Abitur / Mathematics / Curriculum
Analytische Geometrie / Lineare Algebra: structured theory, worked examples, answered practice, and a mastery checklist for Abitur.
CHAPTER PLAN
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Estimated active study time
209 minutes
Objectives, key ideas, and structured theory.
Open →Worked examples that explain every step.
Open →Graded tasks with hints and answer guidance.
Open →Assignments from core work to challenge.
Open →A timed, marked test with solution guidance.
Open →Progress is stored only in this browser and does not require AI credits.
Unit
The essential chapter ideas in a clear sequence before practice.
Mark every given on the diagram and state the target clearly. Start from a relationship that connects exactly those quantities.
Every geometric step needs a property, theorem, or definition. A result without justification is not a complete proof.
Length, area, volume, and angle use different units. Round only at the end and keep sufficient intermediate precision.
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Mathematics
Follow the method step by step and check why every step is valid.
A right triangle has perpendicular sides 6 cm and 8 cm. Find its hypotenuse and area.
Hypotenuse 10 cm, area 24 cm².
Before calculating, explain the key idea from “Vector operations” and which conditions must be checked.
The answer should show not only which rule is used for “Vector operations”, but also why it is valid here.
Analytische Geometrie / Lineare Algebra
Eight graded tasks from core fluency to exam-style application. Work independently before opening a hint or answer.
A 6 cm by 2 cm rectangle: find perimeter and area.
P = 2(l+w), A = lw.
P = 16 cm, A = 12 cm².
A triangle has base 36 cm and height 4 cm. Find its area.
A = 1/2 bh.
A = 1/2 · 36 · 4 = 72 cm².
Two angles in a triangle are 50° and 60°. Find the third.
Triangle angles sum to 180°.
70°.
A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.
Use a²+b²=c².
15 cm.
A circle has radius 5 cm. Find circumference and area in terms of π.
C = 2πr, A = πr².
C = 10π cm, A = 25π cm².
Scale 1:700. What real length in metres does 2 cm represent?
Multiply in cm, then convert to metres.
2 · 700 = 1400 cm = 14 m.
A cuboid is 3 cm × 4 cm × 7 cm. Find its volume.
V = lwh.
V = 84 cm³.
Points A(0,0) and B(6,6). Find AB.
AB = √(Δx²+Δy²).
AB = √72.
Analytische Geometrie / Lineare Algebra
Six distinct assignments from core fluency to challenge, each with an estimated time, hint, and answer guide.
Homework 1: A 9 cm by 3 cm rectangle: find perimeter and area.
P = 2(l+w), A = lw.
P = 24 cm, A = 27 cm².
Homework 2: A triangle has base 25 cm and height 5 cm. Find its area.
A = 1/2 bh.
A = 1/2 · 25 · 5 = 62.5 cm².
Homework 3: Two angles in a triangle are 80° and 20°. Find the third.
Triangle angles sum to 180°.
80°.
Homework 4: A right triangle has legs 12 cm and 16 cm. Find the hypotenuse.
Use a²+b²=c².
20 cm.
Homework 5: A circle has radius 6 cm. Find circumference and area in terms of π.
C = 2πr, A = πr².
C = 12π cm, A = 36π cm².
Homework 6: Scale 1:300. What real length in metres does 3 cm represent?
Multiply in cm, then convert to metres.
3 · 300 = 900 cm = 9 m.
50 minutes / 40 marks
A timed, full-mark self-assessment with model-answer guidance.
Start the timer when ready, work without notes, show every step, and open model answers only after finishing.
1. A 7 cm by 3 cm rectangle: find perimeter and area.
2 marksP = 20 cm, A = 21 cm².
2. A triangle has base 15 cm and height 5 cm. Find its area.
3 marksA = 1/2 · 15 · 5 = 37.5 cm².
3. Two angles in a triangle are 60° and 20°. Find the third.
3 marks100°.
4. A right triangle has legs 12 cm and 16 cm. Find the hypotenuse.
4 marks20 cm.
5. A circle has radius 6 cm. Find circumference and area in terms of π.
4 marksC = 12π cm, A = 36π cm².
6. Scale 1:800. What real length in metres does 3 cm represent?
4 marks3 · 800 = 2400 cm = 24 m.
7. A cuboid is 4 cm × 5 cm × 9 cm. Find its volume.
5 marksV = 180 cm³.
8. Points A(0,0) and B(7,2). Find AB.
5 marksAB = √53.
9. Similar shapes have length scale factor 1:4. If the small area is 3 cm², find the large area.
5 marks3 · 4² = 48 cm².
10. Explain which givens and theorem are needed for a complete argument about “Intersections”.
5 marksFull response: a labelled diagram, givens, an applicable theorem with its conditions, and a logically derived conclusion.
Unit
Curriculum reference sources. Always confirm the teaching sequence with the school and tutor.
Mathematics
Keep each unit in GR/DE: concept, German term, example, exercise, recap.
The structure follows the official textbook layout and is used to organise study.
The areas that usually create mistakes or need extra revision.
Where to start: textbook, daily material, PDFs, videos, and worked examples.
Targeted practice before full tests so coverage is clear.
How to measure progress in this chapter and when it enters a cumulative mock.
What to do after finishing the chapter and how it connects to the next unit.
Note: for the official examinable syllabus of each school year, always confirm with the school, tutor, and current Ministry/IEP announcements.
