You will learn
- Find quotient and remainder.
- Know the remainder is less than the divisor.
- Write the answer as q r r.
- Interpret the remainder in a problem.
Version 0.2 · expanded edition · built from cited public sources Content is based on the official frameworks and sources listed under Sources and standards. No official endorsement is claimed. Written in the style of common primary school assessments. No exam result is promised.
Student lesson
- Find the biggest multiple that fits.
- The leftover is the remainder.
- The remainder is less than the divisor.
- Write the answer as q r r.
Words to know
What is left over.
The whole-number answer.
The number we divide by.
The number divided.
1. Discover: bags of mangoes
100 mangoes are put into bags of 8. How many full bags? How many are left over?
2. Understand: remainders
- 100 ÷ 8 = 12 r4 (8 × 12 = 96; 100 − 96 = 4).
- 47 ÷ 5 = 9 r2.
- The remainder is always less than the divisor.
Meaning depends on the problem: 12 full bags and 4 mangoes left over.
Idea 1: Find the remainder
25 ÷ 4: 4 × 6 = 24 and 25 − 24 = 1. The answer is 6 r1.
Idea 2: Check
61 ÷ 7: 7 × 8 = 56 and 61 − 56 = 5. The answer is 8 r5.
Idea 3: Bigger
100 ÷ 8: 8 × 12 = 96 and 100 − 96 = 4. The answer is 12 r4.
Why less than the divisor? If it were bigger we could make another group.
3. Worked examples
Example 1 (I show you): Remainder
25 ÷ 4
- 4 × 6 = 24.
Answer: 6 r1
Example 2 (We do it together): Remainder
83 ÷ 9
- 9 × 9 = 81.
Answer: 9 r2
Example 3 (You try): Remainder
100 ÷ 8
Show the steps and answer
- 8 × 12 = 96.
Answer: 12 r4
Did you know?
The remainder can be 0, which means an exact division.
Why this matters
Remainders show up whenever we share things that do not split evenly.
4. Explore: leftovers
You need counters.
- Make 47 counters.
- Group into 5s.
- Count groups and left over.
- Write 47 ÷ 5 = 9 r2.
Think about it: what did you notice?
9 groups, 2 left.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- The biggest multiple of ___ that fits is ___.
- The remainder is ___.
- The answer is ___ r ___.
- The remainder is less than ___.
Guided practice
Question 1. 100 ÷ 8?
12 r4.
Question 2. 47 ÷ 5?
9 r2.
Question 3. Can a remainder be bigger than the divisor?
No.
Independent practice
Level 1: Start
- 25 ÷ 4
- 61 ÷ 7
- 83 ÷ 9
- 100 ÷ 8
- 59 ÷ 6
Level 2: Grow
- 37 ÷ 4
- 50 ÷ 6
- 77 ÷ 8
Level 3: Stretch
- 99 ÷ 10
- 200 ÷ 15
- Ben says 47 ÷ 5 = 8 r7. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
100 mangoes in bags of 8. How many full bags and how many left?
Show a sample answer
12 bags and 4 left.
Another one. 62 eggs are packed in boxes of 12. How many full boxes? How many eggs are left?
Show a sample answer
12 × 5 = 60, so 5 full boxes and 2 eggs left.
7. Common mistakes
Fix: Keep dividing.
Fix: Make one more group.
Fix: Write r and the leftover.
8. Lesson summary and revision card
- I can find the quotient and remainder.
- I know the remainder is less than the divisor.
- I can write q r r.
- I can interpret the remainder.
9. Check and master
Answer all 14 questions. You will see feedback after each one.
Your result appears here after you answer every question.
10. Key takeaways
- Quotient and remainder.
- Remainder < divisor.
- Interpret in context.
11. Next lesson
Next lesson: Division using place value. Keep going with the next idea in this course.
Exam practice
These questions are written in the style of common school and international primary assessments. Try them on your own, without looking back. Then check your answers in the marking guide. The marks are shown in brackets.
- Find 25 ÷ 4. [2 marks]
- Find 61 ÷ 7. [2 marks]
- Find 83 ÷ 9. [2 marks]
- Find 100 ÷ 8. [2 marks]
- Ana says 59 ÷ 6 = 8 r11. Is she right? Explain. [3 marks]
Total: 11 marks.
Marking guide and model answers
- 4 × 6 = 24 [1]; 6 r1 [1]
- 7 × 8 = 56 [1]; 8 r5 [1]
- 9 × 9 = 81 [1]; 9 r2 [1]
- 8 × 12 = 96 [1]; 12 r4 [1]
- No [1]. 11 is bigger than 6 [1]. 6 × 9 = 54, so 9 r5 [1].
Score guide: 10 to 11 = secure. 7 to 9 = nearly there, revise the parts you missed. Under 7 = work through the lesson again with a grown-up.
