Using Place Value Strategies to Multiply - Practice Exercises

Master place value multiplication: expanded form, partial products, and standard algorithm with these 10 detailed exercises.

Exercises 1 to 5
1 Expanded Form Strategy: 23×4
Exercise 1
Multiply 23×4 using the expanded form strategy.
Definition:

Expanded form strategy: Breaking apart numbers by place value and multiplying each part separately.

Expanded form method:
  1. Break apart the larger number by place value: 23 = 20 + 3
  2. Multiply each part by the 1-digit number: 20×4 and 3×4
  3. Add the partial products together
Expanded form
23 = 20 + 3
Partial products
20×4 + 3×4
Total
80 + 12
Step 1: Write in expanded form

23 = 20 + 3

Step 2: Multiply each part by 4

20 × 4 = 80

3 × 4 = 12

Step 3: Add the partial products

80 + 12 = 92

Step 4: Verify

Check that 23 × 4 = 92

23 × 4 = 92
Final answer:

23 × 4 = 92

Applied rules:

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

• Place value: 23 = 2 tens + 3 ones

• Partial products: Multiply each place value separately

2 Partial Products: 34×6
Exercise 2
Multiply 34×6 using the partial products strategy.
Definition:

Partial products: Multiplying each digit of the multi-digit number by the 1-digit number, keeping place value.

Expanded form
34 = 30 + 4
Partial products
30×6 + 4×6
Total
180 + 24
Step 1: Write in expanded form

34 = 30 + 4

Step 2: Multiply each part by 6

30 × 6 = 180

4 × 6 = 24

Step 3: Add the partial products

180 + 24 = 204

Step 4: Verify

Check that 34 × 6 = 204

34 × 6 = 204
Final answer:

34 × 6 = 204

Applied rules:

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

• Place value: 34 = 3 tens + 4 ones

• Partial products: Multiply each place value separately

3 Standard Algorithm: 52×3
Exercise 3
Multiply 52×3 using the standard algorithm.
Definition:

Standard algorithm: Multiplying digits from right to left, carrying when needed.

Ones place
2×3 = 6
Tens place
5×3 = 15
Total
156
Step 1: Multiply ones place

2 × 3 = 6

Step 2: Multiply tens place

5 × 3 = 15

Step 3: Combine results

15 tens + 6 ones = 150 + 6 = 156

Step 4: Verify

Check that 52 × 3 = 156

52 × 3 = 156
Final answer:

52 × 3 = 156

Applied rules:

• Place value: Each digit has its place value

• Standard algorithm: Multiply from right to left

• Carrying: When product > 9, carry to next place

4 Expanded Form: 45×7
Exercise 4
Multiply 45×7 using the expanded form strategy.
Definition:

Expanded form strategy: Breaking apart numbers by place value and multiplying each part separately.

Expanded form
45 = 40 + 5
Partial products
40×7 + 5×7
Total
280 + 35
Step 1: Write in expanded form

45 = 40 + 5

Step 2: Multiply each part by 7

40 × 7 = 280

5 × 7 = 35

Step 3: Add the partial products

280 + 35 = 315

Step 4: Verify

Check that 45 × 7 = 315

45 × 7 = 315
Final answer:

45 × 7 = 315

Applied rules:

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

• Place value: 45 = 4 tens + 5 ones

• Partial products: Multiply each place value separately

5 Partial Products: 68×4
Exercise 5
Multiply 68×4 using the partial products strategy.
Definition:

Partial products: Multiplying each digit of the multi-digit number by the 1-digit number, keeping place value.

Expanded form
68 = 60 + 8
Partial products
60×4 + 8×4
Total
240 + 32
Step 1: Write in expanded form

68 = 60 + 8

Step 2: Multiply each part by 4

60 × 4 = 240

8 × 4 = 32

Step 3: Add the partial products

240 + 32 = 272

Step 4: Verify

Check that 68 × 4 = 272

68 × 4 = 272
Final answer:

68 × 4 = 272

Applied rules:

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

• Place value: 68 = 6 tens + 8 ones

• Partial products: Multiply each place value separately

Exercises 6 to 10
6 Standard Algorithm: 73×5
Exercise 6
Multiply 73×5 using the standard algorithm.
Definition:

Standard algorithm: Multiplying digits from right to left, carrying when needed.

