Adding and subtracting integers on the number line
A frog hops past zero into the negatives. Thermometers and bank balances show why minus 3 plus 5 makes exactly 2.
MathematicsSchleswig-Holstein curriculum · Gymnasium
Mathematics – grade 7–9
Grades 7 to 9: the subject requirements group the content of grades 7, 8 and 9 together, so the units are the same for grades 7, 8 and 9 and are shown as one stage. Topics marked * can be deepened in grade 10. In the eight-year track (G8) the content of grades 7–10 is taught in 6–9; G9 is shown here. The numbers are our own.
Unit 1Number and operation
Integers and rational numbers, percentages and interest, square roots, real numbers and powers.
1.1integers
Absolute value and sign, ordering integers on the number line and calculating with them.
1.2rational numbers
Justifying the extension of the number system, calculating with rational numbers and using the laws of arithmetic.
Adding and subtracting integers on the number line
A frog hops past zero into the negatives. Thermometers and bank balances show why minus 3 plus 5 makes exactly 2.
Expression trees: the structure of terms
An expression drawn as a tree: whatever hangs lowest is calculated first. The order of operations stops being a rule and becomes visible.
1.3percentages and interest
Base, percentage value and rate, the percentage bar, capital, interest and compound interest.
Percentage bar
Discounts, VAT, surveys: one single bar holds the base value, the percentage and the part, one above the other.
1.4square roots as an operation
Finding square and cube roots with a calculator and mentally.
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.
1.5real numbers
Irrational numbers as non-terminating, non-recurring decimals, approximation methods such as nested intervals and Heron’s method.
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.
1.6powers
Laws of exponents, negative and fractional exponents and scientific notation.
Powers, powers of ten and orders of magnitude
From a grain of sand to the Earth in steps of ten. Along the way it becomes clear why 10⁶ is a million and how to write tiny numbers.
Unit 2Quantities and measurement
Areas of polygons, the circle and π, calculations with solids, word problems, and central enlargement or the intercept theorems.
2.1area of polygons
Reducing triangles, parallelograms, trapezia, kites and rhombuses to rectangles of equal area, dissecting and completing polygons.
Parallelogram, triangle and trapezium by decomposing
Cut off a triangle and put it on the other side: the parallelogram becomes a rectangle. Triangles and trapeziums work the same way.
2.2simple calculations with circles
Circumference and area of a circle, finding an approximation for π.
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.
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.
2.3calculations with solids
Surface area and volume of cuboids, cubes, prisms, pyramids, cylinders, cones and spheres and of composite solids.
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.
Nets and surface areas of solids
Fold nets of a cube, cuboid, prism, pyramid and cylinder into a solid in 3D and calculate the surface area.
2.4word problems
Finding quantities in real contexts and evaluating results and methods.
No separate simulation is planned for this.
2.5central enlargement or intercept theorems*
Calculating lengths with the intercept theorems or central enlargement; the scale factor enters lengths, areas and volumes linearly, squared or cubed.
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 3Structures and functional relationships
Variables and terms, functions and their representations, proportional, inversely proportional, linear and quadratic functions, and linear and quadratic equations.
3.1variables and terms
Setting up, evaluating and transforming terms, multiplying sums, factorising, binomial formulas and completing the square.
Expression trees: the structure of terms
An expression drawn as a tree: whatever hangs lowest is calculated first. The order of operations stops being a rule and becomes visible.
Binomial formulas with squares
The three binomial formulas as areas: a square splits into smaller squares and rectangles.
3.2functions and their representations
Relationships and functions in words, tables, graphs and terms, the notation f(x), domain and range.
Function or not? Relations and their graphs
Exactly one y for every x? The arrow diagram, table and graph tell you straight away whether a relation is a 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.
3.3proportional functions
Direct proportion with equal quotients and the constant of proportionality.
Direct and inverse proportion
Twice as many apples cost twice as much. Picture, table and graph show together when a relation is proportional, with the rule of three.
3.4inversely proportional functions
Inverse proportion with equal products and its graphs.
Direct and inverse proportion
Twice as many apples cost twice as much. Picture, table and graph show together when a relation is proportional, with the rule of three.
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.
3.5rule of three
Solving real problems with the rule of three and its inverse.
