Big ideas.
Brave explorers.

8+2 Number Bonds and Making Ten The pairs that make ten, and why they unlock everything after 5 minutesSplitting a number into two parts and back again Grades K-2 →

i Complex Numbers Inventing a square root for −1, and what it turns out to do 8 minutesWorking with i, and reading complex numbers as points Grades 10-12 → r∠θ Polar Form and De Moivre's Theorem Multiplying complex numbers is stretching and turning 7 minutesWriting a complex number by size and angle, and using it to take powers and roots Grades 11-12 →

.×. Calculating with Decimals Three different rules about where the point goes 7 minutesAdding, subtracting, multiplying, and dividing decimals Grades 5-7 → 0.1 Decimals as Parts of One Place value carrying on past the point 6 minutesReading, comparing, and placing decimal digits Grades 4-6 →

xⁿ Exponents and Powers Repeated multiplication, and how fast it runs away from you 7 minutesReading powers and using the exponent rules Grades 6-8 → x^½ Fractional Exponents What it means to raise something to the power of a half 7 minutesReading a fractional power as a root, and using the exponent laws on it Grades 9-12 → ℚ Rational and Irrational Numbers Why some decimals stop, some repeat forever, and some never settle down 7 minutesTelling which numbers can be written as a fraction, and which cannot Grades 8-10 → ×10ⁿ Scientific Notation A way to write numbers too big or too small to read 6 minutesWriting and calculating with very large and very small numbers Grades 7-10 → √⬚ Simplifying Radicals Getting the square numbers out from under the root 7 minutesWriting surds in simplest form and rationalising denominators Grades 9-11 → √ Square Roots The side length hiding inside an area 6 minutesFinding roots and estimating the ones that aren't whole Grades 8-10 →

½ Adding Fractions with Unlike Denominators Why you can't just add the tops and bottoms 6 minutesCommon denominators Grades 4-6 → ⅜<½ Comparing and Ordering Fractions Why bigger numbers can mean a smaller fraction 6 minutesDeciding which of two fractions is larger Grades 3-6 → ⅘ Equivalent Fractions Why one half and four eighths are the same piece of pizza 6 minutesRecognising and making fractions that name the same amount Grades 3-5 → ⅓ Fractions on a Number Line A fraction is a place on the line, not just a piece of something 6 minutesPlacing a fraction as a number rather than a picture of pizza Grades 3-5 → ½=50% Fractions, Decimals, and Percents Three costumes worn by the same number 7 minutesConverting freely between the three forms Grades 5-8 → APR Loans, Credit and the Cost of Borrowing Where a monthly payment goes, and why it is mostly interest at first 7 minutesWorking out what a loan actually costs, and why early payments barely touch the balance Grades 10-12 → 2¾ Mixed Numbers and Improper Fractions Two ways to write the same amount of pizza 6 minutesConverting between the two ways of writing more than one whole Grades 4-6 → ×½ Multiplying and Dividing Fractions Why multiplying can make things smaller, and why dividing flips 7 minutesMultiplying fractions, and dividing by flipping Grades 5-7 → %err Percent Error and Accuracy Being 2 cm out matters differently on a pencil than on a road 5 minutesMeasuring how far a measurement or estimate is from the true value, in proportion Grades 6-9 → ↕% Percent Increase and Decrease Discounts, tips, markups, and why they don't cancel 7 minutesApplying and reversing a percent change Grades 6-8 → % Percents as Parts of 100 Why “per cent” literally means “out of 100” 6 minutesFinding a percent of a number Grades 5-7 → %↗ Simple and Compound Interest The difference one word makes to a savings account 7 minutesCalculating interest that does and doesn't build on itself Grades 7-10 →

| | Absolute Value How far from zero, regardless of direction 6 minutesReading absolute value as distance, and solving equations with it Grades 6-9 → −×− Multiplying and Dividing Negatives Why two negatives make a positive 6 minutesHandling signs in multiplication and division Grades 6-8 → − Negative Numbers on a Number Line Use direction and distance to make negatives feel natural 6 minutesComparing and adding integers Grades 6-8 →

a:b=c:d Proportions and Cross Multiplication Two ratios set equal, and the one move that solves them 7 minutesSetting up a proportion from a word problem and solving for the missing value Grades 6-8 → : Ratios and Proportions How to make more of something without changing what it is 7 minutesScaling quantities while keeping the relationship fixed Grades 6-8 → km/h Speed, Distance, and Time A rate problem you already meet every day 6 minutesRearranging one relationship three ways Grades 6-9 →

