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

Five lessons for Year 10 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 twelve bounds cards to match, twelve error cards to sort, 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. Cut the 12 bounds cards before Day 1 and the 12 error cards before Day 2, and keep each set in an envelope.
4. Calculators are used every day. A few rulers, a kitchen scale and a stopwatch make Day 1 real, but every task also works from the sheets.
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 measurement is exact. A length read as 5.0 cm to the nearest millimetre could truly be anything from 4.95 to 5.05 cm: half a unit either side. That uncertainty is the absolute error; divided by the measurement it becomes the percentage error, which tells how good a measurement really is. Significant figures carry the same message: 5.00 cm claims more precision than 5 cm. When measurements are combined, in a perimeter, an area or a speed, their errors combine too, and the result can be less accurate than any one of its parts. In industry, a tolerance sets how much error is acceptable.

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

A measurement keeps the decimal places its accuracy justifies: 2.40 m to the nearest centimetre, not 2.4 m. Bounds are written as a pair, 4.95 cm to 5.05 cm, and percentage errors to two decimal places. A tolerance is written as a target plus or minus an amount, 50 ± 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 finds the bounds of a single measurement; Days 2 and 3 judge its accuracy as a percentage and in significant figures; Day 4 follows the error into calculations; and Day 5 sets limits on it in practice. It builds on Year 9, where students calculated absolute, relative and percentage errors and learned that every measurement is an estimate, and it prepares the treatment of uncertainty in senior science and mathematics.

Materials for the week (one trip)

• From the classroom: calculators, rulers, scissors, envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 bounds cards (Day 1).
• Cut-out sheet 2, one per pair: 12 error cards (Day 2).
• If you have them: a kitchen scale, a stopwatch and a vernier calliper for Days 1 and 5. No other equipment to buy.

Words for the word wall

measurement, accuracy, precision, lower bound, upper bound, absolute error, relative error, percentage error, significant figure, uncertainty, tolerance, calliper, limits of accuracy

Dear families

This week in maths, Year 10 is learning that no measurement is exact: how far off a measurement could be, how errors grow when measurements are combined, and how factories decide what is close enough.

Try this at home

• Ask: a bag says 1 kg. Could it hold 1.004 kg? (Yes: packers weigh to a limit of accuracy.)
• Ask: which is the better measurement, a 2 m plank to the nearest centimetre or a 2 cm bead to the nearest centimetre? (The plank: 0.25% against 25%.)
• Ask: a speed sign of 60 km/h and a car’s speedometer: which might be more precise? (Talk about what each is measured to.)
• Ask: why might two parts that each pass inspection still not fit together? (Their errors can add up.)

My measurements week

Fill one row a day. Write a measurement you saw, what it was measured to, and how far off it could be.

No measurement is exact

A measurement is only as good as the unit it was measured to: the true value lies within half a unit either side. Students find the lower and upper bounds of measurements, read a ruler honestly, and match six measurements to their bounds.

We are learning to

• say what unit a measurement was taken to,
• find its lower and upper bounds, half a unit either side,
• explain why 3.0 m and 3 m are different measurements.

Success criteria

• I can give the bounds of 5.0 cm measured to the nearest 0.1 cm.
• I can explain what a scale reading of 85 g, to the nearest 5 g, really tells us.

You need

Rulers; a kitchen scale or stopwatch if you have one; the bounds cards (cut-out sheet 1), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Using a whole unit either side: a reading to the nearest centimetre is within half a centimetre.
• Dropping a zero: 2.40 m says to the nearest centimetre; 2.4 m says only to the nearest 10 cm.
• Thinking the bounds are mistakes: every honest measurement has them.

Answers

• Part A: the ruler reads 4.7 cm, so the strip is between 4.65 and 4.75 cm. 5.0 cm: 4.95 cm to 5.05 cm; 23 kg: 22.5 kg to 23.5 kg; 4.7 L: 4.65 L to 4.75 L; 380 m: 375 m to 385 m; 1.25 m: 1.245 m to 1.255 m; 60 s: 59.5 s to 60.5 s.
• Part B: 82.5 to 87.5 g; any mass in that range shows 85 g.
• Part C: the card key is with the mini-check answers.
• Part D: 3.0 m is to the nearest 0.1 m, so 2.95 m to 3.05 m; 3 m is to the nearest metre, so 2.5 m to 3.5 m. The first is ten times as precise.

No measurement is exact

A. Read the ruler, then find the bounds

Now fill the table. The first row is done for you.

B. A kitchen scale

C. Match the bounds

Match each measurement card to its bounds card. Write the letters.

