Repeating and Terminating Decimals

By the end of this lesson you will be able to:

1. Convert a fraction to a decimal using long division
2. Tell the difference between terminating and recurring decimals
3. Write recurring decimals using dot or bar notation
4. Convert a terminating decimal to a fraction in lowest terms
5. Round off a decimal to any place up to hundred-thousandths

Five Learning Objectives for This Lesson

1. Write any fraction as a decimal by long division
2. Recognise a terminating and a recurring decimal
3. Write recurring decimals with dot or bar notation
4. Convert a terminating decimal back to a fraction in lowest terms
5. Round off decimals to any place value up to hundred-thousandths

The Tailor's Tape: Two Names for One Length

The tailor measured m of cloth. Her digital tape reads 0.75 m.

• Same length — two different names
• How do we get from one form to the other?

The fraction bar already holds the answer.

Activation: What Does a Fraction Mean?

• means 3 ÷ 4 — three divided by four
• means 1 ÷ 8
• means 2 ÷ 5

The fraction bar IS the division sign.

Which number goes inside the long-division bracket? Which goes outside?

Long Division Recall: Dividing Past the Point

Example: 7 ÷ 4

• Write 7.000 and place the decimal point
• rem 2 → digit 1, carry
• Continue until remainder = 0 or pattern appears

This move applies every time a fraction becomes a decimal.

The Rule: Fraction to Decimal

• Numerator a goes inside the bracket
• Denominator b goes outside
• If , write and place the decimal point

Sanity check: proper fraction → decimal must be less than 1.

Worked Examples: Three Fraction-to-Decimal Conversions

All three stop — remainder reaches 0.

Worked: 1/8 Step by Step to Thousandths

— thousandths

Check-In: Write 3/8 as a Decimal

Work this out in your exercise book.

Set up:

• Which number goes inside the bracket?
• Write 3.000 and begin dividing
• Stop when remainder = 0

Answer: 0.375

Two Divisions — Two Very Different Outcomes

• rem 1
• rem 1
• rem 1 …

Remainder 1 returns every time. This never stops.

Terminating vs. Recurring: The Two Families

• Terminates — remainder reaches 0: ,
• Recurs — a remainder keeps returning: ,

Recurring Notation: Dots and Bars

Dot over the repeating digit:

Bar over the repeating block:

Position matters — the dot marks exactly which digits repeat.

Predictive Rule: Will It Terminate or Recur?

• Only 2s and 5s in denominator → terminates
• Any other prime → recurs

Probe: 1/6 — Predict Before You Divide

6 = 2 × 3 — has a 3 → predicts recurs

Divide to verify: 10÷6=1 r4; 40÷6=6 r4; 40÷6=6 r4 …

Dot over the 6 only — the 1 before it does not repeat.

Guided Practice: T (Terminates) or R (Recurs)?

Reversing Chunk 1: Decimal Back to Fraction

The place name of the last digit gives the denominator:

• → hundredths → denominator 100
• → tenths → denominator 10
• → thousandths → denominator 1000

Rule: Count the Decimal Places

Count places = zeros in denominator, then simplify. (Recurring decimals: out of P7 scope.)

Worked: 0.35 as a Fraction

• Two places → denominator 100
• GCD(35, 100) = 5 → answer:

Check: 20 = 2² × 5 → terminates ✓

NCDC exit-task example — practise until you can reproduce it completely.

Check-In: 0.8 as a Fraction

Write 0.8 as a fraction in lowest terms.

• How many decimal places?
• What is the denominator?
• What is the numerator?
• Simplify

Answer:

Why Rounding? The Timber Merchant's Problem

A board measures 1.4375 m by calculation.

The merchant's tape only shows hundredths: 1.43, 1.44, 1.45 …

• Which hundredth is 1.4375 nearest to?

That decision is rounding — choosing the nearest value at the precision you need.

Naming the Place-Value Columns to Hundred-Thousandths

Worked Example: Rounding 3.14159 Two Ways

Thousandths — next digit (ten-thousandths) = 5 → round up:

Hundredths — next digit (thousandths) = 1 → round down:

Look only at the ONE digit immediately after your rounding place.

Worked: The Carry Case — 2.96 to Tenths

Digit after tenths = 6 ≥ 5 → round up the tenths digit (9):

Write 3.0 — the zero marks the tenths place.

Practice: Try These Two Rounding Cases

1. Round () to hundred-thousandths

2. Round m to hundredths

Answers: (1) 0.66667 — 6th digit is 6, round up the 5th. (2) 1.44 — thousandths digit is 7, round up hundredths.

Unscaffolded: Three Tasks, No Hints

Work through all of these in order in your exercise book:

1. Convert to a decimal. Is it terminating or recurring?
2. Write your answer to task 1 using the correct notation.
3. Write as a fraction in lowest terms.

No scaffolding — apply everything from this lesson.

Key Takeaways: What You Can Do Now

• Fraction → decimal: numerator ÷ denominator
• Remainder = 0 → terminates; cycling remainder → recurs
• Dot/bar over the repeating block
• Only 2s/5s in denominator → terminates
• Decimal → fraction: places → power of 10 → simplify
• Rounding: one digit right; ≥5 rounds up