Percentage bar
Discounts, VAT, surveys: one single bar holds the base value, the percentage and the part, one above the other.
MathematicsRhineland-Palatinate curriculum · Gymnasium
Mathematics – grade 7/8
Grades 7 and 8: the framework curriculum describes both grades together, so the units are the same for grades 7 and 8 and are shown as one stage. It applies to all school types and marks content as basic (B), extension (E) and deepening (V); V prepares for upper secondary, so all three apply at the Gymnasium. The numbers are our own.
Unit 1Number and number sets: percentages and interest
The concept of percentage, basic problems with base, rate and percentage value, and interest, also in more complex contexts.
1.1Percentages and interest
Illustrating percentages with bar and pie charts, solving basic problems, calculating interest, distinguishing percentage points; increased and reduced base values (E).
Unit 2Number and number sets: rational numbers
Representing and ordering negative fractions, absolute value, and all four operations with rational numbers.
2.1Representing rational numbers
Interpreting negative numbers in context, ordering them on the number line, showing them as arrows and understanding absolute value as distance from zero.
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.
2.2Calculating with rational numbers
Calculating changes of state; adding and subtracting with a negative second operand, multiplying and dividing, evaluating terms, using the laws of arithmetic (E).
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.
Unit 3Measurement and quantities
Areas of parallelograms, triangles, trapezia and circles, and volume and surface area of right prisms and cylinders.
3.1Calculations with plane figures
Reducing areas to known figures by dissection and completion, deriving formulas, determining circumference and area of a circle with π as the constant of proportionality.
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.
3.2Calculations with solids
Finding volume and surface area of right prisms and cylinders, working out the principle of base area times height, calculating 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 4Space and shape
Nets and oblique drawings, basic constructions, symmetric and congruent figures, angle theorems and Thales’ theorem, and congruence transformations.
4.1Properties and relations of geometric objects
Nets and oblique drawings of prisms and cylinders, perpendicular and angle bisectors with circumcircle and incircle, symmetric triangles and quadrilaterals, the “house of quadrilaterals” (E), constructing triangles and using congruence criteria.
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.
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.
The family of quadrilaterals
Move one corner and the square becomes a kite. When does a quadrilateral get which name?
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.
4.2Theorems of plane geometry
Angle sum in triangles and quadrilaterals, angle theorems at intersecting lines (E), justifying and applying Thales’ theorem (E), its converse (V).
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.
Thales’s theorem
Drag a point along the semicircle: the angle there is always a right angle.
4.3Geometric transformations
Performing rotations and translations in the coordinate system; examining transformations by axis, centre, angle and arrow, fixed elements and congruence transformations (E).
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.
Unit 5Functional relationships: mappings and functions
Showing and interpreting everyday mappings, functions as unique mappings, proportional, inversely proportional and linear functions.
5.1Basics
Showing mappings in words, tables and graphs and checking whether they are unique; unique mappings as functions with a function term (E).
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.
5.2Special mappings and functions
Recognising direct and inverse proportion and solving with the rule of three; linear functions with slope and y-intercept, graphs as straight lines and hyperbolas (E).
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.
Linear function
Change the slope and the y-intercept of a line and read off the slope triangle and the zero.
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 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 6Functional relationships: terms and equations
Setting up and transforming terms, solving linear equations and inequalities, rearranging formulas, products of sums and binomial formulas.
6.1Terms and equations part I
Setting up and evaluating terms for real problems, solving simple linear equations, also with a balance and the number line; inequalities and the functional view of terms (E).
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.
Solving equations graphically: intersections of graphs
Draw both sides of an equation as graphs. The solutions are where the graphs cross.
6.2Terms and equations part II
Solving formulas with parameters for one variable; expanding products of sums and using binomial formulas in both directions (E).
Binomial formulas with squares
The three binomial formulas as areas: a square splits into smaller squares and rectangles.
Unit 7Data and chance
Analysing data with range, median and box plot, questioning surveys, estimating probabilities from long series of trials and calculating them for Laplace experiments.
7.1Data
Displaying and analysing data, range, median and box plot, spotting misleading displays and checking surveys for representativeness.
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.
7.2Random phenomena
Describing single-stage random experiments, estimating probabilities from relative frequencies over long series of trials, also by computer, and calculating Laplace probabilities.
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.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Rahmenlehrplan und Lehrplan, Gymnasium, Mathematics, grade 7/8