Place value and rounding
Ten counters make a ten, ten tens make a hundred. Once you have seen the bundling, reading millions and rounding them is no longer a mystery.
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129 simulations
Adding and subtracting integers on the number line
A frog hops past zero into the negatives. Thermometers and bank balances show why minus 3 plus 5 makes exactly 2.
Prime factorisation and divisibility
Break numbers down into prime factors, check divisibility rules and find primes with a sieve. This also gives you the gcd and the lcm.
Powers, powers of ten and orders of magnitude
From a grain of sand to the Earth in steps of ten. Along the way it becomes clear why 10⁶ is a million and how to write tiny numbers.
Representing fractions
Pizza, cake or a line: share it out fairly first, then colour some pieces. That is exactly what a fraction means.
Equivalent fractions and comparing
Compare two fractions as bars or circles. When you expand a fraction, the pieces get smaller but the amount stays the same.
Terminating and repeating decimals
Why does 1/4 end as a decimal but 1/3 never does? Long division gives it away as soon as a remainder repeats.
Multiplying and dividing fractions with area models
Two strips overlap in a rectangle: the shared area is the product of the fractions. Dividing by a fraction is shown visually as well.
Percentage bar
Discounts, VAT, surveys: one single bar holds the base value, the percentage and the part, one above the other.
Square roots and Heron’s method
Step by step, a rectangle turns into a square with the same area. The method is almost 2000 years old and reaches √2 surprisingly quickly.
Expression trees: the structure of terms
An expression drawn as a tree: whatever hangs lowest is calculated first. The order of operations stops being a rule and becomes visible.
Binomial formulas with squares
The three binomial formulas as areas: a square splits into smaller squares and rectangles.
Linear equations with a balance model
Solve linear equations on a balance: whatever you do to both sides keeps the scale balanced.
Solving linear systems graphically
Two equations are two lines. Their point of intersection is the solution, which you can read off and check.
Solving equations graphically: intersections of graphs
Draw both sides of an equation as graphs. The solutions are where the graphs cross.
Points, lines and circles: relative positions
Place and read points in a coordinate system, draw segments and lines and check whether they are parallel or perpendicular.
Estimating and measuring angles
Line up the protractor, hit the zero point, read the right scale. The estimation game then shows how well your eye already measures.
The family of quadrilaterals
Move one corner and the square becomes a kite. When does a quadrilateral get which name?
Reflections in a line and in a point
In a line or in a point: the shape flips over, and every construction step with the set square stays visible.
Perpendicular bisector and angle bisector
A point equally far from A and B never lies there by chance: the perpendicular bisector appears before you draw it with compasses.
Angles at intersecting lines
Two lines cross and one of them turns. Which angles stay equal? At parallel lines they are called corresponding and alternate angles.
Angle sum in triangles and polygons
Tear off three corners, put them side by side, and you get a straight line: 180°, for every triangle. A quadrilateral gives 360°.
Congruence and constructing triangles
Construct triangles from three given parts with ruler and compass, step by step as in your exercise book. Sometimes there are no, one or two solutions.
Circumcircle, incircle and special lines in a triangle
Construct the special lines in a triangle. Each set of three always meets in one point, for example at the centre of the circumcircle or the incircle.
Thales’s theorem
Drag a point along the semicircle: the angle there is always a right angle.
Pythagorean theorem
Squares on the sides of a right triangle and a puzzle proof show: a² + b² = c².
Intercept theorems
How tall is the tree without climbing it? Two rays and a parallel line are enough, because the ratios stay the same.
Similar figures: how do area and volume grow?
Enlarge lengths, areas and solids. Doubling the length makes the area four times and the volume eight times as large.
Sine, cosine and tangent in right triangles
In a right triangle, the ratio of two sides depends only on the angle. This gives sine, cosine and tangent, with examples such as ramps and ladders.
Law of sines and law of cosines
Calculate sides and angles in any triangle with the law of sines and the law of cosines. A helper shows which law fits your problem.
