Practise your sums and learn maths

You count apples, animals and dots up to ten and get to know the ten frame. Then you add one or take one away.

Counting to 5 · Counting to 10 · The ten frame · One more · One less · Practice: counting to 10

You count to twenty on the number line. You say which number is more or bigger and put numbers in order from smallest to biggest.

Counting to 20 · One more and one less, up to 20 · More or fewer · Bigger and smaller · Putting numbers in order · Practice: up to 20

You put things together and write it as an addition sum. You practise doubles, number bonds to 10 and short word problems.

Putting together · Adding on the ten frame · Doubles · Friends of ten · Little stories · Practice: adding to 10

You take things away and jump back on the number line. You see that adding and taking away belong together, and you work out short word problems.

Taking away · Jumping back · Taking away from 10 · Adding and taking away belong together · Little stories · Practice: taking away to 10

You see that 13 is a ten and three more. Then you jump forward and back up to 20 and work out doubles and word problems.

Ten and a bit · Adding up to 20 · Taking away up to 20 · Doubles up to 20 · Little stories up to 20 · Practice: up to 20

You work out sums like 8 + 5 and 13 − 5 by going to 10 first and then doing the rest.

Make ten first · Jumping over ten · Back to ten first · Jumping back over ten · Little stories over ten · Practice: crossing ten

The test covers counting and comparing, adding and subtracting to 10 and to 20, and sums that cross ten.

Test: counting and comparing · Test: up to 10 · Test: up to 20 · Test: crossing ten · Test: everything mixed

You count in tens up to 100 and see that 34 is made of 3 tens and 4 ones. You compare numbers and jump ten forward or back.

Counting in tens · Tens and ones · The number line to 100 · More and less · Ten more, ten less · Practice: tens and ones

You add up to 100: tens first, then ones. You learn to cross the ten and to add two two-digit numbers.

Adding tens · Adding ones · Adding tens to a number · Crossing a ten · Two big numbers · Practice: adding to 100

You subtract up to 100: tens and ones separately, and back across the ten. Then you take one two-digit number away from another.

Subtracting tens · Subtracting ones · Tens from a number · Back over a ten · Two two-digit numbers · Practice: subtracting to 100

You see multiplication sums as rows of dots and learn the 2, 10 and 5 times tables. You find out that 3 × 4 is the same as 4 × 3.

Rows of · The table of 2 · The table of 10 · The table of 5 · Turn it around · Practice: the tables of 2, 5 and 10

You learn the 3, 4 and 6 times tables, with tricks like 'the 4 times table is double the 2 times table'. Then you practise them all mixed up.

The table of 3 · The table of 4 · The table of 6 · Little stories with the tables · Tables of 2 to 6 mixed · Practice: tables of 3, 4 and 6

You learn the 7, 8 and 9 times tables, the hardest ones. For 9 you use 10 to help you. Finally you practise all the tables from 1 to 10 mixed together.

The table of 7 · The table of 8 · The table of 9 · Little stories with big tables · All the tables mixed · Practice: tables of 7, 8 and 9

You share things fairly and see that dividing is the times table the other way round. That way you divide by every number from 2 to 9.

Sharing fairly · Dividing is the table the other way round · Dividing by 2, 5 and 10 · Dividing by 3, 4 and 6 · Dividing by 7, 8 and 9 · Practice: times and divide

You read the clock with hands: o'clock and half past, quarter past and quarter to, and in steps of five minutes. You also work out how long something takes.

Whole hours · Half hours · Quarter past and quarter to · Five minutes · How long does it take? · Practice: the clock

You count coins and notes, add up prices and work out change. You also work out how much you are saving.

Coins and notes · Prices together · Paying and change · Little stories about prices · Saving · Practice: money

The test covers numbers up to 100, the tables, dividing, the clock and money.

Test: up to 100 · Test: the tables · Test: dividing · Test: the clock · Test: money · Test: everything mixed

You learn hundreds and thousands, put numbers up to 10,000 on the number line, compare them and round them.

