Use of Place Value in Multiplication - Practice Exercises

Master place value concepts: understanding place value, breaking apart numbers, and multiplying by place value with these 10 detailed exercises.

Exercises 1 to 5
1 Understanding Place Value: 234
Exercise 1
Break apart the number 234 into its place values.
Definition:

Place value: The value of a digit based on its position in a number (ones, tens, hundreds, etc.).

Place value breakdown method:
  1. Identify the digit in each place: 2 in hundreds, 3 in tens, 4 in ones
  2. Calculate the value of each digit: 2×100, 3×10, 4×1
  3. Express as expanded form: 200 + 30 + 4
Hundreds place
2 × 100 = 200
Tens place
3 × 10 = 30
Ones place
4 × 1 = 4
Step 1: Identify each digit and its place

2 is in the hundreds place, 3 is in the tens place, 4 is in the ones place

Step 2: Calculate the value of each digit

2 × 100 = 200, 3 × 10 = 30, 4 × 1 = 4

Step 3: Write in expanded form

234 = 200 + 30 + 4

Step 4: Verify

200 + 30 + 4 = 234 ✓

234 = 200 + 30 + 4
Final answer:

234 = 200 + 30 + 4

Applied rules:

• Place value: Each position has a value (1, 10, 100, 1000...)

• Digit value: Digit × Place value = Actual value

• Expanded form: Sum of each digit's actual value

2 Breaking Apart for Multiplication: 34×5
Exercise 2
Use place value to multiply 34×5.
Definition:

Breaking apart for multiplication: Using place value to multiply each digit separately.

Break apart
34 = 30 + 4
Multiply each
30×5 + 4×5
Total
150 + 20
Step 1: Break apart by place value

34 = 30 + 4 (3 tens and 4 ones)

Step 2: Multiply each part by 5

30 × 5 = 150, 4 × 5 = 20

Step 3: Add the partial products

150 + 20 = 170

Step 4: Verify

Check that 34 × 5 = 170

34 × 5 = 170
Final answer:

34 × 5 = 170

Applied rules:

• Distributive property: a(b+c) = ab + ac

• Place value: 34 = 3 tens + 4 ones

• Partial products: Multiply each place value separately

3 Understanding Place Value: 567
Exercise 3
Break apart the number 567 into its place values.
Definition:

Place value system: Each position in a number has a value that is 10 times the position to its right.

Hundreds place
5 × 100 = 500
Tens place
6 × 10 = 60
Ones place
7 × 1 = 7
Step 1: Identify each digit and its place

5 is in the hundreds place, 6 is in the tens place, 7 is in the ones place

Step 2: Calculate the value of each digit

5 × 100 = 500, 6 × 10 = 60, 7 × 1 = 7

Step 3: Write in expanded form

567 = 500 + 60 + 7

Step 4: Verify

500 + 60 + 7 = 567 ✓

567 = 500 + 60 + 7
Final answer:

567 = 500 + 60 + 7

Applied rules:

• Place value: Each position has a value (1, 10, 100, 1000...)

• Digit value: Digit × Place value = Actual value

• Expanded form: Sum of each digit's actual value

4 Breaking Apart for Multiplication: 48×6
Exercise 4
Use place value to multiply 48×6.
Definition:

Breaking apart for multiplication: Using place value to multiply each digit separately.

Break apart
48 = 40 + 8
Multiply each
40×6 + 8×6
Total
240 + 48
Step 1: Break apart by place value

48 = 40 + 8 (4 tens and 8 ones)

Step 2: Multiply each part by 6

40 × 6 = 240, 8 × 6 = 48

Step 3: Add the partial products

240 + 48 = 288

Step 4: Verify

Check that 48 × 6 = 288

48 × 6 = 288
Final answer:

48 × 6 = 288

Applied rules:

• Distributive property: a(b+c) = ab + ac

• Place value: 48 = 4 tens + 8 ones

• Partial products: Multiply each place value separately

5 Understanding Place Value: 789
Exercise 5
Break apart the number 789 into its place values.
Definition:

Place value system: Each position in a number has a value that is 10 times the position to its right.

Hundreds place
7 × 100 = 700
Tens place
8 × 10 = 80
Ones place
9 × 1 = 9
Step 1: Identify each digit and its place

7 is in the hundreds place, 8 is in the tens place, 9 is in the ones place

Step 2: Calculate the value of each digit

7 × 100 = 700, 8 × 10 = 80, 9 × 1 = 9

Step 3: Write in expanded form

789 = 700 + 80 + 9

Step 4: Verify

700 + 80 + 9 = 789 ✓

789 = 700 + 80 + 9
Final answer:

789 = 700 + 80 + 9

Applied rules:

• Place value: Each position has a value (1, 10, 100, 1000...)

