Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 1 - Exact representations and repeated approximation

Exact representations and repeated approximation

Compare exact and approximate representations and analyse how repeated rounding affects numerical results.

Updated 2026-07-26 - 12 min read

Exact representations and repeated approximation 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.

Distinguish exact and approximate representations

Exact forms preserve the full value; decimal forms marked with ≈ or a stated accuracy communicate approximation. 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 distinguish exact and approximate representations 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: Finite displays often round an underlying value, so retain fractions, radicals or π until an approximation is required.

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

Which is an exact representation of the diagonal of a unit square?

Step 1 - identify the governing idea: Exact forms preserve the full value; decimal forms marked with ≈ or a stated accuracy communicate approximation.

Step 2 - apply it to this evidence: Pythagoras gives √(1² + 1²) = √2 exactly.

Result: √2

The relationship is visible in the working: Pythagoras gives √(1² + 1²) = √2 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:

  • 1.41 — It does not agree with the required relationship: Pythagoras gives √(1² + 1²) = √2 exactly.
  • 1.414 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1.4 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Which statement is correct?

Step 1 - identify the governing idea: Exact forms preserve the full value; decimal forms marked with ≈ or a stated accuracy communicate approximation.

Step 2 - apply it to this evidence: The recurring decimal represents the fraction exactly.

Result: 1/3 = 0.333…

The relationship is visible in the working: The recurring decimal represents the fraction 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:

  • 1/3 = 0.33 — It does not agree with the required relationship: The recurring decimal represents the fraction exactly.
  • 1/3 ≈ 0.333… — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0.333 = 1/3 exactly — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Write 7π/3 to three decimal places.

Step 1 - identify the governing idea: Exact forms preserve the full value; decimal forms marked with ≈ or a stated accuracy communicate approximation.

Step 2 - apply it to this evidence: 7π/3 ≈ 7.330382…, which rounds to 7.330.

Result: 7.330

The relationship is visible in the working: 7π/3 ≈ 7.330382…, which rounds to 7.330. 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:

  • 7.329 — It does not agree with the required relationship: 7π/3 ≈ 7.330382…, which rounds to 7.330.
  • 7.33 exactly — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 21.991 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Track error through repeated calculations

Carry guard digits through intermediate steps and round once at the end unless a measurement or instruction sets a different precision. 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 track error through repeated calculations 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: Small rounding differences can accumulate or be amplified by later operations.

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

Using π = 3.14, estimate 20π. How far is this below 20π to two decimal places?

Step 1 - identify the governing idea: Carry guard digits through intermediate steps and round once at the end unless a measurement or instruction sets a different precision.

Step 2 - apply it to this evidence: 20π ≈ 62.8319 while 20 × 3.14 = 62.8, a difference of about 0.03.

Result: 0.03

The relationship is visible in the working: 20π ≈ 62.8319 while 20 × 3.14 = 62.8, a difference of about 0.03. 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:

  • 0.30 — It does not agree with the required relationship: 20π ≈ 62.8319 while 20 × 3.14 = 62.8, a difference of about 0.03.
  • 0.01 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 3.18 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

A value 2.46 is rounded to one decimal then multiplied by 8. What result is produced?

Step 1 - identify the governing idea: Carry guard digits through intermediate steps and round once at the end unless a measurement or instruction sets a different precision.

Step 2 - apply it to this evidence: 2.46 rounds to 2.5 and 2.5 × 8 = 20.0.

Result: 20.0

The relationship is visible in the working: 2.46 rounds to 2.5 and 2.5 × 8 = 20.0. 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:

  • 19.68 — It does not agree with the required relationship: 2.46 rounds to 2.5 and 2.5 × 8 = 20.0.
  • 19.7 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2.0 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Which strategy best limits accumulated rounding error in a multi-step formula?

Step 1 - identify the governing idea: Carry guard digits through intermediate steps and round once at the end unless a measurement or instruction sets a different precision.

Step 2 - apply it to this evidence: Keeping guard digits preserves information until the requested result.

Result: Store full calculator values and round the final result

The relationship is visible in the working: Keeping guard digits preserves information until the requested result. 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:

  • Round each line to one decimal — It does not agree with the required relationship: Keeping guard digits preserves information until the requested result.
  • Replace every radical with an integer — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Drop all decimal tails — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Judge appropriate numerical accuracy

Report accuracy that matches the input measurements and context; extra decimal places do not create extra certainty. 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 judge appropriate numerical accuracy 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: Precision in reporting must be justified by measurement resolution and model assumptions.

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 length is measured to the nearest centimetre. Which report is most defensible for a calculated length near 2.846 m?

Step 1 - identify the governing idea: Report accuracy that matches the input measurements and context; extra decimal places do not create extra certainty.

Step 2 - apply it to this evidence: Centimetre resolution corresponds to 0.01 m, so two decimal places are appropriate.

Result: 2.85 m

The relationship is visible in the working: Centimetre resolution corresponds to 0.01 m, so two decimal places are appropriate. 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:

  • 2.846000 m — It does not agree with the required relationship: Centimetre resolution corresponds to 0.01 m, so two decimal places are appropriate.
  • 3 m exactly — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2.8 m only — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

A population model predicts 1248.7 people. What contextual report is appropriate?

Step 1 - identify the governing idea: Report accuracy that matches the input measurements and context; extra decimal places do not create extra certainty.

Step 2 - apply it to this evidence: People are counted in whole numbers and the model is approximate.

Result: About 1249 people

The relationship is visible in the working: People are counted in whole numbers and the model is approximate. 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:

  • 1248.7 people exactly — It does not agree with the required relationship: People are counted in whole numbers and the model is approximate.
  • 1248.7000 people — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1200.000 people — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Two methods give 6.32 and 6.3198 from inputs measured to two decimal places. What can be claimed?

Step 1 - identify the governing idea: Report accuracy that matches the input measurements and context; extra decimal places do not create extra certainty.

Step 2 - apply it to this evidence: Both round to 6.32; the extra digits are not independently supported.

Result: The methods agree to two decimal places

The relationship is visible in the working: Both round to 6.32; the extra digits are not independently supported. 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:

  • 6.3198 is certainly true — It does not agree with the required relationship: Both round to 6.32; the extra digits are not independently supported.
  • The methods disagree — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 6.32 has no value — 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 statement is correct?
  2. A value 2.46 is rounded to one decimal then multiplied by 8. What result is produced?
  3. A population model predicts 1248.7 people. What contextual report is appropriate?

Answers

  1. 1/3 = 0.333… — The recurring decimal represents the fraction exactly.
  2. 20.0 — 2.46 rounds to 2.5 and 2.5 × 8 = 20.0.
  3. About 1249 people — People are counted in whole numbers and the model is approximate.

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