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
Grade 7 (nine-year Gymnasium): topic fields of the curriculum; the order is binding only where the subject requires it. The numbers are those of the curriculum.
Unit 1Rational numbers
Negative fractions on the number line, opposite and absolute value, the four operations with rational numbers.
Unit 2Terms, equations and inequalities
Statements and open sentences, domain and solution set, equivalent transformations of equations and inequalities.
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.
Linear equations with a balance model
Solve linear equations on a balance: whatever you do to both sides keeps the scale balanced.
Unit 3Relationships
Relationships in tables and graphs, proportional and inversely proportional relationships with line through the origin and hyperbola, 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 4Geometric constructions
Perpendiculars, perpendicular bisectors and angle bisectors, circumcircle and incircle of triangles.
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.
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.
Unit 5Percentages
Base value, percentage value and rate, increase and decrease, discount and VAT, 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 6Introduction to probability
Random experiments with outcomes and events, Laplace probability, simulations.
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.
Unit 7Angles in figures
Adjacent, vertical, corresponding and alternate angles, angle sums, secant and tangent, Thales’ theorem, converse.
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°.
Thales’s theorem
Drag a point along the semicircle: the angle there is always a right angle.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Lehrplan, Gymnasium, Mathematics, grade 7