cm².
Coordinates
Coordinates are a pair of numbers that give an exact position on a grid, like (3, 5). The first number is how far across (x) and the second is how far up (y).
Factors and multiples
Factors are numbers that divide exactly into a number (the factors of 12 are 1, 2, 3, 4, 6 and 12). Multiples are the numbers in its times table (12, 24, 36…).
Negative numbers
Negative numbers are numbers less than zero, written with a minus sign, like −3. You meet them in cold temperatures, bank overdrafts and floors below ground.
Rotational symmetry
A shape has rotational symmetry if it looks exactly the same after a turn of less than 360° about its centre. The order is how many times it fits onto itself in one full turn.
Two-way tables
A two-way table sorts data by two features at once, with rows for one and columns for the other. Rows and columns add up to totals, so you can fill in gaps and find probabilities.
Venn diagrams
A Venn diagram uses overlapping circles to sort things into groups. The overlap shows what belongs to both groups, and the outside shows what belongs to neither.
Long division
Long division splits a big number into equal groups by repeating four steps: divide, multiply, subtract, bring down. Work from the left, one digit at a time.
Multiplying fractions
To multiply fractions, multiply the numerators (tops) together and the denominators (bottoms) together, then simplify. You do not need a common denominator: 2/3 × 4/5 = 8/15.
Line graphs
A line graph shows how something changes over time. Points are plotted on a grid and joined by a line, so you can see rises, falls and patterns at a glance.
Percentages
A percentage is a number out of 100, written with the sign %. So 25% means 25 out of every 100, which is the same as one quarter.
Pie charts
A pie chart is a circle divided into slices that show parts of a whole. A bigger slice means a bigger share, and all the slices together make 100 percent, or 360 degrees.
Prime numbers
A prime number is a whole number greater than 1 that has exactly two factors, 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13.
Square numbers
A square number is what you get when you multiply a whole number by itself, like 4 × 4 = 16. The first square numbers are 1, 4, 9, 16, 25 and 36.
Transformations: reflection, rotation and translation
Transformations move a shape without changing its size or form. A translation slides it, a reflection flips it over a mirror line, and a rotation turns it around a point.
Volume
Volume is the amount of space a 3D object takes up, measured in cubic units like cm³ or m³. A box 4 cm by 3 cm by 2 cm has a volume of 24 cm³.
Dividing decimals
To divide by a whole number, divide as usual and put the decimal point straight above. To divide by a decimal, multiply both numbers by 10, 100… until the divisor is whole: 7.2 ÷ 0.6 = 72 ÷ 6 = 12.
Dividing fractions
To divide by a fraction, multiply by its reciprocal. Keep the first fraction, change ÷ to ×, flip the second fraction, then simplify. 3/4 ÷ 1/2 = 3/4 × 2/1 = 3/2.
Order of operations
The order of operations is the agreed order for working out calculations: brackets, then powers, then × and ÷, then + and −. UK pupils learn it as BIDMAS; US pupils as PEMDAS.
Around ages 11–16
Secondary
Probability
Probability measures how likely something is to happen, on a scale from 0 (impossible) to 1 (certain). A fair coin landing heads has a probability of 1/2.
Area of a trapezium (trapezoid)
Add the two parallel sides, multiply by the height and halve: A = ½(a + b)h. Parallel sides of 4 cm and 6 cm, 3 cm apart, give an area of 15 cm².
Area of a triangle
The area of a triangle is half of its base times its perpendicular height: A = ½ × b × h. A triangle with base 8 cm and height 5 cm has an area of 20 square centimetres.
Circumference of a circle
The circumference is the distance all the way around a circle. C = πd or C = 2πr, so a circle with a diameter of 10 cm has a circumference of about 31.4 cm.
Direct proportion
Two quantities are in direct proportion when one is always the same multiple of the other, so doubling one doubles the other. The rule is y = kx, where k is the constant of proportionality.
Highest common factor and lowest common multiple
The HCF is the largest number that divides exactly into two numbers. The LCM is the smallest number that both divide into. For 12 and 18, the HCF is 6 and the LCM is 36.
Prime factorisation
Prime factorisation means writing a number as a product of prime numbers. For example, 60 = 2 × 2 × 3 × 5, which can be written as 2² × 3 × 5.
Cube numbers and cube roots
A cube number is a number multiplied by itself three times, like 4³ = 4 × 4 × 4 = 64. The cube root undoes it, so ∛64 = 4.
Mean, median and mode
Mean, median and mode are three kinds of average. The mean shares the total out equally, the median is the middle value in order, and the mode is the most common value.
