You will learn
- Use order and grouping to simplify.
- Use the distributive property.
- Use the identity and zero properties.
- Apply properties to mental maths.
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
- Order does not change a product.
- Grouping does not change a product.
- Split one factor, multiply, then add.
- Times 1 stays the same. Times 0 is 0.
Words to know
Order does not change the product.
Grouping does not change the product.
Split a factor and add.
Times 1 stays the same.
1. Discover: rearranging
Is 4 × 25 easier than 25 × 4? Can you find 7 × 12 using 7 × 10 and 7 × 2?
2. Understand: properties
- Commutative: 6 × 4 = 4 × 6 = 24.
- Associative: (2 × 3) × 5 = 2 × (3 × 5) = 30.
- Distributive: 7 × 12 = 7 × 10 + 7 × 2 = 70 + 14 = 84.
- Identity: any number × 1 is that number.
- Zero: any number × 0 = 0.
Idea 1: Distributive
6 × 15 = 6 × 10 + 6 × 5 = 60 + 30 = 90.
Idea 2: Regroup
2 × 3 × 5: 2 × 5 = 10, then × 3 = 30. 4 × 25 = 100.
Idea 3: Identity and zero
8 × 1 = 8. 0 × 12 = 0.
Why use properties? They make hard sums easy.
3. Worked examples
Example 1 (I show you): Split
6 × 15
- 6 × 10 = 60 and 6 × 5 = 30.
Answer: 90
Example 2 (We do it together): Regroup
2 × 3 × 5
- Multiply 2 × 5 first.
Answer: 30
Example 3 (You try): Zero
0 × 12
Show the steps and answer
- Anything times 0.
Answer: 0
Did you know?
4 × 25 is like 4 quarters making 100.
Why this matters
Properties let us do hard multiplication in our heads.
4. Explore: break it up
You need paper.
- Write 7 × 12.
- Split 12 into 10 + 2.
- Multiply each part.
- Add.
Think about it: what did you notice?
84.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- I split ___ into ___ and ___.
- I multiply each part to get ___ and ___.
- I add to get ___.
- The property used is ___.
Guided practice
Question 1. 7 × 12?
84.
Question 2. 5 × 1?
5.
Question 3. 9 × 0?
0.
Independent practice
Level 1: Start
- 6 × 15 using 10 + 5
- 4 × 25
- 2 × 3 × 5
- 8 × 1
- 0 × 12
Level 2: Grow
- 7 × 14 using 10 + 4
- 5 × 18 using 10 + 8
- 2 × 7 × 5
Level 3: Stretch
- 4 × 27 using 20 + 7
- 25 × 8
- Kofi says 5 × (2 + 3) = 5 × 2 + 3 = 13. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
A hall has 6 rows of 15 seats. How many seats?
Show a sample answer
90.
Another one. A hall has 15 rows of 6 seats. Use 10 + 5 to find the number of seats.
Show a sample answer
6 × 10 = 60 and 6 × 5 = 30, so 90 seats.
7. Common mistakes
Fix: 6 × 4 = 4 × 6.
Fix: Multiply both parts, then add.
Fix: Any number times 0 is 0.
8. Lesson summary and revision card
- I can use order and grouping.
- I can use the distributive property.
- I can use the identity and zero properties.
- I can use properties for mental maths.
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.
- Split to make easier.
- ×1 stays; ×0 is 0.
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.
- Find 6 × 15 using 10 + 5. [2 marks]
- Find 4 × 25. [1 mark]
- Find 2 × 3 × 5 by regrouping. [2 marks]
- Find 8 × 1. [1 mark]
- Ana says 0 × 12 = 12. Is she right? Explain. [3 marks]
Total: 9 marks.
Marking guide and model answers
- 60 and 30 [1]; 90 [1]
- 100 [1]
- 2 × 5 = 10 [1]; 30 [1]
- 8 [1]
- No [1]. Any number times 0 is 0 [1]. 0 × 12 = 0 [1].
Score guide: 8 to 9 = secure. 6 to 7 = nearly there, revise the parts you missed. Under 6 = work through the lesson again with a grown-up.
Printable worksheet
- 7 × 12
- 6 × 15
- 4 × 25
- 2 × 3 × 5
- 0 × 12
- 6 × 15 =
- 4 × 25 =
- 2 × 3 × 5 =
- 8 × 1 =
- 0 × 12 =
Homework
Use the distributive property to find 8 × 14.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| 7 × 12 | 84 | 2 |
| 6 × 15 | 90 | 2 |
| 4 × 25 | 100 | 2 |
| 2 × 3 × 5 | 30 | 2 |
| 0 × 12 | 0 | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 84; 2 90; 3 100; 4 30; 5 0.
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 90; 7 100; 8 30; 9 8; 10 0.
Independent practice answers. Level 1: 1 90; 2 100; 3 30; 4 8; 5 0. Level 2: 1 98; 2 90; 3 70. Level 3: 1 108; 2 200; 3 He must multiply both parts: 5 × 2 + 5 × 3 = 25..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 Use order and grouping to simplify | 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 the distributive 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 |
| O3 Use the identity and zero properties | 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 Apply properties to mental maths | 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: Thinking order matters for multiplication. 6 × 4 = 4 × 6.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Practise with arrays.
- 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 uses multiplication properties to make calculations easier. You do not need to be good at mathematics to help.
- Show how to split a hard number.
- Ask: which property helped?
- Use coins to show 4 × 25.
- 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 order and grouping to simplify; use the distributive property; use the identity and zero properties; apply properties to mental maths.
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. 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 change the order, regroup or split?
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 order matters for multiplication. 6 × 4 = 4 × 6.
13–15. Assessment questions, answer keys, marking scheme
Use the quiz and worksheet on this page. The 9-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. Practise with arrays. 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 3.OA.B.5 and 4.NBT.B.5 (properties of operations); Cambridge Primary Mathematics Stage 4 (multiplication strategies). These sources informed the lesson; no official alignment is claimed.