Example: $2345.678$

Read as: "Two thousand, three hundred forty-five and six hundred seventy-eight thousandths"

• Tenths place: First digit after the decimal point → $\frac{1}{10}$ or $0.1$
• Hundredths place: Second digit after the decimal point → $\frac{1}{100}$ or $0.01$
• Thousandths place: Third digit after the decimal point → $\frac{1}{1000}$ or $0.001$

Moving Left (×10):

Each place value is 10 times greater than the place to its right.

$$0.001 \times 10 = 0.01 \times 10 = 0.1 \times 10 = 1 \times 10 = 10$$

Moving Right (÷10):

Each place value is 10 times smaller than the place to its left.

$$10 \div 10 = 1 \div 10 = 0.1 \div 10 = 0.01 \div 10 = 0.001$$

Example:

In the number $555.555$:

• The $5$ in the hundreds place = $500$ (which is 10 times the tens place)
• The $5$ in the tens place = $50$ (which is 10 times the ones place)
• The $5$ in the ones place = $5$ (which is 10 times the tenths place)
• The $5$ in the tenths place = $0.5$ (which is 10 times the hundredths place)
• The $5$ in the hundredths place = $0.05$ (which is 10 times the thousandths place)
• The $5$ in the thousandths place = $0.005$

Definition:

The regular way we write numbers with digits and a decimal point.

Example:

$3.456$

Definition:

Writing the decimal in words.

Important: The decimal point is read as "and"

Example:

$3.456$ = "Three and four hundred fifty-six thousandths"

Definition:

Breaking down the number by showing the value of each digit.

Formula: Add the value of each digit

Example:

$3.456 = 3 + 0.4 + 0.05 + 0.006$

Definition:

Breaking down the number using fractions to show place values.

Formula: Multiply each digit by its fractional place value

Example:

$$3.456 = (3 \times 1) + (4 \times \frac{1}{10}) + (5 \times \frac{1}{100}) + (6 \times \frac{1}{1000})$$

Definition:

Putting parts together to make a decimal number.

Example:

$2 + 0.3 + 0.04 = 2.34$

$1 + 0.5 + 0.06 + 0.007 = 1.567$

Definition:

Breaking a decimal into its parts in multiple ways.

Example: Decompose $2.45$ in different ways

Way 1: $2 + 0.4 + 0.05$

Way 2: $2 + 0.45$

Way 3: $1 + 1 + 0.4 + 0.05$

Way 4: $1 + 1.45$

Way 5: $2.4 + 0.05$

Steps to Round Decimals:

1. Find the rounding place (ones, tenths, hundredths, etc.)
2. Look at the digit to the right of the rounding place
3. If the digit is 5 or greater (5, 6, 7, 8, 9) → Round UP (add 1)
4. If the digit is less than 5 (0, 1, 2, 3, 4) → Round DOWN (stay the same)
5. Drop all digits after the rounding place

Rule:

Look at the tenths place (first digit after the decimal point)

Examples:

$4.3$ → Look at 3 (less than 5) → Round DOWN → $4$

$7.8$ → Look at 8 (5 or greater) → Round UP → $8$

$12.5$ → Look at 5 (5 or greater) → Round UP → $13$

Rule:

Look at the hundredths place (second digit after the decimal point)

Examples:

$3.42$ → Look at 2 (less than 5) → Round DOWN → $3.4$

$8.67$ → Look at 7 (5 or greater) → Round UP → $8.7$

$5.95$ → Look at 5 (5 or greater) → Round UP → $6.0$ or $6$

Rule:

Look at the thousandths place (third digit after the decimal point)

Examples:

$2.345$ → Look at 5 (5 or greater) → Round UP → $2.35$

$6.782$ → Look at 2 (less than 5) → Round DOWN → $6.78$

$9.999$ → Look at 9 (5 or greater) → Round UP → $10.00$ or $10$

What is a Decimal Number Line?

A number line that shows decimals between whole numbers. It helps visualize the size and position of decimal numbers.

Example: Between 0 and 1

The number line is divided into 10 equal parts.

$0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0$

Example: Between 0 and 0.1

The number line is divided into 10 equal parts (showing hundredths).

$0.00, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10$

• Count the number of spaces between whole numbers or marks
• Each space represents an equal part
• Use the number line to compare decimals (numbers to the right are greater)
• Number lines help visualize rounding by showing which whole number is closer

Common Models:

• Base-10 Blocks:
  • Large cube = 1 whole
  • Flat = 0.1 (one tenth)
  • Rod = 0.01 (one hundredth)
  • Small cube = 0.001 (one thousandth)
• Grid Models: 10×10 grid where each small square = 0.01
• Shaded Diagrams: Showing parts of a whole shaded to represent decimals

Example: Grid Model

If a 10×10 grid has 45 squares shaded:

45 out of 100 squares = $\frac{45}{100}$ = $0.45$

Given: $12.378$

Solutions:

Standard Form: $12.378$

Word Form: Twelve and three hundred seventy-eight thousandths

Expanded Form: $10 + 2 + 0.3 + 0.07 + 0.008$

Expanded Form (fractions): $(1 \times 10) + (2 \times 1) + (3 \times \frac{1}{10}) + (7 \times \frac{1}{100}) + (8 \times \frac{1}{1000})$

Given: $56.847$

Solutions:

Nearest whole: $57$ (look at 8 → round up)

Nearest tenth: $56.8$ (look at 4 → round down)

Nearest hundredth: $56.85$ (look at 7 → round up)

Given: What is the value of the digit 7 in $45.673$?

Solution:

The digit $7$ is in the hundredths place

Its value is: $7 \times 0.01 = 0.07$ or $\frac{7}{100}$