Australian Curriculum v9 / ACiQ Year 8 Mathematics - Unit 1 - Terminating and recurring decimals

Terminating and recurring decimals

Recognise terminating and recurring decimals and convert familiar recurring decimals to fractions using place-value reasoning.

Updated 2026-07-26 - 11 min read

Terminating and recurring decimals is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Recognise terminating and recurring decimals

A rational decimal either terminates or eventually repeats a fixed block of digits. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by recognise terminating and recurring decimals and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Length does not decide the type; a repeating block makes a non-terminating decimal rational.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

Classify 0.625.

Step 1 - identify the governing idea: A rational decimal either terminates or eventually repeats a fixed block of digits.

Step 2 - apply it to this evidence: The decimal ends after three places.

Result: Terminating

The relationship is visible in the working: The decimal ends after three places. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Recurring — It does not agree with the required relationship: The decimal ends after three places.
  • Irrational — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Undefined — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Classify 0.142857142857…

Step 1 - identify the governing idea: A rational decimal either terminates or eventually repeats a fixed block of digits.

Step 2 - apply it to this evidence: The six-digit block 142857 repeats.

Result: Recurring

The relationship is visible in the working: The six-digit block 142857 repeats. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Terminating — It does not agree with the required relationship: The six-digit block 142857 repeats.
  • Irrational — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • An integer — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Which decimal notation represents one third exactly?

Step 1 - identify the governing idea: A rational decimal either terminates or eventually repeats a fixed block of digits.

Step 2 - apply it to this evidence: The digit 3 repeats without end; 0.33 is only an approximation.

Result: 0.333…

The relationship is visible in the working: The digit 3 repeats without end; 0.33 is only an approximation. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 0.33 — It does not agree with the required relationship: The digit 3 repeats without end; 0.33 is only an approximation.
  • 0.3 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0.333 exactly with three digits — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Convert one-digit recurring decimals to fractions

For x = 0.ā, multiply by 10 and subtract the original equation so the recurring tails cancel. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by convert one-digit recurring decimals to fractions and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: The dots mean infinitely many digits: if x = 0.777…, then 10x − x = 7, so x = 7/9.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

Convert 0.555… to a fraction.

Step 1 - identify the governing idea: For x = 0.ā, multiply by 10 and subtract the original equation so the recurring tails cancel.

Step 2 - apply it to this evidence: Let x = 0.555…; then 10x − x = 5, so 9x = 5.

Result: 5/9

The relationship is visible in the working: Let x = 0.555…; then 10x − x = 5, so 9x = 5. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 5/10 — It does not agree with the required relationship: Let x = 0.555…; then 10x − x = 5, so 9x = 5.
  • 555/1000 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1/5 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Convert 0.222… to a fraction.

Step 1 - identify the governing idea: For x = 0.ā, multiply by 10 and subtract the original equation so the recurring tails cancel.

Step 2 - apply it to this evidence: Let x = 0.222…; subtraction gives 9x = 2.

Result: 2/9

The relationship is visible in the working: Let x = 0.222…; subtraction gives 9x = 2. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 1/2 — It does not agree with the required relationship: Let x = 0.222…; subtraction gives 9x = 2.
  • 2/10 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 22/100 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Convert 0.999… to a fraction and simplify.

Step 1 - identify the governing idea: For x = 0.ā, multiply by 10 and subtract the original equation so the recurring tails cancel.

Step 2 - apply it to this evidence: The recurring-decimal method gives 9/9, which equals 1.

Result: 1

The relationship is visible in the working: The recurring-decimal method gives 9/9, which equals 1. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 9/10 — It does not agree with the required relationship: The recurring-decimal method gives 9/9, which equals 1.
  • 99/100 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • A number just below 1 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Compare exact and approximate decimal forms

Compare numbers using equivalent exact forms or enough aligned decimal places, and distinguish truncation from rounding. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by compare exact and approximate decimal forms and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Place value decides the comparison; trailing zeros and recurring notation must be interpreted before comparing.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

Which is larger: 0.7 or 0.6999…?

Step 1 - identify the governing idea: Compare numbers using equivalent exact forms or enough aligned decimal places, and distinguish truncation from rounding.

Step 2 - apply it to this evidence: 0.6999… equals 0.7 exactly.

Result: They are equal

The relationship is visible in the working: 0.6999… equals 0.7 exactly. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 0.7 — It does not agree with the required relationship: 0.6999… equals 0.7 exactly.
  • 0.6999… — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • It cannot be decided — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Round 2.1666… to two decimal places.

Step 1 - identify the governing idea: Compare numbers using equivalent exact forms or enough aligned decimal places, and distinguish truncation from rounding.

Step 2 - apply it to this evidence: The third decimal is 6, so the hundredths digit rounds up.

Result: 2.17

The relationship is visible in the working: The third decimal is 6, so the hundredths digit rounds up. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 2.16 — It does not agree with the required relationship: The third decimal is 6, so the hundredths digit rounds up.
  • 2.166 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2.20 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Order 0.58, 7/12 and 0.58333…

Step 1 - identify the governing idea: Compare numbers using equivalent exact forms or enough aligned decimal places, and distinguish truncation from rounding.

Step 2 - apply it to this evidence: 7/12 has recurring decimal 0.58333…, which is greater than 0.58.

Result: 0.58 < 7/12 = 0.58333…

The relationship is visible in the working: 7/12 has recurring decimal 0.58333…, which is greater than 0.58. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 7/12 < 0.58 < 0.58333… — It does not agree with the required relationship: 7/12 has recurring decimal 0.58333…, which is greater than 0.58.
  • All are equal — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0.58333… < 7/12 < 0.58 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. Classify 0.142857142857…
  2. Convert 0.222… to a fraction.
  3. Round 2.1666… to two decimal places.

Answers

  1. Recurring — The six-digit block 142857 repeats.
  2. 2/9 — Let x = 0.222…; subtraction gives 9x = 2.
  3. 2.17 — The third decimal is 6, so the hundredths digit rounds up.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources