You will learn
- Write factor pairs.
- List pairs in order.
- Know when the pairs meet at a square number.
- Find a missing factor.
Version 0.2 · expanded edition · expert review pending Standards alignment: pending verification. Written in the style of common primary school assessments. No exam result is promised.
Student lesson
- A factor pair multiplies to make the number.
- List pairs in order from 1.
- Stop when the pairs meet.
- Square numbers have a repeated factor.
Words to know
Two factors that multiply to the number.
A number times itself.
The unknown factor.
The answer to multiplication.
1. Discover: arranging chairs
24 chairs can make a 1 by 24, 2 by 12, 3 by 8 or 4 by 6 hall. Are there others?
2. Understand: factor pairs
A factor pair multiplies to the number.
- 24: 1 × 24, 2 × 12, 3 × 8, 4 × 6.
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
- 36: 1×36, 2×18, 3×12, 4×9, 6×6. The pair 6 × 6 repeats, so 36 is a square number.
Missing factor: 7 × ? = 42, so 42 ÷ 7 = 6.
Idea 1: Pairs of 20
1 × 20, 2 × 10, 4 × 5. So the factors are 1, 2, 4, 5, 10, 20.
Idea 2: Pairs of 18
1 × 18, 2 × 9, 3 × 6. Pairs of 30: 1 × 30, 2 × 15, 3 × 10, 5 × 6.
Idea 3: Missing factor and squares
8 × ? = 56, so ? = 7. 25 = 5 × 5, so 25 is a square number.
Why stop? After the pairs meet, we repeat them reversed.
3. Worked examples
Example 1 (I show you): Pairs
Factor pairs of 20
- Try 1, 2, 3, 4, …
Answer: 1 × 20, 2 × 10, 4 × 5
Example 2 (We do it together): Missing
8 × ? = 56
- 56 ÷ 8.
Answer: 7
Example 3 (You try): Square
Is 25 square?
Show the steps and answer
- Is there a number times itself?
Answer: Yes, 5 × 5
Did you know?
A square number has an odd number of factors.
Why this matters
Factor pairs make it easy to list every factor without missing one.
4. Explore: pair finder
You need paper and a pencil.
- Start with 1 × 24.
- Test 2, 3, 4, 5, 6.
- Write each pair found.
- List all factors in order.
Think about it: what did you notice?
8 factors.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- The pairs of ___ are ___.
- The missing factor is ___ because ___.
- ___ is a square number because ___.
- I stop when the pairs ___.
Guided practice
Question 1. Pairs for 24?
1×24, 2×12, 3×8, 4×6.
Question 2. 7 × ? = 42?
6.
Question 3. Is 36 a square number?
Yes, 6 × 6.
Independent practice
Level 1: Start
- Factor pairs of 20
- Factors of 30
- Missing: 8 × ? = 56
- Is 25 a square number?
- Pairs of 18
Level 2: Grow
- Factor pairs of 15
- 6 × ? = 48
- Is 36 a square number?
Level 3: Stretch
- All factors of 48
- Is 100 a square number?
- Ben says 6 × ? = 48 gives 7. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
30 desks are arranged in equal rows. Name the possible shapes.
Show a sample answer
1×30, 2×15, 3×10, 5×6.
Another one. 24 chairs are put in equal rows. List the possible rectangles.
Show a sample answer
1 × 24, 2 × 12, 3 × 8, 4 × 6
7. Common mistakes
Fix: Test up to where pairs repeat.
Fix: Test each number in order from 1.
Fix: 4 × 4 gives only one factor, 4.
8. Lesson summary and revision card
- I can write factor pairs.
- I can list pairs in order.
- I know when the pairs meet at a square.
- I can find a missing factor.
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
- Factor pairs multiply to the number.
- Stop when pairs repeat.
- Squares have a repeated pair.
11. Next lesson
Next lesson: Understanding multiples. 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.
