Measurement error: a week of ready-to-teach maths

Five lessons for Year 9 Measurement. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus two paper rulers and four lines to measure, error sort cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair: sheet 1 has the error sort cards (Day 3), sheet 2 the paper rulers and the lines to measure (Days 1 and 4). Cut the cards before Day 3.
4. Calculators are needed from Day 3. For the estimation games, a stopwatch and a tape measure per group help.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

No instrument reads a true value exactly, so every measurement is an estimate: a length read to the nearest millimetre could be up to half a millimetre out. Absolute error is the size of the gap between a reading and the true value, in the same unit. Relative error divides that gap by the true value, which shows how much it matters: 1 cm is a lot on a pencil and little on a wall. Percentage error is the relative error times 100. Whether an error is acceptable depends on what the measurement is for.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these answers

An absolute error carries the unit of the reading; a relative error has no unit and is written as a decimal; a percentage error is rounded to two decimal places unless it comes out exactly. Where the true value is given in another unit, it is converted first. A reading to the nearest step can be up to half a step out, so 47 mm to the nearest millimetre is written 47 ± 0.5 mm.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 shows that every reading is an estimate with limits; Day 2 measures how far out a reading is; Day 3 asks how much that matters, as a proportion; Day 4 traces the error back to the instrument; and Day 5 judges errors in context. It builds on percentages, ratio and decimal place value from earlier years. It prepares the honest reporting of measurements in science and the modelling with rates, ratio and scale that completes the Year 9 Measurement strand.

Materials for the week (one trip)

• From the classroom: calculators, scissors, envelopes for the cards, this pack printed; a stopwatch and a tape measure per group if you have them.
• Cut-out sheet 1, one per pair: 12 error sort cards (Day 3).
• Cut-out sheet 2, one per pair: two paper rulers and four lines to measure (Days 1 and 4), printed at 100%.
• If you have them: kitchen scales, a measuring jug and a bathroom scale for Day 4. No other equipment to buy.

Words for the word wall

measurement, estimate, true value, reading, division, precision, limits of accuracy, absolute error, relative error, percentage error, tolerance, acceptable, instrument, nearest, half a division, significant, context

Dear families

This week in maths, Year 9 is learning that every measurement is an estimate, and how to say how good an estimate is: the size of the error, and the error as a percentage of the true value. We also ask when an error matters, which depends on the job.

Try this at home

• Weigh a packet of food that shows its mass on the label. How far out is the scale, in grams and as a percentage?
• Ask: guess 20 seconds with your eyes closed while someone times you. What was your percentage error?
• Ask: a 1 cm mistake on a phone screen and on a fence: which matters more, and why?
• Ask: a kitchen scale shows whole grams. How far out could a reading of 250 g be? (Half a gram either way.)

My error diary

Fill one row a day. Write a measurement from class, how far out it was, and the error as a percentage.

Every measurement is an estimate

No reading is exact. An edge falls between two marks, so a reading to the nearest mark could be up to half a division out: every reading stands for a range, from half a step below to half a step above.

We are learning to

• read a length to the nearest mark on two rulers and say the range it stands for,
• find the smallest and largest a reading to the nearest step could be,
• explain why a finer scale gives a narrower range.

Success criteria

• I can write a reading with its range, such as 52 ± 0.5 mm.
• I can explain why the millimetre ruler gives a narrower range than the centimetre ruler.

You need

The worksheet; cut-out sheet 2 (the paper rulers and the lines), one per pair, the rulers cut out; a real ruler to compare.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the board picture again.

Watch for these ideas

• Thinking a digital display is exact: it rounds to its last digit too.
• Giving the range as a whole division either way: 7 cm to the nearest cm is 7 ± 0.5 cm, not 7 ± 1 cm.
• Measuring from the end of the ruler instead of the 0 mark.

