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German Gymnasium Klasse 10 / Mathematics Thüringen / Curriculum

Gleichungen und Modellierung

Gleichungen und Modellierung: structured theory, worked examples, answered practice, and a mastery checklist for German Gymnasium Klasse 10.

Цели обучения

Recognise, explain, and apply “Lineare Gleichungen”.
Recognise, explain, and apply “Quadratische Gleichungen”.
Recognise, explain, and apply “Bruchgleichungen in school scope”.
Recognise, explain, and apply “Exponential equations introduction”.
Recognise, explain, and apply “Ungleichungen”.

Ключевые идеи и проверки

I can expand and factor with a check.
I solve equations using equivalent steps.
Every solution or explanation for “Gleichungen und Modellierung” should include a method, justification, and final check.

Раздел

Основная теория

Ключевые идеи главы в понятной последовательности перед практикой.

01

Variables and equivalence

A variable represents a number or quantity. Every algebraic transformation must preserve the equivalence of the original expression or equation.

02

One valid step at a time

Write each transformation on a new line and apply the same valid operation to both sides. This keeps the solution easy to audit.

03

Substitute and verify

A final value is not a complete solution until it is substituted into the original relation. Verification catches sign and expansion errors.

04

Lineare Gleichungen

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.

05

Quadratische Gleichungen

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.

06

Bruchgleichungen in school scope

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.

07

Exponential equations introduction

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.

08

Ungleichungen

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 Thüringen

Разобранные примеры

Следуйте методу шаг за шагом и проверяйте обоснование каждого шага.

Worked equation example

Solve 3(x - 2) + 3 = 2x + 6.

1Expand the bracket.
2Collect x-terms on one side and constants on the other.
3Divide by the coefficient of x and verify in the original equation.

x = 9

Reasoning example

Before calculating, explain the key idea from “Lineare Gleichungen” 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 “Lineare Gleichungen”, but also why it is valid here.

Gleichungen und Modellierung

Практика с ответами

Восемь заданий от базовых навыков до экзаменационного уровня. Сначала решите самостоятельно.

Базовый навык4 минут

Solve 9x + 2 = 20.

Подсказка

Subtract 2 from both sides, then divide by 9.

Ответ

x = 2.

Базовый навык5 минут

Expand and simplify 5(x + 4) - 4x.

Подсказка

Multiply every term inside the bracket.

Ответ

5x + 20 - 4x = 1x + 20.

Применение6 минут

Factorise 8x + 48.

Подсказка

Take out the common factor 8.

Ответ

8(x + 6).

Применение7 минут

For f(x) = 4x + 3, find f(7) and solve f(x) = 15.

Подсказка

Substitute, then solve the equation.

Ответ

f(7) = 31; x = 3.

Обоснование8 минут

A sequence starts 5, 12, 19. Find its nth term and 10th term.

Подсказка

The common difference is 7.

Ответ

7n + -2; term 10 = 68.

Обоснование9 минут

Solve 3x + 2 > 8.

Подсказка

Use equation steps; the coefficient of x is positive.

Ответ

x > 2.

Экзаменационный10 минут

Solve x + y = 10, x - y = 2.

Подсказка

Add the two equations.

Ответ

x = 6, y = 4.

Экзаменационный12 минут

A learner writes 9(x + 6) = 9x + 6. Identify and correct the error.

Подсказка

The outside factor multiplies every term.

Ответ

9(x + 6) = 9x + 54.

Частые ошибки

Changing a sign without applying it to every term.
Combining terms that are not like terms.
Dividing by an expression that may be zero without stating a restriction.

Проверка понимания

I can expand and factor with a check.
I solve equations using equivalent steps.
I verify the solution in the original relation.

Gleichungen und Modellierung

Домашняя работа по главе

Шесть разных заданий от базового уровня до сложного с временем, подсказками и ответами.

Базовый8 минут

Homework 1: Solve 5x + 3 = 18.

Подсказка

Subtract 3 from both sides, then divide by 5.

Ответ

x = 3.

Базовый10 минут

Homework 2: Expand and simplify 8(x + 5) - 5x.

Подсказка

Multiply every term inside the bracket.

Ответ

8x + 40 - 5x = 3x + 40.

Углубленный12 минут

Homework 3: Factorise 4x + 8.

Подсказка

Take out the common factor 4.

Ответ

4(x + 2).

Углубленный15 минут

Homework 4: For f(x) = 7x + 4, find f(11) and solve f(x) = 32.

Подсказка

Substitute, then solve the equation.

Ответ

f(11) = 81; x = 4.

Сложный18 минут

Homework 5: A sequence starts 6, 9, 12. Find its nth term and 10th term.

Подсказка

The common difference is 3.

Ответ

3n + 3; term 10 = 33.

Сложный20 минут

Homework 6: Solve 6x + 3 > 21.

Подсказка

Use equation steps; the coefficient of x is positive.

Ответ

x > 3.

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. Solve 3x + 3 = 12.

2 баллов

Ответ

x = 3.

2. Expand and simplify 6(x + 5) - 5x.

3 баллов

Ответ

6x + 30 - 5x = 1x + 30.

3. Factorise 9x + 18.

3 баллов

Ответ

9(x + 2).

4. For f(x) = 5x + 4, find f(9) and solve f(x) = 24.

4 баллов

Ответ

f(9) = 49; x = 4.

5. A sequence starts 6, 14, 22. Find its nth term and 10th term.

4 баллов

Ответ

8n + -2; term 10 = 78.

6. Solve 4x + 3 > 15.

4 баллов

Ответ

x > 3.

7. Solve x + y = 12, x - y = 2.

5 баллов

Ответ

x = 7, y = 5.

8. A learner writes 3(x + 2) = 3x + 2. Identify and correct the error.

5 баллов

Ответ

3(x + 2) = 3x + 6.

9. Model this: 4 tickets cost x euros each plus a fixed 6 euros. Write the cost and find it for x = 10.

5 баллов

Ответ

C = 4x + 6; C(10) = 46.

10. Represent “Exponential equations introduction” as an equation, a graph, and a verbal description of the same model.

5 баллов

Ответ

Example y = 9x + 6: a line with gradient 9, intercept 6, meaning “starts at 6 and rises by 9 per unit”.

Раздел

Официальные источники

Источники программы; порядок тем уточняйте в школе и у преподавателя.

Thüringer Schulportal - LehrpläneOfficial Thüringen curricula by stage and subject.

Mathematics Thüringen

German Gymnasium Klasse 10: структурированный маршрут по математике и физике с главами, практикой и следующими шагами.

Назад к предмету

Что покрывает глава

Структура следует учебнику и помогает организовать изучение темы.

Lineare Gleichungen

Quadratische Gleichungen

Bruchgleichungen in school scope

Exponential equations introduction

Ungleichungen

Modelling tasks

Где сфокусироваться

Темы, где чаще возникают ошибки или нужна дополнительная практика.

I can expand and factor with a check.

I solve equations using equivalent steps.

I verify the solution in the original relation.

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.

Solve 9x + 2 = 20.

Expand and simplify 5(x + 4) - 4x.

Factorise 8x + 48.

For f(x) = 4x + 3, find f(7) and solve f(x) = 15.

A sequence starts 5, 12, 19. Find its nth term and 10th term.

Solve 3x + 2 > 8.

Solve x + y = 10, x - y = 2.

A learner writes 9(x + 6) = 9x + 6. Identify and correct the error.

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.