Rounding and estimation: a week of ready-to-teach maths

Five lessons for Year 7 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 rounding cards, estimate match 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, and cut out the rounding cards (Days 1 and 2) and the estimate match cards (Day 4).
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

Rounding replaces a number with the nearest value at a chosen accuracy: 3.7 is closer to 4 than to 3, so to the nearest whole number it rounds to 4. Only the digit straight after the chosen place decides: 5 or more rounds up, 4 or less rounds down, and an exact halfway value rounds up by the usual convention. The accuracy is chosen to suit the situation: cents for money, the nearest thousand for a crowd. Estimation uses rounding on purpose: swap awkward numbers for friendly ones, work out a rough answer, and use it to check that the real answer is about the right size.

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 rounded numbers

Write a rounded answer to the accuracy asked for, zeros included: 2.97 to one decimal place is 3.0, not 3, because the 0 says the tenths were checked. Use “about” or ≈ for an estimate: 19.8 × 4.9 ≈ 100. Calculators come out only to check, after the estimate is written down.

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 makes rounding a question of which neighbour is closer; Day 2 carries the same idea to any decimal place; Day 3 asks which accuracy a situation needs; Day 4 puts rounding to work as an estimate that checks a calculation; and Day 5 extends estimation to division and to judging answers to problems. It builds on the decimals and the number line of earlier years and on last week’s equivalent forms, and it supports every calculation that follows, from fractions and percentages to measurement.

Materials for the week (one trip)

• From the classroom: pencils, scissors, small envelopes or paper clips for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 24 rounding cards and 4 accuracy cards (Days 1 and 2).
• Cut-out sheet 2, one per pair: 12 calculation cards and 12 estimate cards (Day 4).
• Calculators for checking only, on Days 4 and 5. A shop receipt or two for Day 3 (optional).

Words for the word wall

round, nearest, halfway, round up, round down, decimal place, deciding digit, accuracy, estimate, approximately, friendly numbers, compatible numbers, reasonable, overestimate, underestimate

Dear families

This week in maths, Year 7 is rounding decimals to a sensible accuracy and estimating answers before calculating them. An estimate such as 19.8 × 4.9 ≈ 20 × 5 = 100 tells you roughly what the real answer should be, so a slipped decimal point or a wrong key is caught at once.

Try this at home

• At the shops, estimate the total before you reach the checkout, then compare.
• Ask: a cash total of $12.62 is rounded to the nearest 5 cents. What do you pay? ($12.60.)
• Ask: is 2.49 closer to 2 or to 3? (2. It rounds down, even though 2.5 would round up.)
• Ask: about how much petrol for a 410 km trip, if the car uses 7 litres per 100 km?

Estimates this week

Fill one row a day: an estimate you made, then the real answer.

Printed from the free seegongsik Rounding and estimation teaching pack · seegongsik.com/au/y7/number/AC9M7N05/pack

Rounding is about closeness

Rounding chooses the closer neighbour. Students mark numbers on short lines to round them to the nearest whole number, round whole numbers to the nearest ten and hundred, and settle the halfway case and the trap of rounding twice.

We are learning to

• round a decimal to the nearest whole number by finding the closer neighbour,
• round a whole number to the nearest ten or hundred,
• explain the halfway convention, and why 2.49 rounds down.

Success criteria

• I can show on a number line why 4.6 rounds to 5.
• I can say why 6.47 rounds to 6 and not to 7.

You need

The rounding cards from cut-out sheet 1 (one set per pair), the worksheet. No calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Rounding twice: 6.47 to 6.5, then to 7. Only the tenths digit decides: 4, so 6.
• Thinking halfway is closer to the bigger number. It is equally close; rounding up is a convention.
• Rounding 1250 to the nearest hundred as 1200. A 5 rounds up: 1300.

Answers

• Part A: 7.3 rounds to 7, 12.8 rounds to 13, 0.5 rounds to 1, 9.49 rounds to 9, 15.51 rounds to 16, 99.7 rounds to 100.
• Part B: 347: 350 and 300; 1250: 1250 and 1300; 85: 90 and 100; 4996: 5000 and 5000.
• Part C: false (only the tenths digit decides, and 4 rounds down to 2); true (a halfway value rounds up); true (the tenths digit is 0); false (the tenths digit 9 rounds up, to 4).
• Part D: 7.5, 7.6, 7.7, 7.8, 7.9, 8.0, 8.1, 8.2, 8.3, 8.4; the biggest two-place decimal is 8.49.

