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Multiplication

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Chapter Notes

Overview
This chapter explores multiplication in depth, building on basic facts to develop flexible strategies for multiplying numbers. It uses real-life contexts like a dairy farm, butter packets, and milk production to show why multiplication is useful. The big idea is that multiplication can be done in many ways — by breaking numbers apart, doubling and halving, using nearest multiples, or following the standard algorithm — and understanding these strategies helps you solve problems quickly and accurately.
Key Concepts (13)
Multiplication
Multiplication is repeated addition of the same number. For example, 3 × 4 means adding 3 four times: 3 + 3 + 3 + 3 = 12.
e.g. If one cow gives 5 litres of milk, then 3 cows give 3 × 5 = 15 litres.
Multiplicand
The number that is being multiplied. In 5 × 3 = 15, the multiplicand is 5.
e.g. In 12 × 8, the multiplicand is 12.
Multiplier
The number by which we multiply. In 5 × 3 = 15, the multiplier is 3.
e.g. In 12 × 8, the multiplier is 8.
Product
The result obtained after multiplying two or more numbers. In 5 × 3 = 15, the product is 15.
e.g. The product of 6 and 7 is 42.
Order of numbers in multiplication
Changing the order of numbers being multiplied does not change the product. This is called the commutative property of multiplication.
e.g. 2 × 3 = 6 and 3 × 2 = 6. Similarly, 8 × 5 = 40 and 5 × 8 = 40.
Multiplication by 10s and 100s
When we multiply a number by 10, each digit moves one place to the left and a zero is added. When we multiply by 100, each digit moves two places to the left and two zeros are added.
e.g. 4 × 10 = 40, 20 × 10 = 200, 3 × 100 = 300, 10 × 100 = 1000.
Multiplication by 1,000
When we multiply a number by 1,000, each digit moves three places to the left and three zeros are added.
e.g. 2 × 1,000 = 2,000, 5 × 1,000 = 5,000, 10 × 1,000 = 10,000.
Doubling and Halving
A strategy where we double one number and halve the other to make multiplication easier. The product remains the same.
e.g. 18 × 5 = 9 × 10 = 90. Here 18 is halved to 9 and 5 is doubled to 10.
Nearest Multiple
A strategy where we round one number to the nearest ten or hundred, multiply, and then add or subtract the extra part.
e.g. 4 × 19 = 4 × 20 – 4 = 80 – 4 = 76. 14 × 21 = 14 × 20 + 14 = 280 + 14 = 294.
Breaking numbers into parts (Distributive Property)
We can break a number into tens and ones, multiply each part separately, and then add the results.
e.g. 35 × 12 = (30 + 5) × (10 + 2) = 30×10 + 30×2 + 5×10 + 5×2 = 300 + 60 + 50 + 10 = 420.
Place value in multiplication
When multiplying, we multiply each digit of one number by each digit of the other, keeping track of place values (ones, tens, hundreds, etc.).
e.g. 268 × 4: 4 × 8 ones = 32 ones, 4 × 6 tens = 24 tens = 240, 4 × 2 hundreds = 8 hundreds = 800. Total = 800 + 240 + 32 = 1,072.
Multiplication by 0
Any number multiplied by 0 gives 0. For example, 9 × 0 = 0 and 0 × 9 = 0.
e.g. If there are 0 cows, they give 0 litres of milk.
Number pairs
Any two adjacent numbers in a row or a column of a grid are called number pairs.
e.g. In a grid, 8 and 32 are a number pair if they are next to each other.
Formulas & Equations (8)
a × b = b × a
Reads as: a times b equals b times a
Variables: a and b are any two numbers
Use when: When you need to rearrange numbers to make multiplication easier
⚠ Students think order matters and that 3 × 5 is different from 5 × 3, but the product is the same.
a × 10 = a with one zero added
Reads as: a times 10 equals a with one zero added at the end
Variables: a is any whole number
Use when: When multiplying any number by 10
⚠ Forgetting to add the zero, e.g., writing 4 × 10 = 4 instead of 40.
a × 100 = a with two zeros added
Reads as: a times 100 equals a with two zeros added at the end
Variables: a is any whole number
Use when: When multiplying any number by 100
⚠ Adding only one zero, e.g., writing 3 × 100 = 30 instead of 300.
a × 1,000 = a with three zeros added
Reads as: a times 1,000 equals a with three zeros added at the end
Variables: a is any whole number
Use when: When multiplying any number by 1,000
⚠ Adding only two zeros, e.g., writing 5 × 1,000 = 500 instead of 5,000.
a × 0 = 0
Reads as: a times 0 equals 0
Variables: a is any number