1011 Binary and Other Number Bases Place value works the same way whatever number you group by 6 minutesConverting between base ten and other bases, and reading place value in any base Grades 6-9 → LCM Common Factors and Common Multiples Two lists, and the number where they first agree 6 minutesFinding the greatest common factor and the least common multiple of two numbers Grades 4-6 → < Comparing and Ordering Numbers The first digit that differs settles it 5 minutesDeciding which number is bigger, and why Grades K-2 → 123 Counting and How Many Why the last number you say is the answer, and not just the last number 5 minutesCounting a group and knowing what the last number means Grades K-2 → ÷? Divisibility Rules Quick tests that save you the long division 6 minutesTelling whether one number divides another without dividing Grades 4-7 → ≈ Estimating to Check an Answer The five-second calculation that catches the big mistakes 6 minutesGetting close on purpose, then judging whether an answer is sensible Grades 3-5 → 2n Even and Odd Numbers Which numbers pair up exactly, and which always have one left over 5 minutesTelling evens from odds and predicting what happens when you combine them Grades 2-4 → ×÷ Factors, Multiples, and Primes Every number has one recipe, and it explains a lot 7 minutesBreaking numbers into primes, and finding HCF and LCM Grades 5-8 → mod Modular Arithmetic Clock arithmetic, and what it secures 7 minutesWorking with remainders as a number system in their own right Grades 9-12 → →▦ Patterns and Rules Spotting what stays the same when everything is changing 6 minutesFinding the rule behind a sequence and continuing it Grades 2-5 → ≈ Rounding and Estimating Trading exactness for a number you can actually work with 6 minutesChoosing the nearest round number, and knowing when to Grades 3-5 → gcd The Euclidean Algorithm Divide, keep the remainder, repeat — the oldest algorithm still in use 6 minutesFinding a greatest common divisor by repeated division, without factorising Grades 7-10 → 10 Understanding Place Value How a digit’s position changes what it means 5 minutesOnes, tens, and hundreds Grades 1-3 →

+ Regrouping in Addition and Subtraction What "carry the one" is actually carrying 6 minutesCarrying and borrowing across columns Grades 1-3 →

÷ Division as Sharing and Grouping The two different questions that both get answered by dividing 7 minutesSplitting a total fairly, and reading what is left over Grades 3-5 → ×7 Learning the Times Tables There are far fewer facts to learn than the grid suggests 6 minutesRecalling multiplication facts without counting them out Grades 2-4 → ⟌ Long Division The same four steps, repeated until you run out of digits 7 minutesDividing by a single or double digit, one place at a time Grades 4-6 → × Making Sense of Multiplication See multiplication as groups, not a list to memorise 6 minutesEqual groups and arrays Grades 3-5 → ×⬚ Multiplying Bigger Numbers Splitting a hard multiplication into four easy ones 7 minutesMultiplying two-digit numbers with the area model and the column method Grades 4-6 →

() The Order of Operations Why 2 + 3 × 4 is 14 and not 20 6 minutesEvaluating an expression the way everyone else will Grades 5-7 → a(b+c) The Properties of Addition and Multiplication The rules that let you move numbers around without changing the answer 6 minutesRearranging a calculation into an easier one that gives the same answer Grades 3-6 →

…? Turning Words into Sums The step before the arithmetic, which is where it usually goes wrong 7 minutesReading a problem and deciding what to calculate Grades 3-7 →

▭ Area and Perimeter Are Not the Same Thing Two shapes can share a perimeter and still hold different amounts 6 minutesMeasuring inside versus around Grades 3-5 → ½bh Area of Triangles, Parallelograms and Trapezoids Every one of these formulas is a rectangle in disguise 7 minutesFinding the area of any straight-sided shape by cutting it into rectangles Grades 6-8 →

