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.
Mathematics – grade 9
Grade 9: the framework curriculum for grades 1–10 is organised by levels A–H, not by grades; at the Gymnasium grade 9 works at level G; G and H are linked within grades 9/10. Units are the subject areas, sub-units their topics; the numbers are our own.
Unit 1Numbers and operations (level G)
1.1Understanding numbers
π and square roots as irrational numbers, approximations of real numbers, the number sets up to the real numbers.
1.2Understanding operations and calculation strategies
Powers with negative exponents, laws of exponents for integer exponents, the link between powers and roots.
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 2Quantities and measurement (level G)
2.1Understanding quantities and measuring
Unit prefixes from nano to tera, taking measurements from technical drawings.
Matching simulations are listed under Grade 5/6, Grade 7, Grade 10
2.2Calculating with quantities
Sine, cosine and tangent in right triangles, volume and surface area of pyramids, cones and spheres, the sine rule and the cosine rule.
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.
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 3Space and shape (level G)
3.1Geometric objects
Describing composite and hollow solids, oblique drawings of cones, cylinders and pyramids, constructions with the theorems of Thales and Pythagoras.
Pythagorean theorem
Squares on the sides of a right triangle and a puzzle proof show: a² + b² = c².
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.
3.2Geometric transformations
Using transformations in complex contexts, enlarging and reducing solids to scale.
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 4Equations and functions (level G)
4.1Terms and equations
Transforming terms with binomial formulas, solving quadratic equations and linear systems algebraically and graphically and examining their solvability.
Binomial formulas with squares
The three binomial formulas as areas: a square splits into smaller squares and rectangles.
Solving equations graphically: intersections of graphs
Draw both sides of an equation as graphs. The solutions are where the graphs cross.
Solving linear systems graphically
Two equations are two lines. Their point of intersection is the solution, which you can read off and check.
4.2Relationships and functions
Quadratic functions y = a(x + d)² + e, trigonometric functions y = a·sin(bx) and exponential functions with parameters, symmetry and vertex; modelling.
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.
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.
Unit 5Data and chance (level G)
5.1Data
Planning and evaluating surveys independently, recognising trends, exposing manipulation in charts.
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.
5.2Counting strategies and probabilities
Multi-stage experiments with tree diagrams, path rules, two-way tables and urn models, including conditional probabilities.
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.
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.
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.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Rahmenlehrplan, Gymnasium, Mathematics, grade 9