Approximation versus exact: a week of ready-to-teach maths

Five lessons for Year 10 Number. 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 exact-or-approximate cards to sort, twelve match cards that pair an exact calculation with its rounded twin, 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 sort cards before Day 1 and the 12 match cards before Day 3, and keep each set in an envelope.
4. Check that calculators are ready every day. A spreadsheet on a laptop or tablet helps on Days 2 and 3, but every task also works on paper.
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

Many useful numbers cannot be written out in full: π and √2 never end, and one third is 0.333... with the threes going on for ever. A rounded value such as 3.14 or 1.41 is a convenient stand-in, close to the true number but not equal to it, and the gap is at most half a unit in the last place kept. One rounding on its own is harmless. The trouble starts when the rounded value is used again and again: added up, the gap grows in step with the number of uses; multiplied, it grows faster still. Keeping the exact form, the fraction, the surd or the symbol π, through the working and rounding once at the end, removes the build-up altogether. Keeping more digits shrinks the gap but never closes it.

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 numbers

An exact answer keeps its fraction, root sign or π, such as 5π cm² or 3√2 cm. A rounded answer uses ≈, such as 5π ≈ 15.71. The error of a rounding is the size of the gap between the true value and the rounded one, always given as a positive number. Calculators are used every day, always after an estimate, and students write down every rounding they make.

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 separates exact values from rounded ones and measures one rounding; Day 2 repeats a rounded value by adding, and the gap grows in step with the uses; Day 3 repeats it by multiplying, and the gap grows faster; Day 4 asks how many digits a job really needs; and Day 5 puts the habit to work: exact through the middle, rounded once at the end. It builds on Year 9, where students sorted the real numbers into rationals and irrationals, folded repeating decimals into fractions and simplified surds. It prepares the exponential models later in Year 10, measurement error and accuracy, and every calculation in science and finance that runs in stages.

Materials for the week (one trip)

• From the classroom: calculators, scissors, envelopes for the cards, pencils, this pack printed.
• Cut-out sheet 1, one per pair: 12 exact-or-approximate cards (Day 1).
• Cut-out sheet 2, one per pair: 12 match cards (Day 3).
• If you have them: laptops or tablets with a spreadsheet for Days 2 and 3 (one per pair is plenty). No other equipment to buy.

Words for the word wall

exact value, approximation, rounded value, decimal place, significant figure, error, absolute error, relative error, half a unit, upper bound, surd, irrational, recurring decimal, repeated calculation, compound, tolerance, exact form, round at the end

Dear families

This week in maths, Year 10 is learning what happens when we round a number and then keep using the rounded value. Numbers such as π and √2 never end, so we often write them as 3.14 or 1.41. One rounding makes only a tiny error, but used again and again that error grows. Working exactly and rounding once at the end keeps a long calculation honest.

Try this at home

• Ask: a wheel 1 m across turns 100 times. Using 3.14 for π, how far off is the distance? (About 16 cm short.)
• Ask: is 1.41 × 1.41 equal to 2? (No: it is 1.9881. But √2 × √2 is exactly 2.)
• Ask: if a recipe needs one third of a 1 kg bag, should we write 0.33 kg or 1/3 kg? (1/3 is exact; 0.33 is a little short.)
• Ask: when should you round a long calculation? (Once, at the end.)

My exact-or-rounded week

Fill one row a day. Write a number you met, whether it was exact or rounded, and how far off a rounding could be.

Exact or rounded?

A rounded value is a stand-in, close to the true number but not equal to it. Students measure how far six roundings are off, find the most a rounding can be off, and sort exact statements from approximations.

We are learning to

• tell an exact value from a rounded approximation,
• work out the error of a rounding: the size of the gap between the true value and the rounded one,
• state the largest error a rounding can make: half a unit in the last place kept.

Success criteria

• I can find the error of a rounding and give it to two significant figures.
• I can explain why a value rounded to two decimal places is never more than 0.005 out.

You need

Calculators; the exact-or-approximate cards (cut-out sheet 1), one set per pair; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; sort the cards and do Part D next time.

Watch for these ideas

• “3.14 is π.” It is a stand-in about 0.0016 short; the digits of π never stop.
• “Two places can be out by 0.01.” The rounded value is the nearest hundredth, so the gap is at most 0.005.
• “22/7 is the exact value of π.” 22/7 is an exact fraction, but of a different number, 3.142857..., only close to π.