Rounding is about closeness

A. Mark it, then round to the nearest whole number

B. Nearest ten and nearest hundred

C. True or false? Give a reason.

2.49 rounds to 3, because 2.49 rounds to 2.5 and 2.5 rounds to 3.   True □    False □   Reason: ____________________

6.5 rounds to 7 as a whole number.   True □    False □   Reason: ____________________

18.03 rounds to 18.   True □    False □   Reason: ____________________

3.95 rounded to the nearest whole number is 3.   True □    False □   Reason: ____________________

D. Stretch

List every number with one decimal place that rounds to 8. What is the biggest number with two decimal places that rounds to 8? ______________

Rounding to a chosen accuracy

One number rounds differently at different accuracies. Students find the place, read the one digit that decides, carry when a 9 rounds up, and keep the zeros that show the accuracy.

We are learning to

• round a decimal to one, two or three decimal places,
• use the deciding digit, the one straight after the chosen place,
• keep zeros that show the accuracy, as in 2.97 → 3.0.

Success criteria

• I can round 2.718 to the nearest whole, to one place and to two places.
• I can say why 9.996 to two decimal places is 10.00.

You need

The rounding and accuracy cards from cut-out sheet 1 (one set per pair), the worksheet. No calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Looking at the last digit instead of the next one: for 6.248 to one place, the 4 decides, not the 8.
• Dropping the zero: 2.97 to one place is 3.0. Writing 3 hides that the tenths were checked.
• Losing the carry: 9.996 to two places is 10.00, not 9.100.

Answers

• Part A: 5.2869: 5, 5.3, 5.29, 5.287; 0.0748: 0, 0.1, 0.07, 0.075; 12.3051: 12, 12.3, 12.31, 12.305.
• Part B: 2.97 to 1 place = 3.0; 9.996 to 2 places = 10.00; 4.95 to 1 place = 5.0; 0.398 to 2 places = 0.40.
• Part C: 6.248 wrong: the deciding digit is 4, so 6.2; 3.065 right; 12.96 right; 0.457 wrong: the deciding digit is 7, so 0.46; 7.04 wrong: the deciding digit is 4, so 7.0.
• Part D: to round to 4.27 it is from 4.265 up to (not including) 4.275, and all of those round to 4.3; for example 4.266, 4.27 or 4.274. 3.0 says the answer is right to the tenths; 3 says only to the whole number.

Rounding to a chosen accuracy

A. One number, four accuracies

B. When a 9 rounds up

C. Right or wrong?

Underline the deciding digit. Tick the right ones; fix the others.

D. Stretch

A number rounds to 4.3 at one decimal place and to 4.27 at two. Give three numbers it could be. ____________________ Why is 3.0 a different answer from 3 when one decimal place is asked for? ________________________________

Accuracy for the context

The right accuracy depends on the situation. Students choose accuracies for times, money, crowds and lengths, round cash to the nearest 5 cents, and see the cost of rounding too early.

We are learning to

• choose an accuracy that suits a situation, and say why,
• round a cash total to the nearest 5 cents,
• round only the final answer, not the numbers along the way.

Success criteria

• I can say why a swimming time is given to hundredths but a crowd to the nearest thousand.
• I can say what a cash customer pays for a $7.48 total.

You need

The worksheet; a shop receipt or two (optional). Calculators for Part C only.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Thinking more decimal places is always better. 23.68 °C claims more than a thermometer shows.
• Rounding cash to the nearest 10 cents. It is the nearest 5 cents: $8.98 → $9.00, $6.45 stays.
• Rounding every step: 3 × $2.49 is $7.47, not 3 × $2 = $6.

Answers

• Part A (other sensible answers with a reason are fine): 12.47 s; $47.33; 47 000 people; 1.54 m; 23.7 °C; 24 km. Reasons: race results give hundredths of a second; money is paid in cents; a report rounds a crowd to the nearest thousand; a tape measure reads to the nearest centimetre; weather reports give one decimal place; road signs show whole kilometres.
• Part B: $12.62 → $12.60, $8.98 → $9.00, $4.03 → $4.05, $19.97 → $19.95, $6.45 → $6.45, $3.11 → $3.10.
• Part C: rounding first gives $6.00 but the bill is $7.47; 4.6 × 3.4 = 15.64, about 16, while rounding first gives 15. Round only the final answer.
• Part D: about 500 × $12 = $6000, an overestimate, because both numbers were rounded up; exactly 487 × $11.95 = $5819.65.

Accuracy for the context

A. Choose a sensible accuracy

B. Paying cash

Round each total to the nearest 5 cents.