D. Stretch: the same length?

Absolute and percentage error

The absolute error is half the unit a measurement was taken to; the percentage error compares that with the measurement itself. Students work out both, decide which of two measurements is more accurate, and sort twelve measurements by percentage error.

We are learning to

• find the absolute error of a measurement,
• find its percentage error: the absolute error as a percentage of the measurement,
• use percentage error to compare the accuracy of different measurements.

Success criteria

• I can find the percentage error of 12 cm, to the nearest centimetre.
• I can explain why the same absolute error matters less on a bigger measurement.

You need

Calculators; the error cards (cut-out sheet 2), one set per pair; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Using the whole unit as the absolute error: it is half the unit.
• Dividing the measurement by the error: divide the error by the measurement, then multiply by 100.
• Mixing units: write the error and the measurement in the same unit first.

Answers

• Part A: 12 cm: absolute 0.5 cm, 4.17%; 2.40 m: absolute 0.005 m, 0.21%; 750 g: absolute 5 g, 0.67%; 8 s: absolute 0.5 s, 6.25%.
• Part B: both are ±0.5 m, but 10 m is 5.00% out and 1000 m only 0.05%, so the 1000 m measurement is more accurate.
• Part C: the card key, with each percentage error, is with the mini-check answers.
• Part D: 0.5 ÷ L < 0.01 needs L > 50 cm: anything longer than 50 cm measured to the nearest centimetre is under 1% out.

Absolute and percentage error

A. Two kinds of error

Percentage error = absolute error ÷ measurement × 100. Give percentages to two decimal places. The first row is done for you.

B. Which is more accurate?

C. Sort by error

Work out the percentage error on each error card and sort the cards into three piles. Write the letters.

D. Stretch: long enough

Significant figures

Significant figures show how precise a measurement is: 5.00 cm claims more than 5 cm. Students count and round significant figures, and give a calculated result to no more figures than its measurements justify.

We are learning to

• count the significant figures in a measurement,
• round a value to a given number of significant figures,
• give a calculated result to no more significant figures than its least precise measurement.

Success criteria

• I can say that 0.0350 has three significant figures, and why the zeros at the front do not count.
• I can explain why 7.2 ÷ 3.1 should be given as 2.3, not 2.322580645.

You need

Calculators; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Counting the zeros at the front: in 0.0350 they only place the decimal point.
• Dropping a significant zero at the end: 20.40 has four significant figures.
• Copying every calculator digit: a result is no more precise than its measurements.

Answers

• Part A: 4.07: 3; 0.0350: 3; 5.00: 3; 0.6: 1; 20.40: 4; 1200: 2, 3 or 4: the zeros may only hold place.
• Part B: 3.14159 to 3 significant figures is 3.14; 0.07865 to 2 significant figures is 0.079; 98 450 to 3 significant figures is 98 500; 6.995 to 3 significant figures is 7.00.
• Part C: 7.2 ÷ 3.1 = 2.322580645... Both measurements have two significant figures, so the speed is 2.3 m/s. 3.6 × 2.15 = 7.74, kept to two significant figures: 7.7 cm².
• Part D: 1200 could have two, three or four significant figures; 1.2 × 10³ shows two and 1.20 × 10³ shows three, plainly.

Significant figures

A. Count the significant figures

The first is done for you.

B. Round

C. Honest answers

D. Stretch: what does 1200 mean?

Errors that combine

When measurements are combined, their errors combine too. Students find the bounds of a perimeter, an area, a speed and a difference, choosing the right pair of bounds each time, and see the result is less precise than its parts.

We are learning to

• find the bounds of a sum and of a product from the bounds of its measurements,
• choose the bounds that make a quotient or a difference as big, or as small, as it can be,
• explain how combining measurements spreads the error.

Success criteria

• I can find the bounds of the area of a room 6.2 m by 3.5 m.
• I can explain why the greatest speed uses the greatest distance and the least time.

You need

Calculators; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Using both upper bounds for a quotient: the greatest speed divides the greatest distance by the least time.
• Subtracting the bounds the same way: the greatest difference takes the least from the greatest.
• Rounding the bounds early: keep them exact until the end.

Answers

• Part A: perimeter from 2(6.15 + 3.45) = 19.2 m to 19.6 m; area from 6.15 × 3.45 = 21.2175 to 22.1875 m².
• Part B: least speed 399.5 ÷ 52.45 ≈ 7.617 m/s; greatest 400.5 ÷ 52.35 ≈ 7.650 m/s.
• Part C: least difference 15.15 − 8.75 = 6.4 cm; greatest 15.25 − 8.65 = 6.6 cm. Each length is ±0.05; the difference is ±0.1.
• Part D: the errors of a sum or a difference add; for a product or quotient the percentage errors roughly add, so the result is less precise than either measurement.