Printable worksheet
- 25 ÷ 4
- 61 ÷ 7
- 83 ÷ 9
- 100 ÷ 8
- 59 ÷ 6
- 100 ÷ 8 =
- 47 ÷ 5 =
- 25 ÷ 4 =
- 61 ÷ 7 =
- 83 ÷ 9 =
Homework
Make up two remainder problems and solve them.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| 25 ÷ 4 | 6 r1 | 2 |
| 61 ÷ 7 | 8 r5 | 2 |
| 83 ÷ 9 | 9 r2 | 2 |
| 100 ÷ 8 | 12 r4 | 2 |
| 59 ÷ 6 | 9 r5 | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 6 r1; 2 8 r5; 3 9 r2; 4 12 r4; 5 9 r5.
Quiz answers: 1 A, 2 B, 3 C, 4 A, 5 B, 6 A, 7 B, 8 C, 9 A, 10 B, 11 C, 12 A, Challenge 1 C, Challenge 2 C.
Extra worksheet answers (items 6 onward): 6 12 r4; 7 9 r2; 8 6 r1; 9 8 r5; 10 9 r2.
Independent practice answers. Level 1: 1 6 r1; 2 8 r5; 3 9 r2; 4 12 r4; 5 9 r5. Level 2: 1 9 r1; 2 8 r2; 3 9 r5. Level 3: 1 9 r9; 2 13 r5; 3 7 is bigger than 5. 5 × 9 = 45, so the answer is 9 r2..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 Find quotient and remainder | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O2 Know the remainder is less than the divisor | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O3 Write the answer as q r r | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O4 Interpret the remainder in a problem | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
Courseversity badge thresholds (overall quiz score): Gold = 90–100%; Silver = 80–89.9%; Bronze = 70–79.9%. A badge is never a full record of progress.
If a learner struggles
- Find the misconception: A remainder as big as the divisor. Keep dividing.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Group counters to see leftovers.
- Guided practice: work through one example together and let the learner try the next one.
- Reassess: use the quiz again or the worksheet.
- Move on when the learner shows Silver-level evidence independently.
For parents and guardians
In this lesson your child divides with remainders and writes them correctly. You do not need to be good at mathematics to help.
- Share sweets that do not split evenly.
- Ask: what is left over?
- Check: quotient × divisor + remainder.
- Ask “how do you know?” after each answer, and listen.
- Praise effort, not only right answers. Mistakes are how we learn.
- Ten to fifteen minutes a day is better than one long session.
Teacher package
1–5. Objectives, prerequisites, materials, preparation, duration
Lesson objectives. Learners will: find quotient and remainder; know the remainder is less than the divisor; write the answer as q r r; interpret the remainder in a problem.
Prerequisite knowledge. Lesson 3.
Materials. Counters (stones, seeds or bottle tops), paper, pencils, board or chalkboard. Low-resource options: use stones, seeds, sticks, bottle tops or drawings in the sand or on the board instead of printed materials.
Preparation. Gather the counters and draw the example pictures on the board before the lesson.
Suggested duration. 45–60 minutes. It can be split into two shorter sessions.
6–9. Teaching procedure, explanation notes, guided and independent activities
Introduction (5 min). Start with the Discover situation and let learners suggest ideas before you teach.
Explanation (10 min). Teach the Understand section with real objects first, then pictures, then symbols and words.
Guided activity (10 min). You need counters. Learners follow the Explore steps in pairs.
Independent activity (10 min). Learners complete the independent practice and the worksheet.
Assessment and reflection (5 min). Ask each learner to say one thing they learned and how they know it.
Key questions. What is the biggest multiple that fits? What is left?
Teacher explanation notes. Use the concrete, pictorial, abstract order: real objects first, then pictures, then symbols and words. Return to real objects whenever a learner needs support.
10–12. Differentiation, formative assessment, common misconceptions
Support. Use real objects and work with a partner; say each step aloud together. Extension. Ask learners to make up their own example and explain it to a partner.
Formative assessment. Listen to learner explanations, ask “how do you know?”, and check the independent work.
Misconceptions. A remainder as big as the divisor. Keep dividing.
13–15. Assessment questions, answer keys, marking scheme
Use the quiz and worksheet on this page. The 11-mark exam practice scheme, the 10-mark scheme and all answers are in the answer sections above. Re-check every answer and the marking scheme before publication.
16–19. Remediation, enrichment, homework, teacher reflection
Remediation. Group counters to see leftovers. Enrichment. Invite learners to find or create three more examples at home or in the classroom and explain them. Homework. See above.
Teacher reflection. After teaching, record: which concepts learners understood; which misconceptions occurred; which activities worked; which learners need more practice; and what to improve next time.
Sources and standards
This lesson is based on the sources below, and no official endorsement is claimed. Reference starting points named in the Courseversity standard: Common Core State Standards for Mathematics, Cambridge Primary and the IB Primary Years Programme. Where a standard code is named, it points to the official framework.
Official standards for this topic area. Common Core State Standards for Mathematics (official site): Grade 4 » Operations & Algebraic Thinking; Grade 4 » Number & Operations in Base Ten. These pages list the standards that cover this topic area; read them alongside the lesson.
Research used to shape this lesson. Common Core 4.NBT.B.6 (remainders); Cambridge Primary Mathematics Stage 4 (division with remainders). These sources informed the lesson.