C. Rounding too early

Three drinks cost $2.49 each. Round each price to the nearest dollar and add: ________ Now find the real total: ________ What went wrong? ______________________

Work out 4.6 × 3.4 and round the answer to the nearest whole number: ________ Now round 4.6 and 3.4 first, then multiply: ________

D. Stretch

A school sells 487 tickets at $11.95 each. Estimate the takings. Is your estimate more or less than the real amount, and how can you tell without working it out? ____________________________

Estimating to check

An estimate swaps awkward numbers for friendly ones to show roughly what an answer should be. Students estimate before calculating, use estimates to catch wrong answers, and match calculations to their estimates.

We are learning to

• estimate a sum, difference or product with friendly numbers,
• use an estimate to decide whether an answer is about the right size,
• spot a slipped decimal point from an estimate.

Success criteria

• I can estimate 8.9 × 6.1 as 9 × 6 = 54 before calculating.
• I can explain why 7.9 × 5.1 cannot be about 4.

You need

The estimate match cards from cut-out sheet 2 (one set per pair), calculators for checking only, the worksheet.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Trusting the calculator over the estimate. A key slip can make 40.29 into 4.029; the estimate catches it.
• Picking friendly numbers that are no easier, such as 4.8 → 4.75.
• Expecting an estimate to equal the answer. It only needs to be about the right size.

Answers

• Part A: 4.8 × 21 ≈ 5 × 20 = 100 (exactly 100.8); 0.52 × 39 ≈ 0.5 × 40 = 20 (exactly 20.28); 61.2 + 19.7 ≈ 60 + 20 = 80 (exactly 80.9); 9.9 × 9.9 ≈ 10 × 10 = 100 (exactly 98.01); 204 − 98.6 ≈ 200 − 100 = 100 (exactly 105.4); 3.05 × 72 ≈ 3 × 70 = 210 (exactly 219.6).
• Part B: 7.9 × 5.1: wrong, it is 40.29; 48.6 ÷ 5.2: right; 19.6 × 31: right; 0.98 × 450: wrong, it is 441; 612.5 + 98.7: wrong, it is 711.2.
• Part C: answers come from the match cards; check the estimate and the exact value on each row.
• Part D: more than 12: 5.97 × 2.03 = 12.1191. 6 × 2.03 = 12.18, and taking away 0.03 × 2.03 (about 0.06) still leaves more than 12.

Estimating to check

A. Estimate first, then check

B. Catch the mistakes

Use an estimate, not a calculator, to decide which answers cannot be right.

C. From the match game

D. Stretch

Without a calculator, is 5.97 × 2.03 more or less than 12? Explain how you know. ________________________________

Estimates that make sense

Divisions are estimated with compatible numbers, numbers that divide easily. Students estimate before they calculate in money, fuel and measurement problems, and decide whether each answer makes sense, including when to round up.

We are learning to

• estimate a division with compatible numbers,
• judge whether an answer to a problem is reasonable,
• round an answer to suit the question, up or down.

Success criteria

• I can estimate 238 ÷ 4.1 as 240 ÷ 4 = 60.
• I can say when an answer has to be rounded up, whatever the digit.

You need

The worksheet, calculators for checking, and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 3; start the second with problem 4.

Watch for these ideas

• Rounding each number on its own: 600 ÷ 29 is no easier. Choose numbers that go together.
• Rounding a count of boxes or buses down. If 41.67 boxes are needed, it takes 42.
• Thinking dividing always makes smaller. Dividing by 0.8 makes a number bigger.

Answers with working

1. about $88 ÷ 4 = $22; exactly $87.40 ÷ 4 = $21.85.
2. about 7 × 4 = 28 L; exactly 6.8 × 4.12 = 28.016 L, about 28.0 L.
3. 600 ÷ 3 = 200; exactly 213.10..., about 213.1.
4. about 2.4 ÷ 0.3 = 8; 8 whole pieces, since 8 × 0.29 = 2.32 m and 9 pieces need 2.61 m.
5. 40 ÷ 0.8 = 50, so 4.95 is too small by a factor of 10; it is 49.5.
6. (a) $7.38, (b) 7.38 m, (c) about 7.
7. about 1000 ÷ 25 = 40; exactly 42 boxes, as 41 boxes hold only 984 cans.
8. answers vary; for example 99.6 ÷ 2.1 ≈ 100 ÷ 2 = 50, exactly 47.428....

Estimates that make sense

Write the estimate first, then the exact answer. Say whether the answer makes sense.

