- distinguish exact values from approximations
- use = for equality and ≈ for an approximation
- round and truncate accurately
- explain why repeated use of rounded values can change a final result
- compare exact and approximate calculation pathways
- choose sensible final accuracy for a context
Year 10 Mathematics · AC9M10N01
Exact Values, Approximation and Rounding Error
Small rounding choices can become meaningful after a value is reused many times. Learn when to keep fractions, π, roots or full calculator precision, how rounding differs…
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Learning goalsSay it simply
Key conceptTeach from the board
Example 1 — exact versus rounded
Compare (√2)² with (1.41)².
Exact route: (√2)² = 2.
Rounded route: 1.41² = 1.9881.
The method is not wrong; the difference comes from replacing √2 with an approximation before the calculation.
Example 2 — repeated distance
A wheel has diameter 0.72 m and turns 250 times. Distance = 250 × 0.72π = 180π m ≈ 565.49 m.
If π is replaced by 3.14 first, the result is 565.20 m. That is about 0.29 m lower. One small approximation was repeated through the calculation.
Example 3 — repeated multiplication
A quantity starts at 100 and is multiplied by 1.037 twenty times.
Using full precision: 100(1.037)²⁰ ≈ 206.8117, so to 1 decimal place the final answer is 206.8.
If the running value is rounded to 1 decimal place after every multiplication, the final value is 206.6. Repeated early rounding has created a difference of 0.2.
Example 4 — exact circle calculation
A circle has radius 7 cm. Its exact area is A = πr² = 49π cm². Only when a decimal is requested should you evaluate it: 49π ≈ 153.94 cm² to 2 d.p.
Example 5 — rounding versus truncation
For x = 12.4867, rounding to 2 d.p. gives 12.49; truncating to 2 d.p. gives 12.48. The two approximations fall on different sides of the exact value.
Example 6 — when approximation is appropriate
A calculated walking distance is 3.846217 km. Reporting 3.846217 km suggests unrealistic precision for an ordinary route estimate. A final answer such as 3.85 km may be more meaningful. Keeping precision during the calculation and choosing sensible precision for the report are different decisions.
Worked examplesWe do
Worked examples
AC9M10N01 - Exact Values, Approximation and Rounding Error
Example 1
Example 1 — exact versus rounded Compare (√2)² with (1.41)². Exact route: (√2)² = 2. Rounded route: 1.41² = 1.9881. The method is not wrong; the difference comes from replacing √2 with an approximation before the calculation.
Example 2
Example 2 — repeated distance A wheel has diameter 0.72 m and turns 250 times. Distance = 250 × 0.72π = 180π m ≈ 565.49 m . If π is replaced by 3.14 first, the result is 565.20 m. That is about 0.29 m lower . One small approximation was repeated through the calculation.
Example 3
Example 3 — repeated multiplication A quantity starts at 100 and is multiplied by 1.037 twenty times. Using full precision: 100(1.037)²⁰ ≈ 206.8117 , so to 1 decimal place the final answer is 206.8 . If the running value is rounded to 1 decimal place after every multiplication, the final value is 206.6 . Repeated early rounding has created a difference of 0.2.
Example 4
Example 4 — exact circle calculation A circle has radius 7 cm. Its exact area is A = πr² = 49π cm² . Only when a decimal is requested should you evaluate it: 49π ≈ 153.94 cm² to 2 d.p.
Curriculum examplesCopied content
Australian Curriculum:AC9M10N01, Year 10 Number — students examine the effect of approximating real numbers in repeated calculations and compare approximate results with exact representations.
Victoria:VC2M10N01, Level 10 Number — a direct alignment covering approximation, repeated calculations, rounding/truncation and comparison with exact representations.
NSW:Stage 5. The closest overlap is MA5-MAG-C-01 (significant-figure accuracy in measurement) with supporting Path content MA5-IND-P-02 where surds provide exact representations. NSW does not package the repeated-rounding emphasis into a single direct equivalent outcome, so this is a partial/supporting mapping rather than a one-to-one match.
Alignment explanation: The lesson's exact-versus-approximate comparisons and repeated calculations directly satisfy AC9M10N01 and VC2M10N01. Its rounding decisions connect to NSW Stage 5 accuracy, while the use of roots as exact forms supports Indices Path content; the lesson does not claim that those NSW outcomes are identical to the Australian descriptor.
| Lesson component | Australian Curriculum | Victoria | NSW |
|---|---|---|---|
| Exact vs approximate representations | AC9M10N01 | VC2M10N01 | MA5-IND-P-02 supporting |
| Rounding and truncation | AC9M10N01 | VC2M10N01 | MA5-MAG-C-01 partial |
| Repeated-calculation error | AC9M10N01 | VC2M10N01 | Stage 5 supporting connection; no forced equivalent |
| Reasoning about appropriate accuracy | AC9M10N01 + proficiency | VC2M10N01 | MAO-WM-01 + relevant Stage 5 content |
Questions and answersWith answers
- Why keep π or a square root instead of replacing it immediately?
- It preserves the exact mathematical value and avoids adding rounding error before later operations.
- Is 0.333… exact for 1/3?
- Yes, when the dots mean the 3s continue forever. A terminating value such as 0.33 is only an approximation.
- What is the difference between rounding and truncation?
- Rounding uses the next digit to choose the nearest value; truncation removes later digits without adjusting the retained digit.
- When should I round?
- Usually once, at the final step, to the accuracy requested or justified by the context.
- Why can repeated rounding be important?
- Each approximation changes the input to later operations, so differences can accumulate or be magnified.
Practice and reviewReady for practice
- “More decimal places always means exact.” No. 1.41421356 is still only an approximation to √2.
- “3.14 equals π.” Use π = π and 3.14 ≈ π.
- “Round after every calculator step.” Usually keep full precision and round the final result.
- “Exact means realistic measurement.” A formula can be evaluated exactly even when measured inputs have limited accuracy.
- “A tiny error never matters.” Multiplication, powers and repeated operations can accumulate or magnify small differences.
Curriculum alignmentStart here
- distinguish exact values from approximations
- use = for equality and ≈ for an approximation
- round and truncate accurately
- explain why repeated use of rounded values can change a final result
- compare exact and approximate calculation pathways
- choose sensible final accuracy for a context
SkillrHub paraphrases curriculum requirements for teaching clarity. Use the official curriculum sites for authoritative wording and updates.
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