Expanded Notation with Decimals: Mastering Decimal Place Value in 4th Grade

Learn to write decimal numbers in expanded notation: break down decimals into their place value components including tenths and hundredths.

Decimal Expanded Notation Worksheets: Problems 1 to 3
1 Tenths Place
Worksheet 1
Write 3.7 in expanded notation.
Definition:

Decimal expanded notation: Writing a decimal number as the sum of each digit multiplied by its place value, including tenths and hundredths.

Method for decimal expanded notation:
  1. Identify the place value of each digit (ones, tenths, hundredths)
  2. Multiply each digit by its place value
  3. Write the sum of these products
Ones
3
× 1 = 3
Decimal Point
.
Tenths
7
× 1/10 = 0.7
3
.
7
3.7 = 3 + 0.7
Additional Examples for Practice:

• 2.4 = 2 + 0.4

• 5.9 = 5 + 0.9

• 1.3 = 1 + 0.3

Step 1: Identify place values

In 3.7: 3 is in ones place, 7 is in tenths place

Step 2: Multiply each digit by its place value

3 × 1 = 3; 7 × 1/10 = 0.7

Step 3: Write the sum

Combine all values: 3 + 0.7

3.7 = 3 + 0.7
Final answer:

3.7 = 3 + 0.7

Applied rules:

• Decimal point: Separates whole numbers from fractional parts

• Tenths place: First digit after decimal point represents tenths (1/10)

• Expanded form: Express number as sum of place values

2 Tenths and Hundredths
Worksheet 2
Write 4.25 in expanded notation.
Definition:

Decimal with hundredths: When a decimal has two digits after the decimal point, the second digit is in the hundredths place.

Ones
4
× 1 = 4
Decimal Point
.
Tenths
2
× 1/10 = 0.2
Hundredths
5
× 1/100 = 0.05
4
.
2
5
4.25 = 4 + 0.2 + 0.05
Additional Examples for Practice:

• 1.34 = 1 + 0.3 + 0.04

• 6.78 = 6 + 0.7 + 0.08

• 2.56 = 2 + 0.5 + 0.06

Step 1: Identify place values

In 4.25: 4 is in ones place, 2 in tenths, 5 in hundredths

Step 2: Multiply each digit by its place value

4 × 1 = 4; 2 × 1/10 = 0.2; 5 × 1/100 = 0.05

Step 3: Write the sum

Combine all values: 4 + 0.2 + 0.05

4.25 = 4 + 0.2 + 0.05
Final answer:

4.25 = 4 + 0.2 + 0.05

Applied rules:

• Hundredths place: Second digit after decimal represents hundredths (1/100)

• Place value progression: Each position is 1/10 of the previous one

• Decimal precision: More digits after decimal mean more precise values

3 Decimal with Zero
Worksheet 3
Write 7.03 in expanded notation.
Definition:

Decimal with zero in tenths: When the tenths place has a zero, it still holds a place in the decimal number and is included in expanded form.

Ones
7
× 1 = 7
Decimal Point
.
Tenths
0
× 1/10 = 0.0
Hundredths
3
× 1/100 = 0.03
7
.
0
3
7.03 = 7 + 0.0 + 0.03
Additional Examples for Practice:

• 5.08 = 5 + 0.0 + 0.08

• 9.02 = 9 + 0.0 + 0.02

• 3.07 = 3 + 0.0 + 0.07

Step 1: Identify place values

In 7.03: 7 is in ones place, 0 in tenths, 3 in hundredths

Step 2: Handle the zero

Even though 0 × 1/10 = 0.0, we include it in the expanded form to maintain decimal place structure

Step 3: Calculate and write

7 × 1 = 7; 0 × 1/10 = 0.0; 3 × 1/100 = 0.03

7.03 = 7 + 0.0 + 0.03
Final answer:

7.03 = 7 + 0.0 + 0.03

Applied rules:

• Zero inclusion: Include zeros in expanded notation to maintain decimal place structure

• Place holding: Zeros hold positions even when they contribute zero value

• Completeness: Every digit position should be represented

Decimal Place Value and Expanded Notation Rules
3.45 = 3 + 0.4 + 0.05
Decimal Expanded Notation
Ones Place
Digit × 1
Whole units
Tenths Place
Digit × 1/10
One-tenth parts
Hundredths Place
Digit × 1/100
One-hundredth parts
Decimal Point
Separates whole from fractional
Separator symbol
Key definitions:

Decimal Number: A number that contains a decimal point separating the whole number part from the fractional part.