Ones place
3×5 = 15
Tens place
7×5 = 35
Carry and total
365
Step 1: Multiply ones place

3 × 5 = 15, write 5, carry 1

Step 2: Multiply tens place

7 × 5 = 35, add carried 1 = 36

Step 3: Combine results

36 tens + 5 ones = 365

Step 4: Verify

Check that 73 × 5 = 365

73 × 5 = 365
Final answer:

73 × 5 = 365

Applied rules:

• Place value: Each digit has its place value

• Standard algorithm: Multiply from right to left

• Carrying: When product > 9, carry to next place

7 Expanded Form: 84×6
Exercise 7
Multiply 84×6 using the expanded form strategy.
Definition:

Expanded form strategy: Breaking apart numbers by place value and multiplying each part separately.

Expanded form
84 = 80 + 4
Partial products
80×6 + 4×6
Total
480 + 24
Step 1: Write in expanded form

84 = 80 + 4

Step 2: Multiply each part by 6

80 × 6 = 480

4 × 6 = 24

Step 3: Add the partial products

480 + 24 = 504

Step 4: Verify

Check that 84 × 6 = 504

84 × 6 = 504
Final answer:

84 × 6 = 504

Applied rules:

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

• Place value: 84 = 8 tens + 4 ones

• Partial products: Multiply each place value separately

8 Partial Products: 92×3
Exercise 8
Multiply 92×3 using the partial products strategy.
Definition:

Partial products: Multiplying each digit of the multi-digit number by the 1-digit number, keeping place value.

Expanded form
92 = 90 + 2
Partial products
90×3 + 2×3
Total
270 + 6
Step 1: Write in expanded form

92 = 90 + 2

Step 2: Multiply each part by 3

90 × 3 = 270

2 × 3 = 6

Step 3: Add the partial products

270 + 6 = 276

Step 4: Verify

Check that 92 × 3 = 276

92 × 3 = 276
Final answer:

92 × 3 = 276

Applied rules:

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

• Place value: 92 = 9 tens + 2 ones

• Partial products: Multiply each place value separately

9 Standard Algorithm: 57×8
Exercise 9
Multiply 57×8 using the standard algorithm.
Definition:

Standard algorithm: Multiplying digits from right to left, carrying when needed.

Ones place
7×8 = 56
Tens place
5×8 = 40
Carry and total
456
Step 1: Multiply ones place

7 × 8 = 56, write 6, carry 5

Step 2: Multiply tens place

5 × 8 = 40, add carried 5 = 45

Step 3: Combine results

45 tens + 6 ones = 456

Step 4: Verify

Check that 57 × 8 = 456

57 × 8 = 456
Final answer:

57 × 8 = 456

Applied rules:

• Place value: Each digit has its place value

• Standard algorithm: Multiply from right to left

• Carrying: When product > 9, carry to next place

10 Expanded Form: 49×7
Exercise 10
Multiply 49×7 using the expanded form strategy.
Definition:

Expanded form strategy: Breaking apart numbers by place value and multiplying each part separately.

Expanded form
49 = 40 + 9
Partial products
40×7 + 9×7
Total
280 + 63
Step 1: Write in expanded form

49 = 40 + 9

Step 2: Multiply each part by 7

40 × 7 = 280

9 × 7 = 63

Step 3: Add the partial products

280 + 63 = 343

Step 4: Verify

Check that 49 × 7 = 343

49 × 7 = 343
Final answer:

49 × 7 = 343

Applied rules:

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

• Place value: 49 = 4 tens + 9 ones

• Partial products: Multiply each place value separately

Complete Summary: Place Value Strategies for Multiplication

Multiplication Strategies 🧮

📊
Expanded Form
Break apart by place value: 23 = 20 + 3
Partial Products
Multiply each part: 20×4 + 3×4 = 80 + 12
Standard Algorithm
Multiply right to left: 2×4, then 20×4
Combine Results
Add partial products: 80 + 12 = 92
Verify Answer
Check reasonableness of result
Expanded Form Strategy
23
Original
20+3
Expanded
×4
Multiplied
80+12
Partial
92
Total
Strategy Comparison
Expanded: 23×4 = (20+3)×4 = 80+12 = 92
Standard: 23×4 = 92 (right-to-left)
Both strategies use place value!
Master All Strategies! 🌟
a(b+c) = ab + ac
Distributive Property
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

Standard Algorithm: Traditional method of multiplication from right to left

Distributive Property: Multiplication distributes over addition

Strategy Selection Guide:
  1. Expanded form: Best for understanding place value concepts
  2. Partial products: Good for visual learners and checking work
  3. Standard algorithm: Most efficient for quick calculations
  4. Choose based on: Problem complexity and personal preference
Tip 1: Always break apart by place value first.
Tip 2: Keep track of zeros when multiplying by tens.
Tip 3: Always verify your answer makes sense.
Tip 4: Practice with different strategies to build fluency.
Common Errors: Forgetting place value, missing partial products, incorrect carrying.
Success Strategy: Practice place value understanding, check work systematically.
Multiplication Properties:

• Distributive: a(b+c) = ab + ac

• Associative: (ab)c = a(bc)

• Commutative: ab = ba

• Identity: a×1 = a

Place Value Concepts & Strategies
23 × 4 = (20 + 3) × 4 = 20×4 + 3×4 = 80 + 12 = 92
Expanded Form Example
Strategy Types:

Expanded Form: Break apart by place value and multiply each part

Partial Products: Multiply each place value separately

Standard Algorithm: Traditional right-to-left multiplication

When to Use Each Strategy:
  1. Expanded form: When learning place value concepts
  2. Partial products: When checking standard algorithm work
  3. Standard algorithm: When speed and efficiency are needed
  4. Any strategy: As long as it's accurate and understood
Strategy 1: Start with expanded form to build understanding.
Strategy 2: Use partial products as a bridge to standard algorithm.
Strategy 3: Always verify with estimation (23×4 ≈ 20×4 = 80).

Strategy 4: Practice all strategies to build flexibility.
Key Concepts: All strategies rely on understanding place value.
Real World: These strategies apply to calculating costs, measurements, etc.
Problem-Solving Applications:

• Shopping: Calculating total cost of multiple items

• Area: Finding area of rectangles (length × width)

• Time: Converting between units (minutes to hours)

• Measurement: Converting between metric units

Questions & Answers

Question: Why do we need different strategies for multiplication? Can't we just learn one way?

Answer: Great question! Different strategies serve different purposes:

  • Understanding: Expanded form helps you see why multiplication works
  • Checking: Partial products let you verify your standard algorithm
  • Speed: Standard algorithm is fastest once you master it
  • Flexibility: Some problems are easier with certain strategies

Learning multiple strategies builds deeper understanding and gives you options when one method doesn't work well. It's like having different tools in your math toolbox!

Plus, understanding why the standard algorithm works makes it less likely you'll make mistakes.

Question: My child is getting confused between the different multiplication strategies. How can I help them choose the right one?

Answer: Here's a simple guide to help your child choose:

  • When learning: Start with expanded form to build understanding
  • When checking: Use partial products to verify answers
  • When confident: Standard algorithm for speed
  • When stuck: Go back to expanded form for clarity

Encourage your child to practice all strategies but let them use the one that feels most comfortable for each problem. The goal is accuracy and understanding, not forcing a specific method.

Remind them that all strategies use the same mathematical principles - they're just different ways of organizing the work.

Question: How can I help my students understand the distributive property in relation to these multiplication strategies?

Answer: The distributive property is the foundation of place value strategies:

  • Concrete example: 23×4 = (20+3)×4 = 20×4 + 3×4
  • Visual model: Use arrays to show how (a+b)×c = a×c + b×c
  • 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 numbers

Emphasize that breaking apart numbers by place value (like 23 = 20 + 3) is just applying the distributive property. This makes multiplication of larger numbers manageable.

The distributive property connects all place value multiplication strategies!