Answers

• Part A, printed at 100%: ruler A reads 6 cm, 12 cm, 4 cm, 10 cm (each ± 0.5 cm); ruler B reads 6.3 cm, 11.7 cm, 4.2 cm, 9.8 cm (each ± 0.05 cm).
• Part B: 47 mm: 46.5 to 47.5 mm; 3.2 kg: 3.15 to 3.25 kg; 120 cm: 115 to 125 cm; 15 s: 14.5 to 15.5 s; 2.75 L: 2.725 to 2.775 L.
• Part C: yes. 6 cm on ruler A means 5.5 to 6.5 cm, and 6.3 cm on ruler B sits inside that range, more precisely.
• Part D: the totals run from 7.5 + 4.5 = 12 cm to 8.5 + 5.5 = 14 cm: the two half-centimetres add up.

Every measurement is an estimate

A. Two rulers, four lines

Measure each line on cut-out sheet 2 with ruler A (whole centimetres only) and ruler B (millimetres). Read to the nearest mark, then give the range.

B. Limits of a reading

Each reading is to the nearest step shown. Give the smallest and largest the true value could be.

C. Both right?

One student reads Line 1 as 6 cm with ruler A. Another reads it as 6.3 cm with ruler B. Can both be right? Explain with the ranges.

D. Stretch: adding estimates

Two lengths are 8 cm and 5 cm, each to the nearest centimetre. What are the smallest and largest their total could be?

Absolute error

Absolute error is the size of the gap between a reading and the true value, |measured − true|. It is never negative, and it carries the unit of the measurement, so both values must be in the same unit first.

We are learning to

• find the absolute error of a reading, with its unit,
• convert the true value into the reading’s unit before subtracting,
• decide which of two readings is closer to the true value.

Success criteria

• I can find the absolute error of a reading when the true value is given in a different unit.
• I can say which of two readings is closer to the true value, and by how much.

You need

The worksheet; a stopwatch and a tape measure per group; kitchen scales and a labelled packet if you have them.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the game; start the second with Part A.

Watch for these ideas

• Writing a negative absolute error: the gap is a size, so it is always positive.
• Subtracting values in different units: change 0.5 kg to 500 g first.
• Leaving off the unit: an absolute error of 2 means nothing without seconds or grams.

Answers

• Part A: 49.6 cm against 50 cm: 0.4 cm; 2.07 m against 2 m: 0.07 m; 58 s against 1 min = 60 s: 2 s; 510 g against 0.5 kg = 500 g: 10 g; 3.35 L against 3.5 L: 0.15 L.
• Part B: answers vary; check that each error is a positive gap with its unit.
• Part C: 2.43 m is 0.07 m out and 2.56 m is 0.06 m out, so 2.56 m is closer; 742 g is 8 g out and 759 g is 9 g out, so 742 g is closer.
• Part D: it depends on what was measured: 3 mm is a lot on a 2 cm screw and very little on a 2 m door. That is tomorrow’s idea.

Absolute error

A. How far out?

Find each absolute error. Change the true value into the reading’s unit first if you need to.

B. Estimation game

Estimate first, then measure. Find how far out each estimate was.

C. Which is closer?

D. Stretch: is 3 mm good?

A measurement has an absolute error of 3 mm. Is that a good measurement? Explain why the answer depends on what was measured.

Relative and percentage error

Relative error divides the absolute error by the true value: it has no unit, and it shows how much the gap matters. Percentage error is the relative error times 100. The same 1 cm is 10% of a pencil and 0.1% of a classroom wall.

We are learning to

• find the relative error of a reading as a decimal with no unit,
• turn a relative error into a percentage error,
• compare measurements by their percentage errors rather than their absolute errors.

Success criteria

• I can find the percentage error of a reading from its absolute error and its true value.
• I can explain why the same absolute error matters more on a smaller object.

You need

The worksheet; calculators; the error sort cards (cut-out sheet 1), one set per pair, cut out.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the sort; start the second with Part A.

Watch for these ideas

• Dividing by the reading instead of the true value.
• Forgetting to multiply by 100, so 0.03 is written as 0.03%.
• Giving relative error a unit: cm ÷ cm leaves a plain number.