Counting in hundreds · Hundreds, tens and ones · Thousands · The number line to 10,000 · Comparing and ordering · Rounding

You add and subtract up to 1,000 by splitting numbers, including crossing the hundred. Then you learn written methods: sums set out in columns.

Adding by splitting · Crossing the hundred · Subtracting by splitting · Back over the hundred · Column arithmetic · Practice: adding and subtracting to 1,000

You learn the 11 and 12 times tables and multiply by 10, 100 and 1,000. You do bigger multiplications like 23 × 4 by splitting them.

The tables of 11 and 12 · Times 10, 100 and 1,000 · Tens times ones · Two digits times one · Two-digit numbers times 10 and 100 · Practice: multiplying big

You use the times tables up to 12 to divide and find out what is left over: the remainder. You divide by 10 and 100 and split bigger numbers.

All the tables backwards · What is left over · How many times does it fit · Dividing by 10 and 100 · Bigger numbers · Practice: dividing with a remainder

You learn halves, quarters and thirds, and what the numerator and denominator are. You add and subtract fractions, find equivalent fractions and take a fraction of a number.

Half, quarter, third · Numerator and denominator · Adding fractions · Subtracting fractions · Equivalent fractions · A fraction of a number · Practice: fractions

You work with metres, centimetres, kilometres and millimetres, with kilograms and grams, litres and millilitres, and with hours, minutes and seconds.

Metres and centimetres · Kilometres and millimetres · Kilograms and grams · Litres and millilitres · Hours, minutes and seconds · Practice: measuring

The test covers numbers up to 10,000, adding and subtracting, multiplying and dividing, fractions and measuring.

Test: numbers to 10,000 · Test: adding and subtracting · Test: multiplying and dividing · Test: fractions · Test: measuring · Test: everything mixed

You learn tenths and hundredths as decimals. You compare decimals, add them, subtract them, multiply and divide by 10 and 100, and round.

Tenths · Hundredths · Comparing decimals · Adding decimals · Subtracting decimals · Hundredths: adding and subtracting · Times and divided by 10 and 100 · Rounding and practice

You learn what percent means and convert between fractions, decimals and percentages. You work out any percentage, including discounts and a price going up.

What per cent means · Fractions, decimals and percentages · 50%, 25%, 10% and 1% · Any percentage · Discount · Going up · Practice: percentages

You learn what a ratio is and work with the ratio table. By working out the value for 1 first, you adjust a price or a recipe.

What a ratio is · The ratio table: back to 1 · Scaling up · Through 1 · Practice: ratios

You get to know numbers below zero, like the temperature in winter. You add and subtract on the number line, also with negative numbers.

Below zero · Adding on the line · Going below zero · Adding a negative · Subtracting a negative · Practice: negative numbers

You work out the perimeter of rectangles and squares, and the area with length × width. You learn when you need which of the two.

Perimeter: all the way round · The square · Area: counting squares · Area = length × width · Perimeter or area? · Practice: perimeter and area

You read bar charts, pictograms and pie charts. You find totals and differences and read off percentages.

The bar chart · Totals and differences · The pictogram · The pie chart · Practice: reading charts

The test covers decimals, percentages, ratios, negative numbers, perimeter and area, and charts.

Test: decimals · Test: percentages · Test: ratios · Test: negative numbers · Test: perimeter and area · Test: charts · Test: everything mixed

You learn the order of operations: multiplication comes before plus and minus, brackets first. You also multiply and divide with negative numbers.

Times before plus · Times before minus · Brackets first · Times with negative numbers · Dividing with negative numbers · Practice: order of operations

You learn that a letter stands for a number and substitute numbers into formulas. You collect like terms and expand brackets.

