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.
Mathematics – grade 10
Grade 10: the framework curriculum for grades 1–10 is organised by levels A–H, not by grades; at the Gymnasium grade 10 works at level H, the entry requirements for the qualification phase. Units are the subject areas, sub-units their topics; the numbers are our own.
Unit 1Numbers and operations (level H)
1.1Understanding numbers
Representing numbers to suit the situation, using integers, rational and real numbers appropriately, nesting intervals for square roots or π.
Matching simulations are listed under Grade 5, Grade 6, Grade 7, Grade 8, Grade 9
1.2Understanding operations and calculation strategies
Powers with rational exponents and laws of roots, logarithms as the inverse of taking powers.
Unit 2Quantities and measurement (level H)
2.1Understanding quantities and measuring
Radians and degrees on the unit circle, approximating figures with curved boundaries.
Unit circle
Sine, cosine and tangent on the unit circle, and how the sine curve grows out of it.
2.2Calculating with quantities
Volume of oblique prisms, cylinders and pyramids with Cavalieri’s principle, the cosine rule also for angles.
Cavalieri’s principle
Push a stack of coins sideways: the shape changes, but the volume stays the same. This is Cavalieri's principle.
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 H)
3.1Geometric objects
Justifying properties of geometric objects with symmetry, angle theorems, trigonometric relations and the theorems of Thales and Pythagoras.
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 4Equations and functions (level H)
4.1Terms and equations
Completing the square, exponential equations with logarithms, higher-degree equations and linear systems in three variables, for example to reconstruct quadratic functions.
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.
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.
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, trigonometric, power and exponential functions with parameters, inverse functions, average and local rate of change of polynomial functions and a sketch of the derivative.
Quadratic function
Stretch, reflect and shift parabolas and read off the vertex and the zeros.
General sine function
Change the amplitude, period and shift of a sine curve with sliders or by dragging.
Unit circle
Sine, cosine and tangent on the unit circle, and how the sine curve grows out of it.
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.
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.
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.
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 H)
5.1Data
Analysing and interpreting averages and measures of spread such as range and box width.
Matching simulations are listed under Grade 6, Grade 7, Grade 8, Grade 9
5.2Counting strategies and probabilities
Counting with factorials and binomial coefficients, estimating probabilities from relative frequencies and simulations and predicting frequencies.
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.
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.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Rahmenlehrplan, Gymnasium, Mathematics, grade 10