Australian Curriculum v9 / ACiQ Year 9 Mathematics - Unit 1 - Real, rational and irrational numbers

Real, rational and irrational numbers

Recognise rational and irrational numbers, compare real values and use approximations without confusing them with exact forms.

Updated 2026-07-26 - 12 min read

Real, rational and irrational numbers is part of the Year 9 curriculum because students must do more than carry out a familiar calculation. They need to choose a relationship, represent it accurately, explain the result and decide whether it makes sense. The sections below develop those decisions through explicit rules, worked examples, misconception repair and transfer.

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.

Classify rational and irrational numbers

Integers, fractions, terminating decimals and recurring decimals are rational. Values such as √2 and π are irrational because their decimal expansions neither terminate nor recur.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by classify rational and irrational numbers 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: Recurring decimals continue forever but are rational because they can be expressed as fractions.

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 0.272727…

Step 1 - identify the governing idea: A rational number can be written as a fraction of integers with a non-zero denominator; an irrational number cannot.

Step 2 - apply it to this evidence: The block 27 repeats, so the decimal can be written as a fraction.

Result: Rational

The relationship is visible in the working: The block 27 repeats, so the decimal can be written as a fraction. 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:

  • Irrational — It does not agree with the required relationship: The block 27 repeats, so the decimal can be written as a fraction.
  • Not a real number — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Rational only after rounding — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Classify √49 and √50.

Step 1 - identify the governing idea: A rational number can be written as a fraction of integers with a non-zero denominator; an irrational number cannot.

Step 2 - apply it to this evidence: √49 = 7, while √50 is not the square root of a perfect square.

Result: √49 is rational; √50 is irrational

The relationship is visible in the working: √49 = 7, while √50 is not the square root of a perfect square. 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:

  • Both are rational — It does not agree with the required relationship: √49 = 7, while √50 is not the square root of a perfect square.
  • Both are irrational — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • √49 is irrational; √50 is rational — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Which value is irrational: -3, 5/8, 0.125 or π?

Step 1 - identify the governing idea: A rational number can be written as a fraction of integers with a non-zero denominator; an irrational number cannot.

Step 2 - apply it to this evidence: The other three values are integers, fractions or terminating decimals.

Result: π

The relationship is visible in the working: The other three values are integers, fractions or terminating decimals. 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:

  • -3 — It does not agree with the required relationship: The other three values are integers, fractions or terminating decimals.
  • 5/8 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0.125 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Compare and approximate real values

Bounding a square root between consecutive square numbers provides a dependable estimate. Rounding too early can reverse close comparisons or accumulate error.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by compare and approximate real values 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: √10 is between 3.1 and 3.2, but it is not exactly 3.1; use a stated approximation such as √10 ≈ 3.162.

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

Between which consecutive integers does √70 lie?

Step 1 - identify the governing idea: Keep exact forms while reasoning, then use enough decimal places to compare or communicate the required accuracy.

Step 2 - apply it to this evidence: 64 < 70 < 81, so 8 < √70 < 9.

Result: 8 and 9

The relationship is visible in the working: 64 < 70 < 81, so 8 < √70 < 9. 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 and 8 — It does not agree with the required relationship: 64 < 70 < 81, so 8 < √70 < 9.
  • 9 and 10 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 35 and 36 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Order 3.14, π and 22/7 from least to greatest.

Step 1 - identify the governing idea: Keep exact forms while reasoning, then use enough decimal places to compare or communicate the required accuracy.

Step 2 - apply it to this evidence: π ≈ 3.14159 and 22/7 ≈ 3.14286.

Result: 3.14 < π < 22/7

The relationship is visible in the working: π ≈ 3.14159 and 22/7 ≈ 3.14286. 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:

  • π < 3.14 < 22/7 — It does not agree with the required relationship: π ≈ 3.14159 and 22/7 ≈ 3.14286.
  • 22/7 < π < 3.14 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 3.14 < 22/7 < π — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

A square has area 30 cm². Give its side length to 2 decimal places.

Step 1 - identify the governing idea: Keep exact forms while reasoning, then use enough decimal places to compare or communicate the required accuracy.

Step 2 - apply it to this evidence: The side is √30 cm ≈ 5.477… cm, which rounds to 5.48 cm.

Result: 5.48 cm

The relationship is visible in the working: The side is √30 cm ≈ 5.477… cm, which rounds to 5.48 cm. 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:

  • 15.00 cm — It does not agree with the required relationship: The side is √30 cm ≈ 5.477… cm, which rounds to 5.48 cm.
  • 5.47 cm — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 900.00 cm — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Solve problems with exact and approximate values

A problem about a diagonal may naturally produce a square root. Keep the root exact during later calculations and round only the final measured result.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by solve problems with exact and approximate values 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: A finite calculator display is usually a rounded approximation; label it with ≈ and a unit or accuracy.

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

A square has side 5 m. What is its diagonal exactly and to 2 decimal places?

Step 1 - identify the governing idea: Exact forms preserve mathematical information; approximations serve measurement and communication when their accuracy is stated.

Step 2 - apply it to this evidence: Pythagoras gives √(25+25)=√50=5√2.

Result: 5√2 m, approximately 7.07 m

The relationship is visible in the working: Pythagoras gives √(25+25)=√50=5√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:

  • 10 m exactly — It does not agree with the required relationship: Pythagoras gives √(25+25)=√50=5√2.
  • √10 m, approximately 3.16 m — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 25√2 m — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Which is better for an exact algebraic proof: 1/3 or 0.33?

Step 1 - identify the governing idea: Exact forms preserve mathematical information; approximations serve measurement and communication when their accuracy is stated.

Step 2 - apply it to this evidence: 1/3 is exact while 0.33 is a rounded approximation.

Result: 1/3

The relationship is visible in the working: 1/3 is exact while 0.33 is a rounded 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: 1/3 is exact while 0.33 is a rounded approximation.
  • They are exactly identical — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Use 33 instead — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

A calculator gives √7 = 2.64575131. Report to 3 significant figures.

Step 1 - identify the governing idea: Exact forms preserve mathematical information; approximations serve measurement and communication when their accuracy is stated.

Step 2 - apply it to this evidence: The first three significant digits are 2, 6 and 4; the next digit 5 rounds 4 upward.

Result: 2.65

The relationship is visible in the working: The first three significant digits are 2, 6 and 4; the next digit 5 rounds 4 upward. 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:

  • 2.64 — It does not agree with the required relationship: The first three significant digits are 2, 6 and 4; the next digit 5 rounds 4 upward.
  • 2.646 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2.645 — 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 √49 and √50.
  2. Order 3.14, π and 22/7 from least to greatest.
  3. Which is better for an exact algebraic proof: 1/3 or 0.33?

Answers

  1. √49 is rational; √50 is irrational — √49 = 7, while √50 is not the square root of a perfect square.
  2. 3.14 < π < 22/7 — π ≈ 3.14159 and 22/7 ≈ 3.14286.
  3. 1/3 — 1/3 is exact while 0.33 is a rounded approximation.

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