Цели обучения
- Recognise, explain, and apply “Dreisatz”.
- Recognise, explain, and apply “Tabelle”.
- Recognise, explain, and apply “Graph”.
German Grade 6 / Mathematics / Proportionalität und Prozent
Verhältnisse und proportionale Zuordnung
Verhältnisse und proportionale Zuordnung: structured theory, worked examples, answered practice, and a mastery checklist for German Grade 6.
Ключевые идеи и проверки
- I move between formula, table, and graph.
- I interpret slope, intercepts, and key features.
- Every solution or explanation for “Verhältnisse und proportionale Zuordnung” should include a method, justification, and final check.
Раздел
Основная теория
Ключевые идеи главы в понятной последовательности перед практикой.
Input-output relationship
A function maps each allowed input to one output. Formula, table, and graph are different representations of the same relationship.
Parameters and graph
Parameters change slope, position, amplitude, or rate of change. Read the key features before plotting individual points.
Domain and interpretation
The domain states which inputs are allowed. In a real model, also check whether the output is physically meaningful.
Dreisatz
Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.
Tabelle
Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.
Graph
Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.
Mathematics
Разобранные примеры
Следуйте методу шаг за шагом и проверяйте обоснование каждого шага.
Worked arithmetic example
6 equal groups contain 8 objects each. How many objects are there, and how can the answer be checked?
- 1Write repeated addition: 8 + 8 + 8 + 8 + 8 + 8.
- 2Connect it to 6 × 8.
- 3Check with division: 48 ÷ 6 = 8.
48
Thinking and explanation example
Explain “Dreisatz” in your own words and use a picture, objects, a table, or a simple measurement to support it.
- 1State what is known and what is being found.
- 2Choose one clear representation.
- 3Explain what it shows and check that the answer is sensible.
A complete explanation connects the correct idea, a clear representation, and a sensible check.
Verhältnisse und proportionale Zuordnung
Практика с ответами
Восемь заданий от базовых навыков до экзаменационного уровня. Сначала решите самостоятельно.
Базовый навык3 минут
Calculate 5 + 6 and show two representations connected with “Dreisatz”.
Подсказка
Use a number line, place value, or objects.
Ответ
5 + 6 = 11. Valid representations include number-line jumps and grouped objects.
Базовый навык4 минут
3 equal groups contain 8 objects each. Find the total and write an addition and multiplication statement.
Подсказка
Add 8 a total of 3 times.
Ответ
8 + 8 + 8 = 3 · 8 = 24.
Применение4 минут
Continue 4, 9, 14, ... for three more terms and explain the rule.
Подсказка
Check how much each term increases.
Ответ
19, 24, 29; +5.
Применение5 минут
A rectangle has sides 7 cm and 2 cm. Find its perimeter and explain why four sides are counted.
Подсказка
Opposite sides are equal.
Ответ
Perimeter = 7 + 2 + 7 + 2 = 18 cm.
Обоснование5 минут
Out of 12 objects, 3 are shaded. What fraction is shaded and how does it simplify?
Подсказка
Put shaded objects over the total.
Ответ
3/12 = 1/4.
Обоснование6 минут
The measurements are 6, 6, 6, and 6. Find the greatest, least, and range.
Подсказка
Range = greatest value minus least value.
Ответ
Greatest 6, least 6, range 0.
Экзаменационный7 минут
Anna has 27 cards, gives away 9, then shares the rest into 2 equal groups. How many are in each group?
Подсказка
Subtract first, then divide.
Ответ
27 - 9 = 18; (18) : 2 = 9.
Экзаменационный8 минут
A learner says 5 + 5 = 11. Find the error, correct it, and check the correction.
Подсказка
Check with a number line or inverse subtraction.
Ответ
5 + 5 = 10; 10 - 5 = 5.
Частые ошибки
- Confusing f(x) with the product f times x.
- Plotting without a scale or key points.
- Ignoring domain and model restrictions.
Проверка понимания
- I move between formula, table, and graph.
- I interpret slope, intercepts, and key features.
- I state domain and relevant restrictions.
