Exact answers versus approximations | Year 9

01 / LEARN

The idea, step by step

Exact values retain all the information

An exact answer has not been rounded. A fraction, root or multiple of pi can describe a value more accurately than a long decimal copied from a calculator. The side of a square with area fifty square centimetres is exactly the square root of fifty centimetres.

An approximate answer is useful in measuring and decision-making. Use an approximately-equal sign when rounding changes the value. Include the requested unit and precision: nearest centimetre and two decimal places are different instructions.

\sqrt{50} cm \approx 7.07 cm

Round once at the end

Retain exact values or calculator memory during a multi-step calculation. Rounding an intermediate value can change the final answer, especially if the value is multiplied many times. State an exact expression first where useful, then evaluate it.

A root sign does not guarantee an irrational answer. The square root of forty-nine is exactly seven. Conversely, every finite calculator display for pi is an approximation, regardless of how many digits it shows.

A common trap

Writing $π = 3.14$ means the values are exactly equal.

Instead: Use $π \approx 3.14$ . Use the pi key or exact expression until the final rounding step.

02 / WORKED EXAMPLES

See the thinking

Example 1

Exact and measured square sides

A square has area $50 {cm}^{2}$ . Give its side exactly and to two decimal places.

$s^{2} = 50$ and $s > 0$ , so $s = \sqrt{50} cm$ .
$\sqrt{50} = 7.071067 \dots$ . Round the final value.

\sqrt{50} cm \approx 7.07 cm

Example 2

A circumference involving pi

A circle has radius $3 cm$ . Give its circumference exactly and to two decimal places.

$C = 2 π r = 6 π cm$ .
$6 π = 18.849555 \dots$ .

6 π cm \approx 18.85 cm

Check your understanding

Which sign belongs between $\sqrt{9}$ and $3$ : $=$ or $\approx$ ?

Show answer

$=$ .

$3^{2} = 9$ , so no rounding occurs.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A square has area $23 m^{2}$ . Write its exact positive side.
2 marks
Round $\sqrt{23}$ to two decimal places.
2 marks
A circle has diameter $7 m$ . Give its exact circumference.
2 marks
Round the circumference for diameter $7 m$ to one decimal place.
2 marks
A student uses $1.4$ for $\sqrt{2}$ and obtains $10 \sqrt{2} \approx 14$ . Find the result to two decimal places without rounding early.
3 marks
For radius $50 cm$ compare $C = 100 π$ with $100 (3.14)$ . How much smaller is the second value, to two decimal places?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A square has area $23 m^{2}$ . Write its exact positive side.
2 marks
The positive number whose square is $23$ is $\sqrt{23}$ .
$\sqrt{23} m$ .
Round $\sqrt{23}$ to two decimal places.
2 marks
$\sqrt{23} = 4.795831 \dots$ ; the third decimal rounds $4.79$ to $4.80$ .
$4.80$ .
A circle has diameter $7 m$ . Give its exact circumference.
2 marks
$C = π d = 7 π m$ .
$7 π m$ .
Round the circumference for diameter $7 m$ to one decimal place.
2 marks
$7 π = 21.991148 \dots$ , giving $22.0 m$ .
$22.0 m$ .
A student uses $1.4$ for $\sqrt{2}$ and obtains $10 \sqrt{2} \approx 14$ . Find the result to two decimal places without rounding early.
3 marks
$10 \sqrt{2} = 14.142135 \dots$ .
Round the complete calculation, not the root alone.
$14.14$ .
For radius $50 cm$ compare $C = 100 π$ with $100 (3.14)$ . How much smaller is the second value, to two decimal places?
3 marks
$100 π - 314 = 0.159265 \dots cm$ .
$0.16 cm$ smaller.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Year 9 real-number classification and applications; root bounds and sensible digital approximation. Full surd manipulation is developed in Year 10.

Official curriculum source ↗

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