Converting units with a place-value chart
3.5 km is 3500 m, but 3.5 m² is 35,000 cm². The unit table shows why the decimal point jumps twice as far for areas.
Scale: map and reality
Measure on a map with a ruler and use the scale to work out the real distance.
Perimeter and area of rectangles
Count the squares or walk once around? An ant makes the difference between area and perimeter tangible.
Parallelogram, triangle and trapezium by decomposing
Cut off a triangle and put it on the other side: the parallelogram becomes a rectangle. Triangles and trapeziums work the same way.
Nets and surface areas of solids
Fold nets of a cube, cuboid, prism, pyramid and cylinder into a solid in 3D and calculate the surface area.
Prism and cylinder: surface area and volume
Layer by layer a prism grows upwards, and the curved surface of a cylinder unrolls into a rectangle.
Pyramid, cone and sphere
Pour three times and the prism is full: a pyramid holds exactly one third. Cones and spheres can be turned and measured in 3D.
Cavalieri’s principle
Push a stack of coins sideways: the shape changes, but the volume stays the same. This is Cavalieri's principle.
Volume of cuboids with unit cubes
First a row of cubes, then a layer, then the whole cuboid. Length times width times height follows naturally from filling it.
Circumference, area of a circle and the number π
A wheel rolls once and travels a bit more than three times its diameter. Cut into pieces, the circle becomes a rectangle and the area formula follows.
Direct and inverse proportion
Twice as many apples cost twice as much. Picture, table and graph show together when a relation is proportional, with the rule of three.
Function or not? Relations and their graphs
Exactly one y for every x? The arrow diagram, table and graph tell you straight away whether a relation is a function.
Function plotter with custom terms
Up to three of your own functions at once, with a table of values, zeros and intersections. The tool for anything that can be a graph.
Linear function
Change the slope and the y-intercept of a line and read off the slope triangle and the zero.
Hyperbolas and their asymptotes
The hyperbola gets ever closer to its asymptotes and yet never touches them. Shifting and stretching show what the parameters do.
Poles and asymptotes of rational functions
A zero or a pole, a hole or a leap to infinity? Drag the terms of the fraction and the graph shows the difference.
Quadratic function
Stretch, reflect and shift parabolas and read off the vertex and the zeros.
Parabola through three points
Drag three points and the matching parabola always passes through all of them. The equation is worked out step by step.
Optimisation problems with parabolas
Which enclosure has the largest area? Picture and graph show the maximum, first by trying, then by exact calculation.
Power functions y = a · xⁿ
Even or odd, positive or negative: y = a · xⁿ changes with every exponent, and only a few points always stay the same.
Unit circle
Sine, cosine and tangent on the unit circle, and how the sine curve grows out of it.
General sine function
Change the amplitude, period and shift of a sine curve with sliders or by dragging.
Logarithms as the inverse of exponentiation
The logarithm answers the question: to what power do I raise the base to get this number? With graphs, rules and exponential equations.
Exponential growth and decay
Water lilies on a pond, compound interest and decay: exponential growth compared with linear growth.
The natural exponential function
Why is the number e so special? For the exponential function, the graph and its derivative are exactly the same.
Polynomial functions: zeros, multiplicity and end behaviour
Does the graph cross the x-axis or only touch it? The multiplicity of the zero decides, the highest power decides what happens far out.
Shifting, stretching and reflecting graphs
Shift, stretch, reflect: every move of the graph leaves its mark in the function's equation.
Limits and asymptotes
What happens to a graph when x gets very large? A point runs outwards and a magnifier shows how close it gets to the asymptote.
Inverse functions: reflecting in y = x
Reflect the graph in y = x and the inverse function appears. It only works if no function value occurs twice.
Counting principle and tree diagrams
How many outfits or ice cream combinations are there? A tree diagram grows step by step and shows why you multiply the options.
Reading charts critically
Show the same data as bar, line and pie charts and more, and discover how charts can mislead.
Mean, median and box plots
One outlier pulls the mean away while the median barely moves. Dragging single data points shows which value is how robust.
