Цели обучения
- Recognise, explain, and apply “Sinus”.
- Recognise, explain, and apply “Kosinus”.
- Recognise, explain, and apply “Tangens”.
- Recognise, explain, and apply “Beliebige Dreiecke in school scope”.
- Recognise, explain, and apply “Körper”.
German Gymnasium Klasse 10 / Mathematics Thüringen / Curriculum
Geometrie
Geometrie: structured theory, worked examples, answered practice, and a mastery checklist for German Gymnasium Klasse 10.
Ключевые идеи и проверки
- I select a theorem from the diagram's givens.
- I justify every step of the solution.
- Every solution or explanation for “Geometrie” should include a method, justification, and final check.
Раздел
Основная теория
Ключевые идеи главы в понятной последовательности перед практикой.
Diagram, givens, target
Mark every given on the diagram and state the target clearly. Start from a relationship that connects exactly those quantities.
Property with justification
Every geometric step needs a property, theorem, or definition. A result without justification is not a complete proof.
Units and precision
Length, area, volume, and angle use different units. Round only at the end and keep sufficient intermediate precision.
Sinus
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Kosinus
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Tangens
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Beliebige Dreiecke in school scope
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Körper
Draw and label every given before choosing a property or formula. Finish with estimation, an inverse operation, or substitution as a check.
Mathematics Thüringen
Разобранные примеры
Следуйте методу шаг за шагом и проверяйте обоснование каждого шага.
Worked geometry example
A right triangle has perpendicular sides 9 cm and 12 cm. Find its hypotenuse and area.
- 1Pythagoras: c² = 9² + 12².
- 2c = √(225) = 15 cm.
- 3Area = 1/2 · 9 · 12.
Hypotenuse 15 cm, area 54 cm².
Reasoning example
Before calculating, explain the key idea from “Sinus” and which conditions must be checked.
- 1Define the idea in one clear sentence.
- 2Connect it to a representation, law, or formula.
- 3State a restriction, unit, or final check that makes the solution valid.
The answer should show not only which rule is used for “Sinus”, but also why it is valid here.
Geometrie
Практика с ответами
Восемь заданий от базовых навыков до экзаменационного уровня. Сначала решите самостоятельно.
Базовый навык4 минут
A 6 cm by 6 cm rectangle: find perimeter and area.
Подсказка
P = 2(l+w), A = lw.
Ответ
P = 24 cm, A = 36 cm².
Базовый навык5 минут
A triangle has base 27 cm and height 3 cm. Find its area.
Подсказка
A = 1/2 bh.
Ответ
A = 1/2 · 27 · 3 = 40.5 cm².
Применение6 минут
Two angles in a triangle are 50° and 50°. Find the third.
Подсказка
Triangle angles sum to 180°.
Ответ
80°.
Применение7 минут
A right triangle has legs 6 cm and 8 cm. Find the hypotenuse.
Подсказка
Use a²+b²=c².
Ответ
10 cm.
Обоснование8 минут
A circle has radius 4 cm. Find circumference and area in terms of π.
Подсказка
C = 2πr, A = πr².
Ответ
C = 8π cm, A = 16π cm².
Обоснование9 минут
Scale 1:700. What real length in metres does 6 cm represent?
Подсказка
Multiply in cm, then convert to metres.
Ответ
6 · 700 = 4200 cm = 42 m.
Экзаменационный10 минут
A cuboid is 3 cm × 3 cm × 6 cm. Find its volume.
Подсказка
V = lwh.
Ответ
V = 54 cm³.
Экзаменационный12 минут
Points A(0,0) and B(6,5). Find AB.
Подсказка
AB = √(Δx²+Δy²).
Ответ
AB = √61.
Частые ошибки
- Using a formula without checking its conditions.
- Confusing linear, square, and cubic units.
- Treating a not-to-scale diagram as a measurement.
Проверка понимания
- I select a theorem from the diagram's givens.
- I justify every step of the solution.
- I use correct units and a final check.
Geometrie
Домашняя работа по главе
Шесть разных заданий от базового уровня до сложного с временем, подсказками и ответами.
Базовый8 минут
Homework 1: A 9 cm by 2 cm rectangle: find perimeter and area.
Подсказка
P = 2(l+w), A = lw.
Ответ
P = 22 cm, A = 18 cm².
Базовый10 минут
Homework 2: A triangle has base 20 cm and height 4 cm. Find its area.
