Solved Exercises on Expanded Notation Questions in Grade 4

Master expanded notation questions: place value decomposition, digit representation, and number composition through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic expanded notation question
Exercise 1
What is the expanded notation of 5,728?
Definition:

Expanded notation: Writing a number to show the value of each digit based on its place value, expressed as the sum of each digit multiplied by its place value.

Expanded Notation Method:
  1. Identify the place value of each digit
  2. Write each digit multiplied by its place value
  3. Join all parts with addition signs
Number
5,728
Expanded Notation
(5×1,000) + (7×100) + (2×10) + (8×1)
Step 1: Identify place values

5 is in thousands place: 5 × 1,000 = 5,000

7 is in hundreds place: 7 × 100 = 700

2 is in tens place: 2 × 10 = 20

8 is in ones place: 8 × 1 = 8

Step 2: Write in expanded notation form

5,728 = (5×1,000) + (7×100) + (2×10) + (8×1)

Step 3: Verify the calculation

(5×1,000) + (7×100) + (2×10) + (8×1) = 5,000 + 700 + 20 + 8 = 5,728 ✓

5,728 = (5×1,000) + (7×100) + (2×10) + (8×1)
Final answer:

5,728 = (5×1,000) + (7×100) + (2×10) + (8×1)

Applied rules:

• Place Value Multiplication: Each digit multiplied by its place value

• Expanded Notation Format: (digit × place value) + (digit × place value)...

• Verification: Sum of parts equals original number

2 Expanded notation with zeros
Exercise 2
Write 8,053 in expanded notation.
Definition:

Zero in expanded notation: Even zeros are included as (0 × place value) = 0

Number
8,053
Expanded Notation
(8×1,000) + (0×100) + (5×10) + (3×1)
Simplified
(8×1,000) + (5×10) + (3×1)
Step 1: Identify place values

8 is in thousands place: 8 × 1,000 = 8,000

0 is in hundreds place: 0 × 100 = 0

5 is in tens place: 5 × 10 = 50

3 is in ones place: 3 × 1 = 3

Step 2: Write in expanded notation form

8,053 = (8×1,000) + (0×100) + (5×10) + (3×1)

Step 3: Simplify by omitting zero terms

8,053 = (8×1,000) + (5×10) + (3×1)

Step 4: Verify the calculation

(8×1,000) + (5×10) + (3×1) = 8,000 + 50 + 3 = 8,053 ✓

8,053 = (8×1,000) + (5×10) + (3×1)
Final answer:

8,053 = (8×1,000) + (5×10) + (3×1)

Applied rules:

• Zero Inclusion: Zeros are written as (0 × place value)

• Simplification: Terms with (0 × place value) can be omitted

• Verification: Sum of parts equals original number

3 Converting expanded notation to standard form
Exercise 3
What number is represented by (4×10,000) + (2×1,000) + (0×100) + (7×10) + (9×1)?
Definition:

Standard form: The regular way we write numbers without showing place values

Expanded Notation
(4×10,000) + (2×1,000) + (0×100) + (7×10) + (9×1)
Standard Form
42,079
Step 1: Calculate each part of the expanded notation

(4×10,000) = 40,000

(2×1,000) = 2,000

(0×100) = 0

(7×10) = 70

(9×1) = 9

Step 2: Add all parts together

40,000 + 2,000 + 0 + 70 + 9 = 42,079

Step 3: Verify by converting back to expanded notation

42,079 = (4×10,000) + (2×1,000) + (0×100) + (7×10) + (9×1) ✓

(4×10,000) + (2×1,000) + (0×100) + (7×10) + (9×1) = 42,079
Final answer:

The number is 42,079

Applied rules:

• Place Value Addition: Sum all expanded notation parts

• Digit Placement: Each part corresponds to a specific place

• Verification: Convert back to ensure accuracy

Expanded Notation Rules and Methods
\(5,728 = (5×1,000) + (7×100) + (2×10) + (8×1)\)
Expanded Notation Example
Ones Place
\(d \times 1\)
Digit × 1
Tens Place
\(d \times 10\)
Digit × 10
Hundreds Place
\(d \times 100\)
Digit × 100
Thousands Place
\(d \times 1,000\)
Digit × 1,000
Ten-Thousands Place
\(d \times 10,000\)
Digit × 10,000
Key definitions:

Expanded Notation: A number expressed as the sum of each digit multiplied by its place value

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

Standard Form: The regular way we write numbers (e.g., 5,728)

Expanded Form: Writing a number as the sum of place values (e.g., 5,000 + 700 + 20 + 8)

