Exact Values, Approximation and Rounding Error — AC9M10N01

Learning goals: what you will learn

• 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

Prerequisite knowledge

You should already be comfortable with decimal place value, significant figures, fractions, square roots, π, percentages, order of operations and calculator use. You do not need advanced surd algebra for the central idea in this lesson.

If rounding itself is still difficult, revise rounding and significant figures first. The Year 10 step is understanding what happens when an approximation is then used again and again.

Exact values and approximations

An exact representation has not been rounded. Examples include 1/3, π, √2 and 49π. A decimal can also be exact when it terminates, such as 0.25 = 1/4.

An approximation is deliberately close to the exact value. For example, √2 ≈ 1.414 and π ≈ 3.14. Once digits have been discarded, later calculations cannot recover the lost information.

Rounding and truncation are different

For 7.9864, rounding to 2 decimal places gives 7.99. Truncating to 2 decimal places gives 7.98. Rounding chooses the nearest value at the requested place; truncation simply cuts off later digits.

The Year 10 calculation habit

Keep exact values or full calculator precision through the working, then round once at the end unless the context requires intermediate rounding. This does not mean every real measurement is exact. It means you should avoid introducing extra numerical error unnecessarily.

Why repeated calculations matter

Suppose the exact value is 1/3. Using 0.33 once is close. But using it in a repeated total gives a visible difference: 300 × (1/3) = 100 exactly, while 300 × 0.33 = 99. The error in one approximation has been multiplied 300 times.

Worked examples

Common misconceptions and corrections

• “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.

Guided practice

1. Classify: Which are exact: 5π, 0.125, √7, 2.646 to 3 d.p.? Think: has the value been rounded?
2. Compare: Evaluate 50π using calculator π and using 3.14. Find the difference.
3. Repeated use: Compare 120 × (2/3) with 120 × 0.67. Explain the difference in one sentence.
4. Decision: A multi-step geometry problem gives an intermediate value of 8.376492. Should you replace it by 8.38 before the next step? Explain.

Independent practice

1. Round 18.7954 to 2 d.p. and truncate it to 2 d.p.
2. Write the exact circumference of a circle with diameter 13 cm, then approximate it to 1 d.p.
3. Compare 400 × (1/7) with 400 × 0.14. Which is larger and by how much?
4. Compare 9(√3)² with 9(1.73)².
5. A wheel has diameter 0.64 m and rotates 900 times. Calculate the distance using π, then using 3.14. Compare the results.
6. A value starts at 250 and is multiplied by 1.06 for 8 periods. Compare rounding only at the end with rounding the running value to the nearest whole number each period.

Reasoning and problem-solving task

Two students estimate a repeated manufacturing cut. A machine should cut pieces of length 25/3 mm. Student A keeps 25/3 throughout a calculation for 120 pieces. Student B uses 8.3 mm for each piece.

1. Find both predicted total lengths.
2. Quantify the difference.
3. Explain why Student B's method can be reasonable for a rough estimate but unsuitable for checking a tight manufacturing tolerance.
4. State what additional information about the measuring/cutting process would be needed before claiming either mathematical total matches the physical total.

Important questions and 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.

Assessment-style questions

1. 2 marks: Explain why 6π is exact but 18.85 is only an approximation to it.
2. 3 marks: A wheel has diameter 0.78 m and turns 600 times. Find the exact distance in terms of π and the distance to the nearest metre.
3. 4 marks: A student calculates 75(√5)² using √5 ≈ 2.24. Compare the student's result with the exact result and explain the source of the difference.
4. 5 marks: A population index starts at 100 and increases by 3.7% for 20 periods. Compare (a) full-precision calculation rounded to 1 d.p. at the end and (b) rounding to 1 d.p. after every period. Comment on what the comparison demonstrates.

Review hint: If a question asks you to compare methods, show both numerical pathways and then explain where the approximation entered. A numerical difference without an explanation is incomplete reasoning.

Exit ticket: mastery check

1. Give one exact and one approximate representation of √10.
2. Explain why 200π and 200 × 3.14 need not give the same result.
3. State the safest rounding strategy for a multi-step calculation.
4. Give one situation in which a sensible final answer should still be rounded even though the calculator shows many digits.

Ready to move on? You should be able to answer all four without guessing and explain the repeated-calculation idea in your own words.

Teacher and parent guidance

For teachers

Emphasise comparison of calculation pathways rather than turning this descriptor into a stand-alone surds unit. Ask students to locate the exact step where information is discarded, quantify the resulting difference and judge whether it matters in context. Calculator spreadsheets are useful for repeated-rounding investigations.

For parents and carers

A helpful question is, “Did you round because the question asked you to, or just because the calculator showed many digits?” Encourage students to keep full working and label approximate answers with ≈. The goal is not to avoid decimals; it is to use them deliberately.

Curriculum alignment

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.

Practice and teaching resources

Related Year 10 Mathematics topics

• AC9M10A01 — algebraic expressions and equations
• AC9M10A03 — exponential relations
• AC9M10A04 — growth and decay modelling
• All Year 10 Mathematics topics

Official curriculum references

SkillrHub paraphrases curriculum requirements for teaching clarity. Use the official curriculum sites for authoritative wording and updates.

• ACARA — Australian Curriculum v9 Mathematics
• VCAA — Mathematics Version 2.0, Level 10
• NSW Curriculum/NESA — Mathematics K–10 (2022), Stage 5