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 (nine-year Gymnasium): the curriculum covers both grades together, so the units are the same for grades 5 and 6 and are shown as one stage. The order of the topic fields is binding only where the subject requires it. The numbers are those of the curriculum.
Unit 1Natural numbers
Data and charts, the decimal system with place value table and rounding, the binary system and Roman numerals, sets, the number line and order.
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.
Reading charts critically
Show the same data as bar, line and pie charts and more, and discover how charts can mislead.
Unit 2Calculating with natural numbers
The four operations and their terms, laws of arithmetic, terms with brackets, powers, figurate numbers.
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.
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 3Quantities in everyday life
Quantities with value and unit, converting between units, points in time and time spans.
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 4Figures and solids
Point, segment, ray and line, parallel and perpendicular, the coordinate system, quadrilaterals, solids with nets and oblique drawings.
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.
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 5Area and volume
Perimeter and area of rectangle and square, units of area and volume, volume and surface area of cuboid and cube.
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 6Divisibility of natural numbers
Divisors and multiples, divisibility rules, prime numbers and prime factorisation, gcd and lcm.
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.
Unit 7Fractions
Fractions as parts, fractions on the number line, simplifying and expanding, comparing, decimals and recurring decimals.
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.
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 8Calculating with fractions
The four operations with fractions and decimals, reciprocals and complex fractions, moving the decimal point.
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 9Circle, angle, symmetry
The circle with radius and diameter, measuring and drawing angles, line, point and rotational symmetry, reflection, rotation and translation, the base angle theorem.
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.
Unit 10Statistical data
Absolute and relative frequency, pie charts, minimum, maximum, range, median and quartiles, box plots.
Reading charts critically
Show the same data as bar, line and pie charts and more, and discover how charts can mislead.
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.
Unit 11Integers
Negative numbers in everyday life, the number line and signs, the coordinate system with four quadrants, calculating with integers.
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.
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.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Lehrplan, Gymnasium, Mathematics, grade 5/6