059i - Rounding Decimals

Lesson 59 — Rounding Decimals to a Given Accuracy

Strand: Number | Descriptor: AC9M7N05 | Duration: 45 minutes

Learning Intentions

• To round decimals to a given number of decimal places.
• To choose an accuracy appropriate to the context.

Success Criteria

I can:

1. Identify the digit that decides a rounding.
2. Round to a stated number of decimal places.
3. Round to the nearest whole number, tenth or hundredth.
4. Choose a sensible accuracy for a given real-world context.

Warmup

(6 minutes — number line rounding, mini whiteboards)

Draw a number line from to , marked in tenths.

1. Mark . Is it nearer or ?
2. Mark . Is it nearer or ?
3. Mark . What is the problem here?
4. Where would sit? Nearer or ?

Answers: 1. Nearer ; 2. Nearer ; 3. It sits exactly halfway — neither is nearer; 4. Nearer , since is from but from .

Resolving Q3 — state the convention. When a value sits exactly halfway, we round up by agreement. This is a rule chosen for consistency, not a mathematical necessity — different fields sometimes use different conventions.

Activities

Activity 1 — Explicit Instruction: the Rounding Procedure (12 min)

The three-step procedure:

1. Find the digit in the place you are rounding to. Underline it.
2. Look at the digit immediately to its right — the deciding digit.
3. Decide: if it is or more, round up. If it is or less, round down (leave the underlined digit unchanged). Then delete everything to the right.

I do — to one decimal place. Round .

The deciding digit is , which is or more, so round up:

I do — to two decimal places. Round .

Careful: to two decimal places, underline the second decimal digit, which is . The deciding digit is :

I do — a carry. Round to one decimal place.

The deciding digit is , so round the up. But , so the carry moves left:

Note the trailing zero. Write , not . The zero shows the accuracy claimed — one decimal place.

Critical point — only the deciding digit matters. Round to two decimal places. Students often chain: ” rounds the to , then that rounds the next to …” This is wrong. Look only at the third decimal digit, , so:

We do: Round to one decimal place: , , , .

(Answers: ; ; ; .)

We do: Round to two decimal places: , , , .

(Answers: ; ; ; .)

Activity 2 — Independent Practice (10 min)

You do — round to the accuracy stated:

1. to 1 d.p.
2. to 2 d.p.
3. to the nearest whole number
4. to 2 d.p.
5. to 1 d.p.
6. to 1 d.p.
7. to 2 d.p.
8. to the nearest whole number
9. to 1 d.p.
10. to 2 d.p.

(Answers: ; ; ; ; ; ; ; ; ; .)

Discuss Q7 and Q10. Both produce leading or trailing zeros that students often drop. and are the correct forms — the notation communicates the accuracy.

Activity 3 — Inquiry: how Accurate is Appropriate? (12 min)

Pairs.

For each situation, decide what accuracy is sensible, and justify.

1. Your height, in metres.
2. The cost of a shopping trip, in dollars.
3. The distance from Sydney to Melbourne, in kilometres.
4. A medicine dose, in millilitres.
5. The time for a m sprint, in seconds.
6. The area of a classroom floor, in square metres.

Socratic scaffolding:

Prompt	Purpose
Understand: what decides “sensible”?	How precisely the quantity can be measured, and how precisely it needs to be known.
Take height. What can a tape measure realistically show?	Centimetres — so two decimal places in metres, e.g. m.
Would ten decimal places help?	No — the measurement is not that precise, so extra digits are meaningless.
Take money. What is the smallest unit?	One cent, so two decimal places.
Take the Sydney–Melbourne distance. Would two decimal places be useful?	No — quoting km implies a precision no one needs or can verify. Nearest kilometre, or even nearest km.
What about medicine?	High precision matters — errors could be dangerous. One or two decimal places.
State the principle.	Accuracy should match both the precision of measurement and the needs of the context.

Sample answers: 1. d.p. in metres (nearest cm); 2. d.p. (cents); 3. Nearest km, or nearest km; 4. – d.p., since accuracy matters for safety; 5. d.p. (hundredths of a second, as in athletics); 6. – d.p., depending on how the measurements were taken.

