Terminating and Recurring Decimals — AC9M8N03

• Recognise terminating and recurring decimal forms.
• Predict decimal behaviour from a fraction in simplest form.
• Use recurring notation accurately.
• Distinguish rational recurring decimals from irrational non-recurring decimals.

Recall equivalent fractions, simplifying fractions, prime factorisation and fraction-to-decimal division.

For a fraction in simplest form, the decimal terminates exactly when the denominator has no prime factors other than 2 and/or 5. This works because powers of 10 contain only factors 2 and 5.

If another prime factor remains, the decimal repeats. For example, 7/25=0.28 terminates, while 7/24=0.291666… recurs because 24=2³×3 contains a factor 3.

Digital tools can expose longer repeating patterns, but the denominator structure explains why the pattern terminates or recurs.

• Testing an unsimplified denominator. Simplify the fraction first.
• All non-terminating decimals are irrational. Recurring decimals are rational.
• Any denominator containing 2 or 5 terminates. It must contain only 2s and 5s after simplification.
• Writing a few repeated digits without notation proves recurrence. Identify the repeating block clearly.

1. Predict whether 3/8 terminates.
2. Simplify 9/30 and predict the decimal type.
3. Classify 5/12 without long division.
4. Write 0.272727… with recurring notation.

1. Classify 7/20, 7/24 and 11/50.
2. Simplify and classify 14/35.
3. Explain why 13/125 terminates.
4. Explain why 4/15 recurs.
5. Write 2/9 as a recurring decimal.
6. Distinguish 0.121212… from 0.1010010001….
7. Create one fraction that terminates and one that recurs, and justify both without division.

A student says 18/45 must recur because 45 contains factor 3. Explain why the conclusion is wrong and give the correct classification using simplest form.

1. When does a rational decimal terminate? When its simplified denominator has only prime factors 2 and/or 5.
2. Why simplify first? Common factors can remove primes that would otherwise give the wrong prediction.
3. Are recurring decimals rational? Yes.
4. What is the role of a digital tool? It can display/verify a pattern; denominator structure supplies the mathematical reason.

1. Without dividing, determine whether 21/60 terminates or recurs.
   Simplify first, then factor the denominator.
2. Explain why 7/25 terminates but 7/24 recurs.
   Compare prime factors in the denominators.
3. Classify 0.142857142857… and explain why its form is rational.
   A repeating block means recurring decimal.

• I simplify before classifying.
• I use denominator prime factors.
• I write recurring notation.
• I distinguish recurring from irrational decimal behaviour.

Exit ticket: Why can a denominator containing 3 still produce a terminating decimal before simplification?

Australian Curriculum v9.0 — AC9M8N03: recognise terminating and recurring decimals, using digital tools as appropriate.

Victorian Curriculum F–10 Version 2.0 — Level 8, VC2M8N03: Partial.

NSW Mathematics K–10 Syllabus (2022) — Stage 4, MA4-FRC-C-01; MAO-WM-01: Partial.

• AC9M8N01 — irrational numbers
• AC9M8N04 — rational operations

• Australian Curriculum v9.0
• Victorian Curriculum F–10 Version 2.0
• NSW Mathematics K–10 Syllabus (2022)

Curriculum equivalents for Terminating and recurring decimals, using digital tools as appropriate

Mapped skill: recognise terminating and recurring decimals, using digital tools as appropriate

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.