Direct and inverse proportion
Twice as many apples cost twice as much. Picture, table and graph show together when a relation is proportional, with the rule of three.
3.6linear equations
Solving equations by trial, inverse operations and equivalent transformations, also as intersections of graphs; simple inequalities.
Linear equations with a balance model
Solve linear equations on a balance: whatever you do to both sides keeps the scale balanced.
Solving equations graphically: intersections of graphs
Draw both sides of an equation as graphs. The solutions are where the graphs cross.
3.7linear systems of equations
Solving systems in two variables, including over- and underdetermined ones.
Solving linear systems graphically
Two equations are two lines. Their point of intersection is the solution, which you can read off and check.
3.8linear functions
Straight lines with slope, slope triangle and intercepts, linear growth, the meaning of the parameters.
Linear function
Change the slope and the y-intercept of a line and read off the slope triangle and the zero.
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.
3.9quadratic functions*
Parabolas with vertex, symmetry and intercepts, standard, vertex and factorised form, shifting, stretching and reflecting.
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.
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.
3.10quadratic equations*
Solving quadratic equations algebraically and graphically.
Solving equations graphically: intersections of graphs
Draw both sides of an equation as graphs. The solutions are where the graphs cross.
Quadratic function
Stretch, reflect and shift parabolas and read off the vertex and the zeros.
Unit 4Space and shape
Angle theorems and geometry of triangles, quadrilaterals and circles, congruence, transformations, Pythagoras’ theorem and solids.
4.1geometry of triangles, quadrilaterals and circles
Adjacent, vertical, corresponding and alternate angles, angle sum, base angle theorem, special lines in triangles and Thales’ theorem.
Angles at intersecting lines
Two lines cross and one of them turns. Which angles stay equal? At parallel lines they are called corresponding and alternate angles.
Angle sum in triangles and polygons
Tear off three corners, put them side by side, and you get a straight line: 180°, for every triangle. A quadrilateral gives 360°.
Circumcircle, incircle and special lines in a triangle
Construct the special lines in a triangle. Each set of three always meets in one point, for example at the centre of the circumcircle or the incircle.
Perpendicular bisector and angle bisector
A point equally far from A and B never lies there by chance: the perpendicular bisector appears before you draw it with compasses.
Thales’s theorem
Drag a point along the semicircle: the angle there is always a right angle.
The family of quadrilaterals
Move one corner and the square becomes a kite. When does a quadrilateral get which name?
4.2congruence criteria
The congruence criteria SSS, SAS, ASA and SSA and constructing triangles.
Congruence and constructing triangles
Construct triangles from three given parts with ruler and compass, step by step as in your exercise book. Sometimes there are no, one or two solutions.
4.3transformation geometry
Line reflection, rotation, point reflection and translation.
Reflections in a line and in a point
In a line or in a point: the shape flips over, and every construction step with the set square stays visible.
4.4area theorems in right triangles
Pythagoras’ theorem and its converse, Pythagorean triples; the altitude and leg theorems for differentiation.
Pythagorean theorem
Squares on the sides of a right triangle and a puzzle proof show: a² + b² = c².
4.5solids
Prisms, pyramids, cylinders, cones and spheres with nets and oblique drawings.
Nets and surface areas of solids
Fold nets of a cube, cuboid, prism, pyramid and cylinder into a solid in 3D and calculate the surface area.
Unit 5Data and chance
Frequencies and probability, multi-stage random experiments and descriptive statistics.
5.1frequency
Absolute and relative frequencies in frequency tables, pie charts and histograms.
Law of large numbers
Toss a coin, a die or a drawing pin a thousand times: the more often you throw, the closer the relative frequency gets to the probability.
Reading charts critically
Show the same data as bar, line and pie charts and more, and discover how charts can mislead.
5.2probability
Probability via the empirical law of large numbers, events and Laplace experiments.
Law of large numbers
Toss a coin, a die or a drawing pin a thousand times: the more often you throw, the closer the relative frequency gets to the probability.
5.3multi-stage random experiments
Two- and multi-stage experiments in tree diagrams with the sum and product rules.
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.
5.4descriptive statistics
Median and arithmetic mean, analysing charts and checking them critically.
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.
Reading charts critically
Show the same data as bar, line and pie charts and more, and discover how charts can mislead.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Fachanforderungen, Gymnasium, Mathematics, grade 7–9