A letter stands for a number · Filling in · Filling in negatives and squares · Like terms · Like terms with numbers · Expanding brackets · Practice: formulas from life

You solve equations like a balance: do the same on both sides. You practise two steps, negative solutions, x on both sides and brackets.

The balance · Times and divided · Two steps · Negative solutions · x on both sides · Equations with brackets · Practice: equations

You plot points on a coordinate grid and turn a formula into a table and a graph. You read off the y-intercept and the gradient of a straight line.

Coordinates · From formula to table · The graph of a formula · Where it starts · How steep · Practice: straight-line graphs

You work with squares, higher powers and powers of ten. You watch out for a minus in front of the base and learn what a root is.

Squares · Cubes and higher powers · Powers of ten · Negative bases · Square roots · Practice: powers and roots

You learn Pythagoras' theorem: a² + b² = c². With it you calculate the hypotenuse or a shorter side of a right-angled triangle.

The right-angled triangle · a² + b² = c² · Finding the long side · Finding a short side · Practice: Pythagoras

The test covers the rules of arithmetic, formulas, equations, linear relationships, powers and roots, and Pythagoras.

Test: order of operations and negatives · Test: letters and formulas · Test: equations · Test: straight-line graphs · Test: powers and roots · Test: Pythagoras · Test: everything mixed

You solve quadratic equations: with x² = a number, with 'a product is zero', by taking out a common factor and by factorising.

x² = a number · A product is zero · A common factor · Two numbers · Everything on one side first · Practice: quadratic equations

You solve two equations with two unknowns at the same time. You use adding, subtracting and substitution.

Two equations, one answer · Adding them up · Taking one away · Substitution · Practice: simultaneous equations

You recognise similar shapes and work out the scale factor. With it you find a missing side, the height of a tree or a distance on a map.

Same shape, other size · The scale factor · A missing side · Shadows and heights · Practice: similar shapes

You name the sides of a right-angled triangle from an angle and work with sine, cosine and tangent. You also calculate an angle from two sides.

Naming the sides · Sine · Cosine · Tangent · Finding the angle · Practice: trigonometry

You work out the mean, median, mode and range. You learn which measure is the fairest when one number is very different from the rest.

The mean · The median · The mode and the range · Which one tells the truth? · Practice: statistics

You work out probabilities by counting favourable and possible outcomes, with dice and marbles. You also work out the probability of 'not' and write probabilities as a percentage.

What a chance is · A bag of marbles · The chance of not · As a percentage · Practice: probability

The test covers quadratic equations, simultaneous equations, similarity, trigonometry, statistics and probability.

Test: quadratic equations · Test: simultaneous equations · Test: similar shapes · Test: trigonometry · Test: statistics · Test: probability · Test: everything mixed

You work with function rules like f(x) = x² − 3x + 2. You compose functions, find the inverse function and read values from a graph.

A function is a rule · Quadratic and other rules · A function of a function · The inverse function · The graph of a function · Practice: functions

You learn the rules for powers, also with exponent 0 and negative exponents. With the growth factor you work out growth over time, like interest and doubling.

The laws of indices · Zero and negative powers · The growth factor · Growth over time · Doubling and halving · Practice: exponential growth

You learn that a logarithm answers the question 'which power?'. You use the rules and solve equations where x is in the exponent.

What a logarithm is · Base 10 and small numbers · The laws of logarithms · Solving with a logarithm · Practice: logarithms

You recognise arithmetic and geometric sequences and work out a term. You also add up the terms to make a series, like Gauss did with 1 to 100.

Arithmetic sequences · Geometric sequences · Which kind? · Adding up an arithmetic series · Adding up a geometric series · Practice: sequences and series

You learn that the derivative gives the slope of a curve. With the power rule you differentiate term by term, find a tangent line and find peaks and valleys.

The gradient of a curve · The power rule · Term by term · The tangent line · Turning points · Practice: the derivative

You learn vectors as steps with components. You add them, multiply them by a number, and calculate their length and the dot product.

