Цели обучения
- Recognise, explain, and apply “Όριο σε σημείο”.
- Recognise, explain, and apply “Πλευρικά όρια”.
- Recognise, explain, and apply “Πράξεις ορίων”.
- Recognise, explain, and apply “Άπειρα όρια”.
- Recognise, explain, and apply “Όρια στο άπειρο”.
Greek Lyceum Grade 3 / Mathematics / Curriculum
Όρια και συνέχεια
Όρια και συνέχεια: structured theory, worked examples, answered practice, and a mastery checklist for Greek Lyceum Grade 3.
Ключевые идеи и проверки
- I connect limit, derivative, and integral with a graph.
- I apply rules with the correct conditions.
- Every solution or explanation for “Όρια και συνέχεια” should include a method, justification, and final check.
Раздел
Основная теория
Ключевые идеи главы в понятной последовательности перед практикой.
Change and approximation
Calculus studies how a quantity changes and how small changes accumulate. Graphical interpretation should accompany the notation.
Conditions before the rule
Check domain, continuity, or differentiability where required. A correct rule cannot be applied mechanically outside its conditions.
Interpret the result
A derivative represents local rate of change and a definite integral represents net accumulation. Always reconnect the result to the original problem.
Όριο σε σημείο
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.
Πλευρικά όρια
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.
Πράξεις ορίων
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.
Άπειρα όρια
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.
Όρια στο άπειρο
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
3 equal groups contain 7 objects each. How many objects are there, and how can the answer be checked?
- 1Write repeated addition: 7 + 7 + 7.
- 2Connect it to 3 × 7.
- 3Check with division: 21 ÷ 3 = 7.
21
Thinking and explanation example
Explain “Όριο σε σημείο” 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.
Όρια και συνέχεια
Практика с ответами
Восемь заданий от базовых навыков до экзаменационного уровня. Сначала решите самостоятельно.
Базовый навык3 минут
Calculate 4 + 3 and show two representations connected with “Όριο σε σημείο”.
Подсказка
Use a number line, place value, or objects.
Ответ
4 + 3 = 7. Valid representations include number-line jumps and grouped objects.
Базовый навык4 минут
5 equal groups contain 7 objects each. Find the total and write an addition and multiplication statement.
Подсказка
Add 7 a total of 5 times.
Ответ
7 + 7 + 7 + 7 + 7 = 5 · 7 = 35.
Применение4 минут
Continue 3, 5, 7, ... for three more terms and explain the rule.
Подсказка
Check how much each term increases.
Ответ
9, 11, 13; +2.
Применение5 минут
A rectangle has sides 6 cm and 4 cm. Find its perimeter and explain why four sides are counted.
Подсказка
Opposite sides are equal.
Ответ
Perimeter = 6 + 4 + 6 + 4 = 20 cm.
Обоснование5 минут
Out of 54 objects, 9 are shaded. What fraction is shaded and how does it simplify?
Подсказка
Put shaded objects over the total.
Ответ
9/54 = 1/6.
Обоснование6 минут
The measurements are 5, 3, 5, and 3. Find the greatest, least, and range.
Подсказка
Range = greatest value minus least value.
Ответ
Greatest 5, least 3, range 2.
Экзаменационный7 минут
Anna has 40 cards, gives away 8, then shares the rest into 4 equal groups. How many are in each group?
Подсказка
Subtract first, then divide.
Ответ
40 - 8 = 32; (32) : 4 = 8.
Экзаменационный8 минут
A learner says 4 + 2 = 7. Find the error, correct it, and check the correction.
Подсказка
Check with a number line or inverse subtraction.
Ответ
4 + 2 = 6; 6 - 2 = 4.
Частые ошибки
- Applying a rule without checking domain or continuity.
- Omitting the integration constant in an indefinite integral.
- Giving a numerical result without graphical or physical interpretation.
Проверка понимания
- I connect limit, derivative, and integral with a graph.
- I apply rules with the correct conditions.
- I interpret the result in context.
Όρια και συνέχεια
Домашняя работа по главе
Шесть разных заданий от базового уровня до сложного с временем, подсказками и ответами.
Базовый5 минут
Homework 1: Calculate 7 + 4 and show two representations connected with “Όριο σε σημείο”.
