Multiplication using equal groups & arrays

Class 2 Class 3 CBSE UP Board

📘 What's inside:

1 What are Equal Groups?
2 Multiplication from Equal Groups
3 What is an Array?
4 Writing Multiplication from Arrays
5 Rows and Columns — Both Ways
6 Word Problems
7 Exam Questions

Concept 1 — What are Equal Groups?

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When objects are arranged into groups where every group has the same number of objects, we call them equal groups. Equal groups are the starting point for multiplication.

Equal groups in real life:

4 boxes, each with 3 pencils → equal groups of 3. 5 bags, each with 6 marbles → equal groups of 6. 3 plates, each with 4 biscuits → equal groups of 4.

▶ Equal Groups vs Unequal Groups

Equal groups: every group has the same count. We can multiply. e.g. 3, 3, 3, 3 → 4 groups of 3.
Unequal groups: groups have different counts. We can only add. e.g. 3, 5, 2 → cannot multiply directly.

⚠ Remember

For multiplication to apply, all groups must have exactly the same number. Even if one group has one more or one less, the groups are unequal and multiplication cannot be used as a shortcut.

▶ Solved Example 1

Look at the groups below. Are they equal? Write the multiplication sentence if they are.
Group A: 🍎🍎🍎 Group B: 🍎🍎🍎 Group C: 🍎🍎🍎 Group D: 🍎🍎🍎

🍎🍎🍎

= 12

4 equal groups of 3

1Each group has 3 apples. All 4 groups have the same count → equal groups.
2Number in each group = 3. Number of groups = 4.
3Multiplication sentence: 3 × 4 = 12.

Yes, equal groups. 3 × 4 = 12

▶ Solved Example 2

Are these equal groups? Group A: 🎈🎈🎈 Group B: 🎈🎈 Group C: 🎈🎈🎈🎈

🎈🎈🎈

🎈🎈

🎈🎈🎈🎈

1Group A has 3, Group B has 2, Group C has 4.
2The counts are different → unequal groups.
3We cannot write a single multiplication sentence. We can only add: 3 + 2 + 4 = 9.

No — these are unequal groups. Cannot multiply directly.

✍ Practice — Concept 1

Q1 Are these equal groups? Group A: ★★★★ Group B: ★★★★ Group C: ★★★★ Write the multiplication sentence if yes. ▼

Answer Each group has 4 stars. All 3 groups have the same count → equal groups.
Multiplication sentence: 4 × 3 = 12

Q2 Which is a set of equal groups?
(a) 5, 5, 5, 5 (b) 3, 4, 3 (c) 6, 7, 6 ▼

Answer (a) 5, 5, 5, 5 → all the same → equal groups. Multiplication: 5 × 4 = 20.
(b) 3, 4, 3 → not all same → unequal.
(c) 6, 7, 6 → not all same → unequal.
Answer: (a)

Q3 Draw 3 equal groups of 5. Write the multiplication sentence. ▼

Answer

⬛⬛⬛⬛⬛

= 15

3 equal groups of 5. Multiplication sentence: 5 × 3 = 15

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Concept 2 — Multiplication from Equal Groups

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Once we identify equal groups, we can write two things: a repeated addition sentence and a multiplication sentence. Both give the same answer.

▶ How to Write Multiplication from Equal Groups

Count the number of objects in each group → this is the first factor.
Count the number of groups → this is the second factor.
Multiplication sentence: objects per group × number of groups = total.
The total can also be found by repeated addition.

▶ Solved Example 3

Write the repeated addition and the multiplication sentence for: 5 equal groups of 4 flowers.

🌻🌻🌻🌻

= 20

5 groups of 4 flowers = 20 flowers

1Objects per group = 4. Number of groups = 5.
2Repeated addition: 4 + 4 + 4 + 4 + 4 = 20.
3Multiplication: 4 × 5 = 20.

Repeated addition: 4+4+4+4+4 = 20 Multiplication: 4 × 5 = 20

▶ Solved Example 4

A picture shows 6 groups of 7 stars. Write the multiplication sentence and find the total.

1Objects per group = 7. Number of groups = 6.
2Multiplication sentence: 7 × 6 = ?
3Repeated addition: 7+7+7+7+7+7 = 42.

7 × 6 = 42. Total stars = 42.

