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.
MathematicsGrade 5–10
Geometry
Construct and investigate figures, symmetry, angles, triangles, similarity and trigonometry dynamically.
Link to the German educational standards (KMK)
- L4Space and shape
- K1Reasoning mathematically
- K4Using mathematical representations
Basic geometric concepts
Grade 5–7
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?
Symmetry and constructions
Grade 5–7
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.
Perpendicular bisector and angle bisector
A point equally far from A and B never lies there by chance: the perpendicular bisector appears before you draw it with compasses.
Angles in figures
Grade 5–7
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.
Angle sum in triangles and polygons
Tear off three corners, put them side by side, and you get a straight line: 180°, for every triangle. A quadrilateral gives 360°.
Triangles
Grade 6–8
Congruence and constructing triangles
Construct triangles from three given parts with ruler and compass, step by step as in your exercise book. Sometimes there are no, one or two solutions.
Circumcircle, incircle and special lines in a triangle
Construct the special lines in a triangle. Each set of three always meets in one point, for example at the centre of the circumcircle or the incircle.
Thales’s theorem
Drag a point along the semicircle: the angle there is always a right angle.
Pythagorean theorem
Grade 7–9
Pythagorean theorem
Squares on the sides of a right triangle and a puzzle proof show: a² + b² = c².
Similarity and intercept theorems
Grade 8–10
Intercept theorems
How tall is the tree without climbing it? Two rays and a parallel line are enough, because the ratios stay the same.
Similar figures: how do area and volume grow?
Enlarge lengths, areas and solids. Doubling the length makes the area four times and the volume eight times as large.
Trigonometry in triangles
Grade 9–10
Sine, cosine and tangent in right triangles
In a right triangle, the ratio of two sides depends only on the angle. This gives sine, cosine and tangent, with examples such as ramps and ladders.
Law of sines and law of cosines
Calculate sides and angles in any triangle with the law of sines and the law of cosines. A helper shows which law fits your problem.