Подсказка
Use a number line, place value, or objects.
Ответ
7 + 4 = 11. Valid representations include number-line jumps and grouped objects.
Базовый6 минут
Homework 2: 6 equal groups contain 3 objects each. Find the total and write an addition and multiplication statement.
Подсказка
Add 3 a total of 6 times.
Ответ
3 + 3 + 3 + 3 + 3 + 3 = 6 · 3 = 18.
Углубленный7 минут
Homework 3: Continue 6, 9, 12, ... for three more terms and explain the rule.
Подсказка
Check how much each term increases.
Ответ
15, 18, 21; +3.
Углубленный8 минут
Homework 4: A rectangle has sides 9 cm and 5 cm. Find its perimeter and explain why four sides are counted.
Подсказка
Opposite sides are equal.
Ответ
Perimeter = 9 + 5 + 9 + 5 = 28 cm.
Сложный10 минут
Homework 5: Out of 10 objects, 5 are shaded. What fraction is shaded and how does it simplify?
Подсказка
Put shaded objects over the total.
Ответ
5/10 = 1/2.
Сложный12 минут
Homework 6: The measurements are 8, 4, 8, and 4. Find the greatest, least, and range.
Подсказка
Range = greatest value minus least value.
Ответ
Greatest 8, least 4, range 4.
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 5 + 4 and show two representations connected with “Όριο σε σημείο”.
2 балловОтвет
5 + 4 = 9. Valid representations include number-line jumps and grouped objects.
2. 6 equal groups contain 8 objects each. Find the total and write an addition and multiplication statement.
3 балловОтвет
8 + 8 + 8 + 8 + 8 + 8 = 6 · 8 = 48.
3. Continue 4, 7, 10, ... for three more terms and explain the rule.
3 балловОтвет
13, 16, 19; +3.
4. A rectangle has sides 7 cm and 5 cm. Find its perimeter and explain why four sides are counted.
4 балловОтвет
Perimeter = 7 + 5 + 7 + 5 = 24 cm.
5. Out of 6 objects, 3 are shaded. What fraction is shaded and how does it simplify?
4 балловОтвет
3/6 = 1/2.
6. The measurements are 6, 4, 6, and 4. Find the greatest, least, and range.
4 балловОтвет
Greatest 6, least 4, range 2.
7. Anna has 54 cards, gives away 9, then shares the rest into 5 equal groups. How many are in each group?
5 балловОтвет
54 - 9 = 45; (45) : 5 = 9.
8. A learner says 5 + 3 = 9. Find the error, correct it, and check the correction.
5 балловОтвет
5 + 3 = 8; 8 - 3 = 5.
Раздел
Официальные источники
Источники программы; порядок тем уточняйте в школе и у преподавателя.
Mathematics
Greek Lyceum Grade 3: структурированный маршрут по математике и физике с главами, практикой и следующими шагами.
Что покрывает глава
Структура следует учебнику и помогает организовать изучение темы.
Όριο σε σημείο
Πλευρικά όρια
Πράξεις ορίων
Άπειρα όρια
Όρια στο άπειρο
Συνέχεια
Θεώρημα Bolzano
Θεώρημα ενδιάμεσων τιμών
Θεώρημα μέγιστης και ελάχιστης τιμής
Где сфокусироваться
Темы, где чаще возникают ошибки или нужна дополнительная практика.
I connect limit, derivative, and integral with a graph.
I apply rules with the correct conditions.
I interpret the result in context.
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.
Calculate 4 + 3 and show two representations connected with “Όριο σε σημείο”.
5 equal groups contain 7 objects each. Find the total and write an addition and multiplication statement.
Continue 3, 5, 7, ... for three more terms and explain the rule.
A rectangle has sides 6 cm and 4 cm. Find its perimeter and explain why four sides are counted.
Out of 54 objects, 9 are shaded. What fraction is shaded and how does it simplify?
The measurements are 5, 3, 5, and 3. Find the greatest, least, and range.
Anna has 40 cards, gives away 8, then shares the rest into 4 equal groups. How many are in each group?
A learner says 4 + 2 = 7. Find the error, correct it, and check the correction.
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.