- Write all the factor pairs of 20. [3 marks]
- List all the factors of 30. [3 marks]
- Find the missing factor: 8 × ? = 56. [2 marks]
- Is 25 a square number? Explain. [2 marks]
- Kofi says 18 has factor pairs 1 × 18, 2 × 9 and 4 × 5. Is he right? Explain. [3 marks]
Total: 13 marks.
Marking guide and model answers
- 1 × 20 [1]; 2 × 10 [1]; 4 × 5 [1]
- 1 × 30 and 2 × 15 [1]; 3 × 10 and 5 × 6 [1]; 1, 2, 3, 5, 6, 10, 15, 30 [1]
- 56 ÷ 8 [1]; 7 [1]
- 5 × 5 = 25 [1]; yes [1]
- No [1]. 4 × 5 = 20, not 18 [1]. The pairs are 1 × 18, 2 × 9 and 3 × 6 [1].
Score guide: 12 to 13 = secure. 8 to 11 = nearly there, revise the parts you missed. Under 8 = work through the lesson again with a grown-up.
Printable worksheet
- Pairs of 20
- Factors of 30
- 8 × ? = 56
- Is 25 a square number? (yes/no)
- Pairs of 18
- Factor pairs of 24 include
- The number of factors of 24 is
- 7 × ? = 42
- 36 is a square number because
- Factors of 30:
Homework
List the factor pairs of 28 and 36.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| Pairs of 20 | 1×20, 2×10, 4×5 | 2 |
| Factors of 30 | 1, 2, 3, 5, 6, 10, 15, 30 | 2 |
| 8 × ? = 56 | 7 | 2 |
| Is 25 a square number? (yes/no) | yes | 2 |
| Pairs of 18 | 1×18, 2×9, 3×6 | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 1×20, 2×10, 4×5; 2 1, 2, 3, 5, 6, 10, 15, 30; 3 7; 4 yes; 5 1×18, 2×9, 3×6.
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 3 × 8; 7 8; 8 6; 9 6 × 6 = 36; 10 1, 2, 3, 5, 6, 10, 15, 30.
Independent practice answers. Level 1: 1 1 × 20, 2 × 10, 4 × 5; 2 1, 2, 3, 5, 6, 10, 15, 30; 3 7; 4 yes, 5 × 5 = 25; 5 1 × 18, 2 × 9, 3 × 6. Level 2: 1 1 × 15, 3 × 5; 2 8; 3 yes, 6 × 6. Level 3: 1 1, 2, 3, 4, 6, 8, 12, 16, 24, 48; 2 yes, 10 × 10; 3 6 × 7 = 42. 6 × 8 = 48, so the missing factor is 8..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 Write factor pairs | 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 List pairs in order | 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 Know when the pairs meet at a square number | 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 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 |
Proposed Courseversity thresholds for this page: Bronze = at least 8 of 12 core questions correct; Silver = all 12; Gold = all 12 plus both challenge questions. A badge is never a full record of progress.
If a learner struggles
- Find the misconception: Stopping before finding all pairs. Test up to where pairs repeat.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Build rectangles from counters.
- 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 finds factor pairs and square numbers. You do not need to be good at mathematics to help.
- Make rectangles from counters to find pairs.
- Ask: which pair have you not tried?
- Look for square numbers.
- 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 factor pairs; list pairs in order; know when the pairs meet at a square number; find a missing factor.
Prerequisite knowledge. Lesson 1.
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 pairs multiply to this number? Where do the pairs meet?
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. Stopping before finding all pairs. Test up to where pairs repeat.
13–15. Assessment questions, answer keys, marking scheme
Use the quiz and worksheet on this page. The 13-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. Build rectangles from counters. 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
Alignment is pending verification and no external 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. Specific standard codes will be added once checked against the official documents.
Research used to shape this lesson. Common Core 4.OA.B.4 (factor pairs); Cambridge Primary Mathematics Stage 4 and 5 (factors and square numbers). These sources informed the lesson; no official alignment is claimed.