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.
MathematicsRhineland-Palatinate curriculum · Gymnasium
Mathematics – grade 5/6
Orientation stage (grades 5 and 6): the framework curriculum describes both grades together and in no fixed order, so the units are the same for grades 5 and 6 and are shown as one stage. Units are the content areas of the key ideas L1–L5, sub-units their sections; the numbers are our own. Optional extras marked (+) are not compulsory.
Unit 1Number and number sets: natural numbers
Extending the number range beyond one million, mental, written and estimated calculation, structuring terms and examining divisibility.
1.1Representing natural numbers
Reading and writing large numbers, explaining the place-value system, ordering, rounding and showing numbers on the number line; the binary system as an optional extra.
1.2Calculating with natural numbers
Mental and written methods, simple powers, estimating, structuring terms with calculation trees, solving simple equations and using the laws of arithmetic.
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.
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.
1.3Divisibility
Applying divisibility rules for 2, 3, 4, 5, 9 and 10 and meeting prime numbers as the building blocks of natural 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.
Unit 2Number and number sets: integers
Interpreting negative numbers from everyday life, ordering them on the number line and calculating changes of state.
2.1Representing integers
Interpreting signed numbers in contexts such as temperature or bank balance, ordering them on the number line and plotting them in the coordinate system.
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.
2.2Calculating with integers
Solving word problems as a change from a starting state to an end state, evaluating simple terms and solving equations; the second operand stays positive.
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.
Unit 3Number and number sets: fractions
Building a sound concept of fractions, writing them as fractions, decimals and percentages, and calculating with positive fractions.
3.1Representing fractions
Interpreting fractions as measures and operators, calculating parts, writing them as fractions, decimals and percentages, expanding, simplifying, comparing and ordering.
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.
Percentage bar
Discounts, VAT, surveys: one single bar holds the base value, the percentage and the part, one above the other.
3.2Calculating with fractions
Adding, subtracting, multiplying and dividing fractions and decimals, reading the fraction operator as a factor and applying it in real contexts.
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 4Measurement and quantities
Measuring as comparing with a unit: length, mass, time, angle, area and volume; formulas for rectangle and cuboid; calculating with quantities and scale.
4.1The principle of measuring
Measuring and estimating lengths, masses, time spans, angles, areas and volumes with units; deriving perimeter and area of the rectangle and volume and surface area of the cuboid.
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.
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.
4.2Calculating with quantities
Converting quantities between units, using prefixes and decimal notation, rounding sensibly in real contexts, applying scales and formulas.
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.
Unit 5Space and shape
Recognising and creating symmetric figures, drawing angles and circles, describing solids and quadrilaterals, nets and oblique drawings, coordinate system.
5.1Symmetric figures
Line and rotational symmetry, parallel and perpendicular, lines, segments and distance, estimating, measuring and classifying angles, drawing circles.
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.
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.
5.2Shapes and relations in space and the plane
Recognising cubes, cuboids, pyramids, cylinders, cones, spheres and prisms, describing and drawing quadrilaterals, making nets and oblique drawings, working in the coordinate system.
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.
The family of quadrilaterals
Move one corner and the square becomes a kite. When does a quadrilateral get which name?
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.
Unit 6Functional relationships
Not taught separately in the orientation stage: sequences, terms, the operator model, formulas, scale and data displays prepare the functional view within the other key ideas.
No separate simulation is planned for this.
Unit 7Data and chance
Counting with tree diagrams, and collecting data, analysing it with frequencies and the mean and showing it in charts.
7.1Counting strategies
Developing counting methods in concrete situations such as combination locks and showing them in tree diagrams.
Counting principle and tree diagrams
How many outfits or ice cream combinations are there? A tree diagram grows step by step and shows why you multiply the options.
7.2Data
Planning and analysing surveys, absolute and relative frequency, the arithmetic mean, reading charts and checking them for manipulation.
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): Rahmenlehrplan und Lehrplan, Gymnasium, Mathematics, grade 5/6