£ Counting Money and Making Change Place value with coins instead of columns 6 minutesTotalling coins and working out change Grades 1-4 → 🕐 Telling Time and Elapsed Time Two hands, two different jobs, and a number system that isn't tens 6 minutesReading a clock and working out how long something takes Grades 1-4 →

cm Converting Units of Measure Why the answer changes size when the unit does 6 minutesMoving between millimetres, centimetres, metres, and kilometres Grades 3-5 → kg Mass and Capacity The same prefixes, doing the same job on different quantities 6 minutesConverting between grams and kilograms, millilitres and litres Grades 3-6 →

◓ Cylinders, Cones, and Spheres Three formulas that are really one formula and two fractions 7 minutesFinding the volume of round solids Grades 8-11 → ⬚ Surface Area of a Prism How much wrapping paper, rather than how much space 6 minutesAdding up the faces of a solid Grades 6-8 → ▣ Volume of a Rectangular Prism Why volume multiplies three numbers instead of two 6 minutesCounting the cubes that fill a box Grades 5-7 →

⬠ Angles Inside a Polygon Why every triangle you cut a shape into is worth 180° 6 minutesFinding interior and exterior angle sums for any polygon Grades 6-9 → ∥ Angles Made by Parallel Lines One angle is enough to know all eight 7 minutesFinding unknown angles in a diagram without measuring Grades 7-9 → ∠ Measuring and Naming Angles An angle measures turn, not length 6 minutesClassifying angles and measuring them in degrees Grades 4-6 →

⌒ Arc Length and Sector Area Every circle formula, scaled down by the fraction of a turn you kept 7 minutesMeasuring a slice of a circle instead of the whole thing Grades 9-11 → ⊙ Circle Theorems A handful of facts that make circle diagrams solvable 8 minutesFinding angles in circles using the standard results Grades 9-11 → ○ Circumference and Area of a Circle Where pi comes from, and why one formula squares the radius 6 minutesUsing radius to find distance around and space inside Grades 6-8 →

◠ Conic Sections Four curves, one cone, and a single number that tells them apart 8 minutesRecognising circles, ellipses, parabolas, and hyperbolas Grades 11-12 →

d Distance and Midpoint on the Grid Pythagoras, wearing coordinates 6 minutesMeasuring between two points using their coordinates Grades 8-11 → (,) Plotting Points on the Coordinate Plane How two numbers pin down one exact spot 6 minutesReading and writing ordered pairs Grades 5-7 → (r,θ) Polar Coordinates A second address system, better suited to anything round 7 minutesLocating points by distance and angle instead of across and up Grades 11-12 → x²+y² The Equation of a Circle Pythagoras, applied to every point at once 6 minutesWriting and reading a circle's equation on the grid Grades 10-12 →

△(x,y) Coordinate Proofs Put the shape on a grid and the proof becomes arithmetic 7 minutesProving a geometric theorem with algebra by placing the figure on axes Grades 9-11 → ∴ Writing a Geometric Proof The difference between "it looks true" and "it is true" 8 minutesBuilding an argument that a geometric claim must be true Grades 9-11 →

◇ Classifying Shapes and Finding Symmetry What makes a square a square, and a rectangle too 6 minutesNaming 2D shapes by their properties and spotting lines of symmetry Grades 2-5 → ⊥ Constructions with Compass and Straightedge Two tools, no measuring, and every arc carries a proof with it 7 minutesBisecting a segment or an angle with compass arcs, and saying why it works Grades 9-11 → ◌ Locus — Paths Defined by a Rule Every point obeying one rule, and the shape they turn out to make 6 minutesDescribing the set of all points satisfying a distance condition Grades 9-11 → ⬢ Naming 3D Shapes What to count, and the pattern hiding in the counts 6 minutesClassifying solids by faces, edges, and vertices Grades 2-6 → ⧉ Nets and Cross-Sections Two ways of turning a 3D object into 2D shapes you can measure 6 minutesFlattening a solid out, and slicing through it Grades 6-8 →

1:50 Scale Drawings and Maps Shrinking the world onto a page without distorting it 6 minutesConverting between drawing and reality using a scale Grades 6-8 →

⤢ Dilations and Similarity The fourth transformation, and the only one that changes size 6 minutesEnlarging or shrinking a shape from a centre point, and finding the scale factor Grades 8-10 → ⇄ Slides, Flips, and Turns Three ways to move a shape without changing it 7 minutesTranslating, reflecting, and rotating a shape Grades 7-9 →

