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 5/6: the curriculum describes both grades together, so the units are the same for grades 5 and 6, and both are shown as one stage. Units are the content areas with their focal points; the curriculum does not number them, the numbers are our own.
Unit 1Numbers and quantities
Natural numbers and divisibility, first equations, fractions and decimals, quantities with their units, and calculating with non-negative rational numbers.
1.1Natural numbers
Reading, ordering and rounding large numbers, mental and written arithmetic, powers, divisors and prime factors; first negative numbers on the number line.
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.
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.
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.
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.
1.2Equations and inequalities
The terms variable, equation and solution; solving simple equations and inequalities by reasoning or with inverse operations, and checking the result.
No separate simulation is planned for this.
1.3Fractions
Showing fractions as parts of a whole, expanding and simplifying them, comparing and adding like fractions, decimals in the place-value chart.
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.
1.4Quantities
Measuring and estimating length, area, volume, mass, time and money and converting between units; scale.
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.
Scale: map and reality
Measure on a map with a ruler and use the scale to work out the real distance.
1.5Non-negative rational numbers
All four operations with fractions and decimals, converting between both notations and meeting recurring decimals; comparing number sets.
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.
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.
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 2Space and shape
Basic geometric concepts and transformations, angles at lines, perimeter, area and volume, and triangles and quadrilaterals.
2.1Basic geometric concepts and transformations
Points, lines and segments, parallel and perpendicular, coordinate system, measuring and drawing angles; line symmetry, reflection, translation and rotation.
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.
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.
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.
2.2Angle relationships
Recognising adjacent, vertical, corresponding and alternate angles and using them to calculate angles and justify statements.
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.
2.3Perimeter, area and volume
Perimeter and area of rectangles, nets and oblique drawings of cuboids, surface area and volume of cuboids and cubes.
Perimeter and area of rectangles
Count the squares or walk once around? An ant makes the difference between area and perimeter tangible.
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.
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.
2.4Triangles
Types of triangle, the angle sum, congruence criteria and constructions; altitudes, angle bisectors and perpendicular bisectors with incircle and circumcircle; perimeter and area.
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.
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.
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.
2.5Quadrilaterals
Types of quadrilateral and how they relate, properties of sides, diagonals and angles, constructing quadrilaterals, perimeter and area of special quadrilaterals.
The family of quadrilaterals
Move one corner and the square becomes a kite. When does a quadrilateral get which name?
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.
Unit 3Relationships and functions
Describing relationships in everyday life and mathematics and recognising direct and inverse proportion.
3.1Relationships, direct and inverse proportion
Showing relationships in tables, graphs and arrow diagrams, testing them for proportionality and using 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.
Unit 4Data and chance
Collecting data, presenting it clearly and analysing it with measures such as the mean.
4.1Collecting, displaying and analysing data
Planning surveys, recording data in tally charts and frequency tables, and drawing and reading bar charts.
Reading charts critically
Show the same data as bar, line and pie charts and more, and discover how charts can mislead.
4.2Statistical measures
Calculating and interpreting the mean, together with mode, median and range.
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.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Fachlehrplan, Gymnasium, Mathematics, grade 5/6