Use when: When any number is multiplied by zero
⚠ Students sometimes think a × 0 = a, but the correct product is always 0.
If a × b = c, then (a ÷ 2) × (b × 2) = c
Reads as: If a times b equals c, then half of a times double of b also equals c
Variables: a and b are numbers, c is the product
Use when: When one number is even and you want to simplify multiplication, especially with 5, 25, 50
⚠ Students forget to apply the operation to both numbers, e.g., only halving one number without doubling the other.
a × (b + c) = a × b + a × c
Reads as: a times the sum of b and c equals a times b plus a times c
Variables: a, b, c are any numbers
Use when: When breaking a number into parts to multiply easily
⚠ Forgetting to multiply a with both b and c, e.g., writing 4 × (10 + 5) = 4 × 10 + 5 instead of 4 × 10 + 4 × 5.
a × (b – c) = a × b – a × c
Reads as: a times the difference of b and c equals a times b minus a times c
Variables: a, b, c are any numbers
Use when: When using the nearest multiple strategy, e.g., 4 × 19 = 4 × (20 – 1)
⚠ Forgetting to subtract a × c, e.g., writing 4 × 19 = 4 × 20 – 1 instead of 4 × 20 – 4 × 1.
How It Works
Multiplying by 10, 100, 1,000 using place value
  1. Identify the number to be multiplied.
  2. Count the number of zeros in the multiplier (10 has 1 zero, 100 has 2 zeros, 1,000 has 3 zeros).
  3. Write the original number and add that many zeros to its right.
  4. The new number is the product.
💡 The zeros are added at the end, not in the middle. For example, 45 × 10 = 450, not 405.
Doubling and Halving strategy
  1. Look at the two numbers to be multiplied.
  2. Choose one number to halve (it should be even) and the other to double.
  3. Write the new pair: half of the first number and double the second number.
  4. Multiply the new pair. The product is the same as the original.
💡 You must apply both operations — halve one AND double the other. Doing only one changes the product.
Nearest Multiple strategy
  1. Round one number to the nearest ten or hundred that is easy to multiply.
  2. Multiply the other number by this rounded number.
  3. If you rounded up, subtract the extra part (the difference × the other number).
  4. If you rounded down, add the missing part (the difference × the other number).
💡 Always check whether you added or subtracted the extra part. For 4 × 19, 19 is rounded to 20 (up), so subtract 4 × 1 = 4.
Breaking numbers into parts (Distributive Property) — Nida's method
  1. Write each number as a sum of tens and ones (e.g., 35 = 30 + 5, 12 = 10 + 2).
  2. Draw a grid or table with the parts of one number as rows and the parts of the other as columns.
  3. Multiply each row part by each column part and fill the grid.
  4. Add all the products in the grid to get the final answer.
💡 Make sure to multiply every part of the first number with every part of the second number. Do not miss any combination.
Standard multiplication algorithm (Mili's father's method)
  1. Write the larger number on top and the smaller number below, aligned to the right.
  2. Multiply the top number by the ones digit of the bottom number, writing the result below the line.
  3. Multiply the top number by the tens digit of the bottom number, writing the result one place to the left (add a zero at the end).
  4. Add the two results together to get the final product.
💡 When multiplying by the tens digit, the result must be shifted one place to the left (or a zero is added at the end). This is because you are actually multiplying by 10 times that digit.
Solving word problems using multiplication
  1. Read the problem carefully and identify what is being asked.
  2. Find the two quantities that need to be multiplied (e.g., number of rows × seats per row).
  3. Choose a multiplication strategy (breaking numbers, doubling/halving, nearest multiple, or standard algorithm).
  4. Perform the multiplication and write the answer with the correct unit.
💡 Always check if the answer makes sense in the real-world context. For example, number of people cannot be a fraction.
Exam Traps — Don't Make These Mistakes
⚠ Forgetting to add zeros when multiplying by 10, 100, 1,000
✗ 4 × 10 = 4, 3 × 100 = 30, 5 × 1,000 = 500
→ The student forgot to add the correct number of zeros. Multiplying by 10 adds one zero, by 100 adds two zeros, by 1,000 adds three zeros.
✓ 4 × 10 = 40, 3 × 100 = 300, 5 × 1,000 = 5,000
⚠ Confusing doubling and halving — only doing one operation
✗ 18 × 5 = 9 × 5 = 45 (only halved 18, forgot to double 5)
→ The doubling and halving strategy requires both operations to keep the product the same. Halving only one number changes the product.
✓ 18 × 5 = (18 ÷ 2) × (5 × 2) = 9 × 10 = 90
⚠ Nearest multiple — adding instead of subtracting when rounding up
✗ 4 × 19 = 4 × 20 + 4 = 80 + 4 = 84
→ When we round 19 up to 20, we have added an extra 1. So we must subtract 4 × 1 = 4, not add it.
✓ 4 × 19 = 4 × 20 – 4 = 80 – 4 = 76
⚠ Breaking numbers — missing one of the partial products
✗ 35 × 12 = (30 × 10) + (5 × 2) = 300 + 10 = 310
→ When breaking both numbers, you must multiply each part of the first number by each part of the second number. The missing products are 30 × 2 = 60 and 5 × 10 = 50.
✓ 35 × 12 = (30 × 10) + (30 × 2) + (5 × 10) + (5 × 2) = 300 + 60 + 50 + 10 = 420
⚠ Misplacing digits in the standard algorithm
✗ When multiplying 268 × 4, they write 4 × 8 = 32, write 2 and carry 3, then 4 × 6 = 24, add 3 = 27, write 7 and carry 2, then 4 × 2 = 8, add 2 = 10, answer 1072. But they forget to align place values in multi-digit multiplication.
→ For a single-digit multiplier, the method is correct. But for two-digit multipliers like 453 × 13, students often write the second partial product (10 × 453) directly below without shifting it one place left, leading to an incorrect sum.
✓ When multiplying by the tens digit, always shift the result one place to the left (or add a zero at the end). For 453 × 13: 3 × 453 = 1359, then 10 × 453 = 4530 (shifted left), then add: 1359 + 4530 = 5889.
⚠ Thinking a × 0 = a
✗ 9 × 0 = 9
→ Any number multiplied by zero equals zero, not the number itself. Zero groups of something means nothing.
✓ 9 × 0 = 0
Compare & Contrast
Strategy for simplifying multiplication
Doubling and Halving
Nearest Multiple
Doubling and halving changes both numbers (one doubled, one halved) while keeping the product the same. Nearest multiple changes only one number to a round number, then adjusts by adding or subtracting the extra part.
Method of multiplying two numbers
Breaking numbers into parts (Distributive Property)
Standard multiplication algorithm
Breaking numbers into parts explicitly shows all partial products (e.g., 30×10, 30×2, 5×10, 5×2) and then adds them. The standard algorithm compresses these steps by multiplying digit by digit and carrying over, but the underlying mathematics is the same.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (3 marks), 1 LA (5 marks)
Most asked: Multiplication by 10, 100, 1,000 · Doubling and Halving strategy · Nearest Multiple strategy · Breaking numbers into parts (Distributive Property) · Word problems involving multiplication (e.g., rows × seats, cost × quantity)
Don't skip: Word problems requiring multiplication with multi-digit numbers (e.g., 35 rows × 42 seats, 297 cows × 18 kg fodder); Applying doubling and halving or nearest multiple to find products efficiently
One-Line Revision (read aloud before the exam)
  1. Multiplication is repeated addition: 3 × 4 = 4 + 4 + 4 = 12.
  2. Changing the order of numbers in multiplication does not change the product: 2 × 3 = 3 × 2 = 6.
  3. Any number multiplied by 0 equals 0: 9 × 0 = 0.
  4. Multiplying by 10 adds one zero: 4 × 10 = 40.
  5. Multiplying by 100 adds two zeros: 3 × 100 = 300.
  6. Multiplying by 1,000 adds three zeros: 5 × 1,000 = 5,000.
  7. Doubling and halving: halve one number and double the other, product stays the same: 18 × 5 = 9 × 10 = 90.
  8. Nearest multiple: round to nearest ten, multiply, then add or subtract the extra: 4 × 19 = 4 × 20 – 4 = 76.
  9. Break numbers into tens and ones to multiply easily: 35 × 12 = (30 × 10) + (30 × 2) + (5 × 10) + (5 × 2) = 420.
  10. In the standard algorithm, when multiplying by the tens digit, shift the result one place left (add a zero).
  11. Always check if your answer makes sense in a word problem — number of people, litres, etc. cannot be fractions.
  12. Use the commutative property to rearrange numbers for easier multiplication: 5 × 18 = 18 × 5.
  13. The product of two numbers is the same even if you use different strategies — choose the easiest one.
  14. In a grid method, multiply every part of the first number by every part of the second number.

