Learning objectives
- Recognise, explain, and apply “αριθμογραμμή”.
- Recognise, explain, and apply “σύγκριση”.
- Recognise, explain, and apply “κλάσμα ποσότητας”.
Greek Primary Grade 3 / Mathematics / Κλάσματα και δεκαδικοί
Ισοδύναμα απλά κλάσματα
Ισοδύναμα απλά κλάσματα: structured theory, worked examples, answered practice, and a mastery checklist for Greek Primary Grade 3.
CHAPTER PLAN
Learn, practise, and check your progress
Follow the steps in order or jump directly to the part you need.
—/5
completed
Estimated active study time
140 minutes
1. Understand
Objectives, key ideas, and structured theory.
Open →2. Follow the method
Worked examples that explain every step.
Open →3. Practise
Graded tasks with hints and answer guidance.
Open →4. Consolidate
Assignments from core work to challenge.
Open →5. Check
A timed, marked test with solution guidance.
Open →
Progress is stored only in this browser and does not require AI credits.
Key ideas and checks
- I can convert safely between number forms.
- I can explain why the chosen operation is appropriate.
- Every solution or explanation for “Ισοδύναμα απλά κλάσματα” should include a method, justification, and final check.
Unit
Core theory
The essential chapter ideas in a clear sequence before practice.
Represent before calculating
Write the number in a useful form, check sign and place value, and only then choose the operation. A clear representation reduces mechanical errors.
Equivalent forms
Fractions, decimals, percentages, ratios, and powers can describe the same quantity. Changing form is a solving tool, not a separate trick.
Reasonableness check
Estimate before and after the exact calculation. The sign, order of magnitude, and unit must agree with the problem context.
αριθμογραμμή
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.
σύγκριση
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.
κλάσμα ποσότητας
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
Worked examples
Follow the method step by step and check why every step is valid.
Worked arithmetic example
5 equal groups contain 12 objects each. How many objects are there, and how can the answer be checked?
- 1Write repeated addition: 12 + 12 + 12 + 12 + 12.
- 2Connect it to 5 × 12.
- 3Check with division: 60 ÷ 5 = 12.
60
Thinking and explanation example
Explain “αριθμογραμμή” 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.
Ισοδύναμα απλά κλάσματα
Practice with answers
Eight graded tasks from core fluency to exam-style application. Work independently before opening a hint or answer.
Core fluency3 minutes
Calculate 9 + 5 and show two representations connected with “αριθμογραμμή”.
Hint
Use a number line, place value, or objects.
Answer guide
9 + 5 = 14. Valid representations include number-line jumps and grouped objects.
Core fluency4 minutes
2 equal groups contain 5 objects each. Find the total and write an addition and multiplication statement.
Hint
Add 5 a total of 2 times.
Answer guide
5 + 5 = 2 · 5 = 10.
Application4 minutes
Continue 8, 12, 16, ... for three more terms and explain the rule.
Hint
Check how much each term increases.
Answer guide
20, 24, 28; +4.
Application5 minutes
A rectangle has sides 4 cm and 6 cm. Find its perimeter and explain why four sides are counted.
Hint
Opposite sides are equal.
Answer guide
Perimeter = 4 + 6 + 4 + 6 = 20 cm.
Reasoning5 minutes
Out of 21 objects, 7 are shaded. What fraction is shaded and how does it simplify?
Hint
Put shaded objects over the total.
Answer guide
7/21 = 1/3.
Reasoning6 minutes
The measurements are 3, 5, 3, and 5. Find the greatest, least, and range.
Hint
Range = greatest value minus least value.
Answer guide
Greatest 5, least 3, range 2.
Exam style7 minutes
Anna has 12 cards, gives away 6, then shares the rest into 1 equal groups. How many are in each group?
Hint
Subtract first, then divide.
Answer guide
12 - 6 = 6; (6) : 1 = 6.
Exam style8 minutes
A learner says 9 + 4 = 14. Find the error, correct it, and check the correction.
Hint
Check with a number line or inverse subtraction.
Answer guide
9 + 4 = 13; 13 - 4 = 9.
Common mistakes
- Dropping brackets or using the wrong order of operations.
- Converting a fraction or percentage without checking equivalence.
- Answering without an estimate, unit, or sign check.
Mastery check
- I can convert safely between number forms.
- I can explain why the chosen operation is appropriate.
- I check the result with estimation or an inverse operation.
Ισοδύναμα απλά κλάσματα
Chapter homework
Six distinct assignments from core fluency to challenge, each with an estimated time, hint, and answer guide.
