Greek Primary Grade 6 / Mathematics / Αριθμοί και αλγεβρική σκέψη

Μεταβλητές και απλές εξισώσεις

Μεταβλητές και απλές εξισώσεις: structured theory, worked examples, answered practice, and a mastery checklist for Greek Primary Grade 6.

Цели обучения

Recognise, explain, and apply “άγνωστος”.
Recognise, explain, and apply “ισότητα”.
Recognise, explain, and apply “αντίστροφες πράξεις”.
Recognise, explain, and apply “έλεγχος λύσης”.

Ключевые идеи и проверки

I can expand and factor with a check.
I solve equations using equivalent steps.
Every solution or explanation for “Μεταβλητές και απλές εξισώσεις” should include a method, justification, and final check.

Раздел

Основная теория

Ключевые идеи главы в понятной последовательности перед практикой.

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.

άγνωστος

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.

ισότητα

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.

αντίστροφες πράξεις

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.

έλεγχος λύσης

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

5 equal groups contain 10 objects each. How many objects are there, and how can the answer be checked?

1Write repeated addition: 10 + 10 + 10 + 10 + 10.
2Connect it to 5 × 10.
3Check with division: 50 ÷ 5 = 10.

Thinking and explanation example

Explain “άγνωστος” 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.

Μεταβλητές και απλές εξισώσεις

Практика с ответами

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

Базовый навык3 минут

Calculate 7 + 5 and show two representations connected with “άγνωστος”.

Подсказка

Use a number line, place value, or objects.

Ответ

7 + 5 = 12. Valid representations include number-line jumps and grouped objects.

Базовый навык4 минут

2 equal groups contain 3 objects each. Find the total and write an addition and multiplication statement.

Подсказка

Add 3 a total of 2 times.

Ответ

3 + 3 = 2 · 3 = 6.

Применение4 минут

Continue 6, 10, 14, ... for three more terms and explain the rule.

Подсказка

Check how much each term increases.

Ответ

18, 22, 26; +4.

Применение5 минут

A rectangle has sides 9 cm and 6 cm. Find its perimeter and explain why four sides are counted.

Подсказка

Opposite sides are equal.

Ответ

Perimeter = 9 + 6 + 9 + 6 = 30 cm.

Обоснование5 минут

Out of 15 objects, 5 are shaded. What fraction is shaded and how does it simplify?

Подсказка

Put shaded objects over the total.

Ответ

5/15 = 1/3.

Обоснование6 минут

The measurements are 8, 5, 8, and 5. Find the greatest, least, and range.

Подсказка

Range = greatest value minus least value.

Ответ

Greatest 8, least 5, range 3.

Экзаменационный7 минут

Anna has 8 cards, gives away 4, then shares the rest into 1 equal groups. How many are in each group?

Подсказка

Subtract first, then divide.

Ответ

8 - 4 = 4; (4) : 1 = 4.

Экзаменационный8 минут

A learner says 7 + 4 = 12. Find the error, correct it, and check the correction.

Подсказка

Check with a number line or inverse subtraction.

Ответ

7 + 4 = 11; 11 - 4 = 7.

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.

Μεταβλητές και απλές εξισώσεις

Домашняя работа по главе

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

Базовый5 минут

Homework 1: Calculate 3 + 6 and show two representations connected with “άγνωστος”.

Подсказка

Use a number line, place value, or objects.

Ответ

3 + 6 = 9. Valid representations include number-line jumps and grouped objects.

Базовый6 минут

Homework 2: 3 equal groups contain 6 objects each. Find the total and write an addition and multiplication statement.

Подсказка

Add 6 a total of 3 times.

Ответ

6 + 6 + 6 = 3 · 6 = 18.

Углубленный7 минут

Homework 3: Continue 9, 14, 19, ... for three more terms and explain the rule.

Подсказка

Check how much each term increases.

Ответ

24, 29, 34; +5.

Углубленный8 минут

Homework 4: A rectangle has sides 5 cm and 2 cm. Find its perimeter and explain why four sides are counted.

Подсказка

Opposite sides are equal.

Ответ

Perimeter = 5 + 2 + 5 + 2 = 14 cm.

Сложный10 минут

Homework 5: Out of 32 objects, 8 are shaded. What fraction is shaded and how does it simplify?

Подсказка

Put shaded objects over the total.

Ответ

8/32 = 1/4.

Сложный12 минут

Homework 6: The measurements are 4, 6, 4, and 6. Find the greatest, least, and range.

Подсказка

Range = greatest value minus least value.

Ответ

Greatest 6, least 4, range 2.

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 8 + 6 and show two representations connected with “άγνωστος”.

2 баллов

Ответ

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 баллов

Ответ

4 + 4 + 4 = 3 · 4 = 12.

3. Continue 7, 12, 17, ... for three more terms and explain the rule.

3 баллов

Ответ

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 баллов

Ответ

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 баллов

Ответ

6/24 = 1/4.

6. The measurements are 9, 6, 9, and 6. Find the greatest, least, and range.

4 баллов

Ответ

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 баллов

Ответ

15 - 5 = 10; (10) : 2 = 5.

8. A learner says 8 + 5 = 14. Find the error, correct it, and check the correction.

5 баллов

Ответ

8 + 5 = 13; 13 - 5 = 8.

Раздел

Официальные источники

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

Ινστιτούτο Εκπαιδευτικής Πολιτικής (ΙΕΠ)Official curriculum programmes and announcements.Ψηφιακό Σχολείο / Διαδραστικά Σχολικά ΒιβλίαOfficial school textbooks and available digital material.

Mathematics

Greek Primary Grade 6: 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.

άγνωστος

ισότητα

αντίστροφες πράξεις

έλεγχος λύσης

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.