Подсказка
A = 1/2 bh.
Ответ
A = 1/2 · 20 · 4 = 40 cm².
Углубленный12 минут
Homework 3: Two angles in a triangle are 80° and 60°. Find the third.
Подсказка
Triangle angles sum to 180°.
Ответ
40°.
Углубленный15 минут
Homework 4: A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.
Подсказка
Use a²+b²=c².
Ответ
15 cm.
Сложный18 минут
Homework 5: A circle has radius 5 cm. Find circumference and area in terms of π.
Подсказка
C = 2πr, A = πr².
Ответ
C = 10π cm, A = 25π cm².
Сложный20 минут
Homework 6: Scale 1:300. What real length in metres does 2 cm represent?
Подсказка
Multiply in cm, then convert to metres.
Ответ
2 · 300 = 600 cm = 6 m.
50 минут / 40 баллов
Полный тест по главе
Полный тест с таймером, баллами и образцами ответов.
Start the timer when ready, work without notes, show every step, and open model answers only after finishing.
1. A 7 cm by 2 cm rectangle: find perimeter and area.
2 балловОтвет
P = 18 cm, A = 14 cm².
2. A triangle has base 12 cm and height 4 cm. Find its area.
3 балловОтвет
A = 1/2 · 12 · 4 = 24 cm².
3. Two angles in a triangle are 60° and 60°. Find the third.
3 балловОтвет
60°.
4. A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.
4 балловОтвет
15 cm.
5. A circle has radius 5 cm. Find circumference and area in terms of π.
4 балловОтвет
C = 10π cm, A = 25π cm².
6. Scale 1:800. What real length in metres does 2 cm represent?
4 балловОтвет
2 · 800 = 1600 cm = 16 m.
7. A cuboid is 4 cm × 4 cm × 8 cm. Find its volume.
5 балловОтвет
V = 128 cm³.
8. Points A(0,0) and B(7,6). Find AB.
5 балловОтвет
AB = √85.
9. Similar shapes have length scale factor 1:3. If the small area is 3 cm², find the large area.
5 балловОтвет
3 · 3² = 27 cm².
10. Explain which givens and theorem are needed for a complete argument about “Sinus”.
5 балловОтвет
Full response: a labelled diagram, givens, an applicable theorem with its conditions, and a logically derived conclusion.
Раздел
Официальные источники
Источники программы; порядок тем уточняйте в школе и у преподавателя.
Mathematics Thüringen
German Gymnasium Klasse 10: структурированный маршрут по математике и физике с главами, практикой и следующими шагами.
Что покрывает глава
Структура следует учебнику и помогает организовать изучение темы.
Sinus
Kosinus
Tangens
Beliebige Dreiecke in school scope
Körper
Oberfläche
Volumen
Ähnlichkeit
Maßstab
Где сфокусироваться
Темы, где чаще возникают ошибки или нужна дополнительная практика.
I select a theorem from the diagram's givens.
I justify every step of the solution.
I use correct units and a final check.
Источники, ежедневные материалы и ресурсы
С чего начать: учебник, daily material, PDFs, videos и worked examples.
Start from the official textbook or specification referenced on the subject page.
Use the notes and examples as support, not as a replacement for the official syllabus.
Check the current syllabus version before exam preparation.
Практика по подтемам
Целевая практика перед full tests, чтобы coverage было понятным.
A 6 cm by 6 cm rectangle: find perimeter and area.
A triangle has base 27 cm and height 3 cm. Find its area.
Two angles in a triangle are 50° and 50°. Find the third.
A right triangle has legs 6 cm and 8 cm. Find the hypotenuse.
A circle has radius 4 cm. Find circumference and area in terms of π.
Scale 1:700. What real length in metres does 6 cm represent?
A cuboid is 3 cm × 3 cm × 6 cm. Find its volume.
Points A(0,0) and B(6,5). Find AB.
Mocks и проверка прогресса
Как измерять прогресс по главе и когда включать ее в общий mock.
Start with untimed practice by subtopic.
Move to a short timed checkpoint only after completing the mastery checklist.
Record each error with the correct method and revisit it after 48 hours.
Следующий шаг
Что делать после главы и как она связана со следующим разделом.
Complete the practice without support.
Explain the core method aloud in under two minutes.
Continue to the next chapter or request targeted tutor support.
Примечание: официальную экзаменационную программу каждого года всегда уточняйте в школе, у преподавателя и по текущим официальным источникам.