Core5 minutes
Homework 1: Calculate 5 + 6 and show two representations connected with “αριθμογραμμή”.
Hint
Use a number line, place value, or objects.
Answer guide
5 + 6 = 11. Valid representations include number-line jumps and grouped objects.
Core6 minutes
Homework 2: 3 equal groups contain 8 objects each. Find the total and write an addition and multiplication statement.
Hint
Add 8 a total of 3 times.
Answer guide
8 + 8 + 8 = 3 · 8 = 24.
Stretch7 minutes
Homework 3: Continue 4, 9, 14, ... for three more terms and explain the rule.
Hint
Check how much each term increases.
Answer guide
19, 24, 29; +5.
Stretch8 minutes
Homework 4: A rectangle has sides 7 cm and 2 cm. Find its perimeter and explain why four sides are counted.
Hint
Opposite sides are equal.
Answer guide
Perimeter = 7 + 2 + 7 + 2 = 18 cm.
Challenge10 minutes
Homework 5: Out of 12 objects, 3 are shaded. What fraction is shaded and how does it simplify?
Hint
Put shaded objects over the total.
Answer guide
3/12 = 1/4.
Challenge12 minutes
Homework 6: The measurements are 6, 6, 6, and 6. Find the greatest, least, and range.
Hint
Range = greatest value minus least value.
Answer guide
Greatest 6, least 6, range 0.
35 minutes / 30 marks
Full chapter test
A timed, full-mark self-assessment with model-answer guidance.
Start the timer when ready, work without notes, show every step, and open model answers only after finishing.
1. Calculate 3 + 6 and show two representations connected with “αριθμογραμμή”.
2 marksAnswer guide
3 + 6 = 9. Valid representations include number-line jumps and grouped objects.
2. 3 equal groups contain 6 objects each. Find the total and write an addition and multiplication statement.
3 marksAnswer guide
6 + 6 + 6 = 3 · 6 = 18.
3. Continue 9, 14, 19, ... for three more terms and explain the rule.
3 marksAnswer guide
24, 29, 34; +5.
4. A rectangle has sides 5 cm and 2 cm. Find its perimeter and explain why four sides are counted.
4 marksAnswer guide
Perimeter = 5 + 2 + 5 + 2 = 14 cm.
5. Out of 32 objects, 8 are shaded. What fraction is shaded and how does it simplify?
4 marksAnswer guide
8/32 = 1/4.
6. The measurements are 4, 6, 4, and 6. Find the greatest, least, and range.
4 marksAnswer guide
Greatest 6, least 4, range 2.
7. Anna has 21 cards, gives away 7, then shares the rest into 2 equal groups. How many are in each group?
5 marksAnswer guide
21 - 7 = 14; (14) : 2 = 7.
8. A learner says 3 + 5 = 9. Find the error, correct it, and check the correction.
5 marksAnswer guide
3 + 5 = 8; 8 - 5 = 3.
Unit
Official sources and verification
Curriculum reference sources. Always confirm the teaching sequence with the school and tutor.
Mathematics
Start with theory and core examples, then move to worked exercises and short test-style practice.
What this chapter covers
The structure follows the official textbook layout and is used to organise study.
αριθμογραμμή
σύγκριση
κλάσμα ποσότητας
Where to focus
The areas that usually create mistakes or need extra revision.
I can convert safely between number forms.
I can explain why the chosen operation is appropriate.
I check the result with estimation or an inverse operation.
Sources, daily material, and resources
Where to start: textbook, daily material, PDFs, videos, and 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.
Practice by subtopic
Targeted practice before full tests so coverage is clear.
Calculate 9 + 5 and show two representations connected with “αριθμογραμμή”.
2 equal groups contain 5 objects each. Find the total and write an addition and multiplication statement.
Continue 8, 12, 16, ... for three more terms and explain the rule.
A rectangle has sides 4 cm and 6 cm. Find its perimeter and explain why four sides are counted.
Out of 21 objects, 7 are shaded. What fraction is shaded and how does it simplify?
The measurements are 3, 5, 3, and 5. Find the greatest, least, and range.
Anna has 12 cards, gives away 6, then shares the rest into 1 equal groups. How many are in each group?
A learner says 9 + 4 = 14. Find the error, correct it, and check the correction.
Mocks and progress checks
How to measure progress in this chapter and when it enters a cumulative 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.
Next step
What to do after finishing the chapter and how it connects to the next unit.
Complete the practice without support.
Explain the core method aloud in under two minutes.
Continue to the next chapter or request targeted tutor support.
Note: for the official examinable syllabus of each school year, always confirm with the school, tutor, and current Ministry/IEP announcements.