Answers

• Part A: π ≈ 3.14, error 0.0016; √2 ≈ 1.41, error 0.0042; 1/3 ≈ 0.33, error 0.0033; 2/3 ≈ 0.67, error 0.0033; √3 ≈ 1.732, error 0.000051; 1/7 ≈ 0.143, error 0.00014. The largest error possible is 0.005 for two places and 0.0005 for three.
• Part B: 1 E, 2 A, 3 E, 4 A, 5 A, 6 E (the statement π = 22/7 is false: it is only close). Part C: the card key is with the mini-check answers.
• Part D: 2.449 to two places is 2.45, then to one place 2.5; straight to one place it is 2.4, which is right, because 2.449 is nearer 2.4. Round the original value, never a rounded one.

Exact or rounded?

A. How far off?

Use a calculator. Give each error to two significant figures. The first row is done for you.

B. Exact or an approximation?

Write E for exact or A for an approximation beside each statement.

C. Sort the cards

Sort the twelve cards into two piles, then write the letters.

D. Stretch: rounding a rounded number

Errors that add up

A wheel 1 m across rolls π metres a turn. With 3.14 for π every turn is 0.16 cm short, and the same error is added again each turn. Students tabulate and graph the gap, then meet a fence rail that comes out too long.

We are learning to

• work out the gap left by a rounded value that is added again and again,
• graph the gap against the number of uses and describe its shape,
• predict the gap for any number of uses from the gap for one.

Success criteria

• I can find the gap after any number of turns from the gap for one turn.
• I can explain why the graph of gap against turns is a straight line through the origin.

You need

Calculators; the worksheet, one per student; a spreadsheet on a laptop or tablet if you have one.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; do the fence and Part D next time.

Watch for these ideas

• “The error is too small to matter.” One turn is 0.16 cm short; 100 turns are 15.93 cm short.
• “The gap doubles every turn.” Each turn adds the same amount, so double the turns, double the gap. Day 3 meets faster growth.
• “The roundings will cancel out.” Every turn uses the same 3.14, so every error is in the same direction.

Answers

• Part A: gaps 0.16, 0.32, 0.80, 1.59, 3.19, 7.96, 15.93 cm for 1, 2, 5, 10, 20, 50, 100 turns (exact 6.2832 m and rounded 6.28 m for 2 turns, and so on).
• Part B: a straight line through (0, 0), rising 0.16 cm a turn. 1. 70 × 0.0015927 m ≈ 11.15 cm. 2. 1 ÷ 0.0015927 ≈ 627.9, so 628 turns.
• Part C: exact 25 × 2/3 = 50/3 = 16.666... m; Sam orders 16.75 m, 8.33 cm too long (0.33 cm a panel, 25 times).
• Part D: 3.1416 is 0.0000073 m too long a turn, so 136,121 turns. Only π kept exact makes the gap zero.

Errors that add up

A. The wheel

A wheel 1 m across rolls π m each turn. Fill the table with a calculator. The first row is done for you.

B. Graph the gap

Plot each gap from Part A against its number of turns, and join the points if they line up.

C. The fence

D. Stretch: more digits

Errors that multiply

Multiplying by a rounded value carries its error into every factor: 1.41 is 0.0042 short of √2, 1.41² is 0.0119 short of 2 and 1.41¹⁰ is 0.94 short of 32. Students compare powers, match exact calculations to rounded twins, and weigh a rounded rate against a rounded balance.

We are learning to

• compare the exact and rounded results of a repeated multiplication,
• use the percentage gap to compare errors of very different sizes,
• decide which rounding in a calculation does the most damage.

Success criteria

• I can work out the gap and the percentage gap for a power of a rounded value.
• I can explain why rounding a number that is multiplied again and again matters more than rounding a result once.

You need

Calculators; the match cards (cut-out sheet 2), one set per pair; the worksheet; a spreadsheet if you have one.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; do the savings and Part D next time.

Watch for these ideas

• “The gap stays the size of the first rounding.” Each factor’s shortfall is multiplied by the factors after it.
• “The biggest gap is the worst approximation.” Compare percentages: 2.94% for the tenth power, 0.60% for the square.
• “Rounding the balance is the danger.” Whole dollars moved it $1.20; the rate, $110.71.