Tenths: The first position after the decimal point (1/10 of a whole unit).

Hundredths: The second position after the decimal point (1/100 of a whole unit).

Decimal Place Value: The value of each digit based on its position relative to the decimal point.

Complete methodology:
  1. Identify digits: Look at each digit in the decimal number
  2. Determine place values: Identify the value of each position (ones, tenths, hundredths)
  3. Multiply: Multiply each digit by its place value
  4. Add: Sum all the products to get expanded form
  5. Verify: Check that your expanded form adds up to the original number
Tip 1: Remember that tenths are larger than hundredths (0.1 > 0.01).
Tip 2: Each decimal place is 1/10 of the previous one.
Tip 3: Don't skip zeros - they hold important decimal place positions.
Tip 4: Practice with smaller decimals first, then move to larger ones.
Common errors: Forgetting to include zeros in expanded form, confusing tenths with hundredths, misplacing decimal points.
Memorization trick: "Decimal places get smaller: tenths (0.1), hundredths (0.01), thousandths (0.001)..."
Decimal Expanded Notation Worksheets: Problems 4 to 5
4 Word Problem Application
Worksheet 4
Sarah measured the length of her pencil as 5.42 inches. Write this measurement in expanded notation and explain what each part means.
Definition:

Decimal word problem application: Converting real-world measurements to expanded notation helps understand their precision and component parts.

Ones
5
× 1 = 5
Decimal Point
.
Tenths
4
× 1/10 = 0.4
Hundredths
2
× 1/100 = 0.02
5
.
4
2
5.42 = 5 + 0.4 + 0.02
Additional Examples for Practice:

• Length: 3.75 cm = 3 + 0.7 + 0.05 cm

• Weight: 8.29 kg = 8 + 0.2 + 0.09 kg

• Money: $6.34 = $6 + $0.3 + $0.04

Step 1: Identify the decimal number

The measurement is 5.42 inches

Step 2: Break down by place value

5 in ones place, 4 in tenths place, 2 in hundredths place

Step 3: Calculate expanded form

5 × 1 = 5; 4 × 1/10 = 0.4; 2 × 1/100 = 0.02

Step 4: Interpret the result

This means 5.42 inches = 5 inches + 0.4 inches + 0.02 inches

5.42 = 5 + 0.4 + 0.02
Final answer:

5.42 = 5 + 0.4 + 0.02

This means Sarah's pencil is 5 inches plus 4 tenths of an inch plus 2 hundredths of an inch long.

Applied rules:

• Real-world context: Decimal notation helps measure precise quantities

• Component analysis: Breaking down decimals reveals their structure

• Verification: The sum of components equals the original number

5 Mixed Practice
Worksheet 5
Convert the following decimals to expanded notation: 2.6, 8.05, 1.37, and 4.90.
Definition:

Mixed practice: Working with various decimal numbers helps solidify understanding of place value concepts.

Problem Set Solutions:
Problem 1: 2.6

2.6 = 2 + 0.6

(2 ones + 6 tenths)

Problem 2: 8.05

8.05 = 8 + 0.0 + 0.05

(8 ones + 0 tenths + 5 hundredths)

Problem 3: 1.37

1.37 = 1 + 0.3 + 0.07

(1 one + 3 tenths + 7 hundredths)

Problem 4: 4.90

4.90 = 4 + 0.9 + 0.00

(4 ones + 9 tenths + 0 hundredths)

Step 1: Process each decimal individually

Follow the same method for each number: identify place values, multiply by place value, sum components

Step 2: Verify each conversion

Check that the expanded form adds up to the original decimal

Step 3: Notice patterns

Decimals with trailing zeros (like 4.90) still include those zeros in expanded form

2.6 = 2 + 0.6
8.05 = 8 + 0.0 + 0.05
1.37 = 1 + 0.3 + 0.07
4.90 = 4 + 0.9 + 0.00
Final answers:

2.6 = 2 + 0.6

8.05 = 8 + 0.0 + 0.05

1.37 = 1 + 0.3 + 0.07

4.90 = 4 + 0.9 + 0.00

Applied rules:

• Consistent method: Same process applies to all decimal numbers

• Zero preservation: Zeros maintain the structure of decimal place value

• Verification: Each expanded form should equal the original decimal

Decimal Expanded Notation Summary
3.45 = 3 + 0.4 + 0.05
Decimal Expanded Notation Formula
Key definitions:

Decimal Number: A number that contains a decimal point separating the whole number part from the fractional part.

