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.
Measures, area and volume
Understand units, area, volume and surface area: decompose, complete, tile and rotate solids.
Link to the German educational standards (KMK)
- L2Quantities and measurement
- L4Space and shape
- K3Modelling mathematically
Measures and units
Grade 5–9
Scale: map and reality
Measure on a map with a ruler and use the scale to work out the real distance.
Area
Grade 5–8
Perimeter and area of rectangles
Count the squares or walk once around? An ant makes the difference between area and perimeter tangible.
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.
Solids and surface area
Grade 5–10
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.
Prism and cylinder: surface area and volume
Layer by layer a prism grows upwards, and the curved surface of a cylinder unrolls into a rectangle.
Pyramid, cone and sphere
Pour three times and the prism is full: a pyramid holds exactly one third. Cones and spheres can be turned and measured in 3D.
Cavalieri’s principle
Push a stack of coins sideways: the shape changes, but the volume stays the same. This is Cavalieri's principle.
Volume
Grade 5–6
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.
Circle
Grade 7–9
Circumference, area of a circle and the number π
A wheel rolls once and travels a bit more than three times its diameter. Cut into pieces, the circle becomes a rectangle and the area formula follows.