Pythagorean theorem
Squares on the sides of a right triangle and a puzzle proof show: a² + b² = c².
Mathematics – grade 9/10
Grades 9/10: the teaching modules are to be covered in the given order; the units are the same for grades 9 and 10 and are shown as one stage. Modules 9 and 10 both carry the title “Exponential growth” in the curriculum. The numbers are the module numbers of the curriculum.
Unit 1Pythagoras’ theorem
Pythagoras’ theorem and its converse, calculating side lengths, applications and proof.
Unit 2Solids – pyramid, cone and sphere
Nets and oblique drawings, surface area and volume of pyramid, cone and sphere, Cavalieri’s principle, deriving the volume formulas.
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.
Unit 3Quadratic functions and equations
Quadratic functions in context, forms of the term and the effect of parameters, solving quadratic equations numerically, graphically and algebraically.
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.
Solving equations graphically: intersections of graphs
Draw both sides of an equation as graphs. The solutions are where the graphs cross.
Unit 4Laws of exponents
Scientific notation, explaining and proving the laws of exponents, negative and rational exponents, roots as powers.
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.
Unit 5Probability II
Multi-stage random experiments with path rules, complementary probability and expected value, counting principles, conditional probability with two-way tables and tree diagrams.
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.
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.
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.
Monty Hall problem
Play yourself or simulate thousands of games: is it worth switching doors?
Unit 6Similarity
Similarity and congruence, scale factors for lengths, areas and solids, similar triangles, intercept theorems.
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.
Unit 7Trigonometry
Sine, cosine and tangent in right triangles, calculating angles and sides, relationships in general triangles.
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.
Unit 8Periodic processes
Circular motion and its projections, the sine and cosine functions, the effect of parameters, describing periodic processes.
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.
Unit 9Exponential growth
Telling linear and exponential growth apart, exponential functions in context and how to represent them, the logarithm as the inverse of taking powers.
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.
Unit 10Exponential growth (functions of the form a · bˣ)
Exponential functions of the form f(x) = a · bˣ in context, switching representations and features, logarithms on the calculator.
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.
Unit 11Practice and consolidation
Practice and consolidation across all key ideas and process skills with a view to the exam.
No separate simulation is planned for this.
Unit 12Classes of functions
Comparing known and new functions, polynomial and simple rational functions, zeros and substitution, the square root function as an inverse 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.
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.
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.
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.
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.
Unit 13Introduction to differential calculus
Average and local rate of change, the slope of the tangent from the difference quotient, the derivative function, the tangent equation and angle of slope.
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.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Bildungsplan, Gymnasium, Mathematics, grade 9/10