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Sixteen topics from across our Grade 1–10, GCSE and SAT teaching, each written as a single question with a direct answer and a worked method. They are the explanations our tutors give in lessons, written down.

How to Learn Times TablesMathematicsStart with the 2, 5 and 10 tables, use the swap rule to halve the work, build hard facts from near ones, then drill only the ten or so that stay slow.
How to Do Long DivisionMathematicsDivide, multiply, subtract, bring down — the four-step cycle repeated once per digit. A full worked example of 4728 ÷ 12, plus how to handle remainders.
How to Convert Between Fractions, Decimals and PercentagesMathematicsDivide top by bottom for a decimal, multiply by 100 for a percentage, and reverse both to go back. Worked conversions for 3/8, 0.375, 37.5%, 45% and 9/20.
How to Solve Basic Algebra EquationsMathematicsClear brackets, gather letters on one side and numbers on the other, then divide. A full worked solution of 3(2x + 1) = x + 18, with the check that proves it.
How to Solve Simultaneous EquationsMathematicsScale the equations until one letter's coefficients match, add or subtract to remove it, then substitute back. Full worked solution of 3x + 2y = 19 and 4x − 5y = 10.
How to Solve Quadratic EquationsGCSE MathematicsRearrange to ax² + bx + c = 0, factorise into two brackets, or use the quadratic formula. Worked examples of x² + 5x = 24 and 3x² + 5x − 1 = 0, both checked.
How to Use Pythagoras' TheoremMathematicsAdd the squares to find the hypotenuse, subtract to find a shorter side. Worked examples with a 5-12-13 triangle and a hypotenuse of 10, plus the converse test.
How to Use SOHCAHTOAGCSE MathematicsLabel opposite, adjacent and hypotenuse, tick the two sides that matter, then pick the ratio that uses both. Worked examples finding a side and finding an angle.
What Is Probability and How Do You Calculate It?MathematicsProbability runs from 0 to 1. Count favourable outcomes over the total, use 1 − P(A) for the opposite, multiply for "and", add for "or", adjust without replacement.
What Is Photosynthesis?SciencePlants use light energy to build glucose from carbon dioxide and water, releasing oxygen. The equation, the two stages, and the factors that limit the rate.
What Are the Parts of a Cell and What Do They Do?ScienceNucleus, mitochondria, ribosomes, membrane — what each organelle does, how prokaryotic and eukaryotic cells differ, and what plant cells have that animal cells lack.
How to Read the Periodic TableScienceSeven periods, eighteen groups, ordered by atomic number. What each box tells you, why a group shares its chemistry, and where the metals and non-metals sit.
What Are Newton's Three Laws of Motion?ScienceAn object keeps its velocity unless a resultant force acts; that force equals mass times acceleration; and forces come in equal, opposite pairs on two objects.
How to Structure an EssayEnglishWrite the thesis first, order the reasons, build each paragraph on one idea with evidence and explanation, and close by readdressing the argument you made.
How to Improve Reading ComprehensionEnglishPreview the text, read in short chunks and retell each one, notice where understanding stops, and answer questions by returning to the line that proves it.
What Is on the Digital SAT Math Section?SAT PreparationTwo 35-minute modules, 44 questions, calculator throughout. Algebra and Advanced Math are 35% each; Problem-Solving and Data Analysis and Geometry 15% each.
Place Value ExplainedMathematicsThe 4 in 4,000 and the 4 in 40 are the same digit worth a thousand times more. How columns work, and the misconceptions that survive into Grade 8.
Column Addition with CarryingMathematicsCarrying is not a trick — it is what happens when a column holds ten or more. A worked example of 487 + 356, where the carried digit goes, and how to check.
Column Subtraction with BorrowingMathematicsBorrowing is exchange run backwards — one ten broken into ten ones. A worked example of 604 − 287, how to handle zeros, and the errors that cost most marks.
Number BondsMathematicsNumber bonds are the pairs that make a total instantly. Why bonds to 10 and 20 decide how well column methods work, and how to build genuine recall.
How to Round NumbersMathematicsRounding to the nearest 10, 100 and to decimal places. Which digit decides, why 5 rounds up, and the chained-rounding trap that turns 2.449 into 2.5.
Estimation and Checking AnswersMathematicsHow to estimate before calculating, check afterwards, and spot answers that cannot be right. The single cheapest source of extra marks in any maths exam.
Long MultiplicationMathematicsLong multiplication of 342 × 27, why the second row gets a zero, where carried digits go, and how to check. The method that makes multi-digit multiplication routine.
