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.
MathematicsHesse curriculum · Gymnasium
Mathematics – grade 9/10
Double grade 9/10: the core curriculum describes what should be achieved by the end of grade 10, so the units are the same for grades 9 and 10. Both grades are therefore shown as one stage. The numbers (guiding idea.content area.focus) are our own.
Unit 2.2Solids
Pyramid, cone and sphere as well as solids from technology and everyday life: representation and calculation of volume and surface area.
2.2.1Basic solids (pyramid, cone, sphere)
Getting to know pointed and round solids and their properties.
2.2.2Solids from technology and everyday life
Modelling real objects as combinations of basic solids.
No separate simulation is planned for this.
2.2.3Describing volume and surface area of pyramid, cone, sphere
Justifying volume formulas (e.g. Cavalieri) and applying them.
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.
2.2.4Models, oblique drawings and nets of known solids
Representing pyramids and cones as nets and oblique drawings too.
No separate simulation is planned for this.
Unit 2.3Relationships between geometric objects
Trigonometry: sine, cosine and tangent in right-angled and general triangles.
2.3.1Trigonometric relationships (sin, cos, tan) in right-angled and general triangles
Side ratios as sin, cos, tan; law of sines and law of cosines for any triangle.
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 circle
Sine, cosine and tangent on the unit circle, and how the sine curve grows out of it.
Unit 3.2Measuring processes
Calculations in triangles and polygons in applications, and volume and surface area of pyramid, cone, cylinder and sphere.
3.2.1Calculations in triangles and polygons (applications from technology and physics)
Calculating lengths and angles in applied problems with Pythagoras and trigonometry.
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.
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.
3.2.2Volume and surface area of pyramid, cone, cylinder, sphere
Applying and comparing formulas for volume and 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.
Unit 4.1Assignments (mappings) and their representations
Understanding non-proportional functional relationships and representing them in words, tables and graphs.
4.1.1Basic ideas of non-proportional functional relationships
Recognising when a relationship is a function and how it changes.
No separate simulation is planned for this.
4.1.2Representing assignments in words, tables or graphs
Describing relationships flexibly in different representations.
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 4.2Functions and equations
Solving quadratic equations and linear 2×2 systems; using power, exponential and sine functions (according to the content-area description also quadratic functions) for modelling.
4.2.1Solving quadratic equations
Solving quadratic equations algebraically and graphically.
Quadratic function
Stretch, reflect and shift parabolas and read off the vertex and the zeros.
Solving equations graphically: intersections of graphs
Draw both sides of an equation as graphs. The solutions are where the graphs cross.
Optimisation problems with parabolas
Which enclosure has the largest area? Picture and graph show the maximum, first by trying, then by exact calculation.
4.2.2Solving linear 2×2 systems of equations
Solving systems of two equations algebraically and as intersecting lines.
Solving linear systems graphically
Two equations are two lines. Their point of intersection is the solution, which you can read off and check.
4.2.3Simple power functions
Investigating graphs of y = a·xⁿ.
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.
4.2.4Exponential functions
Describing growth and decay processes with exponential functions.
Exponential growth and decay
Water lilies on a pond, compound interest and decay: exponential growth compared with linear growth.
4.2.5Sine function and its main properties
Developing the sine function on the unit circle; period and amplitude.
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.
4.2.6Representing functions (equation, table, graph)
Switching between term, table and graph, also with software.
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 5.2Dealing with chance
Describing multi-stage random experiments and calculating probabilities.
5.2.1Multi-stage random experiments
Finding probabilities for more than two stages with tree diagrams and path rules.
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.
Galton board
Left or right at every pin, purely by chance. Yet every time the balls pile up into the same bell shape.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Kerncurriculum, Gymnasium, Mathematics, grade 9/10