You will learn
- Read an array as rows and columns.
- Write two multiplication sentences for one array.
- See that order does not change the product.
- Solve problems with arrays.
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
- An array has rows and columns.
- Rows go across, columns go down.
- One array gives two multiplication sentences.
- Order does not change the product.
Words to know
Objects in rows and columns.
Across.
Down.
Swap the order of factors.
1. Discover: planting rows
A farmer plants maize in 4 rows with 7 plants in each row. How many plants? What if he plants 7 rows of 4?
2. Understand: arrays
An array arranges objects in rows and columns. 4 rows of 7 is 4 × 7 = 28.
Turn the array and you get 7 rows of 4: 7 × 4 = 28. The product is the same. So 4 × 7 = 7 × 4.
6 × 5 = 30, 9 × 3 = 27, 8 × 2 = 16.
Idea 1: Rows
An array with 4 rows of 7 shows 4 × 7 = 28.
Idea 2: Columns
The same array has 7 columns of 4, so 7 × 4 = 28.
Idea 3: Commute
6 × 5 = 5 × 6 = 30. We can swap the order of the factors.
Why swap? It halves the facts to learn.
3. Worked examples
Example 1 (I show you): Rows
4 rows of 7 = ?
- 4 × 7.
Answer: 28
Example 2 (We do it together): Swap
7 × 4 = ?
- Same as 4 × 7.
Answer: 28
Example 3 (You try): Product
6 × 5 = ?
Show the steps and answer
- 6 rows of 5.
Answer: 30
Did you know?
Seats in a cinema, windows on a building and eggs in a tray are arrays.
Why this matters
Arrays show multiplication as a picture.
4. Explore: build an array
You need 28 counters.
- Make 4 rows of 7.
- Write 4 × 7 = 28.
- Turn it to make 7 rows of 4.
- Write 7 × 4 = 28.
Think about it: what did you notice?
4 × 7 = 7 × 4 = 28.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- There are ___ rows of ___.
- The multiplication is ___ × ___.
- Swapping gives ___ × ___.
- Both give ___.
Guided practice
Question 1. 4 rows of 7?
28.
Question 2. 7 rows of 4?
28.
Question 3. 6 × 5?
30.
Independent practice
Level 1: Start
- 4 rows of 7
- 7 × 4
- 6 × 5
- 9 × 3
- 8 × 2
Level 2: Grow
- 3 rows of 8
- 6 × 2
- 5 × 7
Level 3: Stretch
- 9 × 5
- An array has 24 dots in 4 rows. How many are in each row?
- Kofi says 4 × 6 = 24 but 6 × 4 = 10. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
A hall has 6 rows of 8 chairs. How many chairs?
Show a sample answer
48 chairs.
Another one. A hall has 8 rows of 6 chairs. Write two multiplication sentences and the total.
Show a sample answer
8 × 6 = 48 and 6 × 8 = 48
7. Common mistakes
Fix: They give the same product.
Fix: Rows go across, columns go down.
Fix: The order does not change the product.
8. Lesson summary and revision card
- I can read an array.
- I can write two multiplication sentences.
- I can use the swap rule.
- I can solve array problems.
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
- Arrays have rows and columns.
- Turn an array: same product.
- Order does not matter.
11. Next lesson
Next lesson: Multiplication tables. 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 a multiplication for 4 rows of 7. [2 marks]
- Write the swapped multiplication for 7 × 4. [1 mark]
- 6 × 5 = ? [1 mark]
- What do rows and columns mean? [2 marks]
- Ama says 9 × 3 is different from 3 × 9. Is she right? Explain. [3 marks]
Total: 9 marks.
Marking guide and model answers
- 4 × 7 [1] = 28 [1]
- 4 × 7 = 28 [1]
- 30 [1]
- Rows go across [1]; columns go down [1]
- No [1]. We can swap the factors [1]. Both are 27 [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
- 4 rows of 7
- 7 × 4
- 6 × 5
- 9 × 3
- 8 × 2
- 4 rows of 7 =
- 7 × 4 =
- 6 × 5 =
- Rows go ___.
- 6 rows of 8 chairs =
Homework
Make an array with buttons. Write two multiplication sentences.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| 4 rows of 7 | 28 | 2 |
| 7 × 4 | 28 | 2 |
| 6 × 5 | 30 | 2 |
| 9 × 3 | 27 | 2 |
| 8 × 2 | 16 | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 28; 2 28; 3 30; 4 27; 5 16.
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 28; 7 28; 8 30; 9 across; 10 48.
Independent practice answers. Level 1: 1 28; 2 28; 3 30; 4 27; 5 16. Level 2: 1 24; 2 12; 3 35. Level 3: 1 45; 2 6; 3 Order does not change the product. 6 × 4 = 24..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 Read an array as rows and columns | 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 Write two multiplication sentences for one array | 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 See that order does not change the product | 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 Solve problems with arrays | 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 4 × 7 and 7 × 4 give different products. They give the same product.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Build the array with 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 uses arrays to understand multiplication and swapping factors. You do not need to be good at mathematics to help.
- Find arrays at home: eggs, tiles, windows.
- Ask: rows or columns? two sentences?
- Draw arrays with dots.
- 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: read an array as rows and columns; write two multiplication sentences for one array; see that order does not change the product; solve problems with arrays.
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 28 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. How many rows? How many in each row? What is the swapped sentence?
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 4 × 7 and 7 × 4 give different products. They give the same product.
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. Build the array with 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
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.A.1 and 3.OA.B.5 (arrays, commutative property); Cambridge Primary Mathematics Stage 3 (multiplication). These sources informed the lesson.