Verhältnisse und proportionale Zuordnung
Домашняя работа по главе
Шесть разных заданий от базового уровня до сложного с временем, подсказками и ответами.
Базовый5 минут
Homework 1: Calculate 8 + 2 and show two representations connected with “Dreisatz”.
Подсказка
Use a number line, place value, or objects.
Ответ
8 + 2 = 10. Valid representations include number-line jumps and grouped objects.
Базовый6 минут
Homework 2: 4 equal groups contain 4 objects each. Find the total and write an addition and multiplication statement.
Подсказка
Add 4 a total of 4 times.
Ответ
4 + 4 + 4 + 4 = 4 · 4 = 16.
Углубленный7 минут
Homework 3: Continue 7, 13, 19, ... for three more terms and explain the rule.
Подсказка
Check how much each term increases.
Ответ
25, 31, 37; +6.
Углубленный8 минут
Homework 4: A rectangle has sides 3 cm and 3 cm. Find its perimeter and explain why four sides are counted.
Подсказка
Opposite sides are equal.
Ответ
Perimeter = 3 + 3 + 3 + 3 = 12 cm.
Сложный10 минут
Homework 5: Out of 30 objects, 6 are shaded. What fraction is shaded and how does it simplify?
Подсказка
Put shaded objects over the total.
Ответ
6/30 = 1/5.
Сложный12 минут
Homework 6: The measurements are 9, 2, 9, and 2. Find the greatest, least, and range.
Подсказка
Range = greatest value minus least value.
Ответ
Greatest 9, least 2, range 7.
35 минут / 30 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. Calculate 6 + 2 and show two representations connected with “Dreisatz”.
2 балловОтвет
6 + 2 = 8. Valid representations include number-line jumps and grouped objects.
2. 4 equal groups contain 9 objects each. Find the total and write an addition and multiplication statement.
3 балловОтвет
9 + 9 + 9 + 9 = 4 · 9 = 36.
3. Continue 5, 11, 17, ... for three more terms and explain the rule.
3 балловОтвет
23, 29, 35; +6.
4. A rectangle has sides 8 cm and 3 cm. Find its perimeter and explain why four sides are counted.
4 балловОтвет
Perimeter = 8 + 3 + 8 + 3 = 22 cm.
5. Out of 20 objects, 4 are shaded. What fraction is shaded and how does it simplify?
4 балловОтвет
4/20 = 1/5.
6. The measurements are 7, 2, 7, and 2. Find the greatest, least, and range.
4 балловОтвет
Greatest 7, least 2, range 5.
7. Anna has 12 cards, gives away 3, then shares the rest into 3 equal groups. How many are in each group?
5 балловОтвет
12 - 3 = 9; (9) : 3 = 3.
8. A learner says 6 + 6 = 13. Find the error, correct it, and check the correction.
5 балловОтвет
6 + 6 = 12; 12 - 6 = 6.
Раздел
Официальные источники
Источники программы; порядок тем уточняйте в школе и у преподавателя.
Mathematics
German Grade 6: структурированный маршрут по математике и физике с главами, практикой и следующими шагами.
Что покрывает глава
Структура следует учебнику и помогает организовать изучение темы.
Dreisatz
Tabelle
Graph
Где сфокусироваться
Темы, где чаще возникают ошибки или нужна дополнительная практика.
I move between formula, table, and graph.
I interpret slope, intercepts, and key features.
I state domain and relevant restrictions.
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.
Calculate 5 + 6 and show two representations connected with “Dreisatz”.
3 equal groups contain 8 objects each. Find the total and write an addition and multiplication statement.
Continue 4, 9, 14, ... for three more terms and explain the rule.
A rectangle has sides 7 cm and 2 cm. Find its perimeter and explain why four sides are counted.
Out of 12 objects, 3 are shaded. What fraction is shaded and how does it simplify?
The measurements are 6, 6, 6, and 6. Find the greatest, least, and range.
Anna has 27 cards, gives away 9, then shares the rest into 2 equal groups. How many are in each group?
A learner says 5 + 5 = 11. Find the error, correct it, and check the correction.
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.