S3 Maths | Significant Figures and Percentage Error Prep Guide (for tutors) | The error chain of a 12.4 cm reading in 60 minutes

2026-11-06

Tutor lesson-prep guides · Secondary Maths

A ruler reads 12.4 cm, so students assume the answer is exact to one decimal place. The true value could lie between 12.35 and 12.45, and that 0.05 cm is where the error comes from. Significant figures are about how far a number can be trusted, not how many digits were written.

Objectives — Students can: (1) decide how many significant figures a number has (leading zeros never count, trapped zeros do, trailing zeros count inside a decimal), at least ten of twelve right; (2) work out the maximum absolute error from the smallest scale division (divide by two), then the percentage error, answering 0.4% for the 12.4 cm case; (3) explain why rounding happens once at the end and show where chained rounding goes wrong.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Decimals and place value	Which place is the 4 in 12.4?	The tenths place
Rounding	Round 3.14159 to three significant figures	3.14
Percentages	Write 0.05 divided by 12.4 as a percentage	About 0.40%

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-5 min	Entry: three numbers	Write 0.004560, 28050 and 12.40 and ask how many significant figures each has	Answer with a reason
5-18 min	The significant figure rules	Model the three kinds of zero: leading does not count, trapped counts, trailing counts inside a decimal	Do six items and write the reason
18-32 min	Maximum absolute error	Model: a 0.1 cm scale gives an error of 0.1 / 2 = 0.05 cm, so the true range is 12.35 to 12.45	Write the range for three readings
32-45 min	Percentage error	Model 0.05 / 12.4 x 100 = 0.40% and say what the denominator is	Work three items to three significant figures
45-55 min	The chained rounding trap	Model 2.345 becoming 2.35 then 2.4, against the correct single rounding to 2.3	Repair three items
55-60 min	Exit check	Three questions collected and marked at once	Complete independently

03 Teaching script (read it out)

Scene 1: Significant figures count trusted digits, not positions

Split the rule into three sentences. First, start counting at the first non-zero digit on the left. Second, zeros trapped between two non-zero digits count. Third, zeros trailing after a decimal point count, while trailing zeros in a whole number do not unless marked.
So 0.004560 has four (4, 5, 6, 0) and the three leading zeros do not count; 28050 has four because the trailing zero is unmarked; 12.40 has four because the zero is inside a decimal.
Make students circle the first non-zero digit, then ask of every zero: is there a non-zero digit to its left and to its right?

Get the student to: circle the first digit of six numbers, label every zero count or not, and justify each label.

Scene 2: Maximum absolute error is half the smallest division, by definition

A measuring instrument only reads to its smallest division, and the last digit is estimated, so the true value can be off by at most half a division. A ruler marked to 0.1 cm therefore has a maximum absolute error of 0.05 cm, written as 12.4 plus or minus 0.05, that is 12.35 to 12.45.
Many students take 0.1 as the error, doubling the answer. Stress the word half and always ask what the division is.
Percentage error is maximum absolute error divided by the measured value, times 100%, so 0.05 / 12.4 x 100 = 0.40%. The denominator is the measured value because the true value is unknown.

Get the student to explain: why a small object has a larger percentage error – the same 0.05 cm is 0.4% of 12.4 but 4% of 1.24.

Scene 3: Round once, and only at the end

Students round at every step: 2.345 to 2.35 to 2.4, when a single rounding to one decimal place gives 2.3.
Teach one habit: keep one or two extra digits through the working and round only when writing the final answer. That is what the mark scheme rewards.
Also separate “three significant figures” from “three decimal places” out loud, because students mix them constantly; read which one the question asks before writing anything.

Get the student to: work the same question twice, rounding at every step and rounding once at the end, and put both answers side by side.

04 Question types and worked steps

Type	Example	Steps	Answer
Count significant figures	How many significant figures in 0.004560?	Circle the first non-zero digit, then judge each zero	4
Round to significant figures	Round 3.14159 to three significant figures	Count three from the left, the fourth digit decides	3.14
Maximum absolute error	Scale 0.1 cm, reading 12.4 cm, find the maximum absolute error	Divide the division by two	0.05 cm
Percentage error	Same reading, find the percentage error	Maximum absolute error divided by the reading, times 100%	about 0.40%

Sanity check (teach this): Sanity check: leading zeros never count as significant; unmarked trailing zeros in a whole number do not count; the percentage error must be positive and written with a percent sign; the maximum absolute error is half a division and never a whole division; the answer must not claim more significant figures than the data (0.4032% from a 12.4 cm reading is false precision); rounding happens once, at the end.

05 Independent practice (3 questions)

Question	Answer	What to watch
Twelve counts	Count significant figures in twelve numbers	Check leading zeros are excluded
Three error items	Work maximum absolute and percentage error from a scale	Check the division is halved
Three repairs	Repair common chained-rounding errors	Check rounding happens only at the end

06 Common mistakes and what to say

Mistake	Student writes	Say this
Leading zeros counted	Says 0.004560 has seven significant figures	The three zeros in front only place the decimal point; they are not trusted digits. Start at the 4: four figures.
Full division used as the error	Gives 0.1 cm as the maximum absolute error	The last digit you read is estimated, so it can be off by half a division at most. Half of 0.1 is 0.05.
Rounding at every step	Turns 2.345 into 2.4 through two roundings	You rounded twice. One rounding to one decimal place gives 2.3, not 2.4.
False precision	Answers 0.4032258% from a 12.4 cm reading	The reading carries three significant figures, so eight is invented precision. Answer 0.40%.

07 Homework (paper, section and question numbers)

Secondary Maths chapter index — All chapters in this subject
Prep guide master index — Similar chapters in other subjects
Redo this chapter — Repeat the error calculation with scales of 0.1, 0.5 and 1 cm and compare the percentage errors

08 Exit check (last 5 minutes)

Question	Answer	Decision
How many significant figures in 0.004560	Four	“Seven” means return to act one
How is the maximum absolute error found	Half the smallest division	“Equal to the division” means return to act two
How often do you round	Once, at the end	“At every step” means return to act three

09 Tutor FAQ

How do significant figures differ from decimal places?

Significant figures start at the first non-zero digit; decimal places only count digits after the point. 28050 to three significant figures is 28100, but it has zero decimal places.

Why divide by the measured value and not the true value?

The true value is unknown, so the measured value is the only reference; that is also why percentage error is an estimate.

Do trailing zeros need standard form?

Yes. To show that 28050 has five significant figures, write 2.8050 x 10 to the fourth power, otherwise the reader cannot tell.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.