学习目标
- Recognise, explain, and apply “Ganze Zahlen”.
- Recognise, explain, and apply “Rationale Zahlen”.
- Recognise, explain, and apply “Brüche”.
- Recognise, explain, and apply “Dezimalzahlen”.
- Recognise, explain, and apply “Vergleichen und Ordnen”.
German Gymnasium Klasse 7 / Mathematics / Curriculum
Zahlen und Rechnen
Zahlen und Rechnen: structured theory, worked examples, answered practice, and a mastery checklist for German Gymnasium Klasse 7.
章节计划
学习、练习并检查进度
按顺序完成步骤，或直接跳到你需要的部分。
—/5
已完成
预计主动学习时间
209 分钟
1. 理解
学习目标、核心概念与结构化理论。
打开 →2. 掌握方法
逐步说明的完整例题。
打开 →3. 练习
含提示与答案的分级练习。
打开 →4. 巩固
从基础到挑战的课后任务。
打开 →5. 检查
含计时、分值与解答的章节测试。
打开 →
进度只保存在当前浏览器中，不需要 AI credits。
核心概念与检查
- I can expand and factor with a check.
- I solve equations using equivalent steps.
- Every solution or explanation for “Zahlen und Rechnen” should include a method, justification, and final check.
单元
核心理论
练习前按清晰顺序掌握本章核心概念。
Variables and equivalence
A variable represents a number or quantity. Every algebraic transformation must preserve the equivalence of the original expression or equation.
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.
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.
Ganze Zahlen
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.
Rationale Zahlen
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.
Brüche
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.
Dezimalzahlen
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.
Vergleichen und Ordnen
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
例题讲解
逐步查看方法，并理解每一步为何成立。
Worked equation example
Solve 4(x - 2) + 4 = 3x + 8.
- 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 = 12
Reasoning example
Before calculating, explain the key idea from “Ganze Zahlen” 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 “Ganze Zahlen”, but also why it is valid here.
Zahlen und Rechnen
含答案练习
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。
基础熟练4 分钟
Solve 3x + 3 = 12.
提示
Subtract 3 from both sides, then divide by 3.
答案指南
x = 3.
基础熟练5 分钟
Expand and simplify 6(x + 5) - 5x.
提示
Multiply every term inside the bracket.
答案指南
6x + 30 - 5x = 1x + 30.
应用6 分钟
Factorise 9x + 18.
提示
Take out the common factor 9.
答案指南
9(x + 2).
应用7 分钟
For f(x) = 5x + 4, find f(9) and solve f(x) = 24.
提示
Substitute, then solve the equation.
答案指南
f(9) = 49; x = 4.
推理8 分钟
A sequence starts 6, 14, 22. Find its nth term and 10th term.
提示
The common difference is 8.
答案指南
8n + -2; term 10 = 78.
推理9 分钟
Solve 4x + 3 > 15.
提示
Use equation steps; the coefficient of x is positive.
答案指南
x > 3.
考试题型10 分钟
Solve x + y = 12, x - y = 2.
提示
Add the two equations.
答案指南
x = 7, y = 5.
考试题型12 分钟
A learner writes 3(x + 2) = 3x + 2. Identify and correct the error.
提示
The outside factor multiplies every term.
答案指南
3(x + 2) = 3x + 6.
常见错误
- 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.
掌握度检查
- I can expand and factor with a check.
- I solve equations using equivalent steps.
- I verify the solution in the original relation.
Zahlen und Rechnen
章节作业
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。
基础8 分钟
Homework 1: Solve 6x + 4 = 28.
提示
Subtract 4 from both sides, then divide by 6.
答案指南
x = 4.
基础10 分钟
Homework 2: Expand and simplify 9(x + 6) - 6x.
提示
Multiply every term inside the bracket.
答案指南
9x + 54 - 6x = 3x + 54.
进阶12 分钟
Homework 3: Factorise 5x + 15.
提示
Take out the common factor 5.
答案指南
5(x + 3).
进阶15 分钟
Homework 4: For f(x) = 8x + 5, find f(13) and solve f(x) = 45.
提示
Substitute, then solve the equation.
答案指南
f(13) = 109; x = 5.
挑战18 分钟
Homework 5: A sequence starts 2, 6, 10. Find its nth term and 10th term.
提示
The common difference is 4.
答案指南
4n + -2; term 10 = 38.
挑战20 分钟
Homework 6: Solve 7x + 4 > 32.
提示
Use equation steps; the coefficient of x is positive.
答案指南
x > 4.
50 分钟 / 40 分
完整章节测试
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. Solve 4x + 4 = 20.
2 分答案指南
x = 4.
2. Expand and simplify 7(x + 6) - 6x.
3 分答案指南
7x + 42 - 6x = 1x + 42.
3. Factorise 3x + 9.
3 分答案指南
3(x + 3).
4. For f(x) = 6x + 5, find f(11) and solve f(x) = 35.
4 分答案指南
f(11) = 71; x = 5.
5. A sequence starts 2, 11, 20. Find its nth term and 10th term.
4 分答案指南
9n + -7; term 10 = 83.
6. Solve 5x + 4 > 24.
4 分答案指南
x > 4.
7. Solve x + y = 14, x - y = 2.
5 分答案指南
x = 8, y = 6.
8. A learner writes 4(x + 3) = 4x + 3. Identify and correct the error.
5 分答案指南
4(x + 3) = 4x + 12.
9. Model this: 5 tickets cost x euros each plus a fixed 7 euros. Write the cost and find it for x = 12.
5 分答案指南
C = 5x + 7; C(12) = 67.
10. Represent “Prozentrechnung” as an equation, a graph, and a verbal description of the same model.
5 分答案指南
Example y = 3x + 2: a line with gradient 3, intercept 2, meaning “starts at 2 and rises by 3 per unit”.
单元
官方来源与核验
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。
本章覆盖内容
结构跟随教材，用来帮助组织学习。
Ganze Zahlen
Rationale Zahlen
Brüche
Dezimalzahlen
Vergleichen und Ordnen
Grundrechenarten
Vorzeichenregeln
Primzahlen
Teiler und Vielfache
Prozentrechnung
Verhältnisrechnung
Dreisatz
重点关注
通常容易出错或需要更多复习的部分。
I can expand and factor with a check.
I solve equations using equivalent steps.
I verify the solution in the original relation.
来源、日常资料与学习资源
从教材、daily material、PDFs、videos 和 worked examples 开始。
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.
按小主题练习
进入 full tests 前先做 targeted practice，让覆盖情况更清楚。
Solve 3x + 3 = 12.
Expand and simplify 6(x + 5) - 5x.
Factorise 9x + 18.
For f(x) = 5x + 4, find f(9) and solve f(x) = 24.
A sequence starts 6, 14, 22. Find its nth term and 10th term.
Solve 4x + 3 > 15.
Solve x + y = 12, x - y = 2.
A learner writes 3(x + 2) = 3x + 2. Identify and correct the error.
Mocks 与进度检查
如何衡量本章进度，以及何时加入综合 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.
下一步
完成本章后该做什么，以及它如何连接到下一单元。
Complete the practice without support.
Explain the core method aloud in under two minutes.
Continue to the next chapter or request targeted tutor support.
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。