Greek Gymnasium Grade 3 / Mathematics / Μέρος Α: Άλγεβρα
Συστήματα γραμμικών εξισώσεων: structured theory, worked examples, answered practice, and a mastery checklist for Greek Gymnasium Grade 3.
单元
练习前按清晰顺序掌握本章核心概念。
A variable represents a number or quantity. Every algebraic transformation must preserve the equivalence of the original expression or equation.
Write each transformation on a new line and apply the same valid operation to both sides. This keeps the solution easy to audit.
A final value is not a complete solution until it is substituted into the original relation. Verification catches sign and expansion errors.
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.
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.
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.
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.
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
逐步查看方法，并理解每一步为何成立。
5 equal groups contain 10 objects each. How many objects are there, and how can the answer be checked?
50
Explain “Γραμμική εξίσωση με δύο αγνώστους” in your own words and use a picture, objects, a table, or a simple measurement to support it.
A complete explanation connects the correct idea, a clear representation, and a sensible check.
Συστήματα γραμμικών εξισώσεων
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。
Calculate 7 + 5 and show two representations connected with “Γραμμική εξίσωση με δύο αγνώστους”.
Use a number line, place value, or objects.
7 + 5 = 12. Valid representations include number-line jumps and grouped objects.
2 equal groups contain 3 objects each. Find the total and write an addition and multiplication statement.
Add 3 a total of 2 times.
3 + 3 = 2 · 3 = 6.
Continue 6, 10, 14, ... for three more terms and explain the rule.
Check how much each term increases.
18, 22, 26; +4.
A rectangle has sides 9 cm and 6 cm. Find its perimeter and explain why four sides are counted.
Opposite sides are equal.
Perimeter = 9 + 6 + 9 + 6 = 30 cm.
Out of 15 objects, 5 are shaded. What fraction is shaded and how does it simplify?
Put shaded objects over the total.
5/15 = 1/3.
The measurements are 8, 5, 8, and 5. Find the greatest, least, and range.
Range = greatest value minus least value.
Greatest 8, least 5, range 3.
Anna has 8 cards, gives away 4, then shares the rest into 1 equal groups. How many are in each group?
Subtract first, then divide.
8 - 4 = 4; (4) : 1 = 4.
A learner says 7 + 4 = 12. Find the error, correct it, and check the correction.
Check with a number line or inverse subtraction.
7 + 4 = 11; 11 - 4 = 7.
Συστήματα γραμμικών εξισώσεων
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。
Homework 1: Calculate 3 + 6 and show two representations connected with “Γραμμική εξίσωση με δύο αγνώστους”.
Use a number line, place value, or objects.
3 + 6 = 9. Valid representations include number-line jumps and grouped objects.
Homework 2: 3 equal groups contain 6 objects each. Find the total and write an addition and multiplication statement.
Add 6 a total of 3 times.
6 + 6 + 6 = 3 · 6 = 18.
Homework 3: Continue 9, 14, 19, ... for three more terms and explain the rule.
Check how much each term increases.
24, 29, 34; +5.
Homework 4: A rectangle has sides 5 cm and 2 cm. Find its perimeter and explain why four sides are counted.
Opposite sides are equal.
Perimeter = 5 + 2 + 5 + 2 = 14 cm.
Homework 5: Out of 32 objects, 8 are shaded. What fraction is shaded and how does it simplify?
Put shaded objects over the total.
8/32 = 1/4.
Homework 6: The measurements are 4, 6, 4, and 6. Find the greatest, least, and range.
Range = greatest value minus least value.
Greatest 6, least 4, range 2.
35 分钟 / 30 分
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 8 + 6 and show two representations connected with “Γραμμική εξίσωση με δύο αγνώστους”.
2 分8 + 6 = 14. Valid representations include number-line jumps and grouped objects.
2. 3 equal groups contain 4 objects each. Find the total and write an addition and multiplication statement.
3 分4 + 4 + 4 = 3 · 4 = 12.
3. Continue 7, 12, 17, ... for three more terms and explain the rule.
3 分22, 27, 32; +5.
4. A rectangle has sides 3 cm and 2 cm. Find its perimeter and explain why four sides are counted.
4 分Perimeter = 3 + 2 + 3 + 2 = 10 cm.
5. Out of 24 objects, 6 are shaded. What fraction is shaded and how does it simplify?
4 分6/24 = 1/4.
6. The measurements are 9, 6, 9, and 6. Find the greatest, least, and range.
4 分Greatest 9, least 6, range 3.
7. Anna has 15 cards, gives away 5, then shares the rest into 2 equal groups. How many are in each group?
5 分15 - 5 = 10; (10) : 2 = 5.
8. A learner says 8 + 5 = 14. Find the error, correct it, and check the correction.
5 分8 + 5 = 13; 13 - 5 = 8.
单元
课程参考来源；教学顺序请始终与学校和老师确认。
Mathematics
Greek Gymnasium Grade 3: 数学与物理的结构化路线，包含章节、练习和下一步。
结构跟随教材，用来帮助组织学习。
通常容易出错或需要更多复习的部分。
从教材、daily material、PDFs、videos 和 worked examples 开始。
进入 full tests 前先做 targeted practice，让覆盖情况更清楚。
如何衡量本章进度，以及何时加入综合 mock。
完成本章后该做什么，以及它如何连接到下一单元。
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。