A vector is a step · Adding and subtracting vectors · A number times a vector · The magnitude of a vector · The dot product and practice

The test covers functions, exponential growth, logarithms, sequences and series, the derivative and vectors.

Test: functions · Test: exponential growth · Test: logarithms · Test: sequences and series · Test: the derivative · Test: vectors · Test: everything mixed

You learn what a limit is and calculate limits at a hole in the graph and towards infinity. You also see when a limit does not exist.

What a limit is · A hole in the graph · Towards infinity · When a limit does not exist · Practice: limits

You learn proofs: direct, with a counterexample, by contradiction and by induction. Finally you look for the mistake in a 'proof' that 2 = 1.

A direct proof · A counterexample · Proof by contradiction · Proof by induction · Spot the mistake

You differentiate with the product rule and the chain rule and learn the standard derivatives, like that of eˣ. You use the derivative as a rate of change.

The product rule · The chain rule · The standard derivatives · Rates of change · Practice: differentiating

You learn to find antiderivatives and calculate definite integrals. With them you find the area under a graph, also below the x-axis and between two graphs.

The antiderivative · The definite integral · Term by term · The area under a curve · Below the axis and between curves · Practice: integrating

You learn what a matrix is, multiply it by a number, a vector and another matrix, and calculate the determinant.

What a matrix is · A number times a matrix · A matrix times a vector · A matrix times a matrix · The determinant · Practice: matrices

You get to know complex numbers through i² = −1. You add them, multiply and divide them using the conjugate, and find the modulus.

i² = −1 · Adding and multiplying · The conjugate and dividing · The modulus · Practice: complex numbers

You count orderings and combinations and work with the binomial distribution. You calculate expected values and conditional probabilities.

Counting orders · Combinations · The binomial distribution · The expected value · Conditional probability · Practice: probability

The final test covers limits, proofs, differentiation, integration, matrices, complex numbers and probability theory.

Test: limits · Test: proofs · Test: differentiating · Test: integrating · Test: matrices · Test: complex numbers · Test: probability · Test: everything mixed

You work with remainders and modulo n, find the gcd with the Euclidean algorithm and the lcm. You also work with prime numbers and Fermat's little theorem.

Remainders and mod · Calculating mod n · The gcd and Euclid · The lcm · Primes and Fermat · Practice: number theory

You calculate the inverse of a matrix and use it to solve a system of equations. Then you find eigenvalues and eigenvectors, including those of triangular matrices.

The inverse matrix · Solving a system with the inverse · Eigenvalues · Triangular matrices and eigenvectors · Practice: linear algebra 2

You solve differential equations like y′ = k · y for growth and decay. You calculate half-lives and doubling times and use Newton's law of cooling.

What a differential equation is · Growth and decay · Half-life and doubling time · Newton's law of cooling · Practice: growth, decay and cooling

You work with functions of two variables. You calculate partial derivatives and the gradient, find stationary points and work out double integrals.

A function of two variables · Partial derivatives · The gradient and a flat point · Double integrals · Practice: several variables

You add up infinite geometric series and decide whether a series converges or diverges, using the ratio test among other things. You also build Taylor polynomials.

Infinite geometric series · Converge or not · Convergence tests · Taylor polynomials · Practice: series

You get to know graphs: vertices, edges and degrees. You use the handshake theorem, count the edges of complete graphs and look for Euler paths.

Points and lines · The handshake lemma · Complete graphs · Euler paths and trees · Practice: graphs

You learn what a group is through addition and multiplication modulo n. You find the order of elements and use Lagrange's theorem.

What a group is · ℤₙ: adding modulo n · ℤₙ*: multiplying modulo n · Lagrange's theorem · Practice: groups

The test covers number theory, linear algebra, differential equations, functions of several variables, series, graphs and groups.

Test: number theory · Test: linear algebra · Test: differential equations · Test: several variables · Test: series · Test: graphs · Test: groups · Test: everything mixed