At a glance. One lesson of about 60 minutes. Students round to a place and to decimal places, estimate by rounding, and decide whether an answer is reasonable.
Calculator: scientific calculator. A scientific calculator is useful for the exact answer to check against an estimate. The estimate itself is done by hand, and the reason is that a calculator only does what is typed.
What students will learn
Know
- Rounding to the nearest 10, 100 and to decimal places.
- Look at the digit just right of the place; 5 or more rounds up.
- An estimate rounds each number first, then calculates.
Understand
- Rounding finds the nearest friendly number, and halfway goes up.
- An estimate checks the size of an answer, not its exact value.
- Round once, from the original number.
Do
- Round whole numbers and decimals to a given place.
- Estimate a calculation by rounding.
- Decide whether a calculated answer is reasonable.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: rounding and estimating in familiar situations (questions 1 to 5, 7 and 8).
- Criterion B. Investigating patterns: the second figure. Ask when the estimate is above and when it is below, and why.
- Criterion C. Communicating: explaining why an answer is or is not reasonable, with the estimate (questions 6 and 9).
- Criterion D. Applying mathematics in real-life contexts: checking calculations with an estimate before trusting them.
The one thing to do with the figure
Ask for a prediction before the first slide. “365 to the nearest 10: 360 or 370?” Take both, then show that halfway goes up. Then move to 349 to the nearest 100.
On the second figure, ask when the estimate is much too big and when much too small. Students should see that rounding both numbers the same way pushes the product in the same direction.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with predictions. |
| 10 min | Key idea, decimal places and the second figure. |
| 10 min | Worked example; students finish the faded one in pairs. Calculator for the exact answer only. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 0.6; 64; 180. |
|---|---|
| Worked example | Estimate 3000; exact 2976. |
| Faded example | Estimate 120; exact 121.36. |
| 1. Nearest hundred | B, 3800. A looks at the wrong digit, C rounds to the nearest 10, D to the nearest 1000. |
| 2. Round to 10 | 470. The ones digit is 5. |
| 3. Round to a decimal place | 7.5. The hundredths digit is 6. |
| 4. Estimating a product | 1000. 50 × 20. |
| 5. Estimating a quotient | 20. 600 ÷ 30. |
| 6. Spotting a slip | A. B misrounds 4.9; C uses addition. |
| 7. Carrying when rounding | 8.00. To two decimal places both zeros stay: they say the answer is correct to the hundredth. 7.996 is 4 thousandths below 8. |
| 8. Range of a rounded number | 454. The smallest is 445. |
| 9. Rounding twice | B. A and D accept the chain of roundings; C answers to two decimal places. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Looks at the wrong digit or rounds to the wrong place. | Underline the place being rounded to, then circle the digit to its right. |
| 460 in question 2 | Rounds a 5 down. | Mark the halfway point on a number line. |
| 7.4 in question 3 | Cuts off instead of rounding. | Ask which tenth 7.46 is nearer to on a number line. |
| 1050 or 980 in question 4 | Rounds only one number. | Cross out both original numbers and write the rounded ones. |
| 2 or 200 in question 5 | Loses or adds a zero. | Cancel zeros in matching pairs. |
| B or C in question 6 | Misrounds 4.9 or changes the operation. | Ask what 4.9 is closest to. |
Questions worth asking
- Why does half round up? It is a convention, so everybody gets the same answer. It could have gone the other way.
- Is the estimate bigger or smaller than the exact answer? It depends on which way each number was rounded.
- When is rounding to the nearest 10 too rough? When the numbers are small: 14 rounds down to 10, losing 4, which is a large share of 14.
- Why round once from the original? Because each rounding throws information away.
Differentiation
A way in for anyone stuck
- Use a zoomed number line with the halfway point marked.
- Underline the digit to be rounded and circle the digit that decides.
- Do estimates with multiples of 10 and 100 before decimals.
Stretch
- Question 8, then: what is the smallest whole number that rounds to 450 to the nearest 10? And how many whole numbers round to 450? (445, and 10 of them.)
- How many whole numbers round to 3500 to the nearest 100? (100: from 3450 to 3549.)
Students learning in English as an additional language
“Round” has many meanings in English. Here it means replace by a nearby, simpler number. Show it on a number line each time.
| Word | Plain meaning | Example |
|---|---|---|
| round | replace a number by a nearby friendly number | 365 rounds to 370 to the nearest 10 |
| estimate | an approximate answer found by rounding | 49 × 21 is about 1000 |
| nearest | the closest of the possible choices | the nearest 10 to 43 is 40 |
| decimal place | a place to the right of the decimal point | 7.46 has two decimal places |
| reasonable | close to what you would expect | 1871.8 is not reasonable for 38.2 × 4.9 |
Sentence starters for explaining aloud or in writing:
- … is between … and …, and it is nearer … because …
- My estimate is … because I rounded … to …
- The answer is / is not reasonable because …
Two things not to say
“Round up if it's 5 or more, and that is all.” Say which digit decides, and why.
“Just use a calculator.” Use it to check, and use the estimate to check the calculator.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- RoundingRound 2678 to the nearest hundred.
2700. - Earlier workIs 87 prime? Say why.
No: 87 = 3 × 29.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- StadiumA stadium has 48 612 seats. Round the number to the nearest thousand.
49 000. - TicketsEstimate the cost of 19 tickets at 11.90 dirhams each.
20 × 12 = 240 dirhams.
3Challenge
Reasoning, generalising and working backwards.
- RangeA number with two decimal places rounds to 7.4 to one decimal place. What is the largest it could be?
7.44, because 7.45 would round up to 7.5. - SquaresEstimate 7.8². (Earlier work: use squares you know.)
8² = 64, so about 64 (exact 60.84).
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Draw the zoomed number line on the board and let students place the number.