∼ Similar Triangles Same shape, different size, and what that lets you measure 7 minutesUsing a scale factor to find missing lengths Grades 8-10 → △• The Four Centres of a Triangle A triangle has no single centre, and each of the four is useful for something different 7 minutesTelling the centroid, circumcentre, incentre and orthocentre apart by how each is built Grades 9-11 → △ The Pythagorean Theorem Connect the sides of a right triangle with one powerful rule 6 minutesRight-triangle side lengths Grades 8-10 → ≅ When Two Triangles Must Be Identical How little you need to know before the whole triangle is fixed 7 minutesDeciding whether given measurements force one triangle or allow several Grades 8-10 →

↑θ Angles of Elevation and Depression Measuring things you cannot reach, with one angle and one distance 6 minutesTurning a real height-and-distance situation into a right triangle and solving it Grades 9-11 → ∿ Graphs of Sine and Cosine The shape of anything that repeats 7 minutesReading amplitude, period, and shift from a trig equation Grades 10-12 → sin⁻¹ Inverse Trig Functions Why the calculator picks one angle out of infinitely many 7 minutesUsing arcsin, arccos and arctan, and knowing which single angle each returns Grades 10-12 → sec Secant, Cosecant and Cotangent Three more ratios, each one over a ratio you already know 6 minutesUsing the three reciprocal ratios and the identities built from them Grades 10-12 → sin Sine, Cosine, and Tangent Three ratios that depend only on the angle 8 minutesFinding missing sides and angles in a right triangle Grades 9-11 → sinx=k Solving Trigonometric Equations Why the calculator gives one answer when there are many 7 minutesFinding every solution in a given range, not just the calculator's one Grades 11-12 → △° The Sine and Cosine Rules Trigonometry for the other triangles 8 minutesSolving triangles that have no right angle Grades 10-12 → ◷ The Unit Circle How a point going round a circle becomes a wave 8 minutesExtending sine and cosine beyond right triangles Grades 10-12 → sin²+cos² Trigonometric Identities Equations that are true for every angle, and what they're for 7 minutesUsing identities to rewrite trigonometric expressions Grades 11-12 →

a·b The Dot Product Two vectors in, one number out, and that number is the angle in disguise 7 minutesMultiplying two vectors to get a number, and using its sign to read the angle Grades 11-12 → → Vectors Quantities that need a direction as well as a size 7 minutesAdding vectors and reading magnitude and direction Grades 10-12 → (x,y,z) Vectors in Three Dimensions One more coordinate, almost no new rules, and one genuinely new operation 7 minutesWorking with vectors that have a third component, and using the cross product Grades 11-12 →

a<x<b Compound Inequalities Two conditions at once, and whether you need both or either 6 minutesSolving inequalities joined by AND or OR, and graphing the solution Grades 9-11 → ◺ Linear Programming The continuous optimum is at a corner; whole-item plans need an integer check 7 minutesMaximising something subject to constraints Grades 11-12 → ∩○ Nonlinear Systems of Equations Where a line meets a curve, and why there can be two answers, one, or none 7 minutesSolving a system where one equation is a curve, and knowing how many answers to expect Grades 9-12 → F→C Rearranging Formulas Solving for a letter when every other letter stays a letter 6 minutesMaking any letter the subject of a formula Grades 8-11 → ✗ Solutions That Aren't Solutions Why squaring both sides can hand you an answer that doesn't work 7 minutesChecking answers back in equations where the algebra can invent them Grades 9-12 → x Solving Linear Equations Isolate the unknown without losing the equality 7 minutesKeeping equations balanced Grades 7-9 → < Solving and Graphing Inequalities When the answer is a whole stretch of the number line 7 minutesSolving inequalities and showing the solution on a number line Grades 7-9 → { Systems of Two Equations Where two lines meet, and what it means when they never do 8 minutesFinding the values that satisfy two equations at once Grades 8-11 →

