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.
MathematicsHesse curriculum · Gymnasium
Mathematics – grade 5/6
Double grade 5/6: the core curriculum describes what should be achieved by the end of grade 6, so the units are the same for grades 5 and 6. Both grades are therefore shown as one stage. The numbers (guiding idea.content area.focus) are our own.
Unit 1.1Numbers
Building solid number sense: natural numbers, first negative numbers, fractions in their different meanings and decimals. Numbers are compared, ordered, rounded and examined for divisibility.
1.1.1Natural numbers
Reading, writing and understanding large numbers in the place-value system.
1.1.2Building an understanding of negative numbers
First ideas of numbers below zero, e.g. temperature or account balance.
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.
1.1.3Fractions as part of a whole, as part of several wholes, as a measure and to describe ratios
Interpreting and representing fractions in their different basic meanings.
Representing fractions
Pizza, cake or a line: share it out fairly first, then colour some pieces. That is exactly what a fraction means.
1.1.4Decimals (terminating, repeating) and reasons for termination or periodicity
Converting fractions to decimals and explaining when the expansion terminates or repeats.
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.
1.1.5Simple percentages
Understanding parts as percentages and interpreting simple percentage statements.
Percentage bar
Discounts, VAT, surveys: one single bar holds the base value, the percentage and the part, one above the other.
1.1.6Comparing and ordering natural numbers and fractions (common fractions, decimals)
Ordering numbers by size, also by expanding and simplifying fractions.
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.
1.1.7Rounding natural numbers and decimals
Rounding numbers sensibly to a given place.
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.
1.1.8Divisibility, divisors and multiples (gcd, lcm, prime numbers)
Divisibility rules, prime numbers, greatest common divisor and least common multiple.
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.
1.1.9Representations (number ray, pie chart)
Visualising numbers and parts on the number ray and in pie charts.
No separate simulation is planned for this.
Unit 1.2Operations and their properties
Confident calculation with natural numbers and fractions using the laws of arithmetic, plus first basic tasks with fractions and percentages.
1.2.1Basic operations and arithmetic laws for natural numbers and fractions
Carrying out the four basic operations with whole numbers and fractions and using laws such as commutativity and distributivity.
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.
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.
1.2.2Strategies for efficient calculation
Rearranging and grouping calculations cleverly to make them easier.
No separate simulation is planned for this.
1.2.3Basic tasks in fraction and percentage calculation
Finding part, whole and rate for fractions and percentages.
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.
Percentage bar
Discounts, VAT, surveys: one single bar holds the base value, the percentage and the part, one above the other.
Unit 2.1Plane figures
Recognising, constructing and examining basic figures such as square, rectangle, triangle and circle for symmetry; first orientation in the coordinate system.
2.1.1Basic figures (square, rectangle, triangle, circle) and composite shapes
Naming and describing basic figures and combining them into new shapes.
Perimeter and area of rectangles
Count the squares or walk once around? An ant makes the difference between area and perimeter tangible.
2.1.2Constructing figures and patterns
Drawing figures and patterns with ruler, set square and compass.
No separate simulation is planned for this.
2.1.3Symmetry properties (line and rotational symmetry) of basic figures
Recognising and creating lines of symmetry and rotational symmetry in basic figures.
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.
2.1.4Cartesian coordinate system in the first quadrant
Plotting and reading points with positive coordinates.
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 2.2Solids
Cuboids, cubes and composite solids are explored using models, oblique drawings and nets; volume and surface area are described.
2.2.1Basic solids (cuboid, cube) and composite solids
Recognising cuboids and cubes and describing composite solids.
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.
2.2.2Describing volume and surface area of the basic solids
Capturing volume with unit cubes and surface area via the faces.
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.
2.2.3Models, oblique drawings and nets of the basic solids
Building solids, drawing oblique views and folding or identifying 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.
Unit 2.3Relationships between geometric objects
Positional relationships such as parallel and perpendicular, and mappings of figures (rotation, reflection, translation) are described.
2.3.1Technical terms parallel, perpendicular, distance
Recognising and drawing parallel and perpendicular lines and measuring distances.
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.
2.3.2Movements of figures: rotations, reflections, translations
Rotating, reflecting and translating figures and comparing the results.
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 3.1Working with quantities
Developing a feel for quantities, estimating and approximating, and converting units with prefixes.
3.1.1Sense of quantities
Knowing typical benchmarks for lengths, masses, times etc.
No separate simulation is planned for this.
3.1.2Unit square, unit cube
Measuring areas and volumes by covering/filling with unit squares or cubes.
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.
3.1.3Benchmarks, estimates and rough calculations
Estimating quantities with reasons and checking results by rough calculation.
No separate simulation is planned for this.
3.1.4Rounding
Rounding quantities appropriately for the situation.
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.
3.1.5Converting quantities
Converting between units of one quantity.
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.
3.1.6Unit prefixes
Understanding and using prefixes such as milli-, centi-, kilo-.
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.
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 3.2Measuring processes
Measuring and calculating length, mass, money, time and angle as well as perimeter, area and volume of simple figures and solids.
3.2.1Length
Measuring and converting lengths, e.g. on maps with a 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.
3.2.2Mass/weights
Measuring masses and stating them in suitable units.
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.
3.2.3Currency/money
Calculating with amounts of money in euros and cents.
No separate simulation is planned for this.
3.2.4Time span
Distinguishing and calculating points in time and time spans.
No separate simulation is planned for this.
3.2.5Angle
Estimating, measuring and drawing 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.
3.2.6Area and perimeter of square and rectangle
Finding perimeter and area of rectangles and squares.
Perimeter and area of rectangles
Count the squares or walk once around? An ant makes the difference between area and perimeter tangible.
3.2.7Volume and surface area of the basic solids
Calculating volume and surface area of cuboids and cubes.
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.
Unit 4.1Assignments (mappings) and their representations
First experiences with assignments between quantities and representing them in words, tables and charts.
4.1.1Basic ideas of assigning quantities
Understanding that one quantity is assigned to another (e.g. amount → price).
No separate simulation is planned for this.
4.1.2Representing assignments in charts, tables and in words
Describing assignments in tables, charts and in words.
No separate simulation is planned for this.
Unit 5.1Statistical surveys and their evaluation
Planning and evaluating own surveys, presenting data in lists and diagrams and describing them with simple statistics.
5.1.1Surveys and data collection (planning, carrying out and statistical evaluation)
Carrying out a data collection from the question to the evaluation.
No separate simulation is planned for this.
5.1.2Statistics (frequencies, median, arithmetic mean, range)
Determining and interpreting frequencies, mean, 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.
5.1.3Presenting data (lists and diagrams)
Presenting and reading data in tally charts, tables and diagrams.
Reading charts critically
Show the same data as bar, line and pie charts and more, and discover how charts can mislead.
Unit 5.2Dealing with chance
Recognising chance in everyday life and building first ideas of probability via frequencies.
5.2.1Random phenomena in everyday situations
Recognising and describing random events in everyday life.
No separate simulation is planned for this.
5.2.2Different notions of probability
Getting to know probability as expectation (symmetry) and as long-run relative frequency.
Law of large numbers
Toss a coin, a die or a drawing pin a thousand times: the more often you throw, the closer the relative frequency gets to the probability.
5.2.3Absolute and relative frequencies
Determining frequencies from experiments and stating them as proportions.
Law of large numbers
Toss a coin, a die or a drawing pin a thousand times: the more often you throw, the closer the relative frequency gets to the probability.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Kerncurriculum, Gymnasium, Mathematics, grade 5/6