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
Grade 5: the framework curriculum for grades 1–10 is organised by levels A–H, not by grades; schools that start in grade 5 work at level C–D in grade 5. In Berlin the Gymnasium usually starts in grade 7; some start in grade 5. Units are the subject areas, sub-units their topics; the numbers are our own.
Unit 1Numbers and operations (levels C–D)
1.1Understanding numbers
Large natural numbers in the place-value chart, divisibility rules, primes, multiples and divisors; fractions as parts, expanding, simplifying, comparing and ordering them on the number line; decimals.
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.
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.
1.2Understanding operations and calculation strategies
Understanding and carrying out the four operations with fractions, fractions as operators, order of operations and brackets, written methods and estimating.
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 2Quantities and measurement (levels C–D)
2.1Understanding quantities and measuring
Distinguishing, measuring and estimating lengths, areas, volumes and angles, converting units and giving quantities as decimals.
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.
2.2Calculating with quantities
Calculating the perimeter of polygons, the area of rectangles and the volume of cuboids, also composite ones; quantities in word problems.
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 3Space and shape (levels C–D)
3.1Geometric objects
Describing solids, angles and triangles, quadrilaterals and how they relate (“house of quadrilaterals”), angles at intersecting lines, nets, drawing in the coordinate system.
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?
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.
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.
3.2Geometric transformations
Recognising congruent figures, drawing reflections and translations, making tessellations.
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 4Equations and functions (levels C–D)
4.1Terms and equations
Representing situations by numerical terms and equations, variables as placeholders, using the laws of arithmetic and solving simple equations with the inverse operation.
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 5Data and chance (levels C–D)
5.1Data
Recording measurements and showing them in tables and charts, finding minimum, maximum and range.
Matching simulations are listed under Grade 6, Grade 7, Grade 8, Grade 9
5.2Counting strategies and probabilities
Solving counting problems systematically, giving relative frequencies of single-stage experiments as fractions.
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.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Rahmenlehrplan, Gymnasium, Mathematics, grade 5