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.
Mathematics – grade 7/8
Grades 7/8: teaching modules of the core curriculum, in any order within the two grades; the units are therefore the same for grades 7 and 8 and are shown as one stage. The numbers are the module numbers of the curriculum.
Unit 1Positive and negative rational numbers
Negative numbers in everyday life and on the number line, sign and absolute value, ordering, the four operations with sign rules.
Unit 2Functions and the rule of three
Measuring and showing dependent quantities, proportional and inversely proportional relationships, switching between words, table, graph and term, the rule of three.
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.
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.
Unit 3Percentages and interest
Base value, percentage value and rate on a percentage strip, discounts, percentage increase and decrease, simple and compound interest.
Percentage bar
Discounts, VAT, surveys: one single bar holds the base value, the percentage and the part, one above the other.
Unit 4Probability I
Laplace experiments, counting outcomes, complementary probability, two-stage tree diagrams, expected value and fair games, computer simulation.
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.
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.
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.
Unit 5Triangle geometry
Angles at intersecting lines, the angle sum, types of triangle, altitudes, perpendicular and angle bisectors with circumcircle and incircle, constructions and congruence, 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°.
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.
Thales’s theorem
Drag a point along the semicircle: the angle there is always a right angle.
Unit 6Terms and equations I
Setting up and interpreting variables and terms in different contexts.
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.
Unit 7Quadrilaterals
Types of quadrilateral and their properties, quadrilaterals in the coordinate system, deriving the area of parallelogram and trapezium.
The family of quadrilaterals
Move one corner and the square becomes a kite. When does a quadrilateral get which name?
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.
Unit 8Linear functions
Uniform change, constant rate of change and starting value, slope and y-intercept, terms and graphs of linear functions.
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.
Unit 9Circle
Measuring and calculating the circumference and area of circles, finding π by experiment, parts of circles, approximating curved areas.
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.
Unit 10Terms and equations II
Setting up linear equations and solving them by equivalent transformations, non-linear equations and inequalities, rearranging terms, binomial formulas.
Linear equations with a balance model
Solve linear equations on a balance: whatever you do to both sides keeps the scale balanced.
Binomial formulas with squares
The three binomial formulas as areas: a square splits into smaller squares and rectangles.
Solving equations graphically: intersections of graphs
Draw both sides of an equation as graphs. The solutions are where the graphs cross.
Unit 11Solids – prism and cylinder
Nets, oblique drawings and models of prisms and cylinders, surface area and volume, 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.
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 12Statistics
Planning statistical surveys, showing data and reading it critically, measures including median and quartiles, box plots and misleading charts.
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.
Unit 13Linear functions and systems of equations
Setting up systems of linear equations in two variables, solving them graphically and algebraically, examining solvability, digital tools.
Solving linear systems graphically
Two equations are two lines. Their point of intersection is the solution, which you can read off and check.
Unit 14Real numbers
Irrational numbers, approximation procedures on the computer, estimating, ordering and calculating square and cube roots, the number sets ℕ, ℤ, ℚ and ℝ.
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): Bildungsplan, Gymnasium, Mathematics, grade 7/8