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.
MathematicsNorth Rhine-Westphalia curriculum · Gymnasium
Mathematics – grade 7/8
First stage up to the end of lower secondary. The core curriculum does not name exact grades for the first and second stage, so the first stage is shown as the double grade 7/8 (G9) with the same units for grades 7 and 8; both are shown as one stage. The numbers are our own.
Unit 5Arithmetic/algebra
Rational numbers and sign rules, variables and transforming terms, binomial formulas, and solving linear equations, systems with two variables and simple fractional equations.
5.1Extending the number system
Representing and ordering rational numbers on the number line; why number systems are extended.
5.2Term and variable
Variables as changing quantities, placeholders and unknowns; setting up and purposefully transforming terms.
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.
5.3Laws and rules
Sign rules, laws of arithmetic for rational numbers and binomial formulas.
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.
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.
5.4Solution methods
Algebraic and graphical methods for linear equations, linear systems with two variables and simple fractional equations.
Linear equations with a balance model
Solve linear equations on a balance: whatever you do to both sides keeps the scale balanced.
Solving linear systems graphically
Two equations are two lines. Their point of intersection is the solution, which you can read off and check.
Solving equations graphically: intersections of graphs
Draw both sides of an equation as graphs. The solutions are where the graphs cross.
Unit 6Functions
Proportional and inversely proportional relationships, the concept of a function, linear functions with slope and intercept, and percentages and interest.
6.1Proportional and inversely proportional relationships
Rule of assignment, graph, table, equal quotients and equal products, constant of proportionality and rule of three; functions as unique assignments.
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.
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.
6.2Linear functions
Function term, graph, table and verbal form, intercepts, slope and slope triangle, also with a function plotter.
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.
6.3Percentages and interest
Base value, percentage value and rate, percentage change and growth factor, interest.
Percentage bar
Discounts, VAT, surveys: one single bar holds the base value, the percentage and the part, one above the other.
Unit 7Geometry
Areas of triangles, quadrilaterals and composite figures, angle theorems, congruence theorems and Thales’ theorem, and constructions with compass and ruler.
7.1Perimeter and area
Triangles, quadrilaterals and composite figures, height and base.
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.
7.2Geometric theorems
Adjacent, vertical, corresponding and alternate angles, interior, exterior and base angle theorems, congruence theorems 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°.
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.
Thales’s theorem
Drag a point along the semicircle: the angle there is always a right angle.
7.3Construction
Triangles, perpendicular bisectors, medians, angle bisectors, incircle, circumcircle, Thales circle and centroid.
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 8Stochastics
One- and two-stage random experiments with tree diagrams, estimating probabilities from long series of trials, Laplace probability and path rules.
8.1Probabilities and random experiments
One- and two-stage random experiments and tree diagrams.
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.
8.2Rules of probability
Empirical law of large numbers, Laplace probability and path rules.
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.
8.3Concepts
Outcome, event and probability.
No separate simulation is planned for this.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Kernlehrplan, Gymnasium, Mathematics, grade 7/8