✍ Practice — Concept 2

Q4 Write repeated addition and multiplication for: 4 groups of 6 mangoes. ▼

Answer Repeated addition: 6 + 6 + 6 + 6 = 24
Multiplication: 6 × 4 = 24

Q5 A picture shows 🚵🚵🚵 / 🚵🚵🚵 / 🚵🚵🚵 / 🚵🚵🚵 / 🚵🚵🚵 (5 groups of 3 cars). Write the multiplication sentence. ▼

Answer Objects per group = 3. Groups = 5.
Repeated addition: 3+3+3+3+3 = 15
Multiplication: 3 × 5 = 15

Q6 How many equal groups of 8 are in 40? Write the multiplication sentence. ▼

Answer 8 + 8 + 8 + 8 + 8 = 40 → 8 is added 5 times.
8 × 5 = 40. There are 5 equal groups of 8 in 40.

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Concept 3 — What is an Array?

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An array is an arrangement of objects in equal rows and equal columns forming a rectangle. Arrays give us a clear visual picture of multiplication.

Row: a horizontal line (left to right). Column: a vertical line (top to bottom).
In any array, every row has the same number of objects, and every column has the same number of objects.

▶ Reading an Array

Count the number of rows.
Count the number of objects in each row (columns).
Multiplication sentence: columns × rows = total OR rows × columns = total. Both give the same product.

▶ Solved Example 5 — Reading a 3 × 4 Array

The array below has 3 rows and 4 columns. Write the multiplication sentence.

⬤

3 rows 4 columns

Row 1 (cyan): 4 dots

Row 2 (gold): 4 dots

Row 3 (green): 4 dots

Total = 3 × 4 = 12

1Count rows: 3 rows.
2Count columns (objects per row): 4 columns.
3Multiplication: 4 × 3 = 12 or 3 × 4 = 12.

3 rows × 4 columns = 12

▶ Solved Example 6 — A 2 × 5 Array

Draw a 2-row, 5-column array and write the multiplication sentence.

⬤

2 rows5 columns

Row 1: 5 dots

Row 2: 5 dots

Total = 2 × 5 = 10

2 × 5 = 10

✍ Practice — Concept 3

Q7 An array has 4 rows and 3 columns. How many objects are there? Write the multiplication sentence. ▼

Answer

⬤

4 rows, 3 columns. Multiplication: 3 × 4 = 12. There are 12 objects.

Q8 Write two multiplication sentences from this array: 5 rows, 2 columns. ▼

Answer

⬤

Reading by rows: 2 × 5 = 10
Reading by columns: 5 × 2 = 10
Both give 10 — commutative property!

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Concept 4 — Writing Multiplication from Arrays

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Every array gives us two multiplication sentences — one by counting rows first, and one by counting columns first. Both always give the same product. This is another way to see the commutative property.

▶ Solved Example 7 — Two Sentences from One Array

Write two multiplication sentences from this 4-row, 6-column array.

⬤

4 rows6 columns

Reading across rows:
6 × 4 = 24

Reading down columns:
4 × 6 = 24

Both give 24 ✓

1Reading rows: 6 in each row, 4 rows → 6 × 4 = 24.
2Reading columns: 4 in each column, 6 columns → 4 × 6 = 24.

6 × 4 = 4 × 6 = 24

▶ Solved Example 8 — Identifying Rows and Columns

An array has 3 rows and 7 columns. Write both multiplication sentences and find the product.

⬤

3 rows7 columns

7 × 3 = 21

3 × 7 = 21

7 × 3 = 3 × 7 = 21

✍ Practice — Concept 4

Q9 Write two multiplication sentences from an array with 5 rows and 4 columns. ▼

Answer

⬤

4 × 5 = 20 and 5 × 4 = 20

Q10 Raju says a 6-row, 3-column array gives 6 × 3 = 18 and also 3 × 6 = 18. Is he correct? ▼

Answer Yes, Raju is correct.
Reading across: 3 per row, 6 rows → 3 × 6 = 18.
Reading down: 6 per column, 3 columns → 6 × 3 = 18.
This shows the commutative property of multiplication.