Profit and loss
Profit = selling price − cost price; loss = cost price − selling price. Profit percentage = profit ÷ cost price × 100. Buy at $20, sell at $25: profit $5, which is 25%.
Ratio
A ratio compares two or more amounts, written with a colon. A ratio of 2:3 means for every 2 of one thing there are 3 of the other.
Reciprocals
The reciprocal of a number is 1 divided by that number, so the two multiply to make 1. To find the reciprocal of a fraction, flip it upside down: the reciprocal of 3/4 is 4/3.
Speed, distance and time
Speed is distance divided by time. So speed = distance ÷ time, distance = speed × time, and time = distance ÷ speed. A car that goes 150 km in 2 hours travels at 75 km/h.
Surface area
Surface area is the total area of all the outside faces of a 3D shape. For a cube with side a, it is 6a², because a cube has six equal square faces.
Algebra
Algebra is maths that uses letters, like x or n, to stand for numbers we don't know yet or numbers that can change. For example, if n + 3 = 10, then n = 7.
Equations
An equation says two things are equal, using an = sign, like 2x + 1 = 9. Solving it means finding the value of the letter that makes it true, here x = 4.
Pi (π)
Pi (π) is the number you get when you divide any circle's circumference by its diameter. It is about 3.14, and its digits go on forever without repeating.
Angles in parallel lines
When a line crosses two parallel lines, corresponding (F) and alternate (Z) angles are equal and co-interior (C) angles add to 180°. Vertically opposite angles are always equal.
Area of a circle
The area of a circle is pi times the radius squared: A = πr². A circle with radius 5 cm has an area of about 78.5 square centimetres.
Simplifying expressions
To simplify an expression, expand any brackets, then collect like terms (terms with exactly the same letters and powers). For example, 3x + 2y + 5x − y simplifies to 8x + y.
Frequency polygons
A frequency polygon is a line graph of a grouped frequency table. Plot each class's frequency above the class midpoint and join the points with straight lines.
Histograms
A histogram shows grouped numerical data as touching bars on a number scale, one bar per class. When the classes have different widths, the area of each bar shows the frequency.
Inequalities
An inequality compares two values with <, >, ≤ or ≥, like x + 2 > 5. Solve it like an equation, but flip the sign if you multiply or divide both sides by a negative number.
Powers and indices
A power (also called an index or exponent) is a small raised number showing how many times to multiply a number by itself. So 2⁴ means 2 × 2 × 2 × 2 = 16.
Relative frequency
Relative frequency is how often something happens compared with the total, frequency ÷ total. If 6 of 20 students walk to school, the relative frequency is 6/20 = 0.3, or 30%.
Sample space
A sample space is the list of every possible outcome of an experiment. Flipping a coin has the sample space {H, T}; rolling a dice has {1, 2, 3, 4, 5, 6}.
Square roots
The square root of a number is the value that, multiplied by itself, gives that number. The square root of 25 is 5, because 5 × 5 = 25. The sign is √.
Congruent triangles
Congruent triangles are exactly the same shape and size, with all matching sides and angles equal. You can prove it with SSS, SAS, ASA, AAS or RHS.
Distance formula
The distance between (x₁, y₁) and (x₂, y₂) is d = √((x₂ − x₁)² + (y₂ − y₁)²). It is Pythagoras' theorem used on a coordinate grid.
Functions (maths)
A function is a rule that gives exactly one output for each input. In f(x) = 2x + 3, the input 4 gives the output 2 × 4 + 3 = 11.
Pythagoras' theorem
Pythagoras' theorem says that in a right-angled triangle, a² + b² = c², where c is the longest side (the hypotenuse). For sides 3 and 4, the hypotenuse is 5.
Quartiles and interquartile range
Quartiles split ordered data into four equal parts. The interquartile range is the upper quartile minus the lower quartile, IQR = Q₃ − Q₁: the spread of the middle half.
Scientific notation
Scientific notation writes very large or very small numbers as a number from 1 to 10 times a power of 10. For example, 3,500,000 is 3.5 × 10⁶. In the UK it is called standard form.
Slope (gradient)
Slope (called gradient in the UK) measures how steep a line is. It is the rise divided by the run, m = (y₂ − y₁) ÷ (x₂ − x₁), and it can be positive, negative, zero or undefined.
Volume of a cylinder
Multiply the area of the circular base by the height: V = πr²h. A cylinder with radius 5 cm and height 12 cm holds about 942 cm³, just under a litre.