3x Variables and Like Terms What a letter is standing in for, and which terms can combine 7 minutesWriting expressions and collecting the terms that match Grades 6-8 →

f∘g Composing Functions Feeding one machine's output into the next, and why swapping them changes the answer 6 minutesEvaluating and building f(g(x)), and knowing why the order matters Grades 10-12 → x→y Domain and Range What you are allowed to put in, and what can possibly come out 6 minutesWorking out which inputs a function accepts and which outputs it can produce Grades 9-11 → 2ˣ Exponential Growth and Decay What happens when the rate of change depends on the amount 8 minutesRecognising and using multiplicative growth Grades 9-12 → f⁻¹ Inverse and Composite Functions Doing one thing after another, and getting back to where you started 8 minutesChaining functions together and undoing them Grades 10-12 → log Logarithms The question "ten to the what?" written as an operation 8 minutesReading a logarithm as an exponent, and using the log laws Grades 10-12 → (t) Parametric Equations Letting a third variable drive both coordinates 7 minutesDescribing a curve by where a point is at each moment Grades 11-12 → {f Piecewise Functions One function, different rules in different places 6 minutesReading and evaluating a function defined in sections Grades 10-12 → x² Reading Quadratic Functions Read the story told by a parabola and its equation 7 minutesVertex, direction, and zeros Grades 9-12 → f(x)+k Shifting and Stretching Graphs One graph, and every version of it you'll ever need 7 minutesPredicting a graph's move from a change to its equation Grades 9-12 → ↗ Straight-Line Graphs and y = mx + b Two numbers are enough to draw any straight line 7 minutesGraphing a line from its equation and reading one off a graph Grades 8-10 → f(x) What Makes a Function One input, one output, no exceptions 7 minutesDeciding whether a relation is a function, and reading function notation Grades 8-10 → e e and the Natural Logarithm The one base where the growth rate equals the amount you already have 7 minutesUsing e for continuous growth and ln to undo it Grades 10-12 →

∝ Direct and Inverse Proportion One goes up and so does the other — or exactly the opposite 7 minutesRecognising the two ways quantities can vary together Grades 8-11 → / Slope as Rise over Run What the number in front of x is actually measuring 6 minutesReading and computing the steepness of a line Grades 7-9 → y=mx+b Writing the Equation of a Line Three forms of the same line, and when each one saves you work 7 minutesProducing the equation of a line from a point and a slope, or from two points Grades 8-10 →

[ ] Matrices as Transformations A grid of numbers that moves every point at once 8 minutesMultiplying matrices and reading what one does to the plane Grades 11-12 →

÷(x−a) Dividing Polynomials Long division, with x in place of digits 7 minutesDividing by a linear factor and using the remainder Grades 10-12 → xⁿ Polynomials and Their Shape What the highest power tells you before you plot anything 8 minutesPredicting a polynomial's graph from its degree and factors Grades 10-12 → ÷x Rational Expressions and Asymptotes What happens to a function where its denominator hits zero 8 minutesSimplifying algebraic fractions and finding where they break Grades 10-12 →

∎ Direct Proof, Contradiction and Contrapositive Three ways to get from "if" to "then", and how to tell which one fits 7 minutesChoosing a proof strategy and writing an argument that actually closes Grades 10-12 → ⇒n+1 Proof by Induction Knock the first domino, and make sure each one topples the next 8 minutesProving a statement for every whole number at once Grades 11-12 →

(x+p)² Completing the Square Forcing a quadratic into a shape you can solve by taking a square root 7 minutesRewriting a quadratic so its vertex and its roots are readable Grades 9-11 → ( )( ) Factoring Quadratics Why two brackets beat one expanded expression 8 minutesTurning a quadratic into a product, and reading its roots off Grades 9-11 → ±√ The Quadratic Formula and the Discriminant The method that works when factoring won't 8 minutesSolving any quadratic, and knowing in advance how many solutions exist Grades 9-11 →

Σ Adding Up a Sequence How Gauss added 1 to 100 in seconds 7 minutesSumming arithmetic and geometric series with a formula Grades 10-12 → … Arithmetic and Geometric Sequences Two kinds of pattern, and the formulas that skip the counting 7 minutesFinding the rule and jumping to any term Grades 9-11 → △1 Pascal's Triangle and the Binomial Theorem Where the coefficients come from, and why they're also combinations 7 minutesExpanding a bracket raised to a power without multiplying it out Grades 10-12 → aₙ₋₁ Recursive Sequences Defining a sequence by how it moves rather than where it lands 6 minutesReading and writing a rule that defines each term from the one before Grades 9-11 →