Roman NumeralsMathematicsI, V, X, L, C, D, M and the subtraction rule that makes IX nine and XI eleven. How to read and write Roman numerals, and why they show what place value gave us.
Negative NumbersMathematicsWhy subtracting a negative adds, how the number line settles every argument, and the sign errors that cost marks in algebra for years afterwards.
Order of Operations (BIDMAS)MathematicsBrackets, indices, division and multiplication, addition and subtraction. Why the last four are two pairs read left to right, not four ranked steps.
Prime Numbers and FactorsMathematicsWhat makes a number prime, why 1 is not, how to build a factor tree, and how prime factorisation makes HCF and LCM straightforward instead of guesswork.
HCF and LCMMathematicsHighest common factor and lowest common multiple using prime factorisation and Venn diagrams, plus how to decide which one a word problem is asking for.
Equivalent FractionsMathematicsMultiplying or dividing top and bottom by the same number changes how a fraction looks, not what it is worth. Simplifying, comparing and the errors in between.
Adding and Subtracting FractionsMathematicsWhy denominators must match before you add, how to find the lowest common denominator, and how to handle mixed numbers without losing the whole-number part.
Multiplying FractionsMathematicsMultiply the tops, multiply the bottoms — no common denominator needed. Why 'of' means multiply, how cancelling first saves work, and why the answer gets smaller.
Dividing FractionsMathematicsKeep, change, flip — and the reason it works. Why dividing by a fraction usually makes the answer bigger, with worked examples and the errors to avoid.
Decimal Place ValueMathematicsDecimals continue the same place value columns to the right of the point. Why 0.5 is bigger than 0.25, and how to compare decimals reliably.
Ratio and ProportionMathematicsHow to share an amount in a given ratio, when to use the unitary method, and the difference between a part-to-part ratio and a fraction of the whole.
Percentage Increase and DecreaseMathematicsMultipliers make percentage change one step instead of two. How to increase, decrease, find a percentage change, and undo one with a reverse percentage.
Simple and Compound InterestMathematicsSimple interest is calculated on the original amount; compound interest on the growing balance. The formulas, the difference, and depreciation.
Indices and PowersMathematicsAdd indices to multiply, subtract to divide, multiply to raise a power to a power. Why anything to the power zero is 1 and what a negative index means.
Standard FormMathematicsHow to write numbers as A × 10ⁿ where A is between 1 and 10, convert back, and multiply or divide without a calculator.
SurdsMathematicsA surd is a root that cannot be written exactly as a fraction. How to simplify using square factors, add like surds, and rationalise a denominator.
Expanding BracketsMathematicsMultiply every term inside by the term outside. Single brackets, double brackets, the negative-outside trap, and how to expand three brackets without losing terms.
Factorising QuadraticsMathematicsFind two numbers that multiply to c and add to b. How to factorise when a is not 1, the difference of two squares, and what to do when it will not factorise.
Rearranging FormulaeMathematicsDo the same thing to both sides, in reverse order of operations. How to change the subject when the letter appears twice or sits inside a root.
Solving InequalitiesMathematicsSolve inequalities like equations, with one exception — multiplying or dividing by a negative flips the sign. Number line notation and integer solutions.
Sequences and the nth TermMathematicsFind the nth term of a linear sequence from its common difference, handle quadratic sequences with second differences, and recognise the standard special sequences.
Straight Line Graphs (y = mx + c)MathematicsWhat m and c actually mean, how to find a gradient from two points, and how to sketch a line in seconds without a table of values.
Solving Simultaneous Equations GraphicallyMathematicsThe solution to a pair of simultaneous equations is the point where their graphs intersect. How to plot, read off, and what parallel or identical lines mean.
Perimeter and AreaMathematicsPerimeter is the distance around, area is the space inside. Formulas for rectangles, triangles, parallelograms and trapeziums, plus compound shapes and units.
Area and Circumference of a CircleMathematicsWhich formula uses the radius and which uses the diameter, how to tell area from circumference, and how to handle answers left in terms of π.
Volume of SolidsMathematicsVolume of a prism is cross-section times length. The formulas for cylinders, pyramids, cones and spheres, plus surface area and unit conversion traps.
Angles on Lines and at PointsMathematicsAngles on a straight line add to 180°, at a point to 360°. Vertically opposite, corresponding, alternate and co-interior angles, and how to reason with them.
Angles in PolygonsMathematicsInterior angles sum to (n − 2) × 180, exterior angles always total 360. How to find each angle in a regular polygon and work backwards from an angle to n.