Extension discussion — false precision. If you measure a room as m by m and multiply, the calculator gives . But the original measurements were only to one decimal place, so claiming hundredths of a square metre overstates what you know. A sensible answer is about .

Checks for Understanding

(5 minutes — exit ticket)

1. Round to 1 d.p.: (a) (b) (c) .
2. Round to 2 d.p.: (a) (b) (c) .
3. Round to the nearest whole number.
4. Reasoning. A student rounds to two decimal places and writes . Explain the error.
5. Reasoning. Why would it be unhelpful to give the distance between two cities to three decimal places?

Answers: 1. (a) (b) (c) ; 2. (a) (b) (c) ; 3. ; 4. The student rounded from right to left in a chain. Only the third decimal digit () decides, so the answer is ; 5. It implies a precision that cannot be measured and that nobody needs — distances between cities are not known or useful to the nearest metre.

Common Misconceptions

Misconception	How to pre-empt it
Chain rounding from the right.	Address explicitly with . Only the deciding digit matters.
Dropping a trailing zero: writing instead of .	The zero states the accuracy. Require it in every answer.
Rounding to because ” can’t go up”.	The carry moves left: .
Counting decimal places from the wrong end.	Count rightwards from the decimal point. Underline the target digit first.
Believing rounding always makes a number smaller.	It moves to whichever nearby value is closer, up or down.
Reporting more digits than the measurement supports.	The false-precision discussion in Activity 3.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A number rounded to one decimal place is . What is the smallest it could be? The largest?

Answer

The smallest is (which rounds up). The largest is anything just under — conventionally written as being the upper bound, though itself rounds to . So .

E2 (AMC Junior style). Round to three decimal places.

Answer

, so to 3 d.p. it is . (This is the classic approximation to , accurate to two decimal places.)

E3 (Challenge). A number rounds to to one decimal place and to to two decimal places. Is this possible? Explain.

Answer

Rounding to 2 d.p. gives , so the number lies in . Rounding that range to 1 d.p. gives ✓. So yes — for example, rounds to and to .

E4 (Challenge). A rectangle measures cm by cm, each to one decimal place. Calculate the area and state it to a sensible accuracy.

Answer

. But since the measurements are only to one decimal place, quoting hundredths overstates the precision. A sensible answer is about , or at most.

E5 (Reasoning challenge). Explain why rounding and both “up” means that rounding does not always balance out.

Answer

Always rounding halves upwards introduces a small upward bias when many values are rounded. Some fields use “round half to even” (, ) precisely to remove this bias. Year 7 uses the round-half-up convention, but knowing why alternatives exist is worthwhile.

Homework

1. Round to 1 d.p.: (a) (b) (c) (d) (e) .
2. Round to 2 d.p.: (a) (b) (c) (d) (e) .
3. Round to the nearest whole number: (a) (b) (c) (d) .
4. Round to (a) 1 d.p. (b) 2 d.p. (c) 3 d.p.
5. Round to (a) 1 d.p. (b) 2 d.p. (c) 4 d.p.
6. State a sensible accuracy for each, with a reason: (a) the mass of a person in kg (b) the price of a car (c) the length of a pencil in cm (d) the time for a swim race.
7. A number rounds to to one decimal place. What is the smallest it could be?
8. Reasoning. A student rounds to 2 d.p. and writes . Explain the error and give the correct answer.
9. Reasoning. Explain why and carry different information when reporting a measurement.
10. Challenge. A rectangle is cm by cm, each to 1 d.p. Find the area, then state it to a sensible accuracy and explain your choice.

Answers: Q1 — (a) (b) (c) (d) (e) . Q2 — (a) (b) (c) (d) (e) . Q3 — (a) (b) (c) (d) . Q4 — : (a) (b) (c) . Q5 — (a) (b) (c) . Q7 — . Q8 — chain rounding from the right; only the third decimal digit () decides, giving . Q9 — claims accuracy to the nearest hundredth; claims only the nearest tenth. Q10 — ; since inputs are to 1 d.p., report about or .