You will learn
- Use the commutative property.
- Use multiplication by 1 and by 0.
- Split a factor to multiply (distributive).
- Use properties to find facts.
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
- We can swap factors.
- Times 1 leaves a number the same.
- Times 0 makes 0.
- Split a factor into easier parts.
Words to know
The order of the factors does not change the product.
Multiplying by 1 leaves the number unchanged.
Anything times 0 is 0.
Split a factor into easier parts.
1. Discover: clever shortcuts
You know 5 × 8 = 40. Can you use that to find 8 × 5? And 6 × 8?
2. Understand: properties of multiplication
- Order: 5 × 8 = 8 × 5 = 40.
- Times 1: 9 × 1 = 9.
- Times 0: 7 × 0 = 0.
- Split a factor: 6 × 7 = 5 × 7 + 1 × 7 = 35 + 7 = 42.
- Group the factors: 3 × 4 × 2 = 3 × 8 = 24.
Idea 1: Swap
5 × 8 = 8 × 5. This is the commutative property.
Idea 2: 1 and 0
9 × 1 = 9. 12 × 0 = 0.
Idea 3: Split
6 × 7 = 5 × 7 + 1 × 7 = 35 + 7 = 42. This is the distributive property.
Why split? Easy facts build harder ones.
3. Worked examples
Example 1 (I show you): Times 1
9 × 1 = ?
- One group of 9.
Answer: 9
Example 2 (We do it together): Times 0
12 × 0 = ?
- No groups.
Answer: 0
Example 3 (You try): Split
6 × 7 = ?
Show the steps and answer
- 5 × 7 = 35.
- 1 × 7 = 7.
Answer: 42
Did you know?
Multiplying by 1 is called the identity property.
Why this matters
Properties make mental maths quicker.
4. Explore: shortcut lab
You need paper and a pencil.
- Show 5 × 8 and 8 × 5.
- Show 9 × 1 and 7 × 0.
- Split 6 × 7 as 5 × 7 + 1 × 7.
- Check 3 × 4 × 2 two ways.
Think about it: what did you notice?
40; 9; 0; 42; 24.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- The property is ___.
- I split ___ into ___ and ___.
- The two parts are ___ and ___.
- The product is ___.
Guided practice
Question 1. 8 × 5, if 5 × 8 = 40?
40.
Question 2. 7 × 0?
0.
Question 3. 6 × 7 = 5 × 7 + 1 × 7?
35 + 7 = 42.
Independent practice
Level 1: Start
- 9 × 1
- 12 × 0
- 6 × 7 by splitting
- 3 × 4 × 2
- 8 × 6 using 8 × 5 + 8
Level 2: Grow
- 7 × 1
- 25 × 0
- 7 × 6 using 7 × 5 + 7
Level 3: Stretch
- 4 × 5 × 2
- 9 × 6 using 9 × 5 + 9
- Ben says 8 × 0 = 8. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
A shop has 6 shelves of 7 tins. Use 5 × 7 + 1 × 7 to find the total.
Show a sample answer
42 tins.
Another one. Find 6 × 8 by splitting 8 into 5 and 3.
Show a sample answer
6 × 5 = 30 and 6 × 3 = 18, so 48
7. Common mistakes
Fix: Any number times 0 is 0.
Fix: Anything times 0 is 0.
Fix: Add the two products.
8. Lesson summary and revision card
- I can swap factors.
- I know times 1 and times 0.
- I can split a factor.
- I can use properties to find facts.
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
- Order does not matter.
- ×1 and ×0.
- Split a factor.
11. Next lesson
Next lesson: Multiplying by multiples of ten. 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.
- What is 9 × 1? [1 mark]
- What is 12 × 0? [1 mark]
- Find 6 × 7 by splitting. [3 marks]
- 3 × 4 × 2 = ? [2 marks]
- Ama says 5 × 8 is not the same as 8 × 5. Is she right? Explain. [3 marks]
Total: 10 marks.
Marking guide and model answers
- 9 [1]
- 0 [1]
- 5 × 7 = 35 [1]; 1 × 7 = 7 [1]; 42 [1]
- 12 [1] × 2 = 24 [1]
- No [1]. We can swap factors [1]. Both are 40 [1].
Score guide: 9 to 10 = secure. 6 to 8 = nearly there, revise the parts you missed. Under 6 = work through the lesson again with a grown-up.
Printable worksheet
- 9 × 1
- 12 × 0
- 6 × 7 (split)
- 3 × 4 × 2
- 8 × 6
- 5 × 8 = 8 × 5 is the ___ property.
- 9 × 1 =
- 7 × 0 =
- 3 × 4 × 2 =
- 8 × 6 = 8 × 5 + 8 =
Homework
Pick three facts and find them using a property. Show a family member.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| 9 × 1 | 9 | 2 |
| 12 × 0 | 0 | 2 |
| 6 × 7 (split) | 42 | 2 |
| 3 × 4 × 2 | 24 | 2 |
| 8 × 6 | 48 | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 9; 2 0; 3 42; 4 24; 5 48.
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 commutative; 7 9; 8 0; 9 24; 10 48.
Independent practice answers. Level 1: 1 9; 2 0; 3 42; 4 24; 5 48. Level 2: 1 7; 2 0; 3 42. Level 3: 1 40; 2 54; 3 Anything times 0 is 0..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 Use the commutative property | 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 by 1 and by 0 | 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 Split a factor to multiply (distributive) | 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 Use properties to find facts | 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 any number times 0 is that number. Any number times 0 is 0.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Use arrays to see why order does not matter.
- 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 learns properties that make multiplication easier. You do not need to be good at mathematics to help.
- Ask your child to split a hard fact into two easy ones.
- Say: what happens times 0 or times 1?
- Use counters.
- 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: use the commutative property; use multiplication by 1 and by 0; split a factor to multiply (distributive); use properties to find facts.
Prerequisite knowledge. Lessons 2 and 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 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. Can I swap? Can I split? What is times 1 or 0?
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 any number times 0 is that number. Any number times 0 is 0.
13–15. Assessment questions, answer keys, marking scheme
Use the quiz and worksheet on this page. The 10-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 arrays to see why order does not matter. 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.5 (properties of operations); Cambridge Primary Mathematics Stage 3 (multiplication). These sources informed the lesson.