Place value and rounding
Ten counters make a ten, ten tens make a hundred. Once you have seen the bundling, reading millions and rounding them is no longer a mystery.
Mathematics – grade 5/6
Grades 5 and 6: the curriculum describes requirements at the end of grade 6, so the units are the same for grades 5 and 6 and are shown as one stage. The order of the modules is not fixed. The codes are the module numbers of the curriculum (A arithmetic/algebra, G geometry, F functions, S stochastics).
Unit A1Natural numbers
Natural numbers in the place value table and on the number line, the meaning of the four operations, mental and written arithmetic, divisors and prime numbers.
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.
Prime factorisation and divisibility
Break numbers down into prime factors, check divisibility rules and find primes with a sieve. This also gives you the gcd and the lcm.
Powers, powers of ten and orders of magnitude
From a grain of sand to the Earth in steps of ten. Along the way it becomes clear why 10⁶ is a million and how to write tiny numbers.
Unit A2Fractions
Fractions in rectangle and strip diagrams and on the number line, equivalent fractions by simplifying and expanding, ordering and comparing, first calculations with fractions.
Representing fractions
Pizza, cake or a line: share it out fairly first, then colour some pieces. That is exactly what a fraction means.
Equivalent fractions and comparing
Compare two fractions as bars or circles. When you expand a fraction, the pieces get smaller but the amount stays the same.
Multiplying and dividing fractions with area models
Two strips overlap in a rectangle: the shared area is the product of the fractions. Dividing by a fraction is shown visually as well.
Unit A3Decimals
Decimals in the place value table and on the number line, the four operations mentally and in writing, rounding and estimating.
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.
Terminating and repeating decimals
Why does 1/4 end as a decimal but 1/3 never does? Long division gives it away as soon as a remainder repeats.
Unit A4Working with quantities
Estimating and measuring quantities and converting units.
Converting units with a place-value chart
3.5 km is 3500 m, but 3.5 m² is 35,000 cm². The unit table shows why the decimal point jumps twice as far for areas.
Unit A5Rational numbers
Negative numbers in everyday life, extending the number system to ℚ, ordering, sign rules and the four operations.
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.
Unit A6Further fraction arithmetic
Ratios as fractions, multiplying and dividing fractions with their meaning.
Multiplying and dividing fractions with area models
Two strips overlap in a rectangle: the shared area is the product of the fractions. Dividing by a fraction is shown visually as well.
Unit G1Shapes in the plane and in space I
Naming and classifying quadrilaterals, solids with nets and oblique drawings, models of solids.
The family of quadrilaterals
Move one corner and the square becomes a kite. When does a quadrilateral get which name?
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 G2Basic geometric terms
Describing figures and solids with basic terms, perpendicular and parallel lines.
The family of quadrilaterals
Move one corner and the square becomes a kite. When does a quadrilateral get which name?
Unit G3Perimeter, area, volume and surface area I
Measuring with units, perimeter and area of rectangles, volume and surface area of cuboids.
Perimeter and area of rectangles
Count the squares or walk once around? An ant makes the difference between area and perimeter tangible.
Volume of cuboids with unit cubes
First a row of cubes, then a layer, then the whole cuboid. Length times width times height follows naturally from filling it.
Unit G4Angles
Naming angles, types of angle, estimating, measuring and drawing angles.
Estimating and measuring angles
Line up the protractor, hit the zero point, read the right scale. The estimation game then shows how well your eye already measures.
Unit G5Angle sum and angle theorems
The angle sum in triangles, adjacent, vertical and corresponding angles.
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°.
Unit G6Geometric transformations
Translating figures in the coordinate system and reflecting them in lines parallel to the axes.
Points, lines and circles: relative positions
Place and read points in a coordinate system, draw segments and lines and check whether they are parallel or perpendicular.
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 F1Scale
Estimating quantities with reference objects, reading quantities from scale drawings, drawing to scale.
Scale: map and reality
Measure on a map with a ruler and use the scale to work out the real distance.
Unit F2Distance–time graphs
Coordinate systems with suitable scales, plotting and reading pairs of values, drawing and interpreting distance–time graphs.
Points, lines and circles: relative positions
Place and read points in a coordinate system, draw segments and lines and check whether they are parallel or perpendicular.
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.
Unit S1Statistics I
Planning surveys, data in lists and tables, bar charts, median and range.
Reading charts critically
Show the same data as bar, line and pie charts and more, and discover how charts can mislead.
Unit S2Probability I
Random experiments and first 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): Bildungsplan, Gymnasium, Mathematics, grade 5/6