Answers

• Part A: exact 2, 4, 8, 16, 32; gaps 0.0119, 0.0475, 0.1420, 0.3774, 0.9407 (0.60%, 1.19%, 1.77%, 2.36%, 2.94%). Both grow with the power.
• Part B: rounded answers 1.9881, 0.99, 31.4, 5.9858, 4.02, 25.1763; gaps 0.0119, 0.0100, 0.0159, 0.0142, 0.0200, 0.1763. The fourth power drifts most: its rounding is used four times. Letters: the card key.
• Part C: exact $2,844.20; whole dollars $2,843.00; rate 5% $2,954.91. The rate matters far more: it is multiplied in every year.
• Part D: about 20 × 0.30% ≈ 5.96%; the calculator gives 5.79%, a little less, because each shortfall is taken from a smaller number.

Errors that multiply

A. Powers of √2

Work out each exact power, then the same power of 1.41 to four decimal places. The first row is done for you.

B. Match the twins

Match each exact calculation card to its rounded twin. Record the letters, the rounded answer and the gap.

C. Two roundings, one account

Mia saves $2,000 at 4.5% a year for 8 years. Find the final balance three ways.

D. Stretch: twenty factors

How many digits?

Each extra digit of π cuts the error, usually about tenfold, but never to zero: π is irrational. Students tabulate π to 0 to 5 places, then choose the fewest digits for three long jobs.

We are learning to

• describe how the error changes as more digits are kept,
• explain why no decimal that stops can equal π,
• choose the number of digits a repeated calculation needs from its tolerance.

Success criteria

• I can find the error of π to a given number of places and check it against half a unit in the last place.
• I can work out the fewest digits that keep a repeated calculation inside a tolerance.

You need

Calculators with a π key; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; choose the digits and do Part D next time.

Watch for these ideas

• “Enough digits make the error zero.” π never ends or repeats, so every decimal that stops is off.
• “Each digit is ten times better.” Not always: 3.142 is 0.00041 out, 3.1416 only 0.0000073.
• “Use every digit on the screen.” Use what the job needs; the calculator’s π is rounded too.

Answers

• Part A: 3, 3.1, 3.14, 3.142, 3.1416, 3.14159; errors 0.14, 0.042, 0.0016, 0.00041, 0.0000073, 0.0000027; largest possible 0.5, 0.05, 0.005, 0.0005, 0.00005, 0.000005. None is zero.
• Part B: 1. Usually about ten times, not always. 2. 0.0000005. 3. π never stops or repeats, so no rounding equals it.
• Part C: 4 places, 0.07 cm (3 give 4.07 cm); 3 places, 57.03 cm (2 give 2.23 m); 5 places, 2.65 cm (4 give 7.35 cm).
• Part D: 1. The error in π must stay under 0.000000001: 9 places, 3.141592654. 2. 3.142 cuts off about 0.000593, near halfway, so it lands 0.00041 away; 3.1416 cuts off about 0.0000927, nearly a whole unit, so it lands 0.0000073 away.

How many digits?

A. π to more places

Use the π key. Give each error to two significant figures. The first row is done for you.

B. Never zero

C. Choose the digits

Total error = turns × diameter × error in π. For each job find the fewest decimal places of π that keep the total error under the tolerance.

D. Stretch

Exact through the middle

The week’s habit in one rule: keep fractions, surds and π exact through the working and round once, at the end. Students solve three problems both ways, find the slip in a worked solution, place one third on a zoomed line, and play the self-check quiz.

We are learning to

• carry fractions, surds and π exactly through a calculation of several steps,
• round once, at the end, to a sensible number of places,
• compare the exact route with the round-as-you-go route and explain the difference.

Success criteria

• I can keep a surd or π through a calculation and round only the final answer.
• I can say how far a round-as-you-go answer drifted and which step caused it.

You need

Calculators; the worksheet, one per student; the unit’s self-check quiz on the board for the game.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; do the zoomed line, Part D and the quiz next time.

Watch for these ideas

• “Keeping it exact means no calculator.” The calculator is used once, on the final exact expression.
• “Two places at every step gives a two-place answer.” Each rounding adds its own error, so the last digit can be wrong.
• “0.333 and one third are the same point.” Zoomed in, 0.333 sits about 0.00033 to the left of one third.