Answers

• Part A: 2 ÷ 80 = 0.025 = 2.5%; 6 ÷ 400 = 0.015 = 1.5%; 0.6 ÷ 12 = 0.05 = 5%; 0.05 ÷ 2.5 = 0.02 = 2%; 45 ÷ 1500 = 0.03 = 3%.
• Part B: the card key is with the mini-check answers.
• Part C: the book is 0.5 ÷ 20 = 2.5% out and the student 2 ÷ 160 = 1.25%: the student’s measurement is relatively better, though its absolute error is larger.
• Part D: yes. A guess of 3 kg for a 1 kg bag is 2 kg out: 2 ÷ 1 = 2 = 200%.

Relative and percentage error

A. Relative and percentage error

Relative error = absolute error ÷ true value. Percentage error = relative error × 100.

B. Error sort

Work out each card’s percentage error, then write its letter in its pile.

C. Which is better?

Compare a book 20 cm tall, measured 0.5 cm out with a student 160 cm tall, measured 2 cm out. Which measurement is relatively better? Use percentage errors.

D. Stretch: more than 100%?

Can a percentage error be more than 100%? Give an example, or explain why not.

Instruments and the largest error

A reading to the nearest division can be up to half a division out, so the instrument sets the largest absolute error. Dividing it by the reading gives the largest percentage error: the same ruler does far better, in percentage terms, on long lengths than on short ones.

We are learning to

• find the largest absolute error of a reading from the instrument’s smallest division,
• find the largest percentage error of a reading,
• choose an instrument that keeps a measurement’s error under a limit.

Success criteria

• I can give the largest error of a reading on kitchen scales marked every 10 g.
• I can choose an instrument that keeps a measurement’s error under 1%.

You need

The worksheet; calculators; the paper rulers (cut-out sheet 2); kitchen scales, a measuring jug and bathroom scales if you have them.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the board picture again.

Watch for these ideas

• Taking the whole division as the largest error: it is half a division.
• Dividing by the division instead of the reading for the percentage.
• Thinking a finer instrument has no error at all: it has a smaller one.

Answers

• Part A: 0.5 mm; 2.5 g; 0.005 s; 12.5 mL; 0.05 kg.
• Part B: 0.5 ÷ 120 × 100 ≈ 0.42%; 2.5 ÷ 80 × 100 ≈ 3.13%; 0.005 ÷ 9.58 × 100 ≈ 0.05%; 12.5 ÷ 150 × 100 ≈ 8.33%.
• Part C: 50 mL divisions: 25 ÷ 400 = 6.25%; 10 mL divisions: 5 ÷ 400 = 1.25%; 2 mL divisions: 1 ÷ 400 = 0.25%. Only the 2 mL cylinder keeps it under 1%.
• Part D: on a millimetre ruler, 0.5 mm is 5% of a 10 mm length but only about 0.17% of 300 mm: the error stays the same size while the length grows.

Instruments and the largest error

A. The largest error of an instrument

A reading can be up to half the smallest division out. Give each instrument’s largest error, with its unit.

B. The largest percentage error of a reading

Largest percentage error = largest error ÷ reading × 100. Round to two decimal places.

C. Choose the instrument

You need to measure 400 mL of water with an error under 1%. You can use a jug marked every 50 mL, a jug marked every 10 mL, or a measuring cylinder marked every 2 mL. Which will do? Show the largest percentage error of each.

D. Stretch: long or short?

Explain why a millimetre ruler gives a smaller percentage error on a long length than on a short one. Use two lengths of your own.

Error in context

Whether an error is acceptable depends on what the measurement is for. A job states a limit, as an absolute error, a percentage or a direction, and a reading passes or fails against it. An honest report gives a measurement together with its error.

We are learning to

• decide whether a reading meets a limit given as a percentage, an absolute error or a direction,
• compare estimates fairly with percentage errors,
• report a measurement with its largest error.

Success criteria

• I can decide whether a reading meets a limit given as a percentage or as an absolute error.
• I can write a measurement with its error, such as 58 ± 0.5 cm.

You need

The worksheet; calculators; tape measures, a stopwatch and bathroom scales for the challenge; the unit’s self-check quiz.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the challenge; start the second with Part A and the quiz.