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Concept 5 — Rows and Columns — Both Ways

```

An array can always be read two ways. Reading across (rows) gives one multiplication sentence. Reading down (columns) gives the other. The product is always the same.

▶ Solved Example 9 — Same Array, Two Readings

Show how a 5 × 6 array can be read two ways.

⬤

Reading across rows:
6 in each row, 5 rows
6 × 5 = 30

Reading down columns:
5 in each column, 6 columns
5 × 6 = 30

6 × 5 = 5 × 6 = 30

✍ Practice — Concept 5

Q11 An array has 8 rows and 3 columns. Write both multiplication sentences and check that both give the same product. ▼

Answer Reading across: 3 per row, 8 rows → 3 × 8 = 24.
Reading down: 8 per column, 3 columns → 8 × 3 = 24.
Both give 24. ✓

Q12 True or False: A 9-row, 4-column array and a 4-row, 9-column array give different products. ▼

Answer False. Both give the same product.
9 × 4 = 36 and 4 × 9 = 36.
This is the commutative property — the order of factors does not change the product.

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Concept 6 — Word Problems using Equal Groups and Arrays

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Many real-life situations involve equal groups or arrays. We identify the size of each group and the number of groups, then multiply.

▶ Solved Example 10 — Equal Groups Problem

A teacher arranges chairs into 6 equal rows with 8 chairs in each row. How many chairs are there in all?

1Chairs per row = 8. Number of rows = 6.
2This is an array: 6 rows × 8 columns.
3Multiplication: 8 × 6 = 48.

💺

There are 48 chairs in all.

▶ Solved Example 11 — Find Total from Groups

A sweet shop packs 9 sweets in each box. There are 7 boxes. How many sweets in total?

1Sweets per box = 9. Number of boxes = 7.
2Multiplication: 9 × 7 = 63.
3Check: 9+9+9+9+9+9+9 = 63. ✓

There are 63 sweets in total.

✍ Practice — Concept 6

Q13 A garden has 7 rows of plants with 6 plants in each row. How many plants are there? ▼

Answer Plants per row = 6. Rows = 7.
Multiplication: 6 × 7 = 42
There are 42 plants in the garden.

Q14 A stamp album has 8 pages. Each page holds 5 stamps. How many stamps can the album hold? ▼

Answer Stamps per page = 5. Pages = 8.
Multiplication: 5 × 8 = 40
The album can hold 40 stamps.

Q15 A box of chocolates is arranged as an array of 4 rows and 9 columns. Write both multiplication sentences and find the total chocolates. ▼

Answer Reading across: 9 × 4 = 36
Reading down: 4 × 9 = 36
Total chocolates = 36

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★ Key Points to Remember

Equal groups have the same number of objects in every group. Multiplication applies only to equal groups.
Multiplication sentence from equal groups: objects per group × number of groups = product.
An array is an arrangement of objects in equal rows and equal columns.
Every array gives two multiplication sentences (rows × columns and columns × rows) — both give the same product.
This is the commutative property: a × b = b × a.
In a row, objects are arranged left to right. In a column, objects are arranged top to bottom.

Exam Style — Class 2 & 3

5 Questions on Equal Groups & Arrays

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Q1 Are these equal groups? Write the multiplication sentence if yes.
Group 1: 🍌🍌🍌🍌 Group 2: 🍌🍌🍌🍌 Group 3: 🍌🍌🍌🍌 Group 4: 🍌🍌🍌🍌 Group 5: 🍌🍌🍌🍌 ▼

Answer Each group has 4 mangoes. All 5 groups are equal.
Multiplication sentence: 4 × 5 = 20
Total mangoes = 20.

Q2 Draw an array for 3 × 8. Write both multiplication sentences. ▼

Answer 3 rows, 8 columns:

⬤

8 × 3 = 24 and 3 × 8 = 24

Q3 An array has 42 objects arranged in 6 rows. How many columns does it have? Write the multiplication sentence. ▼

Answer Total = 42. Rows = 6. Let columns = c.
Multiplication: c × 6 = 42.
We know 7 × 6 = 42, so c = 7.
Multiplication sentence: 7 × 6 = 42. The array has 7 columns.

Q4 A farmer plants saplings in 9 rows with 6 saplings in each row. How many saplings does he plant in all? Draw a simple array and write the multiplication sentence. ▼

Answer Saplings per row = 6. Rows = 9.

🌿

Multiplication sentence: 6 × 9 = 54
The farmer plants 54 saplings in all.

Q5 Priya says a 5-row, 7-column array gives 5 × 7 = 35. Rohan says the same array gives 7 × 5 = 35. Who is correct? Explain. ▼

Answer Both are correct.
Priya reads across: 7 per row, 5 rows → 7 × 5 = 35.
Rohan reads down: 5 per column, 7 columns → 5 × 7 = 35.
The same array gives two valid multiplication sentences, both with product 35.
This is the commutative property of multiplication: a × b = b × a.

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📄 Free Worksheet

Practise equal groups, arrays, rows and columns, and word problems with our free printable worksheet for Class 2 and 3. No login required.

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