Decimal Place Value: The value of a digit based on its position relative to the decimal point (ones, tenths, hundredths, etc.).

Expanded Form: A decimal number written as the sum of each digit multiplied by its place value.

Tenths: The first position after the decimal point, representing 1/10 of a whole unit.

Hundredths: The second position after the decimal point, representing 1/100 of a whole unit.

Complete methodology:
  1. Read the decimal: Identify each digit and its position relative to the decimal point
  2. Assign place values: Determine the value each digit position represents (ones, tenths, hundredths)
  3. Multiply digits by place values: Calculate each digit's contribution
  4. Write as addition: Combine all components with plus signs
  5. Verify: Check that the sum equals the original decimal
Tip 1: Use a place value chart to organize your work systematically.
Tip 2: Remember: each decimal place is 1/10 of the previous one (tenths, hundredths, thousandths...).
Tip 3: Practice reading decimals aloud to reinforce place value understanding.
Tip 4: Check your work by adding the expanded components to see if you get the original decimal.
Common errors: Forgetting to include zeros in expanded form, confusing tenths with hundredths, misplacing decimal points.
Memory aid: "Decimal places get smaller as you move right: tenths (0.1), hundredths (0.01), thousandths (0.001)...".
Decimal Place Value System:

• Ones place: 1 = 1 (to the left of decimal)

• Tenths place: 1 = 0.1 (first after decimal)

• Hundredths place: 1 = 0.01 (second after decimal)

• Pattern: Each place value is 1/10 of the previous one

• Decimal point: Separates whole numbers from fractional parts

Simple Decimal
2.3 = 2 + 0.3
Ones + Tenths
Decimal with Hundredths
4.56 = 4 + 0.5 + 0.06
Ones + Tenths + Hundredths
Decimal with Zero
7.04 = 7 + 0.0 + 0.04
Ones + Tenths + Hundredths
Mixed Practice
1.80 = 1 + 0.8 + 0.00
Ones + Tenths + Hundredths

Questions & Answers

Question: Why do we need to learn expanded notation with decimals? When will I ever use it in real life?

Answer: Great question! Decimal expanded notation is incredibly useful in many real-life situations:

  • Money calculations: Understanding $4.75 means $4 + $0.70 + $0.05
  • Measurements: Understanding 3.45 meters means 3 meters + 45 centimeters
  • Science experiments: Precise measurements often require decimal understanding
  • Shopping: Comparing prices and calculating change

Expanded notation builds the foundation for understanding how decimal numbers work and prepares you for more advanced math concepts.

Question: My child gets confused about tenths versus hundredths. How can I help them understand the difference?

Answer: Here are effective strategies to help distinguish tenths from hundredths:

  • Position counting: Tenths is the first digit after the decimal, hundredths is the second
  • Size comparison: Show that 0.1 (one tenth) is much larger than 0.01 (one hundredth)
  • Visual aids: Use grids divided into 10 or 100 parts to show the difference
  • Real examples: Use money ($0.10 vs $0.01) to demonstrate the size difference

Example: In 0.37, the 3 is in the tenths place (0.3 = 3 tenths) and the 7 is in the hundredths place (0.07 = 7 hundredths).

Practice with examples like 0.4 vs 0.04 to reinforce the difference.

Question: Are there more decimal places beyond hundredths? What comes next?

Answer: Yes! The decimal place value system continues beyond hundredths:

  • Thousandths: Third digit after decimal (1/1000 or 0.001)
  • Ten-thousandths: Fourth digit after decimal (1/10000 or 0.0001)
  • Pattern continues: Each place is 1/10 of the previous one

Example: In 0.1234:

1 is in tenths place (0.1)

2 is in hundredths place (0.02)

3 is in thousandths place (0.003)

4 is in ten-thousandths place (0.0004)

For now, focus on tenths and hundredths, but know that the pattern continues indefinitely.