学习目标
- Recognise, explain, and apply “gleiche Gruppen”.
- Recognise, explain, and apply “Reihen”.
- Recognise, explain, and apply “Tauschaufgaben”.
German Grade 2 / Mathematics / Multiplikation und Division
Einmaleins-Grundlagen
Einmaleins-Grundlagen: structured theory, worked examples, answered practice, and a mastery checklist for German Grade 2.
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 “Einmaleins-Grundlagen” 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.
gleiche Gruppen
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.
Reihen
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.
Tauschaufgaben
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 arithmetic example
4 equal groups contain 11 objects each. How many objects are there, and how can the answer be checked?
- 1Write repeated addition: 11 + 11 + 11 + 11.
- 2Connect it to 4 × 11.
- 3Check with division: 44 ÷ 4 = 11.
44
Thinking and explanation example
Explain “gleiche Gruppen” 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.
Einmaleins-Grundlagen
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3 分钟
Calculate 8 + 4 and show two representations connected with “gleiche Gruppen”.
提示
Use a number line, place value, or objects.
答案指南
8 + 4 = 12. Valid representations include number-line jumps and grouped objects.
基础熟练4 分钟
6 equal groups contain 4 objects each. Find the total and write an addition and multiplication statement.
提示
Add 4 a total of 6 times.
答案指南
4 + 4 + 4 + 4 + 4 + 4 = 6 · 4 = 24.
应用4 分钟
Continue 7, 10, 13, ... for three more terms and explain the rule.
提示
Check how much each term increases.
答案指南
16, 19, 22; +3.
应用5 分钟
A rectangle has sides 3 cm and 5 cm. Find its perimeter and explain why four sides are counted.
提示
Opposite sides are equal.
答案指南
Perimeter = 3 + 5 + 3 + 5 = 16 cm.
推理5 分钟
Out of 12 objects, 6 are shaded. What fraction is shaded and how does it simplify?
提示
Put shaded objects over the total.
答案指南
6/12 = 1/2.
推理6 分钟
The measurements are 9, 4, 9, and 4. Find the greatest, least, and range.
提示
Range = greatest value minus least value.
答案指南
Greatest 9, least 4, range 5.
考试题型7 分钟
Anna has 30 cards, gives away 5, then shares the rest into 5 equal groups. How many are in each group?
提示
Subtract first, then divide.
答案指南
30 - 5 = 25; (25) : 5 = 5.
考试题型8 分钟
A learner says 8 + 3 = 12. Find the error, correct it, and check the correction.
提示
Check with a number line or inverse subtraction.
答案指南
8 + 3 = 11; 11 - 3 = 8.
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.
Einmaleins-Grundlagen
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5 分钟
Homework 1: Calculate 4 + 5 and show two representations connected with “gleiche Gruppen”.
提示
Use a number line, place value, or objects.
答案指南
4 + 5 = 9. Valid representations include number-line jumps and grouped objects.
基础6 分钟
Homework 2: 2 equal groups contain 7 objects each. Find the total and write an addition and multiplication statement.
提示
Add 7 a total of 2 times.
答案指南
7 + 7 = 2 · 7 = 14.
进阶7 分钟
Homework 3: Continue 3, 7, 11, ... for three more terms and explain the rule.
提示
Check how much each term increases.
答案指南
15, 19, 23; +4.
进阶8 分钟
Homework 4: A rectangle has sides 6 cm and 6 cm. Find its perimeter and explain why four sides are counted.
提示
Opposite sides are equal.
答案指南
Perimeter = 6 + 6 + 6 + 6 = 24 cm.
挑战10 分钟
Homework 5: Out of 27 objects, 9 are shaded. What fraction is shaded and how does it simplify?
提示
Put shaded objects over the total.
答案指南
9/27 = 1/3.
挑战12 分钟
Homework 6: The measurements are 5, 5, 5, and 5. Find the greatest, least, and range.
提示
Range = greatest value minus least value.
答案指南
Greatest 5, least 5, range 0.
35 分钟 / 30 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. Calculate 9 + 5 and show two representations connected with “gleiche Gruppen”.
2 分答案指南
9 + 5 = 14. Valid representations include number-line jumps and grouped objects.
2. 2 equal groups contain 5 objects each. Find the total and write an addition and multiplication statement.
3 分答案指南
5 + 5 = 2 · 5 = 10.
3. Continue 8, 12, 16, ... for three more terms and explain the rule.
3 分答案指南
20, 24, 28; +4.
4. A rectangle has sides 4 cm and 6 cm. Find its perimeter and explain why four sides are counted.
4 分答案指南
Perimeter = 4 + 6 + 4 + 6 = 20 cm.
5. Out of 21 objects, 7 are shaded. What fraction is shaded and how does it simplify?
4 分答案指南
7/21 = 1/3.
6. The measurements are 3, 5, 3, and 5. Find the greatest, least, and range.
4 分答案指南
Greatest 5, least 3, range 2.
7. Anna has 12 cards, gives away 6, then shares the rest into 1 equal groups. How many are in each group?
5 分答案指南
12 - 6 = 6; (6) : 1 = 6.
8. A learner says 9 + 4 = 14. Find the error, correct it, and check the correction.
5 分答案指南
9 + 4 = 13; 13 - 4 = 9.
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 Grade 2: 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。
gleiche Gruppen
Reihen
Tauschaufgaben
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。
Calculate 8 + 4 and show two representations connected with “gleiche Gruppen”.
6 equal groups contain 4 objects each. Find the total and write an addition and multiplication statement.
Continue 7, 10, 13, ... for three more terms and explain the rule.
A rectangle has sides 3 cm and 5 cm. Find its perimeter and explain why four sides are counted.
Out of 12 objects, 6 are shaded. What fraction is shaded and how does it simplify?
The measurements are 9, 4, 9, and 4. Find the greatest, least, and range.
Anna has 30 cards, gives away 5, then shares the rest into 5 equal groups. How many are in each group?
A learner says 8 + 3 = 12. Find the error, correct it, and check the correction.
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。