German Gymnasium Klasse 7 / Mathematics / Curriculum
Geometrie: structured theory, worked examples, answered practice, and a mastery checklist for German Gymnasium Klasse 7.
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学习目标、核心概念与结构化理论。
打开 →逐步说明的完整例题。
打开 →含提示与答案的分级练习。
打开 →从基础到挑战的课后任务。
打开 →含计时、分值与解答的章节测试。
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单元
练习前按清晰顺序掌握本章核心概念。
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
逐步查看方法，并理解每一步为何成立。
A right triangle has perpendicular sides 15 cm and 20 cm. Find its hypotenuse and area.
Hypotenuse 25 cm, area 150 cm².
Before calculating, explain the key idea from “Winkelarten” and which conditions must be checked.
The answer should show not only which rule is used for “Winkelarten”, but also why it is valid here.
Geometrie
八道分层练习，从基础熟练到考试应用。请先独立完成，再查看提示或答案。
A 5 cm by 6 cm rectangle: find perimeter and area.
P = 2(l+w), A = lw.
P = 22 cm, A = 30 cm².
A triangle has base 24 cm and height 3 cm. Find its area.
A = 1/2 bh.
A = 1/2 · 24 · 3 = 36 cm².
Two angles in a triangle are 40° and 50°. Find the third.
Triangle angles sum to 180°.
90°.
A right triangle has legs 6 cm and 8 cm. Find the hypotenuse.
Use a²+b²=c².
10 cm.
A circle has radius 4 cm. Find circumference and area in terms of π.
C = 2πr, A = πr².
C = 8π cm, A = 16π cm².
Scale 1:600. What real length in metres does 6 cm represent?
Multiply in cm, then convert to metres.
6 · 600 = 3600 cm = 36 m.
A cuboid is 9 cm × 3 cm × 12 cm. Find its volume.
V = lwh.
V = 324 cm³.
Points A(0,0) and B(5,5). Find AB.
AB = √(Δx²+Δy²).
AB = √50.
Geometrie
六道不同作业，从基础到挑战，包含预计用时、提示和答案指南。
Homework 1: A 8 cm by 2 cm rectangle: find perimeter and area.
P = 2(l+w), A = lw.
P = 20 cm, A = 16 cm².
Homework 2: A triangle has base 16 cm and height 4 cm. Find its area.
A = 1/2 bh.
A = 1/2 · 16 · 4 = 32 cm².
Homework 3: Two angles in a triangle are 70° and 60°. Find the third.
Triangle angles sum to 180°.
50°.
Homework 4: A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.
Use a²+b²=c².
15 cm.
Homework 5: A circle has radius 5 cm. Find circumference and area in terms of π.
C = 2πr, A = πr².
C = 10π cm, A = 25π cm².
Homework 6: Scale 1:900. What real length in metres does 2 cm represent?
Multiply in cm, then convert to metres.
2 · 900 = 1800 cm = 18 m.
50 分钟 / 40 分
带计时器、完整分值和参考答案的章节自测。
Start the timer when ready, work without notes, show every step, and open model answers only after finishing.
1. A 6 cm by 2 cm rectangle: find perimeter and area.
2 分P = 16 cm, A = 12 cm².
2. A triangle has base 36 cm and height 4 cm. Find its area.
3 分A = 1/2 · 36 · 4 = 72 cm².
3. Two angles in a triangle are 50° and 60°. Find the third.
3 分70°.
4. A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.
4 分15 cm.
5. A circle has radius 5 cm. Find circumference and area in terms of π.
4 分C = 10π cm, A = 25π cm².
6. Scale 1:700. What real length in metres does 2 cm represent?
4 分2 · 700 = 1400 cm = 14 m.
7. A cuboid is 3 cm × 4 cm × 7 cm. Find its volume.
5 分V = 84 cm³.
8. Points A(0,0) and B(6,6). Find AB.
5 分AB = √72.
9. Similar shapes have length scale factor 1:3. If the small area is 9 cm², find the large area.
5 分9 · 3² = 81 cm².
10. Explain which givens and theorem are needed for a complete argument about “Volumen”.
5 分Full response: a labelled diagram, givens, an applicable theorem with its conditions, and a logically derived conclusion.
单元
课程参考来源；教学顺序请始终与学校和老师确认。
Mathematics
German Gymnasium Klasse 7: 数学与物理的结构化路线，包含章节、练习和下一步。
结构跟随教材，用来帮助组织学习。
通常容易出错或需要更多复习的部分。
从教材、daily material、PDFs、videos 和 worked examples 开始。
进入 full tests 前先做 targeted practice，让覆盖情况更清楚。
如何衡量本章进度，以及何时加入综合 mock。
完成本章后该做什么，以及它如何连接到下一单元。
说明：每学年的正式考试范围请始终以学校、老师和当前官方来源为准。
