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.
Mathematics – grade 5/6
Double grade 5/6: the core curriculum describes learning areas for grades 5 and 6 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 1Working with natural numbers
Calculating confidently mentally and in writing, using rules of arithmetic, representing numbers on the number line and in the place-value chart, divisors, multiples and primes, rounding and estimating.
1.1Calculating with natural numbers
The four operations and their inverses, rules of arithmetic for clever calculation.
1.2Representing and ordering natural numbers
Number line, place-value system and pictures; multiples, divisors and primes.
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.
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.
1.3Rounding and estimating
Rounding, estimation and plausibility checks.
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.
Unit 2Solids and figures
Describing and making solids and figures, parallel and perpendicular lines, oblique drawings and nets, lengths, areas and volumes, and angles and angle sums.
2.1Exploring shapes in space and the plane
Describing basic solids and figures and finding them in the environment.
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.
2.2Parallel and perpendicular lines
Recognising and drawing parallel and perpendicular lines.
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.
2.3Representing solids
Oblique drawings and models of cubes and cuboids, nets.
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.
2.4Finding lengths, areas and volumes
Comparing, estimating and calculating, developing formulas.
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.
2.5Exploring angles
Estimating, measuring and drawing angles; adjacent, vertical and corresponding 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.
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.
2.6Angle sum theorem
Justifying and using the interior angle sum in triangles and quadrilaterals.
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 3Working with fractions
Fractions in everyday life as parts, scales, percentages and ratios, representing, comparing, simplifying and expanding fractions and calculating with simple fractions.
3.1Exploring fractions in everyday life
Parts of a whole, scales, percentages and ratios.
Representing fractions
Pizza, cake or a line: share it out fairly first, then colour some pieces. That is exactly what a fraction means.
Scale: map and reality
Measure on a map with a ruler and use the scale to work out the real distance.
Percentage bar
Discounts, VAT, surveys: one single bar holds the base value, the percentage and the part, one above the other.
3.2Using representations of fractions
Pictorial, verbal, geometric and algebraic; comparing, simplifying and expanding.
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.
3.3Calculating with fractions
The four operations with simple fractions and simple equations by working backwards.
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 4Planning and carrying out statistical surveys
Planning a survey, observation or experiment, stating hypotheses, recording data in tally charts and tables and checking the hypotheses.
4.1Survey or observation
Defining the characteristic and sample, keeping tally charts, checking hypotheses.
No separate simulation is planned for this.
4.2Experiment
Planning the procedure, recording data in tables, checking hypotheses.
No separate simulation is planned for this.
Unit 5Working with decimals
Representing decimals on the number line, in the place-value chart and as fractions, calculating with them, rounding and converting quantities.
5.1Representing decimals
Number line, place-value system and fraction form.
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.
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.
5.2Calculating with decimals
The four operations, also linked to calculating with fractions.
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.
5.3Rounding and estimating
Approximations and estimates.
No separate simulation is planned for this.
5.4Converting quantities
Time, mass, money, length, area and volume.
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 6Symmetry
Line, point, rotational and plane symmetry, translating, reflecting and rotating, classifying triangles and quadrilaterals by symmetry, and the circle, perpendicular bisector and angle bisector as loci.
6.1Describing symmetry
Plane, line, point and rotational symmetry, also in space.
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.
6.2Translations, reflections and rotations
Carrying out transformations in the plane and creating patterns.
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.
6.3Classifying triangles and quadrilaterals by symmetry
Isosceles and equilateral triangles, the hierarchy of quadrilaterals.
The family of quadrilaterals
Move one corner and the square becomes a kite. When does a quadrilateral get which name?
6.4Describing and using circles
The circle as a locus, perpendicular bisectors and angle bisectors as axes of symmetry.
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 7Measures of statistical surveys
Showing frequencies in bar and pie charts and reading them critically, relative frequency, mean, mode and range.
7.1Graphing frequency distributions
Making and judging bar charts, reading pie charts.
Reading charts critically
Show the same data as bar, line and pie charts and more, and discover how charts can mislead.
7.2Comparing two frequency distributions
Relative frequency, mean, mode 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): Kerncurriculum, Gymnasium, Mathematics, grade 5/6