Expanded Notation: Writing a number as the sum of digits multiplied by their place values (e.g., (5×1,000) + (7×100) + (2×10) + (8×1))

Complete methodology:
  1. Identify positions: Recognize place values (ones, tens, hundreds, thousands, etc.)
  2. Write multiplication: Express each digit as (digit × place value)
  3. Join with addition: Combine all parts with plus signs
  4. Verify: Check that the sum equals the original number
Tip 1: Remember the place value pattern: 1, 10, 100, 1,000, 10,000...
Tip 2: Zeros in expanded notation can be written as (0 × place value) or omitted.
Tip 3: Always verify by adding the expanded parts to get the original number.
Tip 4: Use base-10 blocks to visualize place values for better understanding.
Common errors: Forgetting place values, misplacing digits, confusing expanded form with expanded notation.
Memory aid: Think of expanded notation as "digit times place value".
Formulas to know by heart:

• Expanded Notation: Sum of (digit × place value)

• Place value pattern: 1, 10, 100, 1,000, 10,000...

• Verification: Sum of expanded parts = Original number

• Zero rule: (0 × any place value) = 0

Solution: Exercises 4 to 5
4 Comparing expanded notation
Exercise 4
Which is greater: (3×1,000) + (5×100) + (2×10) + (7×1) or (3×1,000) + (4×100) + (9×10) + (8×1)? Show your work.
Definition:

Comparison in expanded notation: Convert both expressions to standard form to compare

First Expression
(3×1,000) + (5×100) + (2×10) + (7×1)
Equals
3,000 + 500 + 20 + 7 = 3,527
Second Expression
(3×1,000) + (4×100) + (9×10) + (8×1)
Equals
3,000 + 400 + 90 + 8 = 3,498
Step 1: Calculate first expression

(3×1,000) = 3,000

(5×100) = 500

(2×10) = 20

(7×1) = 7

Total: 3,000 + 500 + 20 + 7 = 3,527

Step 2: Calculate second expression

(3×1,000) = 3,000

(4×100) = 400

(9×10) = 90

(8×1) = 8

Total: 3,000 + 400 + 90 + 8 = 3,498

Step 3: Compare the results

3,527 vs 3,498

Since 3,527 > 3,498, the first expression is greater.

Step 4: Alternative method - compare by place value

Thousands: Both have 3,000

Hundreds: 500 > 400, so the first is greater

(3×1,000) + (5×100) + (2×10) + (7×1) = 3,527
(3×1,000) + (4×100) + (9×10) + (8×1) = 3,498
3,527 > 3,498
Final answer:

(3×1,000) + (5×100) + (2×10) + (7×1) is greater than (3×1,000) + (4×100) + (9×10) + (8×1)

Because 3,527 > 3,498

Applied rules:

• Comparison method: Convert to standard form to compare

• Place value comparison: Compare digits from left to right

• Verification: Confirm the comparison is correct

5 Missing digit in expanded notation
Exercise 5
If (6×1,000) + (4×100) + (n×10) + (3×1) = 6,473, what is the value of n?
Definition:

Missing digit problem: Finding an unknown digit in expanded notation

Given
(6×1,000) + (4×100) + (n×10) + (3×1) = 6,473
Calculate known parts
6,000 + 400 + (n×10) + 3 = 6,473
Step 1: Calculate the known parts

(6×1,000) = 6,000

(4×100) = 400

(3×1) = 3

So: 6,000 + 400 + (n×10) + 3 = 6,473

Step 2: Simplify the equation

6,000 + 400 + 3 = 6,403

So: 6,403 + (n×10) = 6,473

Step 3: Solve for n

(n×10) = 6,473 - 6,403 = 70

n×10 = 70

n = 70 ÷ 10 = 7

Step 4: Verify the answer

(6×1,000) + (4×100) + (7×10) + (3×1) = 6,000 + 400 + 70 + 3 = 6,473 ✓

n = 7
(6×1,000) + (4×100) + (7×10) + (3×1) = 6,473
Final answer:

The value of n is 7

Because (6×1,000) + (4×100) + (7×10) + (3×1) = 6,000 + 400 + 70 + 3 = 6,473

Applied rules:

• Algebraic reasoning: Treat unknown as variable to solve

• Place value understanding: Each part contributes to total value

• Verification: Substitute back to confirm answer

Complete Summary: Expanded Notation Questions
\(d_n \times 10^n + d_{n-1} \times 10^{n-1} + ... + d_1 \times 10^1 + d_0 \times 10^0\)
General Expanded Notation Formula
Key definitions:

Expanded Notation: A number expressed as the sum of each digit multiplied by its place value

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

Standard Form: The usual way we write numbers (e.g., 5,728)

Expanded Form: Writing a number as the sum of place values (e.g., 5,000 + 700 + 20 + 8)

Expanded Notation: Writing a number as the sum of digits multiplied by their place values (e.g., (5×1,000) + (7×100) + (2×10) + (8×1))

Digit: Any of the symbols 0-9 used to make numbers

Complete methodology:
  1. Identify positions: Recognize the place value of each digit
  2. Multiplication: Express each digit as (digit × place value)
  3. Addition: Join all parts with plus signs
  4. Verification: Add the expanded parts to confirm they equal the original number
Tip 1: Use the place value pattern: 1, 10, 100, 1,000, 10,000...
Tip 2: Draw a place value chart to organize digits before expanding.
Tip 3: Always verify by calculating the sum of expanded notation parts.
Tip 4: Practice reading numbers aloud to reinforce place value understanding.
Common errors: Confusing expanded form with expanded notation, misplacing digits, forgetting zeros.
Memory aids: Use base-10 blocks, draw charts, remember "digit times place value".
Formulas and Rules to Know:

• Place Value Pattern: ..., 10,000, 1,000, 100, 10, 1

• Expanded Notation: Sum of (digit × place value)

• Zero Rule: (0 × any place value) = 0

• Verification: Sum of expanded parts = Original number

• Format: (digit × place value) + (digit × place value) + ...

Expanded Notation Visualization: Number Breakdown
Exercise 6: Understanding Expanded Notation
Consider the following numbers and their expanded notation breakdowns:
3,456 = (3×1,000) + (4×100) + (5×10) + (6×1)
7,082 = (7×1,000) + (0×100) + (8×10) + (2×1)
9,105 = (9×1,000) + (1×100) + (0×10) + (5×1)

Analysis: The chart shows how numbers decompose into expanded notation components.

  • 3,456 = (3×1,000) + (4×100) + (5×10) + (6×1) (Three thousand four hundred fifty-six)
  • 7,082 = (7×1,000) + (0×100) + (8×10) + (2×1) (Seven thousand eighty-two)
  • 9,105 = (9×1,000) + (1×100) + (0×10) + (5×1) (Nine thousand one hundred five)

Questions & Answers

Question: I don't understand why we need to learn expanded notation. Why can't we just write numbers normally?

Answer: Great question! Expanded notation is crucial for understanding how our number system works:

  • It helps you understand what each digit means in a number
  • It builds a foundation for addition and subtraction with regrouping
  • It prepares you for working with larger numbers and decimals later
  • It makes it easier to compare numbers and understand their magnitude
  • It shows the relationship between digits and their place values

For example, in expanded notation (5×1,000) + (7×100) + (2×10) + (8×1), you can see that the 5 represents 5 thousands (5,000), not just 5. This helps you understand the true value of each digit!

Question: My child gets confused when there are zeros in the expanded notation like (8×1,000) + (0×100) + (5×10) + (3×1). How do I explain this?

Answer: Here's a helpful explanation for zeros in expanded notation:

  • In (8×1,000) + (0×100) + (5×10) + (3×1), the (0×100) means there are 0 hundreds
  • Even though 0×100 = 0, we include it to show that there's a place for hundreds (even if it's empty)
  • This helps us see that the number is 8,053: 8 thousands, 0 hundreds, 5 tens, 3 ones
  • The zero keeps the 5 in the tens place and the 3 in the ones place in the right positions

You can think of it like having an empty box in a row of boxes. Even though the box is empty, it still takes up space and keeps the other items in their correct positions!

Sometimes we simplify by writing (8×1,000) + (5×10) + (3×1), but including the zero version shows the complete place value structure.

Question: What's the difference between expanded form and expanded notation? They seem the same to me.

Answer: While similar, there's an important distinction:

  • Expanded Form: Shows the actual value of each place (e.g., 5,000 + 700 + 20 + 8)
  • Expanded Notation: Shows each digit multiplied by its place value (e.g., (5×1,000) + (7×100) + (2×10) + (8×1))

Expanded Form: 5,728 = 5,000 + 700 + 20 + 8

Expanded Notation: 5,728 = (5×1,000) + (7×100) + (2×10) + (8×1)

Expanded notation explicitly shows the multiplication of each digit by its place value, which makes the relationship between digits and their values clearer. It's like showing the "work" behind how we get each part of the expanded form!