Law of large numbers
Toss a coin, a die or a drawing pin a thousand times: the more often you throw, the closer the relative frequency gets to the probability.
Multi-stage experiments and path rules
Draw balls from an urn, with or without replacement. The tree diagram and the path rules show how to calculate probabilities.
Simulating the birthday problem
How likely is it that two people in a class share a birthday? From just 23 people on, it is more likely than not.
Monte Carlo method: estimating π by chance
Random points fall on a square with a circle inside. From the share inside the circle you can estimate π.
Monty Hall problem
Play yourself or simulate thousands of games: is it worth switching doors?
Two-way tables and Venn diagrams
A two-way table, a Venn diagram and a tree diagram show the same data and are linked. Change one and the others follow.
Conditional probability with medical tests
A medical test is positive, but how likely is it that you are really ill? 1000 people show why the answer is often surprising.
Galton board
Left or right at every pin, purely by chance. Yet every time the balls pile up into the same bell shape.
Binomial distribution
The histogram of the binomial distribution: choose a range and read off the probability directly.
One-sided significance test: type I and II errors
A test decides between two hypotheses. Coloured areas show how often it can be wrong.
Normal distribution
How many bottles are underfilled? Under the bell curve every probability is an area, with the sigma rules as a rule of thumb.
From secant to tangent
Let a point Q move towards P: the secant becomes the tangent, and the average slope becomes the derivative.
The derivative function graphically
A tangent slides along the graph and draws its slope as a new graph: the derivative function.
Newton’s method
Approximate a zero with tangent lines. Each step gets you closer, but sometimes it goes wrong.
Monotonicity, extrema and inflection points
Analyse a graph step by step: zeros, maximum and minimum points, inflection points and where it rises or falls.
Optimisation problems
Which box holds the most, which can uses the least metal? Five everyday problems show how to find the best solution with the derivative.
Families of functions
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
From a graph to its antiderivative
The height of one graph becomes the slope of another. If you understand bank balances, the antiderivative comes quicker than you think.
The integral as signed area (rectangle sums)
Fill the area under a graph with rectangles. The more rectangles, the more precisely you get the value of the integral.
Integral function and fundamental theorem
The area under a graph grows as you move along. Its rate of change is exactly the function itself: this is the fundamental theorem.
The integral as total change: from velocity to distance
From the speed of a car you get the distance travelled: the area under the curve shows the total change.
Solids of revolution
The area under a graph rotates about the x-axis and becomes a solid. Its volume is calculated with thin discs.
Taylor polynomials
Term by term a polynomial hugs a function such as sin x ever more closely. That is pretty much how a calculator does it.
Fourier series
Square, sawtooth and other waves are built from many sine waves. You can also listen to them.
Points and solids in 3D coordinates
Place points in 3D coordinates, rotate the view and calculate distances with Pythagoras' theorem.
Adding and scaling vectors (3D)
Drag vector arrows in space, add them and multiply them by numbers. The calculation is shown alongside in column form.
Dot product and cross product
The dot product tells you the angle between two vectors; the cross product gives a perpendicular vector and an area.
Lines and planes in space
A point travels along a line through space, two vectors span a plane. Parametric, normal and coordinate forms all describe the same thing.
Relative positions and distances in space
Do the lines meet, run parallel or pass each other by? In a 3D space you can turn, intersections, angles and distances become visible.
Spheres, lines and planes
A plane cuts the sphere in a circle, touches it at a point or misses it entirely. Including the equation of the sphere and the tangent plane.
Matrices as linear maps
Two arrows decide everything: wherever they point, the matrix rotates, stretches or reflects the whole grid.
Uniform and accelerated motion (distance–time and velocity–time graphs)
A car drives at a steady speed or speeds up. The graphs for distance, speed and acceleration are drawn live.
Free fall
A ball falls on Earth, the Moon or Mars and gets faster every second. In a vacuum even a feather keeps up.
Hooke’s law
Hang weights on a spring and measure how far it stretches. Double the force means double the stretch.
Law of the lever
How can a child lift an adult on a seesaw? With the longer arm. Wheelbarrows and bottle openers use the same trick.