Errors that combine

A. A room

A room is 6.2 m by 3.5 m, each to the nearest 0.1 m.

B. A run

C. A difference

D. Stretch: what spreads most?

Tolerance

A tolerance sets how far a made thing may be from its target and still be accepted. Students accept or reject bolts, see how tolerances add up along a row of tiles, choose an instrument fit for the job, and play the self-check quiz.

We are learning to

• read a tolerance such as 50 ± 0.5 mm as an accepted band,
• decide whether measured parts pass, including parts on the limit,
• explain how tolerances add up when parts are put together.

Success criteria

• I can say which of the bolts pass a tolerance of 50 ± 0.5 mm.
• I can explain why a ruler marked in millimetres cannot check a tolerance of ±0.5 mm.

You need

Calculators; a vernier calliper if you have one; the worksheet, one per student; the self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; do Parts B to D and the quiz next time.

Watch for these ideas

• Rejecting parts on the limit: a tolerance band includes its ends.
• Thinking parts that each pass must fit together: their errors can add up.
• Checking a fine tolerance with a coarse instrument: the instrument must read finer than the tolerance.

Answers

• Part A: the band is 49.5 to 50.5 mm: 49.4 fails, 50.2 passes, 50.5 passes, 49.6 passes, 50.6 fails.
• Part B: each tile is 299 to 301 mm, so a row of 10 is 2990 to 3010 mm: up to 10 mm more or less than 3000 mm. Leave room for the extra.
• Part C: a ruler marked in millimetres reads to ±0.5 mm, as wide as the tolerance itself; a calliper reading to 0.1 mm can decide.
• Part D: the longest shelf is 1.205 m and the narrowest gap 1.205 m: equal, so it always fits, but only just, with no room to slide it in at an angle.

Tolerance

A. Pass or fail?

Bolts must be 50 ± 0.5 mm long. The first is done for you.

B. A row of tiles

C. The right tool

D. Stretch: will it fit?

Measurements and their bounds

For Day 1, one set per pair. Six cards give a measurement and what it was measured to; six give a pair of bounds. Match them.

Teacher note: the key is with the mini-check answers.

How big is the percentage error?

For Day 2, one set per pair. Work out the percentage error of each measurement and sort the cards into under 1%, 1% to 5%, and over 5%.

Teacher note: the key is with the mini-check answers.

What we know: Measurement error and accuracy

Work on your own. A calculator is allowed. Show your working.

1. A length is 8.4 cm to the nearest millimetre. Give its lower and upper bounds. ______________________
2. What is the percentage error of that measurement? ______________________
3. Which is more accurate: 250 g to the nearest 10 g, or 12 g to the nearest gram? Show why. ______________________
4. Round 0.004567 to 2 significant figures. ______________________
5. A room is 4.2 m by 3.6 m, each to the nearest 0.1 m. Give the bounds of its area. ______________________
6. The upper bound of 60 km/h, to the nearest 10 km/h, is, circle one:   55 km/h  ·  60 km/h  ·  65 km/h
7. A sprinter runs 100 m (to the nearest metre) in 12.5 s (to the nearest 0.1 s). Find the greatest and least possible average speed. ______________________
8. Parts must be 30 ± 0.2 mm. Circle every part that passes:   29.85 mm,   30.25 mm,   29.75 mm,   30.2 mm

Answers and marking guide

Answers with working

1. 8.35 cm to 8.45 cm.
2. 0.05 ÷ 8.4 × 100 ≈ 0.60%.
3. 250 g: 2.00%; 12 g: 4.17%. The 250 g reading is more accurate, though its absolute error is bigger.
4. 0.0046.
5. 4.15 × 3.55 = 14.7325 to 4.25 × 3.65 = 15.5125 m².
6. 65 km/h: half of 10 km/h above 60.
7. Greatest 100.5 ÷ 12.45 ≈ 8.07 m/s; least 99.5 ÷ 12.55 ≈ 7.93 m/s.
8. The band is 29.8 to 30.2 mm: 29.85 and 30.2 mm pass (the limit counts).

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and keeps bounds exact until the last step.

Card keys

Bounds cards (Day 1): 5 cm: G and A; 500 g: D and J; 2.40 m: E and L; 70 km/h: B and H; 12.6 s: K and F; 5.0 cm: I and C. Error cards (Day 2): under 1% M (0.25%), R (0.50%), U (0.21%), X (0.56%); 1% to 5% N (1.25%), P (3.57%), Q (2.78%), S (1.00%), W (1.25%); over 5% O (16.67%), T (8.33%), V (10.00%).

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: bounds, absolute and percentage error, significant figures, combining errors, and tolerance.