Circle TheoremsMathematicsThe angle at the centre is twice the angle at the circumference, angles in a semicircle are 90°, and the rest — with the wording examiners expect.
Symmetry and TransformationsMathematicsThe four transformations, what must be stated to describe each fully, and why a missing centre or scale factor loses the mark even when the drawing is right.
Coordinates and GridsMathematicsAlong the corridor then up the stairs — reading and plotting coordinates in all four quadrants, plus midpoint and distance between two points.
Similar Shapes and Scale FactorsMathematicsSimilar shapes have equal angles and proportional sides. Why area scales by k² and volume by k³ — the rule that catches out most students.
BearingsMathematicsAlways measured clockwise from north and always written with three figures. How to find a back bearing and why 045° is not the same as 45°.
VectorsMathematicsA vector has magnitude and direction. Column vector notation, adding and subtracting, scalar multiples, and how parallel vectors prove points lie on a line.
The Sine and Cosine RulesMathematicsSOHCAHTOA only works in right-angled triangles. Which rule to use for any triangle, how to choose between them, and the area formula.
Speed, Distance and TimeMathematicsSpeed equals distance over time, and rearranging it. Unit conversion, average speed over two legs, and reading distance-time graphs.
Mean, Median, Mode and RangeMathematicsHow to calculate each average, why outliers ruin the mean, and how to choose the right one. Includes finding the mean from a frequency table.
Bar Charts and PictogramsMathematicsHow to read a scale correctly, why the key on a pictogram decides every answer, and the labelling that earns marks when drawing a chart.
Scatter Graphs and CorrelationMathematicsReading correlation from a scatter graph, drawing a line of best fit, and why interpolation is reliable while extrapolation is not.
Cumulative FrequencyMathematicsPlot at the upper bound, read the median at half the total, and find the interquartile range. How cumulative frequency leads into box plots.
HistogramsMathematicsFrequency density is frequency divided by class width, and area represents frequency. Why histograms differ from bar charts and how to read one backwards.
Venn Diagrams and SetsMathematicsFill the overlap first, then work outward. Union, intersection, complement and the notation examiners use, plus probability from a Venn diagram.
Probability Tree DiagramsMathematicsMultiply along the branches and add between outcomes. How to handle with and without replacement, and why each set of branches must total 1.
Telling the Time and Reading TimetablesMathematicsReading clocks, converting between 12-hour and 24-hour time, and working out journey durations without slipping into decimal minutes.
Money ProblemsMathematicsWorking with currency means keeping units consistent and always writing two decimal places. Calculating change, comparing value and multi-step budgeting.
GCSE Maths: Foundation or Higher?GCSE MathematicsFoundation caps at grade 5, Higher starts at grade 4. What each tier covers, who each suits, and why the safe choice is not always the lower one.
GCSE Maths Grade BoundariesGCSE MathematicsBoundaries are set after the exam, not before, so a harder paper needs fewer marks. What that means for revision targets and for reading last year's figures.
The GCSE Maths Non-Calculator PaperGCSE MathematicsOne of the three GCSE maths papers is sat without a calculator. Which skills it actually tests, and how to prepare for arithmetic you cannot delegate.
The GCSE Maths Formula SheetGCSE MathematicsSome formulae are printed in the exam and some must be memorised. Which is which, and why knowing the difference changes how you revise.
A GCSE Maths Revision Plan That WorksGCSE MathematicsWhy past papers beat re-reading notes, how to log errors so they stop repeating, and a realistic weekly structure for the final months.
GCSE Maths Problem-Solving QuestionsGCSE MathematicsMulti-step worded questions carry the most marks and the least guidance. How to extract the maths, work from both ends, and earn method marks.
The Digital SAT Reading and Writing SectionSAT PreparationShort passages with one question each, across four content domains. What the section tests, how it is timed, and where students lose the most points.
How the Digital SAT AdaptsSAT PreparationThe test adapts between modules, not between questions. What that means for pacing, for guessing, and for why the first module matters most.
How the SAT Is ScoredSAT PreparationTwo sections of 200–800 combine to a total out of 1600. What percentiles mean, why raw marks are converted, and how to read a score report.
SAT Algebra QuestionsSAT PreparationLinear equations, inequalities and systems make up the largest share of SAT Math. The question shapes that recur and how to translate words into equations.
SAT Problem Solving and Data AnalysisSAT PreparationRatios, rates, percentages, unit conversion and interpreting tables and graphs. The SAT content area most connected to everyday quantitative reasoning.
SAT Advanced MathSAT PreparationQuadratic equations, exponential growth, polynomial and radical expressions, and function notation. The hardest content area and where top scores are decided.