Adding and resolving forces
Two forces pull on one point. Use the parallelogram of forces to find the total force or to split a force into two.
Inclined plane
A block sits on a slope. At what angle does it start to slide? The weight is split into two parts to find out.
Conservation of energy (roller coaster)
A car rides a roller coaster: energy bars show how potential and kinetic energy turn into each other.
Elastic and inelastic collisions
Two gliders collide on an air track. Before and after are compared to see what stays the same.
Circular motion and centripetal force
A ball swings round on a string, a car drives through a bend. Which force keeps them on a circular path?
Planetary orbits and Kepler’s laws
Too fast and the planet flies off, never to return. A little slower and it traces an ellipse around the Sun, exactly as Kepler's laws say.
Horizontal and oblique projectile motion
Which launch angle goes furthest? On Earth it is 45°, and on the Moon the same ball flies six times as far.
Simple pendulum
Pull a pendulum aside, let it go and measure how long one swing takes. What does it depend on? Also with large angles and damping.
Spring pendulum
A mass bounces up and down on a spring. Graphs show position, speed and energy during the oscillation.
Driven oscillation and resonance
A motor drives an oscillating wheel. If the motor hits exactly the right rhythm, the wheel swings especially strongly: resonance.
Transverse and longitudinal waves
A wave travels through a chain of particles. Each particle only oscillates in place, across or along the direction of travel.
Doppler effect
An ambulance drives past and the siren sounds lower. The wave fronts show why. You can also listen to it.
Interference of two circular waves
Two circular waves overlap: in some places they reinforce each other, in others they cancel out.
Standing waves
A wave is reflected at the end and a fixed pattern of nodes and antinodes appears, like on a guitar string.
Simple circuit
Build a circuit with a battery, a switch and a lamp, as a set-up and as a circuit diagram. When the circuit is closed, the lamp lights up.
Series and parallel circuits
Connect lamps in series or in parallel and measure current and voltage at different points.
Ohm’s law
Turn up the voltage and the current rises, always in the same ratio. Point by point the characteristic curve appears, though not quite straight for a lamp.
Electromagnetic induction
Magnet in, the needle swings. Magnet out, it swings the other way. Every generator works on the same principle.
Lorentz force and fine-beam tube
Electrons fly along a glowing circular path in a magnetic field. From the radius you determine a property of the electron.
Field lines of point charges
Two charges and the field rearranges itself. A test charge measures where the force is strongest.
Charging and discharging a capacitor
A capacitor charges through a resistor and discharges again. The graphs show how voltage and current change over time.
Light and shadow
One lamp casts a sharp shadow, two lamps a penumbra. Solar and lunar eclipses form in just the same way.
Reflection at a plane mirror
Laser at the mirror, measure the angles: angle of incidence equals angle of reflection, every time. And why does the candle appear just as far behind the mirror?
Refraction and total internal reflection
A light ray bends when it passes into glass or water. At a shallow angle it cannot get out at all: total internal reflection, as in an optical fibre.
Lenses and image formation
Move a candle in front of a lens and construct the image with three special rays. It becomes sharp on the screen.
Double slit and grating
Laser light passes through a double slit or a grating. A pattern of bright and dark stripes appears on the screen.
Particle model and temperature
Ice melts, water boils, and the temperature stops rising meanwhile. The particles show where the energy goes in between.
Gas laws
Push the piston halfway in and the pressure doubles. Heat the gas and it pushes the piston out. Pressure, volume and temperature are tightly linked.
Radioactive decay and half-life
Many nuclei decay at random and a Geiger counter clicks. Even so, after each half-life almost exactly half are left.
Photoelectric effect
Red light can be as bright as you like, it frees not a single electron. Short-wave light does it at once. A puzzle for waves, not for photons.
Atomic models and line spectra
Every element glows in its own colours. When an electron drops one level, exactly one line appears in the spectrum.
Electron diffraction
Electrons hit a thin layer of graphite and form rings on the screen, just like waves do.
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