• Digit value: Digit × Place value = Actual value

• Expanded form: Sum of each digit's actual value

Exercises 6 to 10
6 Breaking Apart for Multiplication: 53×7
Exercise 6
Use place value to multiply 53×7.
Definition:

Breaking apart for multiplication: Using place value to multiply each digit separately.

Break apart
53 = 50 + 3
Multiply each
50×7 + 3×7
Total
350 + 21
Step 1: Break apart by place value

53 = 50 + 3 (5 tens and 3 ones)

Step 2: Multiply each part by 7

50 × 7 = 350, 3 × 7 = 21

Step 3: Add the partial products

350 + 21 = 371

Step 4: Verify

Check that 53 × 7 = 371

53 × 7 = 371
Final answer:

53 × 7 = 371

Applied rules:

• Distributive property: a(b+c) = ab + ac

• Place value: 53 = 5 tens + 3 ones

• Partial products: Multiply each place value separately

7 Understanding Place Value: 456
Exercise 7
Break apart the number 456 into its place values.
Definition:

Place value system: Each position in a number has a value that is 10 times the position to its right.

Hundreds place
4 × 100 = 400
Tens place
5 × 10 = 50
Ones place
6 × 1 = 6
Step 1: Identify each digit and its place

4 is in the hundreds place, 5 is in the tens place, 6 is in the ones place

Step 2: Calculate the value of each digit

4 × 100 = 400, 5 × 10 = 50, 6 × 1 = 6

Step 3: Write in expanded form

456 = 400 + 50 + 6

Step 4: Verify

400 + 50 + 6 = 456 ✓

456 = 400 + 50 + 6
Final answer:

456 = 400 + 50 + 6

Applied rules:

• Place value: Each position has a value (1, 10, 100, 1000...)

• Digit value: Digit × Place value = Actual value

• Expanded form: Sum of each digit's actual value

8 Breaking Apart for Multiplication: 67×4
Exercise 8
Use place value to multiply 67×4.
Definition:

Breaking apart for multiplication: Using place value to multiply each digit separately.

Break apart
67 = 60 + 7
Multiply each
60×4 + 7×4
Total
240 + 28
Step 1: Break apart by place value

67 = 60 + 7 (6 tens and 7 ones)

Step 2: Multiply each part by 4

60 × 4 = 240, 7 × 4 = 28

Step 3: Add the partial products

240 + 28 = 268

Step 4: Verify

Check that 67 × 4 = 268

67 × 4 = 268
Final answer:

67 × 4 = 268

Applied rules:

• Distributive property: a(b+c) = ab + ac

• Place value: 67 = 6 tens + 7 ones

• Partial products: Multiply each place value separately

9 Understanding Place Value: 823
Exercise 9
Break apart the number 823 into its place values.
Definition:

Place value system: Each position in a number has a value that is 10 times the position to its right.

Hundreds place
8 × 100 = 800
Tens place
2 × 10 = 20
Ones place
3 × 1 = 3
Step 1: Identify each digit and its place

8 is in the hundreds place, 2 is in the tens place, 3 is in the ones place

Step 2: Calculate the value of each digit

8 × 100 = 800, 2 × 10 = 20, 3 × 1 = 3

Step 3: Write in expanded form

823 = 800 + 20 + 3

Step 4: Verify

800 + 20 + 3 = 823 ✓

823 = 800 + 20 + 3
Final answer:

823 = 800 + 20 + 3

Applied rules:

• Place value: Each position has a value (1, 10, 100, 1000...)

• Digit value: Digit × Place value = Actual value

• Expanded form: Sum of each digit's actual value

10 Breaking Apart for Multiplication: 79×8
Exercise 10
Use place value to multiply 79×8.
Definition:

Breaking apart for multiplication: Using place value to multiply each digit separately.

Break apart
79 = 70 + 9
Multiply each
70×8 + 9×8
Total
560 + 72
Step 1: Break apart by place value

79 = 70 + 9 (7 tens and 9 ones)

Step 2: Multiply each part by 8

70 × 8 = 560, 9 × 8 = 72

Step 3: Add the partial products

560 + 72 = 632

Step 4: Verify

Check that 79 × 8 = 632

79 × 8 = 632
Final answer:

79 × 8 = 632

Applied rules:

• Distributive property: a(b+c) = ab + ac

• Place value: 79 = 7 tens + 9 ones

• Partial products: Multiply each place value separately

Complete Summary: Use of Place Value in Multiplication

Place Value Concepts 🧮

📊
Place Value Positions
Ones (1), Tens (10), Hundreds (100), Thousands (1000)
Breaking Apart Numbers
234 = 200 + 30 + 4
Multiplication Strategy
Multiply each place value separately
Combine Results
Add all partial products together
Verify Answer
Check that the sum equals the original number
Place Value Breakdown
234
Original
200
Hundreds
30
Tens
4
Ones
234
Total
Place Value Multiplication
34×5 = (30+4)×5 = 30×5 + 4×5 = 150+20 = 170
Breaking apart makes multiplication easier!
Master Place Value! 🌟
abc = a×100 + b×10 + c×1
Place Value Formula
Key Definitions:

Place Value: The value of a digit based on its position in a number

Expanded Form: Writing a number as the sum of each digit's value

Partial Products: Products found by multiplying each place value separately

Distributive Property: a(b+c) = ab + ac

Breaking Apart: Separating a number by its place values

Place Value Method:
  1. Identify each digit: Determine which digit is in each place
  2. Calculate digit value: Multiply each digit by its place value
  3. Express in expanded form: Write as a sum of each digit's value
  4. For multiplication: Multiply each part separately, then add
Tip 1: Remember the place value pattern: 1, 10, 100, 1000...
Tip 2: When multiplying, break apart first, then multiply each part.
Tip 3: Always verify that your expanded form adds up to the original number.
Tip 4: Practice with different numbers to build place value fluency.
Common Errors: Forgetting place values, misidentifying digit positions, incorrect multiplication.
Success Strategy: Practice place value identification, check work systematically.
Place Value Properties:

• Base 10 System: Each place value is 10 times the previous

• Position Matters: The same digit has different values in different positions

• Zero Placeholder: Zeros maintain place value position

• Distributive Property: Essential for breaking apart for multiplication

Place Value Concepts & Strategies
234 = 2×100 + 3×10 + 4×1
Place Value Example
Place Value Levels:

Ones Place: Value of 1 (rightmost digit)

Tens Place: Value of 10 (second from right)

Hundreds Place: Value of 100 (third from right)

Thousands Place: Value of 1000 (fourth from right)

Breaking Apart for Multiplication:
  1. Identify place values: Separate the number by position
  2. Multiply each part: Multiply each place value by the multiplier
  3. Add results: Sum all partial products
  4. Verify: Check that the answer is reasonable
Strategy 1: Start with simpler numbers to build understanding.
Strategy 2: Use visual models to represent place values.
Strategy 3: Always verify with estimation (23×4 ≈ 20×4 = 80).
Strategy 4: Practice with different multi-digit numbers.
Key Concepts: Place value understanding is fundamental to all arithmetic operations.
Real World: Place value applies to money, measurements, and data analysis.
Real-World Applications:

• Money: $234 = $200 + $30 + $4

• Measurements: 234 cm = 200 cm + 30 cm + 4 cm

• Data: Population counts, scores, quantities

• Estimation: Rounding to nearest place value for quick calculations

Questions & Answers

Question: Why do we need to break apart numbers by place value when multiplying? Can't we just multiply normally?

Answer: Breaking apart by place value helps you understand why multiplication works:

  • Understanding: It shows why the standard algorithm works
  • Accuracy: Reduces errors by handling smaller, simpler multiplications
  • Flexibility: Gives you a backup method when standard algorithm seems difficult
  • Foundation: Builds the base for more complex multiplication later

While you can multiply "normally," understanding place value makes you more flexible and confident with numbers. It's like knowing how a car engine works - you can still drive without it, but understanding helps when things go wrong!

Plus, this method helps you catch mistakes in your standard multiplication.

Question: My child is struggling with identifying place values. How can I help them practice this concept?

Answer: Here are practical ways to practice place value:

  • Use manipulatives: Base-10 blocks to physically represent hundreds, tens, and ones
  • Practice with money: $2.34 has 2 dollars, 3 dimes, 4 pennies
  • Write expanded form: Regularly write numbers like 543 = 500 + 40 + 3
  • Play games: "What number am I?" where you describe place values

Start with smaller numbers and gradually increase difficulty. Focus on the pattern: ones, tens, hundreds, thousands.

Practice regularly with real-world examples like prices, ages, or distances to make it meaningful.

Question: How can I help my students see the connection between place value and multiplication?

Answer: The connection is the distributive property:

  • Concrete example: 23×4 = (20+3)×4 = 20×4 + 3×4
  • Visual models: Use base-10 blocks to show 2 tens and 3 ones each multiplied by 4
  • Real-world connection: If you buy 4 packs of 23 items, you get 4×20 + 4×3 items
  • Pattern recognition: Show that this works for any multi-digit number

Emphasize that place value breaking apart is just applying the distributive property. This makes multiplication of larger numbers manageable and helps students understand the "why" behind the "how."

The better students understand place value, the more successful they'll be with multiplication algorithms.