Watch for these ideas

• Comparing an absolute error with a limit given as a percentage, or the other way round.
• Missing a one-sided limit: a race straight that is long is allowed; one that is short is not.
• Rounding a percentage error before comparing it with the limit.

Answers

• Part A: 738 g is 12 g out (1.60%), within 2%: acceptable; 2.396 m is 0.004 m out (0.17%), within 3 mm: not acceptable; 99.95 m is 0.05 m out (0.05%), not short: not acceptable; 1.488 L is 0.012 L out (0.80%), within 1%: acceptable.
• Part B: answers vary; the best estimate is the one with the smallest percentage error, not the smallest gap.
• Part C: 182 ± 0.5 cm; 13.4 ± 0.05 s; 3.25 ± 0.025 kg.
• Part D: at most 5 ÷ 250 = 2% either way: fine for a cake, but a jeweller paying for silver by the gram would lose up to 2% of its value, so finer scales are worth it.

Error in context

A. Acceptable or not?

Find the error in the form the limit uses, then decide.

B. Estimation challenge

Estimate, then measure. Compare your estimates by percentage error.

C. Report it honestly

Write each measurement with its largest error, in the form reading ± error.

D. Stretch: sugar or silver?

Kitchen scales in 10 g steps weigh 250 g of sugar for a cake, and 250 g of silver for a jeweller. Find the largest percentage error. Is it acceptable for each? Explain.

Error sort cards

For Day 3, one set per pair: 12 cards

Work out each card’s percentage error. Sort the cards into three piles: under 1%, 1% to 5%, over 5%.

Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.

Two paper rulers and four lines

For Days 1 and 4, one per pair

Cut out the two rulers along the dashed lines. Ruler A is marked only in whole centimetres; ruler B in millimetres. Measure each line from its left bar to its right bar with each ruler.

Teacher note: print at 100%. The rulers and lines share this sheet, so they agree with each other even if the printer shrinks the page; the Day 1 plan gives the readings at 100%.

What we know: Measurement error

Work on your own. Use a calculator. Give each error with its unit, or as a percentage.

1. A length is 74 mm to the nearest millimetre. Between what two values does the true length lie? ______________________
2. A bag’s true mass is 2.5 kg and a scale reads 2.46 kg. What is the absolute error? ______________________
3. An absolute error of 3 cm is made on a true length of 150 cm. What is the relative error? ______________________
4. A 40 s interval is timed as 41.2 s. What is the percentage error? ______________________
5. Kitchen scales are marked every 20 g. What is the largest error of a reading? ______________________
6. A jug marked every 10 mL reads 250 mL. What is the largest percentage error? ______________________
7. Which is more significant, a 1 mm error on a 4 cm screw or a 5 cm error on a 20 m fence? Explain with percentages. ______________________
8. A 20 kg bag of sand may be up to 2% light. It weighs 19.5 kg. Is that acceptable? ______________________

Answers and marking guide

Answers with working

1. Half a millimetre either way: 73.5 to 74.5 mm.
2. |2.46 − 2.5| = 0.04 kg.
3. 3 ÷ 150 = 0.02 (no unit).
4. |41.2 − 40| = 1.2, and 1.2 ÷ 40 = 0.03 = 3%.
5. Half a division: 10 g.
6. 5 ÷ 250 × 100 = 2%.
7. The screw: 1 ÷ 40 = 2.5%, the fence 50 ÷ 20000 = 0.25% (in millimetres). The screw’s error is more significant.
8. 0.5 ÷ 20 = 2.5%, more than 2%: not acceptable.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working, a unit on every absolute error and a verdict that names the limit.

Card key

Error sort (Day 3): under 1%: B (0.40%), D (0.70%), E (0.50%), I (0.25%), K (0.44%); 1% to 5%: A (2.40%), F (3.00%), G (4.17%), L (2.00%); over 5%: C (13.33%), H (8.00%), J (8.70%).

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: estimates and limits, absolute error, percentage error, instruments, and error in context.