Problems (7 and 8 are the stretch)

1. Four friends share a bill of $87.40. Estimate each share, then work it out to the nearest cent.
2. A car uses 6.8 L of fuel per 100 km. Estimate the fuel for a 412 km trip, then work it out.
3. Estimate 618 ÷ 2.9 using numbers that divide easily. Then check with a calculator.
4. A plank 2.4 m long is cut into pieces 0.29 m long. About how many pieces? How many whole pieces exactly?
5. Someone says 39.6 ÷ 0.8 = 4.95. Use an estimate to check. Is it right?
6. A calculator shows 7.3846. Round it to suit: (a) an amount in dollars, (b) a length in metres measured with a tape marked in centimetres, (c) a rough count.
7. Stretch. Cans are packed 24 to a box. About how many boxes hold 1000 cans? How many boxes are needed exactly, and why is rounding down wrong here?
8. Stretch. Write a division with decimals whose estimate is about 50. Show the friendly numbers and the exact answer.

Rounding cards and accuracy cards

Rounding cards (Days 1 and 2)

Day 1: turn a card and round it to the nearest whole number. Day 2: turn a rounding card and an accuracy card together.

Accuracy cards (Day 2)

Teacher note: to the nearest whole number, 3.47 → 3, 12.508 → 13, 0.649 → 1, 7.95 → 8, 19.999 → 20, 4.05 → 4, 8.351 → 8, 2.704 → 3, 15.5 → 16, 6.249 → 6, 0.083 → 0, 11.86 → 12, 5.555 → 6, 9.012 → 9, 1.97 → 2, 14.449 → 14, 3.333 → 3, 0.996 → 1, 6.75 → 7, 20.051 → 20, 7.408 → 7, 13.65 → 14, 2.5 → 3, 4.444 → 4.

Estimate match cards

Calculation cards (Day 4)

Estimate cards (Day 4)

Match each calculation to its estimate, saying the friendly numbers aloud. Then check three of them on a calculator.

Teacher note: 29.7 × 3.1 ≈ 30 × 3 = 90 (exactly 92.07); 48 × 52 ≈ 50 × 50 = 2500 (exactly 2496); 6.9 × 8.2 ≈ 7 × 8 = 56 (exactly 56.58); 403 ÷ 19.8 ≈ 400 ÷ 20 = 20 (exactly 20.3535); 0.49 × 61 ≈ 0.5 × 60 = 30 (exactly 29.89); 9.8 + 70.3 ≈ 10 + 70 = 80 (exactly 80.1); 1.98 × 25 ≈ 2 × 25 = 50 (exactly 49.5); 812 ÷ 4.1 ≈ 800 ÷ 4 = 200 (exactly 198.048); 3.14 × 19.9 ≈ 3 × 20 = 60 (exactly 62.486); 98.5 − 61.3 ≈ 100 − 60 = 40 (exactly 37.2); 7.2 × 0.98 ≈ 7 × 1 = 7 (exactly 7.056); 1188 ÷ 2.9 ≈ 1200 ÷ 3 = 400 (exactly 409.655).

What we know: Rounding and estimation

Work on your own. Use a calculator only where a question asks for one.

1. Round 8.47 to the nearest whole number. ________
2. Round 3.0952 to one, two and three decimal places. ______________________
3. Round 9.97 to one decimal place. ________
4. A shop total is $14.67. What does a customer paying cash pay? ________
5. Estimate 51.3 × 0.48. Show your friendly numbers. ______________________
6. Estimate 799 ÷ 3.9. Show your friendly numbers. ______________________
7. A calculator shows 6.2 × 19.8 = 1227.6. Use an estimate to decide whether that can be right. ______________________
8. Choose a sensible accuracy for (a) a person’s height in metres and (b) the crowd at a football final. ______________________

Answers and marking guide

Answers with working

1. 8: the tenths digit 4 rounds down (do not round to 8.5 first).
2. 3.1, 3.10 and 3.095; the zero in 3.10 shows the hundredths were checked.
3. 10.0: the 9 tenths round up and carry into the ones.
4. $14.65: a last digit of 7 rounds down to 5.
5. 50 × 0.5 = 25 (exactly 24.624).
6. 800 ÷ 4 = 200 (exactly 204.87...).
7. It cannot: 6 × 20 = 120, so the answer is about 120, not 1200. The real answer is 122.76; the decimal point has slipped.
8. (a) metres to two decimal places, the nearest centimetre, such as 1.62 m; (b) the nearest thousand or hundred, such as 47 000 people.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and writes an estimate before trusting a calculated answer.

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: the closer neighbour, decimal places, accuracy for the context, estimating to check, and reasonable answers.