SAT Grammar RulesSAT PreparationStandard English Conventions tests a limited set of rules — clause joining, agreement, modifiers, punctuation. The full list and how to study it.
SAT Time ManagementSAT PreparationHow to pace each module, when to flag and move on, and why finishing every question matters more than perfecting any single one.
States of MatterScienceParticle arrangement explains every property of solids, liquids and gases. Melting, boiling, evaporation and why temperature holds steady during a change of state.
The Water CycleScienceHow water moves between ocean, air and land. The stages, what drives them, and why the cycle has no beginning or end.
Food Chains and Food WebsScienceArrows show the direction energy flows, not who eats whom. Producers, consumers, decomposers, and why food chains are rarely more than five links long.
The Human Digestive SystemScienceMouth to intestine, what each organ does, which enzymes break down which food group, and why the small intestine is so long.
RespirationScienceRespiration releases energy from glucose in every living cell. The equations, the difference from breathing, and what happens without enough oxygen.
Atoms, Elements and CompoundsScienceProtons, neutrons and electrons; how elements differ from compounds; and why a mixture can be separated but a compound cannot.
Chemical Reactions and Balancing EquationsScienceAtoms are never created or destroyed, so both sides must match. How to balance step by step, and why you may change coefficients but never formulae.
Acids and BasesScienceThe pH scale, what makes something acidic or alkaline, the neutralisation reaction, and how to name the salt a reaction produces.
Forces and FrictionScienceWhat balanced and unbalanced forces do, why friction is not always unhelpful, and how terminal velocity follows from forces coming into balance.
Energy Transfer and ConservationScienceEnergy is stored and transferred, never used up. The stores, the transfer pathways, conduction, convection and radiation, and what efficiency measures.
Electric CircuitsScienceCurrent, voltage and resistance, how they behave differently in series and parallel, and how to use V = IR without confusing the three quantities.
Magnetism and ElectromagnetsScienceLike poles repel, field lines run north to south, and a current makes a magnetic field. How to make an electromagnet stronger and why that matters.
Light and ReflectionScienceThe law of reflection, why a straw looks bent in water, and the common misconception that eyes emit light rather than receive it.
Sound WavesScienceSound is a longitudinal wave made of compressions and rarefactions. What changes pitch, what changes volume, and why sound cannot cross a vacuum.
The Solar SystemScienceThe eight planets in order, why we have day and night, what causes the seasons, and the difference between rotation and orbit.
Genetics and InheritanceScienceGenes, alleles, dominant and recessive traits, and how to use a Punnett square to predict inheritance ratios.
Evolution and Natural SelectionScienceVariation, competition, survival and inheritance — the mechanism in four steps, plus why organisms do not adapt during their own lifetime.
Ecosystems and BiodiversityScienceThe vocabulary from organism to ecosystem, what affects population size, and why losing one species can destabilise an entire community.
Parts of SpeechEnglishThe eight word classes, how to identify them by the job a word does rather than by the word itself, and why the same word can change class.
Punctuation RulesEnglishWhen each mark is correct, why the comma splice is the most common error in English, and how apostrophes actually work.
Active and Passive VoiceEnglishActive voice puts the doer first; passive puts the receiver first. How to identify each, convert between them, and when the passive is the better choice.
Figurative LanguageEnglishThe main devices, how to tell a simile from a metaphor, and how to write about effect rather than just naming the technique.
Persuasive WritingEnglishHow to build an argument that changes a reader's mind — audience, structure, rhetorical technique, and why addressing the opposing view makes you stronger.
Summarising a TextEnglishSeparating main ideas from supporting detail, writing in your own words, and why a summary is not a shortened copy of the original.
Creative Writing OpeningsEnglishStart in the middle of something, show rather than explain, and avoid the three openings examiners see most often.
Formal Letter WritingEnglishThe conventional layout, why Yours sincerely and Yours faithfully are not interchangeable, and how to keep the register consistent.
Reading a Historical SourceHistoryWho wrote it, when, and why — how to evaluate a source without dismissing it as biased, and why bias makes a source more useful, not less.
Causes and Consequences in HistoryHistoryHow to categorise causes, weigh their relative importance, and write an argument rather than a list of factors.
Map Skills and Grid ReferencesGeographyAlong the corridor and up the stairs — four and six-figure grid references, scale, distance and what contour lines show about the land.
Rivers and ErosionGeographyThe four erosion processes, four transport processes, and how waterfalls, meanders and oxbow lakes form along a river's course.

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