Volume of cones and spheres
The volume of a cone is V = ⅓πr²h and the volume of a sphere is V = ⁴⁄₃πr³, where r is the radius and h the height. Answers are in cubic units, like cm³.
x-intercepts and y-intercepts
The y-intercept is where a graph crosses the y-axis: put x = 0. An x-intercept is where it crosses the x-axis: put y = 0. For y = 2x − 6 they are (0, −6) and (3, 0).
Compound interest and simple interest
Simple interest is paid only on the starting amount: I = P × r × t. Compound interest is also paid on earlier interest: A = P(1 + r)ⁿ. $1,000 at 10% for 2 years: $1,200 simple, $1,210 compound.
Cumulative frequency
Cumulative frequency is a running total of frequencies. Add each frequency to the total so far; the last value equals the total, and a cumulative frequency graph gives the median and quartiles.
Similar triangles
Similar triangles have the same shape but not always the same size. Their angles are equal and their sides are in the same ratio, so one is an enlargement of the other.
Simultaneous equations (systems of equations)
Simultaneous equations (a system of equations in the US) are two or more equations that share unknowns and must all be true at once. Solve them by elimination, substitution or graphing.
Surds (radicals)
A surd is a root that cannot be written as a whole number or fraction, like √2 or √5. In the US they are called radicals. Simplify them with √(ab) = √a × √b, so √12 = 2√3.
Completing the square
Completing the square rewrites x² + bx + c as (x + b/2)² − (b/2)² + c. It solves quadratics, finds the vertex (turning point) of a parabola and proves the quadratic formula.
Quadratic equations and the quadratic formula
A quadratic equation looks like ax² + bx + c = 0. The quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a solves every one, and b² − 4ac tells you how many solutions there are.
The quadratic formula and the discriminant
The quadratic formula x = (−b ± √(b² − 4ac)) / 2a solves any equation ax² + bx + c = 0. The discriminant b² − 4ac tells you whether there are two, one or no real solutions.
Trigonometry
Trigonometry links the angles and sides of right-angled triangles using sine, cosine and tangent. It lets you find a missing side or angle. SOH CAH TOA helps you remember the three ratios.
Vertex of a parabola
The vertex is the turning point of a parabola: its lowest or highest point. For y = ax² + bx + c it is at x = −b/2a; put that x back into the equation to get y.
Vectors
A vector is a quantity with both size (magnitude) and direction, like “5 km north”. In maths it is written as a column of numbers, such as (3, 4), and drawn as an arrow.
Composite functions
A composite function applies one function and then another. f(g(x)), also written (f ∘ g)(x) or fg(x), means do g first, then put the answer into f.
Conditional probability
Conditional probability is the chance of an event A when you already know event B has happened. It is written P(A|B) and found with P(A|B) = P(A and B) ÷ P(B).
Equation of a circle
A circle with centre (a, b) and radius r has the equation (x − a)² + (y − b)² = r². If the centre is the origin, it is simply x² + y² = r².
Inverse functions
An inverse function f⁻¹ undoes f: if f(3) = 7, then f⁻¹(7) = 3. To find it, write y = f(x), swap x and y, then solve for y.
Logarithms
A logarithm answers the question "what power?". log₂ 8 = 3 because 2³ = 8. Logs undo exponents, and their rules turn multiplying into adding.
The sine rule and the cosine rule
The sine rule a/sin A = b/sin B = c/sin C and the cosine rule a² = b² + c² − 2bc cos A find missing sides and angles in any triangle, not just right-angled ones.
Variance and standard deviation
Variance and standard deviation measure how spread out data are around the mean. Variance is the mean of the squared distances from the mean; standard deviation is its square root.
Differentiation (derivatives)
Differentiation finds the rate of change of a function: the gradient of its curve at any point. The power rule says xⁿ differentiates to nxⁿ⁻¹, so x³ becomes 3x².
Matrices
A matrix is a rectangular grid of numbers in rows and columns. You add matrices entry by entry, and you multiply them row by column, which only works when the sizes match.
The binomial distribution
The binomial distribution B(n, p) gives the probability of k successes in n independent trials, each with success probability p: P(X = k) = ⁿCₖ pᵏ (1 − p)ⁿ⁻ᵏ. Its mean is np.
The normal distribution
The normal distribution is a symmetrical, bell-shaped curve described by its mean μ and standard deviation σ. About 68% of values lie within 1σ of the mean, 95% within 2σ and 99.7% within 3σ.
Integration (integrals)
Integration is the reverse of differentiation, and it finds the area under a curve. The power rule says xⁿ integrates to xⁿ⁺¹/(n + 1) + C, so x² becomes x³/3 + C.