P(A|B) Conditional Probability How new information changes the odds 8 minutesUpdating a probability once you know something Grades 10-12 → n! Counting Without Listing Multiplying choices, and when order stops mattering 8 minutesCounting arrangements and selections Grades 9-12 → P(X) Discrete Random Variables A table of outcomes with their chances, and the two numbers that summarise it 7 minutesBuilding a probability distribution and computing its mean and standard deviation Grades 10-12 → E(X) Expected Value The average you'd get per go if you played forever 7 minutesWorking out the long-run average payoff of a random situation Grades 10-12 → P(A∪B) The Addition Rule and Mutually Exclusive Events Add the two chances, then take back what you counted twice 6 minutesFinding the chance of A or B without double-counting the overlap Grades 7-10 → B(n,p) The Binomial Distribution Counting successes when every trial is the same coin 7 minutesFinding the probability of r successes in n trials Grades 11-12 → 1st The Geometric Distribution Counting trials until something finally works 6 minutesFinding how long you wait for the first success, and what that wait averages Grades 10-12 → ? The Probability of a Single Event Turning "probably" into a number between 0 and 1 7 minutesCounting outcomes to measure how likely something is Grades 6-8 → ⑂ Tree Diagrams and Combined Events Multiply along the branches, add between them 7 minutesFinding the probability of two or more things happening Grades 7-10 →

MAD Comparing Spread with Mean Absolute Deviation How to say that one set is more spread out than another 6 minutesMeasuring how far values typically sit from the mean Grades 6-9 → r Correlation and the Line of Best Fit Turning "the points look linear" into two numbers 7 minutesMeasuring the strength of a linear relationship and fitting a line to it Grades 11-12 → ▄▆█ Histograms and Frequency Tables The chart for measurements, and why bar charts aren't it 7 minutesGrouping numerical data and reading its shape Grades 6-9 → ××× Line Plots and Dot Plots Stack one mark per measurement and the shape of the data appears 5 minutesBuilding a line plot from a list of measurements and reading it back Grades 2-5 → x̄ Mean, Median, and Mode Three different answers to "what is a typical value?" 7 minutesChoosing the right average for a set of numbers Grades 5-8 → ⊟ Quartiles and Box Plots What a box plot shows that an average can't 7 minutesSummarising a distribution with five numbers Grades 8-11 → ▥ Reading Bar and Line Graphs What each kind of graph is built to show 6 minutesPulling answers out of a chart, and spotting a misleading one Grades 3-5 → ⁘ Scatter Plots and Association What a cloud of points can and cannot tell you 7 minutesReading the relationship between two measurements Grades 8-10 → σ Spread and Standard Deviation The average alone hides everything that matters 8 minutesMeasuring how far a set of values typically sits from its mean Grades 10-12 → ∩ The Normal Distribution The shape that turns up whenever many small effects add together 8 minutesUsing the bell curve to judge how unusual a value is Grades 11-12 → ⊞ Two-Way Tables Where the rows and columns meet, and what the totals tell you 6 minutesReading a table that sorts by two things at once Grades 7-10 →

± Confidence Intervals What the margin of error in a poll actually claims 8 minutesPutting an honest range around an estimate Grades 11-12 → H₀ Hypothesis Testing What a p-value says, and the four things it doesn't 8 minutesDeciding whether a result is more than chance Grades 11-12 → SE Sampling Distributions and Standard Error The idea underneath every confidence interval and every p-value 7 minutesDescribing how a sample mean varies from sample to sample, and why that variation shrinks Grades 11-12 → χ² The Chi-Square Test How far off the counts are, measured in a way that can be judged 7 minutesComparing observed counts with expected ones to test a claim about categories Grades 11-12 →

A/B Experiments and Observational Studies Only one kind of study can tell you that something caused something else 7 minutesTelling an experiment from an observational study, and saying which can show cause Grades 10-12 → ⊂ Samples, Populations, and Bias How a survey goes wrong before a single number is calculated 7 minutesJudging whether a sample can support its conclusion Grades 9-12 →