MCQs — 10 Questions with Answers

Q1. What is 7 × 10?

  • A. 70 ✓
  • B. 17
  • C. 77
  • D. 107

Answer: A — When multiplying by 10, add one zero to the number, so 7 × 10 = 70.

Q2. Which statement shows the Commutative Property?

  • A. 12 × 5 = 5 × 12 ✓
  • B. 12 + 5 = 5 + 12
  • C. 12 − 5 = 5 − 12
  • D. 12 ÷ 5 = 5 ÷ 12

Answer: A — The Commutative Property of multiplication means the order of numbers doesn't change the answer.

Q3. How many zeros will be added when 5 is multiplied by 100?

  • A. One zero
  • B. Two zeros ✓
  • C. Three zeros
  • D. No zeros

Answer: B — When multiplying by 100, we add two zeros, so 5 × 100 = 500.

Q4. Solve using Doubling and Halving: 16 × 5 = ?

  • A. 70
  • B. 80 ✓
  • C. 90
  • D. 100

Answer: B — Half of 16 is 8, and 8 × 10 = 80, so 16 × 5 = 80.

Q5. A dairy farm packs 12 pouches in each box. How many pouches are in 10 boxes?

  • A. 22 pouches
  • B. 102 pouches
  • C. 120 pouches ✓
  • D. 112 pouches

Answer: C — 12 × 10 = 120 pouches (when multiplying by 10, add one zero).

Q6. Use Nearest Multiple to find 3 × 19

  • A. 54
  • B. 57 ✓
  • C. 60
  • D. 63

Answer: B — 3 × 20 = 60, then subtract 3 × 1 = 3, so 60 − 3 = 57.

Q7. Which multiplication problem is the same as 6 × 14?

  • A. 14 × 7
  • B. 14 × 6 ✓
  • C. 7 × 14
  • D. 12 × 14

Answer: B — By Commutative Property, 6 × 14 = 14 × 6 because the order doesn't change the answer.

Q8. Priya jogs 4 km every day. In 100 days, how many km will she jog?

  • A. 104 km
  • B. 400 km ✓
  • C. 40 km
  • D. 4 km

Answer: B — 4 × 100 = 400 km (when multiplying by 100, add two zeros).

Q9. If you multiply 8 by 1,000, how many zeros will the answer have?

  • A. One zero
  • B. Two zeros
  • C. Three zeros ✓
  • D. Four zeros

Answer: C — 8 × 1,000 = 8,000, which has three zeros after the 8.

Q10. A school auditorium has 35 rows with 42 seats in each row. Which method works best to find total seats?

  • A. Breaking 42 into 40 + 2, then adding products
  • B. Doubling 35 and halving 42
  • C. Using Nearest Multiple of 40
  • D. All methods above work ✓

Answer: D — All three strategies (breaking into parts, doubling and halving, and nearest multiple) can be used flexibly to find 35 × 42.

Flashcards

What happens when you multiply any number by 10?

