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.
Mathematics – grade 9/10
Double grade 9/10: the core curriculum describes learning areas for grades 9 and 10 together, so they are the same for both grades. Each school decides order and scope. The grades are therefore shown as one stage. The sub-areas are the items of the “core” (and, where present, the optional extensions); the numbers are our own and run through all double grades.
Unit 16Tree diagrams and two-way tables
Representing data with two characteristics and two-stage random experiments in tree diagrams and two-way tables and finding unknown probabilities, preparing conditional probability.
16.1Data with two characteristics
Completing entries and switching between representations.
16.2Two-stage random experiments
Representing and analysing them in tree diagrams and two-way tables.
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.
16.3Finding unknown probabilities
Finding and interpreting them using absolute frequencies.
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.
Unit 17Discoveries in right triangles and similarity
Similar triangles, Pythagoras’ theorem and related theorems, roots as the inverse of squaring, sine, cosine and tangent, and the laws of sines and cosines.
17.1Similarity
Recognising similar triangles, similarity theorems, calculating lengths.
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.
17.2Pythagoras’ theorem and related theorems
Justifying and applying them.
Pythagorean theorem
Squares on the sides of a right triangle and a puzzle proof show: a² + b² = c².
17.3Working with roots
Taking roots as an inverse operation, laws, calculating lengths.
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.
17.4Trigonometric relationships
Sine, cosine and tangent in right triangles, tangent as a measure of slope.
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.
17.5Calculations in general triangles
Law of sines and law of cosines.
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 18Quadratic relationships
Quadratic functions in general, factorised and vertex form, solving quadratic equations, optimisation and parabolas of best fit, and the parabola as a locus.
18.1Investigating quadratic functions
Varying parameters and switching between the three forms.
Quadratic function
Stretch, reflect and shift parabolas and read off the vertex and the zeros.
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.
18.2Quadratic equations
Linking solutions to the graph and the structure of the term, the quadratic formula.
Solving equations graphically: intersections of graphs
Draw both sides of an equation as graphs. The solutions are where the graphs cross.
18.3Modelling quadratic relationships
Optimisation problems, finding zeros and parabolas of best fit.
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.
18.4The parabola as a locus
Equal distance from a point and a line.
No separate simulation is planned for this.
Unit 19Calculations with circles and solids
The number π by approximation, circumference, area, arcs and sectors, and surface area and volume of cylinders, pyramids, cones and spheres.
19.1Area and circumference of the circle
The path to π, arc length, sectors and radians.
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.
19.2Measures of selected solids
Surface area and volume of cylinders, pyramids, cones and spheres.
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.
19.3Optional extensions
Deriving the volume of pyramids, cones and spheres and the surface area of cones and spheres.
Cavalieri’s principle
Push a stack of coins sideways: the shape changes, but the volume stays the same. This is Cavalieri's principle.
Unit 20Exponential relationships
Exponential growth and decay iteratively and explicitly, limited growth, exponential functions with parameters, laws of exponents, taking roots and logarithms.
20.1Modelling growth and decay
Distinguishing linear and exponential, combinations and the limit of limited growth.
Exponential growth and decay
Water lilies on a pond, compound interest and decay: exponential growth compared with linear growth.
20.2Investigating exponential functions
Varying parameters, sketching graphs, the equation from two points.
Exponential growth and decay
Water lilies on a pond, compound interest and decay: exponential growth compared with linear growth.
20.3Calculating with powers
Justifying laws of exponents, solving equations also by taking logarithms.
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.
Unit 21Periodic relationships
Sine and cosine on the unit circle, degrees and radians, parameters of the general sine function and modelling periodic processes.
21.1Sine and cosine as periodic functions
Definition on the unit circle, degrees and radians.
Unit circle
Sine, cosine and tangent on the unit circle, and how the sine curve grows out of it.
21.2Investigating the sine function
Parameters of f(x) = a · sin(b · (x − c)) + d.
General sine function
Change the amplitude, period and shift of a sine curve with sliders or by dragging.
Shifting, stretching and reflecting graphs
Shift, stretch, reflect: every move of the graph leaves its mark in the function's equation.
21.3Modelling periodic relationships
Tides, day length, temperature.
General sine function
Change the amplitude, period and shift of a sine curve with sliders or by dragging.
Unit 22Approximation as limiting processes – extending the number system
Comparing limiting processes: approximating square roots and π, 0.9 recurring equals 1, decay and limited growth, the graph of 1/x; irrationality and the real numbers.
22.1Describing limiting processes
Approximating irrational square roots, the number π, exponential decay and the limiting behaviour of 1/x.
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.
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.
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.
22.2Explaining extensions of the number system
A proof of irrationality and the real numbers.
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.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Kerncurriculum, Gymnasium, Mathematics, grade 9/10