Answers

• Part A: 1. 17.8929 as you go; exact (3√2)² = 18 cm², 18.0000; drift 0.1071. 2. 15.7553 as you go; exact π × (√5)² = 5π cm², 15.7080; drift 0.0473. 3. 41.4217 as you go; exact (2√3)³ = 24√3 cm³, 41.5692; drift 0.1475.
• Part B: the first line, where π became 3.14 and 10 ÷ 3.14 was rounded to 3.18. Exact: C = 2π√(10/π) = 2√(10π) ≈ 11.2100 ≈ 11.21 cm, not 11.18 cm.
• Part C: 0.333 is on a tick; one third is a third of the way from 0.3333 to 0.3334. The gap is 0.000333..., and 0.333 is the one that misses.
• Part D: 2 × 1.7321 = 3.4642 and 3.4642³ ≈ 41.5728, a drift of 0.0035 instead of 0.1475. Closer, but only 24√3 is exact.

Exact through the middle

A. Two routes

Solve each problem twice: round as you go, as shown, then keep it exact and round once at the end. Give answers to four decimal places.

B. Spot the slip

A circle has an area of 10 cm². Noor finds its circumference like this:

C. The zoomed line

Mark 0.333 and one third on the line. How far apart are they? Which one is exact?

D. Stretch: more places on the way

Exact or an approximation?

For Day 1, one set per pair. Sort the cards into two piles: statements that are exactly true, and statements that are only close. Be ready to say what makes each approximation fall short.

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

Exact calculations and their rounded twins

For Day 3, one set per pair. Six cards are exact calculations; six are the same calculations done with rounded values. Match each exact card to its twin, then work out how far the twin drifts.

Twelve match cards

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

What we know: Approximation and exact values

Work on your own. Use a calculator for any question. Show your working.

1. Round √7 to 2 decimal places. What is the error, to two significant figures? What is the largest error a two-place rounding can make? ______________________
2. Circle every statement that is exact:   1/6 = 0.17,   √16 = 4,   √12 = 2√3,   π = 3.1416,   √20 = 4.47,   5/8 = 0.625
3. A wheel 0.5 m across turns 400 times. Using 3.14 for π, how far off is the distance, in centimetres? ______________________
4. (√3)⁴ = 9. Work out 1.73⁴, the gap, and the gap as a percentage of 9. ______________________
5. A calculation uses π in five steps. When should you round? Circle one:   only at the end  ·  at every step  ·  after the first step
6. Savings grow at 3.8% a year for 10 years. Which rounding changes the final balance most? Circle one:   rounding the balance to the nearest cent each year  ·  the final answer to the nearest dollar  ·  the rate to 4% before starting
7. A machine uses π 5000 times, each time multiplied by 2 m. Find the fewest decimal places of π that keep the total error under 10 cm. ______________________
8. A rectangle is √2 cm by √8 cm. Find its area exactly. Then find it again with each side rounded to one decimal place first. How far off is the rounded route? ______________________

Answers and marking guide

Answers with working

1. √7 = 2.645751... ≈ 2.65; error 0.0042; at most 0.005.
2. Exact: √16 = 4, √12 = 2√3, 5/8 = 0.625. The others are only close: 1/6 = 0.17, π = 3.1416, √20 = 4.47.
3. Each turn is 0.5 × (π − 3.14) ≈ 0.000796 m short, so 400 turns are 0.3185 m ≈ 32 cm short.
4. 1.73⁴ = 8.9575; gap 0.0425; 0.47% of 9.
5. Only at the end: every earlier rounding adds an error that later steps carry and can grow.
6. Rounding the rate to 4% before starting: the rate is multiplied in every year, so its error compounds; the others change the balance by cents or once.
7. Total error = 5000 × 2 × error in π: 3 places (3.142) give 4.07 m; 4 places (3.1416) give 7.35 cm. So 4 places.
8. Exact: √2 × √8 = √16 = 4 cm². Rounded: 1.4 × 2.8 = 3.92 cm², 0.08 cm² short.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working, and gives the exact answer before the rounded one wherever a question allows both.

Card keys

Exact-or-approximate cards (Day 1): exact A, C, D, F, I, K, L; an approximation B, E, G, H, J. Match cards (Day 3): S and M, P and V, Q and X, N and T, W and R, U and O.

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: exact or rounded, errors that add up, errors that multiply, choosing the digits, and the exact route through a problem.