7.NS.A.2d 7th Grade The Number System

Terminating vs Repeating Decimals

Convert a rational number to a decimal using long division, and know that the decimal form terminates or eventually repeats.

How to explain it

Students convert rational numbers to decimal form using long division, recognize that the decimal form of a rational number either terminates in zeros or eventually repeats, record repeating decimals with bar notation, and move fluently among fraction, decimal, and percent names for the same value in numeric and real-world settings.

The anchor students hold onto: DIVIDE to turn a fraction into a decimal: numerator ÷ denominator. CLASSIFY the result — it always terminates or repeats; a bar marks the repeating digits. SHIFT the point two places right for a percent.

Worked examples

Example 1 Terminating Decimal

[object Object], → decimal → percent

Step 1Divide: 3 ÷ 8 = 0.375

Step 2The division ends — terminating

Step 3Shift two right: 0.375 = 37.5%

Step 4A: 0.375 = 37.5%

Answer0.375 = 37.5%

Example 2 Repeating Decimal

[object Object], → decimal (bar notation)

Step 1Divide: 5 ÷ 6 = 0.8333…

Step 2The 3 never stops — repeating

Step 3Write a bar over the repeating 3

Step 4A: 0.83 with a bar over the 3

Answer0.833… — bar over the 3

Common mistakes

What students write Moving the point one place: 0.45 = 4.5%

The fix Percent means per 100 — shift TWO places: 0.45 = 45%

Try this Marcus converts 0.45 to a percent and writes 0.45 = 4.5% because "you move the decimal point one place." Find his mistake, then show the correct conversion.

What students write Stopping the division early: 1/3 = 0.3 exactly

The fix 1 ÷ 3 never ends — bar the repeat: 0.333…

Try this Jada divides 1 ÷ 3, stops after one digit, and writes 1/3 = 0.3 exactly. Find her mistake, then write the true decimal form of 1/3.

Teacher tip

Head off the two predictable errors before they happen. First: Percent means per 100 — shift TWO places: 0.45 = 45% Second: 1 ÷ 3 never ends — bar the repeat: 0.333…