Solving equations graphically: intersections of graphs
Draw both sides of an equation as graphs. The solutions are where the graphs cross.
Mathematics – grade 9/10
Section 3.3 covers the double grade 9/10; the same content is also listed under grade 10. Simulations are listed only where their grade range fits.
Unit 3.3.1Guiding idea: number – variable – operation
Powers with rational exponents and solving quadratic, radical, power, rational and exponential equations; exponential growth using the compound interest formula.
3.3.1 (1)–(3)Working with powers
Writing numbers in scientific notation, interpreting powers with rational exponents as roots, and justifying and applying the laws of exponents.
No separate simulation is planned for this.
3.3.1 (4)–(15)Solving equations
Solving quadratic equations by formula, the zero-product rule and graphically, and investigating their solvability; also simple rational, radical, power and exponential equations, logarithms and substitution.
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.
3.3.1 (16)–(17)Applying exponential growth
Explaining the terms of interest calculation and using the final-capital formula as exponential growth, solving it for each quantity.
Exponential growth and decay
Water lilies on a pond, compound interest and decay: exponential growth compared with linear growth.
Unit 3.3.2Guiding idea: measuring
The circle with radian measure, arcs and sectors, and surface area and volume of prisms, pyramids, cylinders, cones and spheres; formulas are also justified with Cavalieri’s principle.
3.3.2 (1)–(8)Calculating quantities of figures and solids
Explaining circumference and area of a circle through a limiting process, using radians, calculating arc lengths and sectors, and deriving and calculating lateral surface, surface area and volume of solids.
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.
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.
Cavalieri’s principle
Push a stack of coins sideways: the shape changes, but the volume stays the same. This is Cavalieri's principle.
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.
Unit 3.3.3Guiding idea: space and shape
Representing solids in oblique views and nets; calculating lengths and angles with Pythagoras’ theorem and with sine, cosine and tangent.
3.3.3 (1)Representing solids in drawings
Sketching oblique views and nets of prisms, pyramids, cylinders and cones and converting between them.
No separate simulation is planned for this.
3.3.3 (2)–(4)Proving geometric relationships and working with trigonometric relations
Calculating with Pythagoras’ theorem and its converse, finding sides and angles with sine, cosine and tangent, and deriving identities such as sin² + cos² = 1.
Pythagorean theorem
Squares on the sides of a right triangle and a puzzle proof show: a² + b² = c².
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.
Unit circle
Sine, cosine and tangent on the unit circle, and how the sine curve grows out of it.
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.
Unit 3.3.4Guiding idea: functional relationships
New types of function: quadratic, power, root and exponential functions, polynomial and trigonometric functions; parameters are interpreted as stretches, reflections and translations.
3.3.4 (1)–(7)Working with quadratic functions
Drawing and sketching parabolas, interpreting the parameters of vertex form, switching between standard, vertex and factorised form, and solving applied problems with maxima and minima.
Quadratic function
Stretch, reflect and shift parabolas and read off the vertex and the zeros.
Optimisation problems with parabolas
Which enclosure has the largest area? Picture and graph show the maximum, first by trying, then by exact calculation.
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.
3.3.4 (8)–(13)Working with power and exponential functions
Sketching graphs of power, root and exponential functions, explaining domain, range and inverse function using x² and √x, describing growth with half-life and doubling time, and interpreting parameters.
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.
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.
Exponential growth and decay
Water lilies on a pond, compound interest and decay: exponential growth compared with linear growth.
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.
Shifting, stretching and reflecting graphs
Shift, stretch, reflect: every move of the graph leaves its mark in the function's equation.
3.3.4 (14)–(19)Working with polynomial and trigonometric functions
Finding (including repeated) zeros of polynomial functions and factorising, sketching general sine functions and using them for periodic processes, and investigating end behaviour and symmetry.
Unit circle
Sine, cosine and tangent on the unit circle, and how the sine curve grows out of it.
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.
General sine function
Change the amplitude, period and shift of a sine curve with sliders or by dragging.
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.
Unit 3.3.5Guiding idea: data and chance
Two-way tables, conditional probability and independence; random variables with probability distribution, expected value, variance and standard deviation.
3.3.5 (1)–(8)Understanding and calculating with probabilities
Creating two-way tables and using them for conditional probabilities, checking events for independence, and finding and interpreting distribution, expected value and spread of a random variable.
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.
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.
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.
Monty Hall problem
Play yourself or simulate thousands of games: is it worth switching doors?
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.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Bildungsplan, Gymnasium, Mathematics, grade 9/10