Australian Curriculum v9 / ACiQ Year 8 Mathematics - Unit 1 - Irrational numbers, square roots and pi

Irrational numbers, square roots and pi

Recognise irrational numbers in practical contexts, estimate square roots and distinguish exact values from approximations.

Updated 2026-07-26 - 12 min read

Irrational numbers, square roots and pi 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.

Classify rational and irrational values

A rational number can be written as a fraction of integers; an irrational number cannot and has a non-terminating, non-recurring decimal expansion. 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 classify rational and irrational 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: Recurring decimals continue forever but are rational because they can be written 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 0.363636…

Step 1 - identify the governing idea: A rational number can be written as a fraction of integers; an irrational number cannot and has a non-terminating, non-recurring decimal expansion.

Step 2 - apply it to this evidence: The block 36 repeats, so the number can be expressed as a fraction.

Result: Rational

The relationship is visible in the working: The block 36 repeats, so the number can be expressed 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 36 repeats, so the number can be expressed 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

Which number is irrational: 7/8, √36, 0.125 or √10?

Step 1 - identify the governing idea: A rational number can be written as a fraction of integers; an irrational number cannot and has a non-terminating, non-recurring decimal expansion.

Step 2 - apply it to this evidence: The first three values are rational; 10 is not a perfect square, so √10 is irrational.

Result: √10

The relationship is visible in the working: The first three values are rational; 10 is not a perfect square, so √10 is irrational. 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:

  • 7/8 — It does not agree with the required relationship: The first three values are rational; 10 is not a perfect square, so √10 is irrational.
  • √36 — 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.

Example 1.3

Classify π − 3.

Step 1 - identify the governing idea: A rational number can be written as a fraction of integers; an irrational number cannot and has a non-terminating, non-recurring decimal expansion.

Step 2 - apply it to this evidence: Subtracting the rational number 3 from irrational π leaves an irrational number.

Result: Irrational

The relationship is visible in the working: Subtracting the rational number 3 from irrational π leaves an irrational number. 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:

  • Rational — It does not agree with the required relationship: Subtracting the rational number 3 from irrational π leaves an irrational number.
  • An integer — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Zero — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Estimate square roots using bounds

Bound a non-perfect square between consecutive perfect squares, then refine the estimate only to the accuracy required. 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 estimate square roots using bounds 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: Since 49 < 50 < 64, √50 is between 7 and 8; 7 is a bound, not the exact value.

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: Bound a non-perfect square between consecutive perfect squares, then refine the estimate only to the accuracy required.

Step 2 - apply it to this evidence: Because 64 < 70 < 81, taking square roots gives 8 < √70 < 9.

Result: 8 and 9

The relationship is visible in the working: Because 64 < 70 < 81, taking square roots gives 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: Because 64 < 70 < 81, taking square roots gives 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

Estimate √20 to one decimal place.

Step 1 - identify the governing idea: Bound a non-perfect square between consecutive perfect squares, then refine the estimate only to the accuracy required.

Step 2 - apply it to this evidence: √20 ≈ 4.472, which rounds to 4.5.

Result: 4.5

The relationship is visible in the working: √20 ≈ 4.472, which rounds to 4.5. 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:

  • 4.4 — It does not agree with the required relationship: √20 ≈ 4.472, which rounds to 4.5.
  • 10.0 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 5.0 — 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 90 cm². Estimate its side length to two decimal places.

Step 1 - identify the governing idea: Bound a non-perfect square between consecutive perfect squares, then refine the estimate only to the accuracy required.

Step 2 - apply it to this evidence: The side length is √90 cm ≈ 9.4868 cm, which rounds to 9.49 cm.

Result: 9.49 cm

The relationship is visible in the working: The side length is √90 cm ≈ 9.4868 cm, which rounds to 9.49 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:

  • 45.00 cm — It does not agree with the required relationship: The side length is √90 cm ≈ 9.4868 cm, which rounds to 9.49 cm.
  • 9.48 cm — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 8100.00 cm — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Use pi as an exact constant

Keep π in exact form through working and round only the final result when a decimal approximation is requested. 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 use pi as an exact constant 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: 3.14 is a useful approximation, but π is an irrational constant whose decimal expansion does not terminate.

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 the exact value of half of π?

Step 1 - identify the governing idea: Keep π in exact form through working and round only the final result when a decimal approximation is requested.

Step 2 - apply it to this evidence: Dividing the exact constant π by 2 preserves an exact representation.

Result: π/2

The relationship is visible in the working: Dividing the exact constant π by 2 preserves an exact representation. 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:

  • 1.57 — It does not agree with the required relationship: Dividing the exact constant π by 2 preserves an exact representation.
  • 3.14/2 exactly — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2π — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

A calculation gives 12π cm. Which decimal is correct to two decimal places?

Step 1 - identify the governing idea: Keep π in exact form through working and round only the final result when a decimal approximation is requested.

Step 2 - apply it to this evidence: 12π ≈ 37.699…, which rounds to 37.70 cm.

Result: 37.70 cm

The relationship is visible in the working: 12π ≈ 37.699…, which rounds to 37.70 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:

  • 37.68 cm — It does not agree with the required relationship: 12π ≈ 37.699…, which rounds to 37.70 cm.
  • 12.00 cm — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 144.00 cm — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

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

Step 1 - identify the governing idea: Keep π in exact form through working and round only the final result when a decimal approximation is requested.

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.

Retrieval check

Try these without looking back at the examples.

  1. Which number is irrational: 7/8, √36, 0.125 or √10?
  2. Estimate √20 to one decimal place.
  3. A calculation gives 12π cm. Which decimal is correct to two decimal places?

Answers

  1. √10 — The first three values are rational; 10 is not a perfect square, so √10 is irrational.
  2. 4.5 — √20 ≈ 4.472, which rounds to 4.5.
  3. 37.70 cm — 12π ≈ 37.699…, which rounds to 37.70 cm.

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