●—● Graphs and Networks Dots joined by lines, and the surprising amount that follows 7 minutesModelling a connection problem with vertices and edges, and counting degrees Grades 9-12 → p→q If-Then Statements Why reversing a true statement often makes a false one 7 minutesForming converses and contrapositives, and knowing which ones follow Grades 9-11 → ∪ Sets and Venn Diagrams The diagram that stops you counting the middle twice 7 minutesSorting things into overlapping groups and counting them properly Grades 7-11 → T/F Truth Tables Every possible case in a grid, so nothing can hide 6 minutesBuilding a truth table and using it to test whether two statements are equivalent Grades 9-11 →

f″ Concavity and Points of Inflection The first derivative gives the slope, the second gives the curve 7 minutesReading the bend of a graph from the second derivative Grades 11-12 → max Finding Maximums and Minimums Where the slope is zero, and what to do with that 8 minutesUsing the derivative to find the best value of something Grades 11-12 → dy/dx Implicit Differentiation Differentiating a curve that was never written as y = something 7 minutesFinding a gradient from an equation you can't rearrange for y Grades 11-12 → s(t) Position, Velocity and Acceleration One object on a line, and three functions describing it 7 minutesMoving between position, velocity and acceleration, and telling displacement from distance Grades 11-12 → dx/dt Related Rates When two things are linked, their speeds are linked too 8 minutesFinding how fast one quantity changes from how fast another does Grades 11-12 → dy The Derivative as a Slope How you find the slope of something that is never straight 8 minutesReading the instantaneous rate of change of a curve Grades 11-12 → f′(c) The Mean Value Theorem Somewhere along the way, your speedometer read exactly your average speed 7 minutesUsing the guarantee that some instant matches the average rate over an interval Grades 11-12 → f′ The Rules for Differentiating The shortcuts that replace the limit, and when each applies 8 minutesDifferentiating without going back to first principles Grades 11-12 →

∫ds Arc Length of a Curve Straighten the curve into infinitely many hypotenuses and add them up 6 minutesFinding how long a curve is by integrating tiny straight pieces Grades 11-12 → y′ Differential Equations When you know how fast something changes but not what it is 8 minutesSolving an equation whose unknown is a function, not a number Grades 11-12 → u dv Integration by Parts The product rule, run backwards, with a choice to make 8 minutesIntegrating a product by trading it for an easier integral Grades 11-12 → ∫du Integration by Substitution Renaming the awkward part until the integral becomes one you know 8 minutesReversing the chain rule to integrate composite functions Grades 11-12 → dP/dt Logistic Growth Exponential growth with a brake, and where the brake starts to bite 7 minutesModelling growth that starts exponential and levels off at a ceiling Grades 11-12 → ▏▎▍ Riemann Sums and Numerical Integration Rectangles under a curve, and what happens as they get thin 7 minutesApproximating a definite integral with rectangles or trapezoids, and knowing which way the estimate leans Grades 11-12 → ∫≬ The Area Between Two Curves Top minus bottom, integrated across where they overlap 7 minutesIntegrating a gap rather than a height Grades 11-12 → f̄ The Average Value of a Function Averaging infinitely many values, by integrating and dividing 6 minutesFinding the mean height of a curve over an interval, and the point where it is attained Grades 11-12 → ∫′ The Fundamental Theorem of Calculus Why differentiating and integrating undo each other 8 minutesEvaluating an integral by finding an antiderivative instead of summing rectangles Grades 11-12 → ∫ The Integral as Accumulated Area Adding up infinitely many rectangles, and why that undoes a derivative 8 minutesReading an integral as a total built from infinitely many small pieces Grades 11-12 → ∫πr² Volumes of Revolution Spin a region round a line and integrate the circles it sweeps out 7 minutesFinding the volume of a solid made by spinning a curve about an axis Grades 11-12 →

∘ Continuity and Where It Breaks The three conditions a point has to pass, and the three ways it fails 7 minutesDeciding whether a function can be drawn without lifting the pen Grades 11-12 → 0/0 L'Hôpital's Rule When substitution gives nonsense, differentiate the top and bottom separately 7 minutesEvaluating a limit that first comes out as 0/0 or infinity over infinity Grades 11-12 → lim What a Limit Is The value a function approaches, which need not be the value it takes 8 minutesReading where a function is heading, whether or not it arrives Grades 11-12 →

Σ Taylor and Maclaurin Series How a calculator works out sin(1) when it can only add and multiply 8 minutesRebuilding a function as an infinite polynomial Grades 11-12 → Σ∞ When an Infinite Sum Has an Answer Adding forever and landing on a number 8 minutesTelling a convergent series from a divergent one Grades 11-12 →