You will learn
- Write a fact family.
- Use multiplication to find a division answer.
- Find a missing factor.
- Check a division by multiplying.
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
- Multiplication and division are inverses.
- A fact family has four related sentences.
- Find a missing factor by dividing.
- Check a division by multiplying.
Words to know
The opposite operation.
Related multiplication and division facts.
A number to find in a multiplication.
Make sure it is right.
1. Discover: one array, four facts
An array has 6 rows of 7. Write the multiplication. What division facts can you write?
2. Understand: linking multiplication and division
Multiplication and division are inverse operations. One array gives a fact family:
- 6 × 7 = 42
- 7 × 6 = 42
- 42 ÷ 6 = 7
- 42 ÷ 7 = 6
Missing factor: 8 × __ = 56 means 56 ÷ 8 = 7. __ × 9 = 72 means 72 ÷ 9 = 8.
Idea 1: Fact family
6 × 7 = 42, 7 × 6 = 42, 42 ÷ 6 = 7 and 42 ÷ 7 = 6. They are a fact family.
Idea 2: Missing factor
8 × __ = 56. Think 56 ÷ 8 = 7.
Idea 3: Check
45 ÷ 9 = 5. Check: 5 × 9 = 45. It works.
Why inverse? Division undoes multiplication, like subtraction undoes addition.
3. Worked examples
Example 1 (I show you): Family
6 × 7 = 42. So 42 ÷ 7 = ?
- Use the family.
Answer: 6
Example 2 (We do it together): Missing
8 × __ = 56.
- 56 ÷ 8.
Answer: 7
Example 3 (You try): Check
Check 45 ÷ 9 = 5.
Show the steps and answer
- 5 × 9.
Answer: 45, so it is correct
Did you know?
If you know one multiplication fact, you already know three more.
Why this matters
This link lets us use times tables to divide.
4. Explore: fact family
You need paper and a pencil.
- Draw an array of 6 rows of 7.
- Write the four facts.
- Find 8 × __ = 56.
- Check by dividing.
Think about it: what did you notice?
6 × 7 = 42, 42 ÷ 6 = 7; 8 × 7 = 56.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- The fact family is ___.
- The missing factor is ___.
- I divide ___ by ___.
- I check by ___ × ___.
Guided practice
Question 1. 6 × 7 = 42, so 42 ÷ 6?
7.
Question 2. 8 × __ = 56?
7.
Question 3. 45 ÷ 9?
5.
Independent practice
Level 1: Start
- 6 × 7 = 42, so 42 ÷ 7
- 8 × __ = 56
- __ × 9 = 72
- 45 ÷ 9
- 12 × __ = 60
Level 2: Grow
- 3 × 8 = 24, so 24 ÷ 8 =
- 7 × __ = 49
- 36 ÷ 4
Level 3: Stretch
- 6 × __ = 72
- Write the fact family for 5, 6 and 30
- Kofi says 6 × 7 = 42 so 42 ÷ 6 = 36. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
5 × 8 = 40. How many 8s in 40?
Show a sample answer
5.
Another one. Write the fact family for 4, 9 and 36.
Show a sample answer
4 × 9 = 36, 9 × 4 = 36, 36 ÷ 4 = 9, 36 ÷ 9 = 4
7. Common mistakes
Fix: Use multiplication to find the division.
Fix: Use the times table to divide.
Fix: The biggest number comes first: 42 ÷ 7.
8. Lesson summary and revision card
- I can write a fact family.
- I can use multiplication to divide.
- I can find a missing factor.
- I can check a division.
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
- One array gives four facts.
- Division undoes multiplication.
- Find missing factors by dividing.
11. Next lesson
Next lesson: Division facts. 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.
- 6 × 7 = 42. What is 42 ÷ 7? [1 mark]
- Find the missing factor: 8 × __ = 56. [2 marks]
- Write the fact family for 5, 9 and 45. [3 marks]
- Check 45 ÷ 9 = 5. [2 marks]
- Ali says division is the inverse of addition. Is he right? Explain. [3 marks]
Total: 11 marks.
Marking guide and model answers
- 6 [1]
- 56 ÷ 8 [1] = 7 [1]
- 5 × 9 = 45 [1]; 9 × 5 = 45 or 45 ÷ 5 = 9 [1]; 45 ÷ 9 = 5 [1]
- 5 × 9 [1] = 45 [1]
- No [1]. Division undoes multiplication [1]. Subtraction undoes addition [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
- 6 × 7 = 42, so 42 ÷ 7
- 8 × __ = 56
- __ × 9 = 72
- 45 ÷ 9
- 12 × __ = 60
- 6 × 7 = 42 so 42 ÷ 6 =
- 8 × __ = 56
- __ × 9 = 72
- 45 ÷ 9 =
- Division is the inverse of ___.
Homework
Pick an array at home and write its four facts.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| 6 × 7 = 42, so 42 ÷ 7 | 6 | 2 |
| 8 × __ = 56 | 7 | 2 |
| __ × 9 = 72 | 8 | 2 |
| 45 ÷ 9 | 5 | 2 |
| 12 × __ = 60 | 5 | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 6; 2 7; 3 8; 4 5; 5 5.
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 7; 7 7; 8 8; 9 5; 10 multiplication.
Independent practice answers. Level 1: 1 6; 2 7; 3 8; 4 5; 5 5. Level 2: 1 3; 2 7; 3 9. Level 3: 1 12; 2 5 × 6 = 30, 6 × 5 = 30, 30 ÷ 5 = 6, 30 ÷ 6 = 5; 3 He subtracted. 42 ÷ 6 = 7 because 6 × 7 = 42..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 Write a fact family | 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 Use multiplication to find a division answer | 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 Find a missing factor | 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 Check a division by multiplying | 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: Thinking division and multiplication are unrelated. Use multiplication to find the division.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Use a counter array to read the four facts.
- 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 links multiplication and division using fact families. You do not need to be good at mathematics to help.
- Say a multiplication fact and ask for its three relatives.
- Ask: what do you multiply to get this?
- Use arrays.
- 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: write a fact family; use multiplication to find a division answer; find a missing factor; check a division by multiplying.
Prerequisite knowledge. Lessons 1 and 2.
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 paper and a pencil. 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. Which multiplication fact helps? What is the fact family?
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. Thinking division and multiplication are unrelated. Use multiplication to find the division.
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. Use a counter array to read the four facts. 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 3 » Operations & Algebraic Thinking; Grade 3 » 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 3.OA.B.6 (division as an unknown-factor problem); Cambridge Primary Mathematics Stage 3 (multiplication and division). These sources informed the lesson.