The number becomes 10 times larger and we add one zero to the right of the number.

What is the Commutative Property of Multiplication?

The order of numbers in multiplication doesn't matter — 6 × 13 equals 13 × 6.

How do you multiply by 100?

Add two zeros to the right of the number, or the number becomes 100 times larger.

What is the Doubling and Halving strategy?

Double one number and halve the other to make multiplication easier, especially with 5 and 25.

How do you use Nearest Multiple strategy?

Round one number to the nearest 10 or 100, multiply, then add or subtract the difference.

Solve: 4 × 19 using Nearest Multiple

4 × 20 = 80, then subtract 4 × 1 = 4, so 80 − 4 = 76.

What is 18 × 5 using Halving and Doubling?

Half of 18 is 9, and 9 × 10 = 90, so 18 × 5 = 90.

Multiply: 30 × 20

3 × 2 = 6, then add two zeros = 600.

What is the product when you multiply any number by 1,000?

Add three zeros to the right of the number, making it 1,000 times larger.

Solve using breaking into parts: 14 × 21

14 × 20 = 280, then 14 × 1 = 14, so 280 + 14 = 294.

Important Board Questions

What is 9 × 100? [1 mark]

When multiplying by 100, add two zeros to the right of the number.

Fill in the blank: 15 × 6 = 6 × ___ [1 mark]

Use the Commutative Property where the order of numbers doesn't change the product.

Solve 22 × 5 using the Doubling and Halving strategy. Show your steps. [2 marks]

Half of 22 is 11. Then multiply 11 by 10 to get your answer.

A family produces 35 kg of kitchen waste in 1 month. How much waste will they produce in 12 months? Use any multiplication strategy and show your working. [3 marks]

You need to find 12 × 35. You can use Nearest Multiple (12 × 40 − 12 × 5) or breaking into parts (12 × 30 + 12 × 5).

True or False: 28 × 25 can be solved easily using the Doubling and Halving strategy. Explain why or why not with an example. [2 marks]

The Doubling and Halving strategy works best with numbers like 5 and 25. Half of 28 is 14, and 14 × 50 = 700.

Assertion-Reason Questions for Multiplication

Q1. Assertion (A):

When we multiply 30 by 20, we get 600.

Reason (R):

We can write 30 × 20 as (3 × 10) × (2 × 10) = 3 × 2 × 10 × 10 = 6 × 100 = 600.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason correctly explains WHY 30 × 20 = 600 by breaking down the multiplication into place values and showing the step-by-step calculation.

Q2. Assertion (A):

The commutative property says 8 × 5 = 5 × 8.

Reason (R):

Multiplication always gives the same answer when we change the order of the numbers.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both assertion and reason are true, and the reason correctly explains the commutative property. The order doesn't matter in multiplication.

Q3. Assertion (A):

Halving and doubling strategy works very well for multiplying with 25.

Reason (R):

Because 25 is half of 50, and we can easily double and halve to make the calculation simpler.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true—halving and doubling works great with 25. But the reason is not the correct explanation. It works because 25 × 4 = 100, making our calculation easier, not just because 25 is half of 50.

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Case-Based Questions (CBQ) for Multiplication

Case Study:

Rajesh's mother runs a small milk shop in the village. She sells milk pouches in different arrangements. One arrangement has 12 rows of pouches with 5 pouches in each row. Another arrangement has 10 rows with 6 pouches in each row. Rajesh wants to know which arrangement has more pouches and by how much.

Q1. How many pouches are in the first arrangement (12 rows with 5 pouches each)?

  • A. 17 pouches
  • B. 60 pouches ✓
  • C. 65 pouches
  • D. 7 pouches

Answer: B

Q2. How many pouches are in the second arrangement (10 rows with 6 pouches each)?

  • A. 60 pouches ✓
  • B. 16 pouches
  • C. 4 pouches
  • D. 100 pouches

Answer: A

Q3. Which arrangement has more pouches and by how much?

  • A. First arrangement has 5 more pouches
  • B. Second arrangement has 10 more pouches
  • C. Both have the same number of pouches ✓
  • D. First arrangement has 10 fewer pouches

Answer: C

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