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.
MathematicsLower Saxony curriculum · Gymnasium
Mathematics – grade 7/8
Double grade 7/8: the core curriculum describes learning areas for grades 7 and 8 together, so they are the same for both grades. Each school decides order and scope. The grades are therefore shown as one stage. The sub-areas are the items of the “core” (and, where present, the optional extensions); the numbers are our own and run through all double grades.
Unit 8Working with negative numbers
Negative numbers on the number line, adding and subtracting with the temperature model, multiplying and dividing with sign rules.
8.1Positive and negative numbers on the number line
Representing and ordering numbers.
8.2Adding and subtracting
In real contexts such as temperatures, and with patterns in sequences of calculations.
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.
8.3Multiplying and dividing
Debt model, permanence principle and sign rules.
No separate simulation is planned for this.
8.4Brackets and rules of arithmetic
Signs and operation symbols, calculating cleverly.
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 9Probability
Series of trials with partly symmetric, asymmetric and fully symmetric objects, probability as a prediction of relative frequency, Laplace probability, the law of large numbers, the addition and complement rules.
9.1Trials with partly symmetric and asymmetric objects
Cuboids or drawing pins: probability as a prediction, the law of large numbers qualitatively.
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.
9.2Trials with fully symmetric objects
Distinguishing Laplace probability from relative frequency.
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.
9.3Addition and complement rules
Justifying and applying the rules.
No separate simulation is planned for this.
Unit 10Proportional and inversely proportional relationships
Relationships in words, tables and graphs, proportional and inversely proportional relationships with the rule of three, and percentages and interest.
10.1Understanding relationships
Words, tables and graphs and switching between them.
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.
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.
10.2Proportional relationships
Recognising them, the rule of three, the quotient as an amount per 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.
10.3Inversely proportional relationships
Recognising them, the rule of three, the product as a total.
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.
10.4Percentages and interest
Using the rule of three.
Percentage bar
Discounts, VAT, surveys: one single bar holds the base value, the percentage and the part, one above the other.
Unit 11Lengths, areas and volumes and their terms
Perimeter and area of triangles, parallelograms and trapezia, surface area and volume of prisms, oblique drawings and nets.
11.1Triangle, parallelogram and trapezium
Justifying and using formulas for perimeter and area.
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.
11.2Surface area and volume of prisms
Comparing, estimating, calculating and justifying formulas.
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.
11.3Working with oblique drawings and nets
Comparing and switching between 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 12Elementary transformations of terms
Collecting like terms, expanding and factorising, binomial formulas, and solving simple linear and proportion equations.
12.1Simple transformations of terms
Collecting like terms, expanding and factorising.
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.
12.2Multiplying sums
Expanding brackets, binomial formulas as a special case.
Binomial formulas with squares
The three binomial formulas as areas: a square splits into smaller squares and rectangles.
12.3Solving simple linear and proportion equations
Transforming by hand.
Linear equations with a balance model
Solve linear equations on a balance: whatever you do to both sides keeps the scale balanced.
Unit 13Discoveries in triangles – constructions and special lines
Constructing triangles with the four basic constructions, congruence, Thales’ theorem, and perpendicular bisectors, angle bisectors, medians and altitudes as loci.
13.1Constructing triangles
Four basic constructions and congruence.
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.
13.2Thales’ theorem
Justifying and applying it.
Thales’s theorem
Drag a point along the semicircle: the angle there is always a right angle.
13.3Exploring special lines
Constructing perpendicular bisectors, angle bisectors, medians and altitudes and justifying their intersections; optionally circumcircle and incircle.
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.
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.
Unit 14Single- and multi-stage random experiments
Predicting, carrying out and simulating frequencies, experiencing variability, justifying and using tree diagrams and path rules with absolute frequencies.
14.1Single-stage random experiments
Predicting absolute frequencies and comparing them with series of trials; the law of large numbers.
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.
14.2Two- and multi-stage random experiments
Tree diagrams, drawing with and without replacement, justifying and using path rules.
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.
Simulating the birthday problem
How likely is it that two people in a class share a birthday? From just 23 people on, it is more likely than not.
Unit 15Linear relationships
Linear functions in all representations, slope as a constant rate of change, lines of best fit, and solving linear equations and systems with two variables.
15.1Identifying and representing linear relationships
Text, diagram, table, coordinate system and equation.
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.
15.2Analysing linear functions
Slope triangle, y-intercept, zero, varying parameters and lines through two points.
Linear function
Change the slope and the y-intercept of a line and read off the slope triangle and the zero.
15.3Solving linear equations
Trial, working backwards, by hand and with digital tools.
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.
15.4Linear systems with two variables
Graphically, by substitution and equating; solvability from the graph.
Solving linear systems graphically
Two equations are two lines. Their point of intersection is the solution, which you can read off and check.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Kerncurriculum, Gymnasium, Mathematics, grade 7/8