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.
Mathematics – grade 7/8
Double grade 7/8: the core curriculum describes what should be achieved by the end of grade 8, so the units are the same for grades 7 and 8. Both grades are therefore shown as one stage. The numbers (guiding idea.content area.focus) are our own.
Unit 1.1Numbers
Extension to rational and real numbers (roots, π); numbers are shown on the number line and as decimals, compared and rounded; orientation in the full coordinate system.
1.1.1Rational numbers
Positive and negative fractions and decimals as one number set.
No separate simulation is planned for this.
1.1.2Real numbers (roots and the number π as a proportionality factor)
Getting to know irrational numbers such as square roots and π and approximating them.
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.
1.1.3Representations (number line, decimals)
Representing rational and real numbers on the number line and in decimal notation.
No separate simulation is planned for this.
1.1.4Comparing, ordering and rounding rational and real numbers
Ordering numbers of both signs and rounding sensibly.
No separate simulation is planned for this.
1.1.5Orientation in the two-dimensional coordinate system
Plotting and reading points in all four quadrants.
No separate simulation is planned for this.
Unit 1.2Operations and their properties
Calculating with rational and real numbers, transforming terms with variables and brackets, powers, and percentage and interest calculations in real-life contexts.
1.2.1Calculation methods, arithmetic laws and their combination for rational and real numbers
Applying sign rules and arithmetic laws confidently.
No separate simulation is planned for this.
1.2.2Brackets, binomials
Expanding and factorising brackets and using the binomial formulas.
Binomial formulas with squares
The three binomial formulas as areas: a square splits into smaller squares and rectangles.
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.
1.2.3Powers with rational exponents
Extending power notation to negative and fractional exponents.
No separate simulation is planned for this.
1.2.4Terms and variables
Setting up, structuring and transforming terms with variables.
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.
1.2.5Basic tasks of percentage and interest calculation
Calculating base value, percentage value, rate and interest.
Percentage bar
Discounts, VAT, surveys: one single bar holds the base value, the percentage and the part, one above the other.
1.2.6Percentages with increased and decreased base value
Calculating price increases and discounts using growth or decay factors.
Percentage bar
Discounts, VAT, surveys: one single bar holds the base value, the percentage and the part, one above the other.
1.2.7Percentages of percentages
Combining several successive percentage changes.
No separate simulation is planned for this.
Unit 2.1Plane figures
Quadrilaterals and circles are classified and constructed, also with dynamic geometry software; symmetry and all four quadrants of the coordinate system.
2.1.1Basic figures (parallelogram, trapezium, rhombus, kite, circle)
Getting to know further basic figures and their properties.
The family of quadrilaterals
Move one corner and the square becomes a kite. When does a quadrilateral get which name?
2.1.2Hierarchy of quadrilaterals
Ordering quadrilaterals by properties and justifying relationships between them.
The family of quadrilaterals
Move one corner and the square becomes a kite. When does a quadrilateral get which name?
2.1.3Constructions with drawing instruments and dynamic geometry software
Constructing triangles, perpendicular bisectors, angle bisectors and more.
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.
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.
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.
2.1.4Symmetry properties of figures
Examining line and point symmetry of figures.
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.
2.1.5Cartesian coordinate system in all four quadrants
Representing figures in the full coordinate system.
No separate simulation is planned for this.
Unit 2.2Solids
Prism and circular cylinder: representation by models, oblique drawings and nets, and description of volume and surface area.
2.2.1Basic solids (prism, circular cylinder)
Recognising prisms and cylinders and describing their properties.
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.
2.2.2Describing volume and surface area of the prism
Understanding volume as base times height and surface area via the net.
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.
2.2.3Models, oblique drawings and nets of known solids
Drawing and recognising known solids in different representations.
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 2.3Relationships between geometric objects
Congruence and symmetry, Pythagorean theorem with proofs, Thales' theorem, similarity, central dilation and intercept theorems.
2.3.1Technical terms (congruence, symmetry)
Defining congruence and symmetry, e.g. via congruence theorems.
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.
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.
2.3.2Pythagorean theorem and its converse including exemplary complete proofs
Justifying and proving the theorem and using it to calculate side lengths.
Pythagorean theorem
Squares on the sides of a right triangle and a puzzle proof show: a² + b² = c².
2.3.3Thales' theorem
Discovering and justifying right angles in a semicircle.
Thales’s theorem
Drag a point along the semicircle: the angle there is always a right angle.
2.3.4Similarity, central dilations, intercept theorems
Enlarging/reducing figures to scale and calculating ratios of segments.
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 3.1Working with quantities
Working with composite figures and angle measures, and angle theorems.
3.1.1Composite figures
Dissecting or completing areas into known component figures.
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.
3.1.2Sizes of angles (minutes, seconds)
Stating angles more precisely in degrees, minutes and seconds.
No separate simulation is planned for this.
3.1.3Angle sum theorem and angle theorems at intersecting lines
Using the interior angle sum and vertical, adjacent, 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°.
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.
Unit 3.2Measuring processes
Area and perimeter of further polygons and the circle, and volume and surface area of the prism.
3.2.1Area and perimeter of triangle, parallelogram, trapezium, rhombus, kite, circle
Deriving area formulas by dissection and completion; circumference and area of the circle with π.
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.
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.
3.2.2Volume and surface area of the prism
Calculating volume and surface area of prisms.
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.
Unit 4.1Assignments (mappings) and their representations
Recognising proportional and inversely proportional assignments, solving them with the rule of three and representing them in text, table and graph.
4.1.1Proportional and inversely proportional assignments and their properties
Recognising and using constant quotients or constant products.
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.
4.1.2Rule-of-three methods
Calculating missing values step by step via one unit.
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.
4.1.3Representing proportional and inversely proportional assignments in words, tables and graphs
Switching between text, table of values and graph.
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.
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.
Unit 4.2Functions and equations
Solving linear equations and inequalities, comparing solution methods and investigating linear functions and their properties.
4.2.1Solving linear equations
Solving equations by equivalent transformations.
Linear equations with a balance model
Solve linear equations on a balance: whatever you do to both sides keeps the scale balanced.
4.2.2Comparing approaches to solving linear equations
Contrasting trial-and-error, algebraic transformation and graphical solution.
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.
4.2.3Linear functions and their properties
Interpreting slope and y-intercept and drawing graphs.
Linear function
Change the slope and the y-intercept of a line and read off the slope triangle and the zero.
4.2.4Solving inequalities
Solving linear inequalities and stating solution sets.
No separate simulation is planned for this.
Unit 5.1Statistical surveys and their evaluation
Presenting data in box plots, bar and pie charts and assessing them with measures of centre and spread.
5.1.1Presenting data in diagrams (box plot, bar chart, pie chart) and tables
Choosing, creating and critically reading a suitable type of diagram.
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.
5.1.2Measures of centre and spread
Determining and comparing mean, median, quartiles and range.
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.
Unit 5.2Dealing with chance
Describing and calculating two-stage random experiments with tree diagrams, two-way tables and path rules.
5.2.1Two-stage random experiments
Investigating experiments with two steps and counting outcomes.
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.
5.2.2Tree diagrams and two-way tables
Representing probabilities 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.
5.2.3Path rules
Applying the product and sum 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.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Kerncurriculum, Gymnasium, Mathematics, grade 7/8