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
Section 3.1 covers the double grade 5/6; the same content is also listed under grade 6. Simulations are listed only where their grade range fits.
Unit 3.1.1Guiding idea: number – variable – operation
Understanding, representing and comparing natural numbers, integers and rational numbers and calculating with them confidently, including fractions, decimals and first expressions.
3.1.1 (1)–(11)Exploring number sets
Place value, large numbers, divisibility and prime factors; negative numbers and fractions as parts, ordered on the number line and converted to decimals and percentages.
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.
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.
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.
3.1.1 (12)–(22)Calculating with numbers
Mental and written arithmetic, shifting the decimal point, powers and square numbers, simplifying and expanding fractions and using all four operations with them; sensible rounding and checking results.
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.
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.
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.
3.1.1 (23)–(28)Working with numerical expressions
Writing real situations as numerical expressions, describing their structure with technical terms and evaluating them cleverly using brackets and laws of arithmetic; solving simple problems with unknowns by trial.
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 3.1.2Guiding idea: measuring
Understanding measurement as filling with units, estimating and converting quantities, and calculating perimeter, area, surface area and volume of simple figures and solids.
3.1.2 (1)–(8)Working with quantities
Measuring length, area, volume, mass, time and money, converting units with prefixes and estimating with everyday benchmarks; measuring and estimating angles.
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.
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.
3.1.2 (9)–(15)Calculating quantities of figures and solids
Finding perimeter and area of polygons and the circle (with the number π) and justifying the formulas visually; calculating surface area and volume of cubes and cuboids.
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.
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.
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 3.1.3Guiding idea: space and shape
Recognising geometric shapes in the environment and describing them with technical terms; drawing accurately with set square, ruler and compass, including reflections and representations of solids.
3.1.3 (1)–(7)Identifying and describing geometric objects and relations
Recognising parallel and perpendicular lines, types of angles, symmetries, types of triangles and quadrilaterals and basic solids, and naming their properties.
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.
The family of quadrilaterals
Move one corner and the square becomes a kite. When does a quadrilateral get which name?
3.1.3 (8)–(15)Drawing and constructing geometric objects
Drawing circles, perpendiculars, parallels and angles, finding distances and heights, plotting figures in a coordinate system, carrying out line and point reflections and drawing nets and oblique views of solids.
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.
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.
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 3.1.4Guiding idea: functional relationships
Recognising relationships between numbers and quantities and representing them in words, tables and graphs; calculating with direct and inverse proportion and working with scales.
3.1.4 (1)–(8)Describing relationships
Continuing patterns, plotting and reading points in a coordinate system, switching between representations; solving word problems with the rule of three and drawing and calculating to scale.
Scale: map and reality
Measure on a map with a ruler and use the scale to work out the real distance.
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.
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 3.1.5Guiding idea: data and chance
Collecting data, presenting them in tables and charts and analysing them with simple statistics; checking charts critically for misleading presentation.
3.1.5 (1)–(8)Collecting, presenting and analysing data
Planning surveys, finding absolute and relative frequencies, drawing bar, column and pie charts, determining minimum, maximum, range, median and mean, and judging presentations.
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.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Bildungsplan, Gymnasium, Mathematics, grade 5/6